JOHANNES GUTENBERG-UNIVERSITÄT MAINZ
Department of Physics, Mathematics and Computer Science
Institute of Physics
A journey through nonlinear magneto-optics
by
Dimitra Kanta
A thesis submitted for the degree of
Doctor of natural sciences
June 2024

iii
Declaration of Authorship
I, Dimitra Kanta, declare that the thesis entitled A journey through nonlinear magneto-optical effects
and the work presented in the thesis are both my own, and have been generated by me as the result
of my own original research. I confirm that:
• this work was done wholly or mainly while in candidature for a research degree at this Univer-
sity;
• where any part of this thesis has previously been submitted for a degree or any other qualific-
ation at this University or any other institution, this has been clearly stated;
• where I have consulted the published work of others, this is always clearly attributed;
• where I have quoted from the work of others, the source is always given. With the exception of
such quotations, this thesis is entirely my own work;
• with the oversight of my main supervisor, editorial advice has been sought. No changes of
intellectual content were made as a result of this advice;
• I have acknowledged all main sources of help;
• where the thesis is based on work done by myself jointly with others, I have made clear exactly
what was done by others and what I have contributed myself;
Signed:
Date: 18st October 2024
Supervisor:
Examiner 1:
Examiner 2:
Examiner 3:

v
Abstract
This thesis contributes to the research on nonlinear magneto-optical effects, specifically focusing on 
atomic magnetometry based on nonlinear magneto-optical rotation and mirrorless lasing (based on 
amplified spontaneous emission) –  while encountering also other nonlinear effects, such as  the Kerr 
effect.
Measuring magnetic fields has been of great importance since the bronze age and has been crucial to 
the development of the human civilization [1]: the first magnetometer was the compass, a magneto-
meter that measures only the direction of the earth’s magnetic field and is so important that it has 
a place amongst the four great inventions of ancient China. Measuring the magnitude of a magnetic 
field is a more modern story that starts with Carl Friedrich Gauss measuring the Earth’s magnetic 
field in 1833 [2].
Among the techniques existing nowadays, for precisely measuring magnetic fields, optically pumped 
magnetometry (OPM) stands out for it’s sensitivity, size, robustness and low cost [3], [4], [5]. After 
the pioneers set the foundations more than half a century ago [6], [7], [8], diode laser technology 
allowed optical magnetometers to become a workhorse for magnetometry. OPM magnetometers are 
potentially as sensitive [9] as SQUIDs (Superconducting Quantum Interference Device) [10] and do 
not require cryogenics. Applications span over a wide range of fields: geophysics [11], bio-magnetic 
measurements [12] and fundamental physics [13]. OPM research in recent years has shifted from 
working in the laboratory to applications in the field and a useful step towards commercialization is 
the self-oscillating configuration. The basic operation principle is based on using the detected signal 
to sustain continuous oscillation at the resonant frequency [14] [15]. Such systems have a broad dy-
namic range, can follow field fluctuations and are simple.
Although OPMs and especially SERF (spin exchange relaxation-free) type magnetometers are highly 
sensitive, they need to operate in low fields and hence require magnetic shielding f rom the Earth’s 
field and other noise sources. Being able to measure in the geophysical field range or earth field with 
high sensitivity could open the path to low-cost bio-magnetic measurements [16], space-magnetometry 
[17], non-destructive testing and imaging [18] and magnetometry on rapidly moving platforms [19]. 
This thesis focuses on Earth-field optical magnetometry and addresses challenges arising f rom the 
Earth’s magnetic field by using techniques like spin locking or creating a device free of classic Earth-
field magnetometry issues, such as heading error.
The second part of this thesis is dedicated to mirrorless lasing. Since their invention in the 60s, lasers 
(light amplification by s timulated emission o f r adiation) have p layed a  huge r ole i n many a reas of 
scientific research, industry, and everyday life and continue to g row. There are three principal com-
ponents usually attributed to a laser: a gain medium, a pumping process and a feedback loop [20], 
although there is a debate over whether a feedback loop is always required. Lasing is often dis-
tinguished from processes such as Amplified Spontaneous Emission (ASE), Superradiance (SR) and 
Superflouorescence (SF) [21,22], but this is not the case for the work presented here. Typically, lasers 
follow a conventional structure that includes an optical resonator setup. This setup uses mirrors to 
amplify light over multiple round trips in the gain medium [20]. In mirrorless laser setups, the gain
medium serves as the resonator, and the feedback loop would typically happen through multiple scat-
tering processes in systems with varying degrees of disorder [21]. Optical feedback through scattering
can also create random lasers [23]. The system we are studying does not involve scattering mechan-
isms and we use the term lasing interchangeably with Amplified Spontaneous Emission (ASE). We
define mirrorless lasing as as directed monochromatic emission from an ensemble of atoms or mo-
lecules pumped with a laser light. Experiments in alkali metal vapor have shown gain through the
phenomenon of amplified spontaneous emission (ASE), see, e.g, [24]. This thesis focuses on the phe-
nomenon of amplification of spontaneous emission and degenerate mirrorless lasing in alkali atoms
with magnetically degenerate hyperfine states.
vi
CONTENTS vii
Contents
List of Figures xi
List of Tables xix
List of Abbreviations xxi
List of Symbols xxiii
1 Light-atom interactions 101 3
1.1 A brief history of light-matter interactions . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Atoms and their structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.1 Fine structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.2 Hyperfine structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 Atoms in a magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3.1 Zeeman effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4 Polarized Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4.1 Density Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4.2 Liouville equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4.3 Angular momentum probability surface . . . . . . . . . . . . . . . . . . . . . . 15
1.4.4 Quantum beats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2 Nonlinear light-atom interactions 19
2.1 Linear magneto-optical rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1.1 Faraday effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1.2 Macaluso-Corbino effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 Nonlinear optical effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.1 Classical formalism – macroscopic . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.2 Quantum mechanical formalism – from microscopic to macroscopic . . . . . . 24
2.3 Atomic transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3.1 Rabi frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3.2 Stimulated emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3.3 Optical Bloch equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.4 Optical pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.4.1 Selection rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.4.2 Saturation parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
viii CONTENTS
2.4.3 Mirrorless lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3 Atomic magnetometers 37
3.1 Simple model of an optical magnetometer . . . . . . . . . . . . . . . . . . . . . . . . 37
3.1.1 Atomic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.1.2 Vapor Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.1.3 Pump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.1.4 Probe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.1.5 Shielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.1.6 Noise and Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.1.7 Polarimetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.1.8 Spin relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.2 Polarization moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.3 Important types of atomic magnetometers . . . . . . . . . . . . . . . . . . . . . . . . 49
3.3.1 Mx and Mz magnetometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.3.2 Bell-Bloom magnetometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.3.3 Nonlinear magneto-optical rotation (NMOR) . . . . . . . . . . . . . . . . . . 53
3.3.4 NMOR with modulated light . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.3.5 Spin-exchange-relaxation-free (SERF) magnetometer . . . . . . . . . . . . . . 58
3.4 NLZ in Earth-Field magnetometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.4.1 Nonlinear Zeeman effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.4.2 Heading error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.5 Experimental devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.5.1 Beam modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.5.2 Lock-in amplifier – LIA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.5.3 Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4 All-optical spin locking in alkali-vapor magnetometers 69
4.1 The idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.2 Spin locking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.2.1 Spin Locking with AC Stark shift . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.2.2 Optical Rotation Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.3 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5 Rb magnetometry free from NLZ-effect-related heading error in Earth-field range 85
5.1 The idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.1.1 Strategies to suppress the NLZ-related heading error . . . . . . . . . . . . . . 86
5.2 The Physics behind the idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.3 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.3.1 Pumping and probing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
CONTENTS ix
5.3.2 q-fold symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.3.3 Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.4 Comparison to an orientation magnetometer . . . . . . . . . . . . . . . . . . . . . . . 92
5.4.1 Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.4.2 Heading error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.4.3 Estimation of NuZ-related Heading Error . . . . . . . . . . . . . . . . . . . . 97
5.5 Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6 Degenerate mirrorless lasing–Forward beam 105
6.1 The idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.2 The Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.3 Evidence for degenerate mirrorless lasing in alkali metal vapor: forward beam magneto-
optical experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.3.1 Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.3.2 Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.3.3 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.3.4 Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.3.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.4.1 Ring structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7 Unsolved (lab) mysteries 123
7.1 Degenerate mirrorless lasing- Backward Beam . . . . . . . . . . . . . . . . . . . . . . 123
7.1.1 The Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
7.1.2 In Ashtarak, part I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
7.1.3 Mainz, part I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
7.1.4 Ashtarak, part II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
7.1.5 Mainz, part II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
7.1.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
7.2 Spin locking in Rb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
7.2.1 The idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
7.2.2 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
7.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
7.2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
7.2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
8 Conclusions 143
Bibliography 145

List of Figures
1.1 A brief history of light-matter interaction . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Magnetic moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Spin projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Energy-levels shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.5 Polarized vs unpolarized . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.6 Two-level system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.7 Probability surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.1 Linear Faraday effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2 Macaluso-Corbino effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3 characteristic spectral shape of Malacuso Corbino effect . . . . . . . . . . . . . . . . . 22
2.4 Polarization of an atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.5 A two-level spin-1/2 system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.6 Rabi oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.7 Absorption, spontaneous and stimulated emission . . . . . . . . . . . . . . . . . . . . 28
2.8 Bloch sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.9 Dark state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.1 Illustration of a basic setup of an atomic magnetometer. . . . . . . . . . . . . . . . . 37
3.2 Transition structure of an alkali atom with nuclear spin I = 3/2 such as 23Na, 41K or
87Rb [19]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.3 Vapor cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.4 1. Before pumping 2. Light depletes population at B=0 3. Optical detection when
B̸= 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.5 Lorentzian shape of the refractive-index change of light due to the Macaluso-Corbino
effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.6 The real parts of the refractive indices are creating dispersive curves that are shifted
in frequency due to the external magnetic field. Their difference is proportional to the
linear optical rotation ϕ of the light passing through the medium. . . . . . . . . . . . 42
3.7 Magnetic shield with high permeability. . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.8 Balanced polarimetry setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.9 Spin-exchange mechanism. Red spheres are atoms that collide and after the collision
precess in different directions as indicated by the purple arrows. . . . . . . . . . . . . 46
3.10 Polarization moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.11 Bloch sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
xi
xii LIST OF FIGURES
3.12 Mz magnetometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.13 Mx magnetometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.14 Bell-Bloom magnetometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.15 Magnetic Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.16 NMOR-based magnetometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.17 Three-level Λ-type system. Linearly polarized light couples the ground states to the
excited state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.18 NMOR with modulated light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.19 Angular-momentum polarization surface (AMPS) or alignment, produced by linearly
polarized light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.20 Optical-rotation signal in AM NMOR . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.21 SERF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.22 Breit-Rabi diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.23 EOM with voltage on and off . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.24 AOM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.25 Direct-band-gap structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.26 ECDL Littrow configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.1 NLZ effect in Cs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.2 Probability P (t, 0) for an atom to be found in the initial state. For small amplitudes
of the spin-locking field (Ωrf ≪ ωrev), the probability undergoes quantum beating.
If the amplitude of the spin-locking field is much larger than the NLZ parameter
(Ωrf ≫ ωrev), the atoms remain in the initial state. If the amplitude of the spin-
locking field is equal to the NLZ parameter (Ωrf = 2ωrev), the atoms are partially
locked in the initial state and the populations undergo oscillation with frequency ωlock. 72
4.3 (a) Schematics for atomic levels J = 1 and J ′ = 0 coupled by circularly polarized
light propagating in the x-direction. Only two levels are coupled in the x-basis. (b)
All levels are coupled if one used the z-basis. . . . . . . . . . . . . . . . . . . . . . . . 74
4.4 Theoretical calculations of signal. Theoretical calculated in-phase (top row) and quad-
rature (bottom row) first-harmonic amplitudes of optical rotation signal. The NLZ
effect splits the magnetic resonance into two peaks: (a) without the the spin-locking
field (b) with RF field (c)with intensity modulated light-shift field. For these plots,
the parameters Ωpr/γ = 100, δ /γ = 106, Ω /γ = 104pr p , δp/γ = 103, ∆m/γ = 103,
ωrev/γ = 1.5, Ω mRF /γ = 0.02, ΩLS/γ = 0.02, ∆/γ = 105 are chosen. . . . . . . . . . . 76
LIST OF FIGURES xiii
4.5 Experimental setup for all-optical spin locking: AOM: acousto-optic modulator; EOM:
electro-optic modulator; HWP: half-wave plate; QWP: Quarter-wave plate; PBS: po-
larizing beam splitter; PD: balanced photodetector; LIA: lock-in amplifier; LO: local
oscillator. A partial view of the magnetic shield is shown in the figure. Atoms are con-
tained in a glass cell positioned in the center of the magnetic shield and are pumped
(along −x̂) and probed (along ŷ) by laser beams under a static magnetic field (along
ẑ). The intensity of the light-shift laser beam is sinusoidally modulated with an AOM
at a frequency Ωm, while its polarization is switched between the σ+ and σ− states
every π/Ωm, using an EOM. Inset shows the monitor setup for polarization and amp-
litude of the light-shift beam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.6 (Experimental data) Magnetic-resonance line shape for a modulation frequency of
216,620 Hz as a function of the leading magnetic field along the z axis with applied
light-shift field and pump (red line), without light-shift field and pump (black line),
and without pump field (blue line). The amplitude of magnetic resonance without
light shift is normalized to unity. The power in (a) and maximum power in (b,c)
of the light-shift beam is 200 mW. The inset shows the polarization modulation (a),
intensity modulation (b) and both modulated (c) schemes for the pump and light-shift
field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.7 (a) Observed monitor signal and input intensity modulation signal. Here ϕ1 = −π/2
and ϕ2 = −π/2. The distortion of fictitious RF field is mainly caused by nonlinear
response of the AOM. (b) Phase scanning of ϕ1 and ϕ2. . . . . . . . . . . . . . . . . . 80
4.8 Map of the magnetic-resonance linewidth as a function of the applied light-shift-field
power and detuning. The minimum linewidth is 25.25(6) Hz. . . . . . . . . . . . . . . 81
5.1 Schematic of NLZ-heading-error-free magnetometer. Nonlinear Zeeman effect of 87Rb
ground state 52S1/2 and corresponding magnetic resonance frequencies. In the F = 2
(F = 1) hyperfine state, there are four (two) different ∆m = 1 magnetic resonance
frequencies, labelled with ω1 to ω4 (ω5 to ω6), respectively, and 3 (1) different ∆m = 2
magnetic resonance frequencies, labelled ω′1 to ω
′
3 (ω
′
4), respectively. . . . . . . . . . . 88
5.2 Pump and probe scheme of alignment magnetometer. Energy levels of 87Rb atoms
and excitation scheme (not to scale). Pump beam is a 795 nm laser beam exciting
the 52S1/2F = 2 to 52P ′1/2F = 2 transition, which generates atomic spin polarization
in the 52S1/2F = 1 state via repopulation pumping; the probe beam is a 780 nm
laser tuned to the low frequency side relative to the 52S1/2F = 1 to 52P ′′3/2F = 0
transition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.3 D1-line spectroscopy of rubidium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.4 D2-line spectroscopy of rubidium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
xiv LIST OF FIGURES
5.5 Alignment resonance setup. Pump, a linearly polarized 795 nm laser beam used to
generate the atomic alignment polarization; Probe, a linearly polarized 780 nm laser
beam used to detect the Larmor precession of the atomic alignment polarization via
optical rotation; AOM, acousto-optic modulator used to pulse the pump beam; PBS,
polarizing beam splitter; BS, beam splitter; λ/2, half-wave plate; λ/4, quater-wave
plate; BPD, balanced photodiode; BPF, bandpass filter with central wavelength of
780 nm, which is used to prevent pump from entering the BPD; BD: beam damp;
LIA, lock-in amplifier; PC, personal computer. . . . . . . . . . . . . . . . . . . . . . 92
5.6 Orientation resonance setup. Pump, a circularly polarized 795 nm laser beam used to
generate the atomic orientation polarization; Probe, a linearly polarized 780 nm laser
beam used to detect the Larmor precession of the atomic orientation polarization via
optical rotation; AOM, acousto-optic modulator used to pulse the pump beam; PBS,
polarizing beam splitter; BS, beam splitter; λ/2, half-wave plate; λ/4, quater-wave
plate; BPD, balanced photodiode; BPF, bandpass filter with central wavelength of
780 nm, which is used to prevent pump from entering the BPD; BD: beam damp;
LIA, lock-in amplifier; PC, personal computer. . . . . . . . . . . . . . . . . . . . . . 93
5.7 Magnetic resonances of alignment and orientation polarization with background mag-
netic field of different strengths. (a) Alignment resonance. (b) Orientation resonance.
Background magnetic field is set along z direction, with strength ranging from 20µT
to 50µT. Different magnetic resonances are shifted vertically for clarity. Since the
alignment-resonance frequency is about twice that of orientation, the scale of detun-
ing in (a) is also twice that of (b). The amplitude of the alignment signal is smaller
than that of the orientation signal for the reasons explained in [25]. . . . . . . . . . . 94
5.8 Magnetic resonances of alignment and orientation polarization with background mag-
netic field along different directions. (a) Alignment resonance with Bx = By. (b)
Alignment resonance with Bx = −By. (c) Orientation resonance with Bx = By. (d)
Orientation resonance with Bx = −By. The strength of the background magnetic
field was kept at 30µT. The angle between the background magnetic field and the z
direction is represented as the vertical axis. The broadening of the magnetic reson-
ances at larger field angle is due to the increased magnetic field gradients. Since this
broadening does not affect our conclusion about resonance symmetry, and that the
compensation of such gradients along different directions is complex, these gradients
are not compensated during this experiment. Gradients are not compensated. . . . . 95
5.9 Typical F = 1 and F = 2 alignment magnetic resonance. Background magnetic field
is set to 20µT along the z direction. The insertion shows details about the F = 2
resonance. This magnetic resonance is averaged from four scans. . . . . . . . . . . . . 96
5.10 (a) Photon-shot-noise-limited sensitivity, (b) Life time and (c) optical rotation amp-
litude of alignment-based magnetometry as a function of background magnetic field.
The background magnetic fields are kept along the z direction. The black squares (red
circles) represent data taken before (after) vapor cell heating. . . . . . . . . . . . . . 100
LIST OF FIGURES xv
6.1 Inversion of population between magnetic sublevels. Onset of population inversion
and amplified radiation corresponding to ∆mF = ±1 transitions under excitation of
85Rb D2 line with an intense linearly polarized laser radiation for zero magnetic field
(the quantization axis Z is chosen to be along the laser light polarization). Bars:
population of sublevels (not to scale). Red corresponds to population of excited states
and purple population of ground states. . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.2 Numerical simulations with the ADM package for inversion of populations: B ∥
E magnetic-field dependence of normalized steady state population of ground state
(g|mFg>) and excited state (e|mFe>) Zeeman sublevels linked by ∆mF = 0 trans-
itions 85Rb Fg=3 → Fe=4 (a) and 87Rb Fg=2 → Fe=3 (b), and population inversion
on corresponding mFe → mFg transitions (c,d). The light power is sufficient to provide
population inversion. Γ is the decay rate of excited state, ΩL is the Larmor frequency.
The width of the resonance is not affected by the ground state relaxation rate included
in the model, which is much smaller than Γ. . . . . . . . . . . . . . . . . . . . . . . . 108
6.3 Experimental setup and the measurement configuration in Ashtarak (A) and Mainz
(B). PBS: polarizing beam splitter; BS: beam splitter; λ/2: half-wave plate; λ/4:
quater-wave plate; PD: photodiode; AT: attenuator; P: polarizer; G-T: Glan-Thompson
Polarizer; Iris: I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.4 Dependences of normalized output intensity with orthogonal polarization (see text)
on magnetic field (B ∥ E) for 85Rb Fg=3 → Fe=4 (a,c) and 87Rb Fg=2 → Fe=3 (b,d)
D2 transitions, measured in Ashtarak (a,b) and Mainz (c,d). . . . . . . . . . . . . . . 112
6.5 Input-intensity dependences of (a) absolute (black symbols, left axis) and normalized
(red symbols, right axis) of the output intensity with orthogonal polarization for B =
0; (b) FWHM width of B = 0 resonance for the output orthogonally-polarized light,
retrieved from the measurements presented in Figure 6.4a,b. Solid squares: 85Rb
Fg=3 → Fe=4; open squares: 87Rb Fg=2 → Fe=3. Solid lines are drawn as a guide to
the eye. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.6 Ellipticity in output light: Magnetic-field dependence of the output signal for the
85Rb Fg=3 → Fe=4 transition recorded with a circular polarimeter at different input-
light intensities (same as in Figure 6.4a,b). No normalization; the curves are shifted
vertically by equal amounts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.7 Double-scanning measurement for the output light with polarization parallel (∥E)
and orthogonal (⊥E) to the input laser polarization for incident light intensity of
87.9 mW/cm2. A single slow (compared to the frequency of the spectral scanning) B-
field scan (B ∥ E) in the range of ±0.12 G is combined with fast laser-frequency scans
over 10 GHz covering the Rb D2 line. Upper inset: zoomed output light spectrum in
the region of B = 0; lower inset: zoomed signals of ⊥E output light (lower trace), and
∥E output light in the region of the dip (upper trace). . . . . . . . . . . . . . . . . . 115
6.8 Spatial-profile measurements of the orthogonally polarized component of the forward
beam at zero magnetic field. Upper curve: saturated-absorption spectrum for reference.116
xvi LIST OF FIGURES
6.9 Spatial-profile measurements of the orthogonal component of the forward beam for
different laser frequencies and different values of the magnetic field with a fixed input
beam power of 12 mW. Upper curve: saturated-absorption spectrum for reference. . . 117
6.10 Schematic diagram of entirely-degenerate four-wave mixing process. . . . . . . . . . . 119
7.1 Backward lasing setup in Ashtarak. AT: Attenuator for controlling laser power; PD:
Photodetector; PBS: Polarizing beam splitter; BS: Beam splitter; PD1 is used for de-
tecting the orthogonal polarization of the backward beam, PD2 detects fluorescence
and PD3 is monitoring the frequency of the laser to have a reference on which fre-
quency we pump. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
7.2 Backward-direction in Ashtarak for different magnetic fields. The blue line is the
signal in the backward direction and the yellow line is the fluorescence. Left: Signal
at 0.5 G, Middle: Signal around 0.2 G, Right: Signal at 0 G. . . . . . . . . . . . . . . 125
7.3 Saturation spectroscopy of probe beam. The dark-red line is the pump, the light-
red line is the probe. We monitor both beams to see possible gain in the expected
frequencies on the probe beam, as illustrated at the top right. . . . . . . . . . . . . . 126
7.4 Setup for measuring probe gain in 85Rb. . . . . . . . . . . . . . . . . . . . . . . . . . 127
7.5 Setup for measuring probe gain in 87Rb. . . . . . . . . . . . . . . . . . . . . . . . . . 127
7.6 Example of S and P polarization of light. The incident light contains two types of
polarization: S and P . The P component is parallel to the plane of incidence and the
S component is perpendicular to it. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
7.7 Measuring extinction ratio of PBSi. We test how well linearly polarized our laser with
power I1. I2 is the power of the transmitted beam, I3 the power of the reflected beam
and IA power of the possibly absorbed beam due to the G-T polarizer. The G-T
polarizer is not a cube, but it is shown as such here for simplicity. . . . . . . . . . . . 129
7.8 Forward beam experiment. We replaced all the polarizers with good ones and heated
the cell starting from RT and up to 45◦C. We detected either with a camera or with a
photodetector to compare the data. We operated at zero(0) magnetic field. . . . . . . 131
7.9 Kerr effect/self-focusing. The index of refraction n and the absorption coefficient
in relation to frequency are illustrated for a better understanding of the Kerr/self-
focusing effect. The phenomenon arises when a high-intensity laser beam propagates
through a Rb vapor while the laser frequency is scanned around a resonant frequency
ω0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
7.10 Kerr effect/self-focusing on a laser-viewing card. Pictures 1−8 show the time evolu-
tion of our detected beam without a crossed polarizer: size, shape and intensity are
changing time. In picture 1 and 7 one can see a small black dot in the middle of the
beam indicating self-focusing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
7.11 Backward beam experiment. We replaced all the polarizers with good ones and heated
the cell starting from RT and up to 45◦C. We detected either with a camera or with a
photodetector to compare the data. We operated at zero(0) magnetic field. The beam
in the forward direction is blocked with various materials. . . . . . . . . . . . . . . . . 134
LIST OF FIGURES xvii
7.12 Backward beam experiment in Ashtarak. Backward-direction detection at zero field
in Ashtarak. The blue line in the oscilloscope is the signal in the backward direction
and the yellow line is the fluorescence. . . . . . . . . . . . . . . . . . . . . . . . . . . 135
7.13 Data from Ashtarak, not yet published: Zero magnetic field input power dependence
of the absolute (left panel) and normalized (right panel) backward radiation power.
Solid symbols: amplitudes of Doppler-broadened contours of 87Rb (circles) and 85Rb
(squares) transitions, Open symbols: amplitudes of the narrow peaks located at cor-
responding crossover resonances CO32 (circles) and CO43 (squares). Lines are drawn
to guide the eye. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
7.14 Spin-locking setup in 87Rb. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
7.15 Rb Resonance under Earth field, at 30µT. The resonance and an estimation of the split
is seen on the left and the right side shows the fitting of the X, Y and R components
of the resonance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
7.16 Spin locking in Rb at 55µT for different RF-field amplitudes. . . . . . . . . . . . . . . 141
7.17 Comparison between spin locking in Rb (left) and Cs (right) and how it affects the
off-resonant case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

List of Tables
1.1 Spherical function/harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Radial functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
xix

List of Abbreviations
Abbreviation Description Chapter
AOM Acousto-Optical Modulator §3.3.2, §3.5
ASE Amplified Spontaneous Emission §2.4.3
ECDL External-Cavity Diode Laser §3.5.3
EOM Electro-Optic Modulator §3.3.2, §3.5
FWHM Full Width at Half Maximum §4.4, §6.3.5
LASER Light Amplification by Stimulated Emission of Radiation §3.5.3
LIA Lock-In Amplifier §3.5.2
NLZ NonLinear Zeeman §1.4.4, §3.4.1
NMOR Nonlinear Magneto-Optic Rotation §3.3.3
OPM Optically-Pumped Magnetometer §3.1, §4.1
PC Personal Computer §5.3.3
PD PhotoDetector §4.3, §6.3.4
QM Quantum Mechanics §1.4.3
RF RadioFrequency §3.3.1, §4.1
PN ProjectionNoise §3.1.6
PSN PhotonShotNoise §3.1.6
VLS VectorLight Shift §4.2
PM Polarization Moments §3.2
xxi

List of Symbols
Symbol Description Notes Chapter
µ⃗ magnetic moment §1.2.1
Jz total angular momentum on quantization axis §1.2.1
γ gyromagnetic ratio §1.3.1
ψi the atomic wave function of an atom i §1.4.1
ρmn density matrix §1.4.1
D electric displacement §2.2.1
χ electric susceptibility §2.2.1
η refractive index §2.2.1
P polarization of a medium §2.2.1, §2.2.2
d electric-dipole operator §2.2.2
ΩR Rabi frequency §2.3
ΩL Larmor frequency §2.3, §3.1.4
κ1, κ2 saturation parameters §2.4.2
Γ linewidth of a transition §2.3.2
t2 spin-coherence time §3.1.6
ρκq polarization moments §3.2
xxiii

Acknowledgements
Alongside my journey into nonlinear magneto-optics, another journey was unfolding: my journey as 
a new researcher in a foreign country. The list of names is extensive, and I apologize if I forgot to 
mention anyone. One way or another, you all supported me through tough times, provided comfort 
and advice, inspired me with ideas, and made this journey fun and, as a result, possible.
First, I would like to thank my advisor and professor Dima, for not only being an exceptional 
scientist, but also a compassionate and patient mentor. His unwavering support and the freedom he 
granted me to explore my own path have been truly invaluable. It was a pleasure to have worked 
with you– and of course also make music with you and the spontaneous emission.
Following a loose chronological order I would like to start by thanking my friends Vasilis and Nat-
alia who introduced me to the academic possibilities in Mainz, they let me stay at their place for as 
long as I needed and we eventually developed a deep friendship. It always makes me smile when I 
think of you. I am also grateful to all my colleagues and friends in MPIP who were part of my first 
community in Mainz and always made me feel welcome. Shout out to Christoph, Jan, Malte and 
Maksim for the most amusing coffee breaks. Thank you Simon and Phillip for all our 
fantastic tabletop quests. A big thanks from the bottom of my heart to my late friend Eduard, 
Alexandra, Yu-Jen and especially Marco who supported me through thick and thin. Marco, 
you have been a great friend and I wish you all the best in Spain.
Moving on, I would like to thank Arne, for all the discussions and supervision as well as his tasty 
group dinners. Your efforts to manage and bond this diverse group are really impressive and valued. 
Lykourgos, thank you for introducing me to the group and for always offering aid when I  needed 
it. I want to thank Hector for our very interesting talks spanning from optical magnetometry to 
determinism and remember: your crippling awareness of the self brings all the existential doubt to 
the yard! Roman thank you for your friendship, for bringing always a good mood with you and for 
leaving your ukulele sometimes back home. And of course I cannot forget to thank Marina for her 
upbeat attitude and Joey for his helpful explanations on theoretical physics and on how on earth 
the US election system works. Antoine thank you for being so special, you give Paris an even better 
name.
For his wry humor and his solidarity I am thankful to Jim: I know you have deep down a big heart, 
an essential skill if you work in the penguin-lifting business. A big thank you to James for always 
listening and giving the most entertaining input: you are smart and a team player; science could only 
benefit from people like y ou. I  am also indebted to Giorgos for being the person that I  know I  can 
always count on to help solve any kind of challenge I face. Giorgo thank you for all the tea and gym 
sessions, and generally for motivating me when things got tough. Thank you Till for your grounding 
and relaxed presence and for reminding me– way too often –to write my thesis. Garry thank you 
for the fascinating science-communication discussions, Anna for our philosophical pen and paper 
sessions and Erik for your always calming presence and composed demeanor.
Thank you Lei for being a great office mate. I can’t thank you enough Yinan for being the best lab
mate I could have wished for. Hu is the best? We had such a delightful time together in the lab:
singing Greek and Chinese music, improvising games and smashing batteries!
I would also like to thank Anne, Peter, Nate, Pavel, Alex, Felipe, Xue, Zhiyin as well as the
newer members Olympia, Dani, Razmik, Alexander and many more, for forming an eclectic
and diverse group of people that has been a joy to be part of every day. A huge thank you to Derek,
a truly inspiring physicist and friend whose love for physics and life is contagious. We are lucky to
have you Derek!
Next, I would like to express my appreciation to the backbone of the group, postdocs that had my
back, all exceptionally skilled and capable: to John, Dionysis, Hendrik, Emmanuel, Geoffrey,
Kirill, Oleg thank you for your practical guidance in the lab and your creative input. And thank
you for the fun times as well!
With deep gratitude, I thank Alexander for the opportunity to learn from him about mirrorless
lasers and optics. Your guidance in the lab was very crucial in these first steps I was making in science.
Thank you Guzhi for sharing your expertise with me, for working together on several projects and
for letting me bombard you with questions on a daily basis. Of course, I need to say a big thank you
to Rui for our excellence collaboration and for instilling in my methodology love for detail, planing
and diligently optimizing a scientific project. I would also like to thank our collaborators in Armenia
and Aram’s group who gave me the opportunity to sleuth out the creation of degenerate mirrorless
lasing and visit and work in Armenia.
Finally a special thanks goes to all the behind-the-scenes people: Evi for dealing with the huge
amounts of administrative work required so often, the administrative staff of HIM and, of course,
Lars, who keeps the building running and who had been trying to fix the cold air draft behind my
back in the lab for quite some time. The draft’s origins remain unknown, but thank you regardless
for trying to deal with this and so many other issues.
Many of the already mentioned people have also become my friends. And since scientific research is
similar to any other creative job in that it entails extensive work hours, inevitable setbacks, disagree-
ments, and a continuous creative process that extends beyond regular working hours, friends are a
crucial part of the PhD equation. This is why I need to thank also my friends outside the research
group for filling my after-work life with laughter and ease. A heartfelt thank you to Nora who is
the bravest researcher I have met, thank you for being there since week one. Eleftheria who I love
chatting with, thank you for empathizing with my troubles every time and of course I have to thank
Eleftheria’s dog Lexy who is– literally –the best dog and Felix who is always so easy to be around.
A big thank you to Jonathan, Michi, and Tim for the great company, especially during the dark
times of COVID and for always being up for doing something spontaneous! Thank you Dr(yeah!)
Malte P. for being one of the two people on this Earth wanting to play Anno Domini with me,
Kiera and Rebecca for all the cosy get-togethers in Mainz, Shamitha for all the loud laughs and
Oriane for the most joyful times we had during your last months in Mainz!
A big thank you goes also to my designer friends who might not understand physics, but understand
the efforts I had to make being a physicist: Eva, Meike, Fede, Juliet, Ranim. I would also like to
thank all those I befriended in Stuttgart while juggling thesis and work, you gave me perspective and
daily serenity and I will honestly miss you. Thank you, Melina, Anta and Dialechti, for making
Thessaloniki still feel like home.
Strange as it may seem, I would like to express my gratitude to my daily radio show, Ellinofreneia,
which I listened to every day during the last 6-7 years. In the windowless, chilly, noisy lab, it kept
me grounded and connected to the outside world.
To my close friends who I met in Athens, but are now spread all over the world: thank you Aggelos
for all the humor, conversations, affection and crepe sharing, Andreas for all our adventures both
real and imaginary, Antonis for being my kind tall souidos friend, Giannis for being hilarious,
tolerant and supportive, Grigoris for all the games, music and discovering Athens with you, Nikos
for having shared our best times together and we keep sharing more, Karim for caring for me since
my bob-cut days until now as we stand at the threshold of our second youth and for still putting up
with me– also for being the other person who wants to play Anno Domini, Vaggelis for all the great
trips with you and your daily help with python and Vaso for the awesome times we have when you
don’t drink coffee. For all the precious moments, for shaping and forming one another, for our daily
chats and rants, for helping each other while studying at the SEMFE, for roaming and loving the
best city in the world: I would do it all over again with you!
To my parents: thank you for the love, support and parenting that did not perpetuate stereotypes,
allowing me to always feel confident and free to discover myself. Mom, dad, thank you for everything!
Σας ευχαριστὼ!
To Stela: thank you for your big heart, for living next to me so that not a day passes without
talking, for meticulously listening to my daily happenings and for believing in me when I don’t. I do
not speak as an owl when I say: Thank you for being the best sister!
To Martin: I thank luck (and maybe also Dima) for you moving to the GNOME lab with me and
I thank you for bringing all your wit, kindness, and humor to my life. Thank you for your moral,
emotional and scientific support, and for always knowing how to cheer me up. Thank you for choosing
to be my partner in this journey and many other journeys in the years to come.
Dimitra Kanta
Mainz June 2024

Happiness can be found even in the darkest of times, when one only remembers to
turn on the light.
Albus Percival Wulfric Brian Dumbledore

It looks strange and it looks strange and
it looks very strange; and then suddenly it
doesn’t look strange at all and you can’t un-
derstand what made it look strange in the
first place.
Gertrude Stein
1
Light-atom interactions 101
1.1 A brief history of light-matter interactions
Light-matter interactions cover a substantially wider range of phenomena than light-atom interac-
tions, from the macroscopic world of black holes to the mesoscopic environment of nanophotonics to
the tiniest matter possible in the realm of quantum mechanics. The expression light-matter interac-
tion implies that these two entities are different and interact through some intermediate agent [26].
Interestingly enough though, the two were not considered separately until the 17th century AD. In
Figure 1.1 one can see some important landmarks of light-matter-interaction theories from ancient
Greece to modern times. Of course the theories in antiquity are not using the scientific method
and were part of a philosophical discourse, although it is quite impressive how close some of them
came to our modern understanding of nature without experimental confirmation. During the Golden
Ages of science in Islam, the Book of Optics by Ibn al-Haytham studied properties of light and came
to conclusions that heavily influenced the scholars during the Renaissance in western and central
Europe. Galileo, Fermat, Hooke, Descartes and Newton set important milestones in this journey
and by the 1700s light and matter had been studied extensively –and separately– for two centuries.
In one faction matter was studied by researchers such as Boyle, who working on the laws of gasses
and Dalton, calculating atomic masses, while Lavoisier realized that mass is conserved in chemical
reactions and Mendeleev organizing the different elements in the periodic table. The other side was
in parallel studying light, with Fresnel and Young demonstrating the wave properties of light, disput-
ing Newton’s by-that-time established theory that light consists of particles{II} and culminating with
Maxwell’s equations of electromagnetism. By the end of the 19th century, the black-body radiation
{II}What a fascinating journey this has been!
3
4 Light-atom interactions 101
and Michelson-Morley experiments made clear that the picture of light and matter was not accurate
and new theories were needed to explain nature [26]. The 20th century established modern-physics
theories of quantum mechanics, special and general relativity and returned to a more unified picture
of light and matter. We will concern ourselves with light-atom interactions, since this thesis belongs
to the realm of atomic physics. What is interesting to note at this point is that so far we under-
stand light-micro-world interactions and light-macro-world interactions (through electromagnetism
and general relativity), but we have not managed to connect the two worlds together. Connecting
quantum mechanics to gravity is arguably the holy Grail of modern physics.
Figure 1.1: A brief history of light-matter theories.
Atoms and their structure 5
1.2 Atoms and their structure
When it comes down to it, our experiments are all interactions between light and atoms. Therefore,
this thesis could not begin without properly introducing to the reader the basic properties of atoms
and how these interact with electromagnetic waves. The atoms that interest us in this thesis, belong
to group 1 of the periodic table which includes hydrogen (H), lithium (Li), sodium (Na), potassium
(K), rubidium (Rb), cesium (Cs), and francium (Fr). Since I worked only with Rb and Cs, these will
be the atoms more referred to throughout the next Chapters.
One of the basic principles of quantum mechanics states that electrons are distributed not randomly
around the nucleus of an atom, but in discrete, allowed energy levels. What alkali atoms have in
common is their ground-state {I} electronic configuration: they all have one electron in the outermost
layer, the layer being an s-orbital. This electron has a low binding energy and can be manipulated
with lower energies {II} and visible light.
The simplest atom is the hydrogen atom, which consists of a proton and a singe electron. Hence, the
properties and energy levels of the outermost electron of the alkali atoms can be closely described by
the formalism of the hydrogen atom. By solving the Schrödinger equation for the hydrogen atom in
spherical coordinates, we get the solution of the equation: the spherical harmonics for the angular
part and the radial functions for the radial part. These functions describe how the electron moves
around the nucleus and are characterized by the integers n, l, and m which are called quantum
numbers (See Table 1.1 and 1.2).
Spherical m
0 √
harmonics Yl (θ, ϕ)
l = 0 m = 0 Y0 = 1/( 4π√)
l = 1 m = +1 Y 1,−11 = ±√1/2 3/2π sinθ e±iϕ
l = 1 m=0 Y 01 = 1/2 3/π cosθ
Table 1.1: Hydrogenic wavefunctions: spherical-harmonics part.
Radial functions Rnl(r)
n = 1 l = 0 3/2R10(r) = 2 exp{−r/a0}/a0
n = 2 l = 0 R20(r) = 2(1− r/2a0) exp{−r/2√a0}/(2a
3/2
0)
n = 2 l = 1 R21(r) = 1(r/a ) exp{−r/2a }/ 3 (2a )3/20 0 0
Table 1.2: Hydrogenic wavefunctions: radial part.
For the radial and angular equations to have physically accepted solutions the quantum numbers
must fulfill the following conditions [27]:
n = 1, 2, 3, ... , (1.1)
{I}The lowest allowed energy state of an atom
{II}Lower with respect to the transition energies in other atoms
6 Light-atom interactions 101
0 ≤ l ≤ n− 1 , (1.2)
−l ≤ m ≤ l . (1.3)
These physically acceptable solutions describe the conserved quantities in our atomic system. The n
is the principal quantum number and it determines the allowed energy levels (which are infinite
but distinct). The angular momentum of an electron is defined by l, called the azimuthal quantum
number and m is the magnetic quantum number. These quantities are conserved and precisely
known.
1.2.1 Fine structure
The careful reader will notice at this point that there is a last important number missing, in order to
be able to fully describe the quantum state of an electron: the spin, the intrinsic angular momentum
of (in our case) an electron. This is not derived from the Schrödinger’s equation, but was first added
experimentally and then the Dirac equation theoretically explained.
When talking about angular momenta (spin and orbital) we have to keep in mind that these produce
magnetic fields and as a result there is also a magnetic moment µ associated to those angular mo-
menta. We can imagine the magnetic moment to be a vector perpendicular to the surface the current
loop is defining.
Figure 1.2: Classical view of a magnetic moment produced by an electric current.
An electron can thus can have a magnetic moment associated with its orbital angular momentum
l{I}:
{I}Bold letters represent vectors
Atoms and their structure 7
µl = −
l
glµB (1.4)ℏ
with magnitude [28]:
| | = |e|
√
µl 2 ℏ l(l + 1) , (1.5)m0
and it also has its own angular momentum s (spin) with magnitude:
√
|s| = ℏ s(s+ 1) , (1.6)
with the z-component equal to:
sz = msℏ , (1.7)
and a magnetic moment associated with spin:
µs = −
|s|
gsµB . (1.8)ℏ
In an external magnetic field the spin takes two values as depicted in Figure 1.3, where V is the po-
tential energy of the magnetic dipole, B0 is the external magnetic field, ms is the magnetic quantum
number, g is the g-factor{I}s of the electron (also called electron spin g-factor) characterizing the
magnetic moment and the angular momentum of the electron and µB is the Bohr magneton – a
quantum unit of magnetic dipole moment equal to:
µB =
|e|
2 ℏ ≈ 9.27 · 10
−24 J ≈ 0.93 · 10−20 erg G−1 (1.9)
m0 T
The spin takes only two values and we can observe the z component, but not the x and y. We will
get back to this fundamental principle in the following Sections.
In our experiments we work with multi-electron atoms and not with Hydrogen. Since no two electrons
in an atom can have an identical set of quantum numbers only a specific number of electrons occupy
each of the n electron shells. The total angular momentum of the completed shells{II} is zero.
This is why we are interested in the outermost shells that contain valence electrons{III} and their
quantum numbers, which luckily for us, in alkali atoms is only one electron–with a non-zero angular
momentum, since the shell is not completed.
{I}Measured to be gs = -2.00231930436092(36) [29]
{II}A complete shell contains the maximum allowed number of electrons
{III}Valence electrons are in the outermost shell of an atom
8 Light-atom interactions 101
Figure 1.3: The projection of the electron spin in an external magnetic field takes one of two
values.
The quantum numbers for multi-electron atoms are written with capital letters. The interaction of
the outer electron’s spin S with its orbital momentum L is called spin-orbit coupling and leads to
the fine structure splitting of the excited state. The fine structure can be experimentally observed
by the splitting of the spectral lines of an atom due to the spin-orbit coupling.
If we want to be scientifically precise we have to talk about the angular momentum states as prob-
ability distributions (more about this can be read in [30]). Luckily for us, there is an easier way to
understand and picture the coupling of angular momenta and this is their representation as vectors,
where the total angular momentum equals to:
J = L +S , (1.10)
with its magnitude being:
√
|J | = J(J + 1) ℏ , |L− S| ≤ J ≤ L+ S , Jz = mJℏ , mJ = J, J − 1...− J , (1.11)
where J is the total angular momentum, L is the orbital angular momentum and S is the internal
angular momentum (spin). In Quantum Mechanics it is impossible to know the exact direction
in space of the angular momentum. The maximal information we can have is one projection of the
electron’s angular momentum on an axis; we call that axis the quantization axis [30]. Traditionally
it is chosen along z and we denote the projection on it as Jz.
An energy level of an atom can now be described by its own term symbol which describes all the
quantum numbers and is given by:
Atoms and their structure 9
n2S+1LJ . (1.12)
A small, but important, note: If the spin-orbit coupling is weak, we can make the assumption that the
orbital angular momenta of the individual electrons can add to the total orbital angular momentum
L. This case is called LS-coupling and is not valid for heavier elements where the J-J coupling ap-
proximation is used. J-J coupling becomes a better approximation as the electrons become more
relativistic.
1.2.2 Hyperfine structure
We have known for more than 100 years, that an atom consists not only of electrons, but also a
nucleus –and you can rest assured that the nucleus affects the spectral lines as well. There is an
even smaller splitting than the fine structure occurring due to the interaction of the nuclear angular
momentum (nuclear spin) I and the electronic angular momentum J: their coupling leads to the total
angular momentum F, which can also be depicted in a vector form with:
F = J + I . (1.13)
If the nuclear spin is smaller than or equal to the electron angular momentum, then the number of
possible values of F is 2I + 1. If the nuclear spin is larger then the angular momentum, the number
of possible values of F is 2J + 1 [30].
In this thesis we used three types of alkali atoms: 87Rb, 85Rb and 133Cs. As an example, the 87Rb
ground state can be written as:
52S1/2 (1.14)
We know that 87Rb has nuclear spin I = 3/2 and hence it splits into two F hyperfine energy levels:
F = 1 and F = 2.
It is important to note that there is another shift happening that we have not talked about so far that
is, however, a QED effect. The fine structure does not predict different energies between states with
the same J , but such a difference was experimentally seen in the Hydrogen atom [31]. The Lamb
shift is what causes this split and is an effect arising from the quantization of the electromagnetic
field: In addition to interacting with the proton the electron interacts with vacuum fluctuations. As
a result, distribution of the electron’s position gets blurred over some particular scale, meaning the
electron gets slightly pushed away from the nucleus, lowering their interaction energy. As a result
electrons closer or further from the nucleus will experience energy shifts from this effect differently
and hence the difference between s and p.
10 Light-atom interactions 101
Figure 1.4: Illustration of the energy-levels shift due to the different quantum numbers: each
line represents a specific energy. Each column shows the energy levels explained by models and
corrections of them: Bohr introduced the n principal quantum number. Dirac predicted the electron
spin; when applied to the hydrogen atom, Dirac’s equation contained a potential energy term L ·S,
resulting to the fine structure energy shift. The Lamb shift accounts for a small “push” of the
electrons position away from the nucleus and is related to quantum field theory. The Hyperfine-
structure energy difference is arising from the interaction between I and J . Finally, an external
magnetic field will shift the F states (Zeeman effect).
Atoms in a magnetic field 11
1.3 Atoms in a magnetic field
1.3.1 Zeeman effect
The next step in our understanding of light-atom interactions would be to look at how an external
magnetic field affects the energy states of an atom inside of it.{I}. This step would conclude our
picture of how the energy levels of an atom shift due to different quantum numbers, see Figure 1.4.
In 1896 Zeeman noticed that the spectral lines of atoms are split when the atoms are placed inside a
static magnetic field [27]. The (potential) energy ∆E that an atom gets from an external magnetic
field B = Bẑ is given by [30]:
∆E = −µ ·B . (1.15)
Let us for simplicity assume that S = 0; then the total angular momentum J coincides with the
orbital angular momentum L [30].
There are a few magnetic moments associated with angular momentums, for example µL with the or-
bital angular momentum, µS with the spin angular momentum, µJ with the total magnetic moment
J and the µF related to the total angular momentum F .
Let us calculate this acquired energy shift: For the equation γ = e/(2mec) = µB/ℏ and from equation
1.5 we get that the magnetic moment associated with L is:
µL =
µB
γL = − L , (1.16)
ℏ
where γ is the gyromagnetic ratio and µB is the Bohr magneton.
If the magnetic field is along z, knowing that
Lz = mLℏ , (1.17)
and rewriting (µL)z as −µBmL, the energy shift is:
∆EmL = −(µL)zB = µBBmL , (1.18)
Eventually we want of course to know the energy splitting of the magnetic sublevels, so for the above
case the energy difference between the initial i and final f state is given by:
{I}Atomic states are also split when the atom is under the influence of a static electric field. The system then behaves as
if it has an electric dipole moment that can orient itself to the field. This is called the Stark effect. It can and does to
some extend affect our experiments due to our CW lasers, but not as much as the Zeeman effect
12 Light-atom interactions 101
∆Ef −∆Ei = µBB(mf iL −mL) . (1.19)
Other examples include the magnetic energy shift of a magnetic sublevel mJ for a fine-structure
state, that would now be [30]:
∆EmJ = −(µJ)z ·B , (1.20)
and for a hyperfine-structure state the Zeeman shift would be:
∆EmF = −(µF )z ·B . (1.21)
How do we measure this energy shift? As we will see in Section 2.3 there are rules that determine
which sublevels couple by electric-dipole transitions, called selection rules. To reveal a small teaser
on what the selection rules for the magnetic quantum numbers are:
∆mL = mfL −m
i
L = 0,±1 , (1.22)
which give three (3) possible values for the energy split. This effect is the normal Zeeman effect.
Remember that we assumed the spin to be zero, so this effect can be described and predicted by
non-relativistic quantum mechanics– hence the name normal in contrast to the anomalous Zeeman
effect that takes the spin into account. Section 5.1 will provide a more detailed look into the Zeeman
effect as well as the nonlinear Zeeman (NLZ) effect that appears for higher magnetic fields and is
significant for Earth-field magnetometry.
1.4 Polarized Atoms
So far we have described the atomic states of a single atom. In the lab, however, we use an ensemble of
atoms: one of our cells contains, in room temperature, about 1010 atoms per cm3. If now we want to
model such an ensemble using wave functions we would have to know the wave function of each atom
and add them up, since every atom is described by a different wave function (assuming the atoms
are not all in the same state). This is not only impossible computationally, but most importantly,
we have no way of measuring experimentally each wave function [30]. In our context we call such a
sample unpolarized, meaning there is equal probability to find an atom in any Zeeman sublevel. In
contrast to that, an unequal distribution of different Zeeman levels is called polarization. A sample
can have different degrees of polarization, but a sample with all its atoms having the same internal
quantum state would be in a pure state or else would be fully polarized.
Polarized Atoms 13
Figure 1.5: Spin states in a fully polarized (left) and unpolarized ensemble (right).
1.4.1 Density Matrix
To be able to describe an unpolarized sample we need to generalize the wave function. We introduce
a new operator called the density operator represents an average over the wave functions [30]:
1 ∑N
ρ = |ψi⟩ ⟨ψi| , (1.23)
N
i=1
where N is the number of atoms and ψi the atomic wave function of an atom i. The matrix elements
n,m of the density operator form the density matrix:
ρmn = ⟨m| ρ |n⟩ . (1.24)
The density matrix describes both the polarized and unpolarized ensembles as well as the partially
polarized which, unsurprisingly, is also the real case for our experiments. We get the expectation
value of an observable A by calculating the expectation value: < A >= Tr(ρA). The density matrix
satisfies also the conditions: Tr(ρ) = 1 and ρ = ρ†.
Two extremely important properties arise from the density matrix: coherences and populations. The
diagonal elements of a matrix represent the probability of populations found in specific sublevels. As
population of a state we define the probability of finding an atom in that state. The off-diagonal
elements are called coherences which means they represent superpositions between the basis states
of the density matrix{I}.
{I}A quantum superposition of states happens when a system is in all of these states at the same time, meaning the sum
of these states is a state of its own [32]
14 Light-atom interactions 101
The properties of the density matrix can help us identify the state of polarization of our ensemble
of atoms: If it is unpolarized– and considering for simplicity the two spin states –there is equal
probability to find an atom in any of the two states, up or down. That would translate in "density-
matrix" terms as both diagonal elements being equal to 1/2. The off-diagonal elements would be
zero since there is no connection between the states; the phase between them is random{I} [30]:
[ ]
1/2 0
ρmn = . (1.25)0 1/2
Note that the basis of the density matrix is the one that determines whether the coherences are zero
or not zero (while populations are left unchanged). In Chapter 5 we will get back to that property
which is the key to some of our experiments.
Two-level system
At this point it is useful to introduce the two-level system: “Consider a physical system having two
states whose energies are close together and very different from those of all other states of the system.
Assume that we want to evaluate the effect of an external perturbation (or of internal interactions
previously neglected) on these two states. When the intensity of the perturbation is sufficiently
weak, it can be shown that its effect on the two states can be calculated, to a first approximation,
by ignoring all the other energy levels of the system. All the calculations can then be performed
in a two-dimensional subspace of the state space” [33]. In short, a two-level system consists of two
(2) discrete energy states–usually called the ground and the excited state– separated by a certain
energy/frequency gap ω0. Two parameters that are important here are the detuning ∆ = ω − ω0
and the Rabi frequency ΩR which describes the strength of the atom-field coupling (See Section
2.3.1). It is a useful and simple model for understanding and studying important phenomena that
are characteristic of quantum mechanical systems. We will later see examples of two-level systems
relevant for this thesis.
1.4.2 Liouville equation
The systems we study are evolving in time either due to internal (atoms move and collide, internal
structure) or external interactions (electric or magnetic fields, initial conditions). The rate of change
of the density matrix is often described by a first order differential equation called the Liouville
equation [34]:
(∂/∂t)ρ = (1/iℏ)[H , ρ] + L(ρ) . (1.26)
The term (1/iℏ)[H , ρ] is following from the time-dependent Schrödinger equation. The term L(ρ)
represents the interactions between our system and the environment. These interactions are the
relaxation and the pumping mechanism.
{I}Population represent a real physical property so the diagonal elements have to be positive or zero
Polarized Atoms 15
Figure 1.6: Illustration of a two-level system.
Relaxation– coming to equilibrium with the surroundings –can happen when the system gets de-
polarized when atoms collide with the walls or with each other, due to field gradients, and local
field-differences. We will get back to that in Chapter 3. As far as pumping is concerned, it is the
process by which one polarizes or depolarizes a system and we will talk about it in more detail in
Chapter 2.
1.4.3 Angular momentum probability surface
It is notoriously hard to visualize quantum mechanics. To put it in Heisenberg’s words: “Quantum
theory provides us with a striking illustration of the fact that we can fully understand a connection
though we can only speak of it in images and parables.” I dare to say that images are usually not
included in our understanding of quantum mechanics (QM). The visualization often reduces the
scientific accuracy and researchers try to avoid it all-together, but, on the other hand, it also helps
simplifying and understanding difficult concepts.
Luckily for us, there is a precise technique to visualize angular quantum polarization states of a
density matrix, called angular momentum probabilities. For that, we calculate the probability
of finding a magnetic sublevel along the quantization axis. If we want to find this probability along
another axis we have to rotate the total angular momentum state. If we now plot this probability as
a function of the direction of the quantization axis we get the angular momentum probability
surface, as is illustrated in Figure 1.7 below (the illustration is based on the computated probability
in [30]).
We can see directly from the plot what the rotational symmetry properties of the density matrix
are. For example, in this case, the only preferred direction is along the quantization axis z. The only
direction we can measure the maximal component of the angular momentum is −z with a probability
equal to 1. The symmetry axis might change, but not the symmetry of the surface around that axis.
16 Light-atom interactions 101
Figure 1.7: Illustrated angular-momentum probability surface of the density matrix ρ. The prob-
ability to measure the maximal component of the angular momentum along −z is one (1).
The symmetry arising from these visualizations plays an important role– as we will see in Chapter 2
and 5 –in our experiments.
1.4.4 Quantum beats
Coherences between quantum states are represented by the off-diagonal elements of the density mat-
rix and can be understood as quantum superposition between the basis states. In our case, it is often
useful to work with coherences and hence we choose our quantization axis to be along the magnetic
field. Of course this is a matter of convention, so it should not make a difference in the experiment
what axis we choose. Interestingly though, if the quantization axis is chosen to be along the magnetic
field, there is no change in the populations: what is oscillating then, is the coherence between two
states. So why do we bother with the more complicated idea of coherences? At this point it is useful
to introduce into the discussion the idea of quantum beats.
Quantum beats is the general term used to describe the time evolution of a coherent superposition
of non-degenerate energy eigenstates at a frequency determined by their energy splittings [35], for
example, the splittings happening in the NLZ effect (see Section 3.4.1).
This phenomenon happens if, for example, we have a pump that stretches atoms along y and a
leading magnetic field along z that causes NLZ splitting. We apply a probe beam also polarized
along y that drives an oscillation modulated at the difference of the frequencies between the magnetic
sublevels: it is modulated at different beats. If we choose the quantization axis along the leading field
(z direction) we will have coherences oscillating between the Zeeman sublevels which are, in this case,
the energy eigenstates{I}. What this practically means is that, when we drive a resonance between
two states, this resonance has a clear oscillating frequency, which we can simply imagine as a rotating
polarization surface whose rotation we detect as our signal.
{I}Along the quantization axis the angular momentum eigenstates are also the energy eigenstates
Polarized Atoms 17
Using the language of superposition between different energy eigenstates (here, different mF states
along the magnetic field), helps to work with coherences which will oscillate at a single frequency and
that makes it is easier for us to count the number of oscillations produced by the NLZ effect.
18 Light-atom interactions 101
ich
denke ist
etwas
("Ich" = I, "denke" = think, "ist" = is, "etwas" = something)
Max Bense, Cartesian concrete, 1966
2
Nonlinear light-atom interactions
So far, we talked about atoms, their energy levels, how these shift with magnetic fields and how we
can describe an ensemble of many atoms. We also defined and visualized atomic polarization, but did
not explain how it could be achieved. We surely can make an educated guess: We know that magnetic
and electric fields influence atoms in a plethora of ways and we know that light is an electromagnetic
wave. It comes as no surprise therefore that we use light-atom interactions to manipulate atoms in a
way that suits our research goals.
In this Chapter we will discuss about nonlinear interactions, one of which is the mechanism called
optical pumping that allows us to polarize atoms, and other nonlinear interactions between light and
atoms that are key to the various effects studied throughout this thesis. Keep in mind that in this
Chapter we switched to the International System of Units (SI) due to the nomenclature commonly
used throughout the literature of nonlinear light-atom interactions.
2.1 Linear magneto-optical rotation
2.1.1 Faraday effect
In order to talk about nonlinear effects, it is intuitively helpful to first explain linear effects between
atoms and light. One of the pioneers in electromagnetism was Michael Faraday. He studied a
great amount of phenomena around electromagnetism and electrochemistry, the magneto-optical
Faraday effect being one of them.
19
20 Nonlinear light-atom interactions
A magneto-optical effect is a phenomenon that appears when an electromagnetic wave (in our case
light) propagates through a medium (in our case atoms) whose properties have been altered by an
external magnetic field.
The Faraday effect is one of the most well known linear magneto-optical effects. It occurs when
linearly polarized light that is transmitted through a medium undergoes rotation of its polarization
plane, while a magnetic field is applied along the direction of propagation. Polarization{I} is a
property of light’s wave nature that describes the orientation of the oscillations in space. In linear
polarization, the electric- and magnetic-field vectors are oscillating in a specific plane along the
direction of propagation. The angle of rotation ϕ is proportional to the strength of the magnetic field
B, the length of the sample l and the properties of the medium [36]:
∫ l
ϕ = V Bdl = V lB , (2.1)
0
where V is wavelength, medium and temperature dependent [37]. Figure 2.1 visualizes the Faraday
effect. If the magnetic field is perpendicular to the light’s direction the effect is called Voigt effect
in a gas and Cotton-Mouton effect in liquids [38].
Figure 2.1: Linearly polarized light with electric field E enters the cell from the left. Its polariz-
ation plane rotates after it propagates through the medium with length l, that is inside a magnetic
field along the light’s propagation direction. The polarization of the field E′ is now rotated by an
angle ϕ.
What causes the Faraday effect is the difference of complex refractive indices of the polarization
eigenmodes {II} that light can be decomposed into. For example, two polarization eigenmodes of
light can be left (σ−) and right (σ+) circularly polarized waves [38]. If the refractive index of these
two eigenmodes is different, then they also propagate with different velocities through the medium;
{I}Not to be confused with atomic polarization
{II}In general decomposing a field into eigenmodes is a well established technique in physics. As an example, electron
orbits are eigenmodes of the energy, angular momentum, and spin operators [39]. Polarization eigenmodes, in particular,
is a state of light that during its propagation through a medium can experience other changes, but its polarization is always
invariant.
Linear magneto-optical rotation 21
this phase difference leads to the rotation of the polarization plane. Polarization of light plays a great
role in our systems and we will talk about it in more detail in Section 2.4.1.
2.1.2 Macaluso-Corbino effect
Macaluso and Corbino studied the Faraday effect around the turn of the 19th century. They dis-
covered that the Faraday effect had a distinct resonant character close to the resonance of absorption
lines{I}. That means that a strangely large Faraday rotation is observed when the frequency of
the transmitted light is close to that of two energy states of the atoms in our medium. The main
mechanism of the linear Macaluso-Corbino effect is illustrated in Figure 2.2:
Figure 2.2: Illustration of the Macaluso-Corbino effect.
When linear light– which consists of the two circular components σ+ and σ−– interacts with a two-
level system in zero field, the resonant frequency for the two components is the same. When we
apply a magnetic field however, the ground-state-sublevels’ degeneracy is lifted and now there is a
difference of ΩL between the sublevels. The difference between the two refractive indices, associated
to the polarization, as we will see next, gives a difference in the phase velocities of the two components
which is responsible for the rotation of the plane of polarization of the light and gives a characteristic
spectral shape, [40], see Figure 2.3.
The Maculuso-Corbino effect is not linear in the magnetic field when the splitting of the Zeeman
sublevels becomes comparable to the spectral width of the absorption line, in this case: when ΩL ∝
ℏ Γ. We will discuss later in more detail about spectral widths.
{I}An atom can absorb photons with energies equal to the difference between two energy states; this energy difference we
call absorption line.
22 Nonlinear light-atom interactions
Figure 2.3: Illustration of the characteristic spectral shape of Malacuso Corbino effect and its
width.
2.2 Nonlinear optical effects
A nonlinear light-atom interaction is an interaction between a medium and light in which the
behaviour of the medium (its induced polarization P) depends non-linearly to the electric field E
of the light. We will next investigate how to relate the microscopic theory for a single atom to the
macroscopic observations that we make in the lab by using large ensembles of atoms [41].
2.2.1 Classical formalism – macroscopic
In a medium we can not only talk about one atom, so when we mention polarization P of a medium
with N atoms this equals to the dielectric polarization density or electric dipole moment per unit
volume (P=Np) i.e. the redistribution of charges polarizes the medium. In a material that is dielec-
tric{I} the two charges in it (nucleus and electrons) can get displaced by applying an electric field
E [42].
This creates an electric dipole moment with the electric displacement field –for the whole material–
defined as:
D = ϵ0E+P , (2.2)
where ϵ0 is a constant that represents the ease by which electric field lines are formed in vacuum. If
our medium is linear{II}, homogeneous{III}, nondispersive{IV} and isotropic{V} then:
P = ϵ0χE , (2.3)
{I}Material that can be polarized by an electric field.
{II}P ∝ E
{III}Properties are independent of the position in space.
{IV}Speed of a wave does not depend on frequency of wave.
{V}Properties are independent of the direction in space.
Nonlinear optical effects 23
where χ is the constant of proportionality called electric susceptibility of the medium. That
is, if the assumptions me made about the behaviour of the medium hold true. If the medium is
anisotropic– changes properties based on the direction →−r – then the susceptibility χ is a tensor.
Figure 2.4: A simple illustration depicting the polarization of an atom without a field (a) and the
induced polarization p of an atom when an electric field is applied (b): the minus stands for the
electrons and the plus for the nucleus.
As the electric field E(t) becomes stronger the polarization becomes bigger as well. However, after
the applied field becomes too large or when its frequency is close to a resonance, it starts to be
comparable to the the intra-atomic electric field of the medium [42]. When this happens the electron
cannot follow the oscillation of the electric field E anymore and it responds in an unharmonic way
[43]. The medium’s properties change and its behaviour (susceptibility) is no longer linear, but
depends on the applied light field [44]. We can describe this nonlinearity by a Taylor series expansion
in the electric field:
∑∞
P = ϵ χ(n)En = ϵ [χ(1)E + χ(2)E2 + χ(3) 30 0 E + ...] , (2.4)
n=1
where the expansion coefficients correspond to the χ(n) nth order nonlinear susceptibility, a n+ 1
rank tensor.
The χ(1) processes are linear processes. The χ(2) processes and above are nonlinear, but there is
another important distinction here: χ(2n) processes are not existing in centrosymmetric media, where
all directions are equivalent, because χ(2n) processes require that the sign of (2n)χijk changes when
the axis reverses to -i-j-k. Homogeneous gases, such as our atomic gases, fall into this category.
Hence, in our experiments we work with 3rd order nonlinear processes such as the nonlinear magneto-
optical rotation. This approach is useful for understanding the beginning of a given effect with
increasing light power and works well when we deal with nonresonant light. However, when we have
resonant laser excitation it makes no sense to distinguish between different perturbative orders once
24 Nonlinear light-atom interactions
the transition is saturated, since the Series 2.4 is non-converging and all orders of the expansion are
coupled [40].
The electric displacement D can now be rewritten as:
D = ϵ0E+P = ϵ0E+ ϵ0χE = ϵrϵ0E , (2.5)
where the relative permittivity ϵr = 1 + χ and the refractive index is equal to:
√ √ √
η = ϵrµr = ϵr = 1 + χ , (2.6)
if we ignore magnetic effects. The refractive index has an imaginary and a real part which are asso-
ciated to two physical properties [45]: The imaginary part describes the absorption of the photons
(scattering in all/random directions) and the real part describes the dispersion of the light (its speed
propagation). The susceptibility{I} is related to the refractive index through the above formula and,
as a result, we can infer the absorptive and dispersive properties of the medium by knowing the
susceptibility of a medium.
2.2.2 Quantum mechanical formalism – from microscopic to macroscopic
For an atom in an electric field E(t)êE , with ê being the unit vector along the light-polarization
direction and amplitude E0, the Hamiltonian is given by
H = −dêEE0 , (2.7)
where d is the electric-dipole operator that is written for a charge q as
d = qr , (2.8)
where r is position vector from the charge to a point around it and whose matrix elements are
important for the light-atom interactions and atomic transitions. In our case, q is equal to −e (the
electron-charge magnitude) and r is the position of the electron with respect to the nucleus.
The expectation value of the dipole operator is ⟨d⟩ = Tr(ρd) (see 1.4.1) and the polarization P of a
medium with atomic density n (number of atoms per unit volume of that material) is:
∑
P = nTrρ(E)d = êE dgeρeg + c.c. , (2.9)
g,e
and
{I}The susceptibility is frequency dependant and this dependence is related to the dispersion properties of the material
Nonlinear optical effects 25
dge = ⟨e| d |g⟩ , (2.10)
where d is the electric dipole moment operator, dge is the dipole matrix element related to a transition
from the ground state |g⟩ to the excited state ⟨e|, ρ(E) is the light-dependent density matrix, ρge
refers to the the optical coherence, E0 is the amplitude of the applied electric field, and êE is the unit
vector [38].
We have talked about electric dipoles so far, but other kinds of multipole moments may be created,
such as magnetic dipoles and electric quadrupoles. In order to distinguish between linear and non-
linear effects, we looked at how the properties of our medium P change with the electric light field E.
In order to create a more systematic way of classifying these effects and including all types of atomic
polarization we solve the density matrix with the pertubative approach [30]. We expand the density
matrix ρ(E) in a power series:
∑∞
ρ(E) = ρ(n)En0 . (2.11)
0
the term ρ(0) describes the density matrix in the absence of light. The terms ρ(1) and ρ(2) are
associated with optical effects that are linear and hence are called linear processes. All higher-
ranking terms are called nonlinear processes. The ρ(2) order of perturbation is responsible, amongst
other effects, for the quantum beats. Due to ρ(2) the density matrix is quadratic in the electric field,
but the optical coherences within the ground and exited states through this order are linear in the
electric field. The optical properties of the ensemble are determined by these coherences and so we
consider the effects of this term to be linear. In general, ρ(n) with n odd describe optical coherences,
for example ρ(1) could describe linear absorption. Optical coherences are coherences between the
opposite-parity lower and upper states of the optical transition{I}– see next Section 2.3.1. A ρ(n)
with even n describes density matrix elements with the same parity. More about it can be read
in [30] Chapter 10: Light-atom interaction observed in transmitted light.
This approach is useful if we want to understand the nonlinear optical phenomena, but it is not
applicable in every case. If we have light strong enough to saturate a transition (see Section 2.4.2)
or resonant light, then we need to solve the Liouville equation 1.4.2 which would make our solution
more precise, but computationally and conceptually harder [38]. Keep in mind that the phenomena
and effects that are going to be studied in this thesis are often complicated as they are a combination
of processes of different ρ orders.
{I}If the orbital angular momentum is L, then the parity of the energy level is (−1)L for single-electron atoms.
26 Nonlinear light-atom interactions
2.3 Atomic transitions
2.3.1 Rabi frequency
We saw in Chapter 1 that the density matrix describes both polarized and unpolarized ensembles
and two important properties arising from it are populations and coherences. In this Chapter we go
back to the microscopic level in order to describe some important mechanisms of light interacting
with an atom and more specific with a two-level system. The populations and coherences of the
energy states of an atom can change through light-atom interactions and these changes are described
by atomic transitions. In atomic physics generally, as well as our experiments in particular, we care
immensely about the evolution of an atomic system under the action of a periodic perturbation. For
us this plays a huge role, since we study light-atom interactions, which means we couple a two-level
system with an optical field–– an oscillating field.
Figure 2.5: A two-level spin-1/2 system.
To be able to study such a system and define some important concepts we will consider the simplest
case of a spin–1/2 system. This two-level system has two energy levels (two independent quantum
states): spin projection 1/2 and spin projection −1/2. We now put this system inside a static
magnetic field B0ẑ. Now this system has a lower state and an upper state that correspond to the two
angular-momentum projections on z. The energy difference between the lower state and the upper
state corresponds to an energy interval of:
∆E = ℏΩL = gµBB0 . (2.12)
We consider next a weak oscillating magnetic field with frequency ω transverse to the leading field
B0 that we will call B1 and want to see how the state of this system will change with time. The trick
to remove the time-dependence is to make a frame rotation to a frame where B1 is stationary along
y (or x). In the rotating frame the time evolution of the spin state is just Larmor precession around
the effective field along ẑ [30].
Atomic transitions 27
In the resonant case, where B1 rotates at the Larmor frequency ΩL, we practically don’t care about
the leading field B0 and the system’s evolution is just Larmor precession around B1. If ω ̸= ΩL (and
assuming B1 = B1ŷ) then we have Larmor precession around:
(
− ℏ
)
ω
B0 ẑ +B1ŷ , (2.13)
gµB
where (B ℏω0 − gµ ) is the effective field along z. Let us go back to the resonant case again, where theB
spin state just rotates around y. At time t, we define the Rabi frequency as the frequency of Larmor
precession in the field B1:
ΩR = gµB1/ℏ , (2.14)
[ ]
0
If we apply the rotation matrix to our system’s wavefunction ψ(t = 0) = we get:
1
[ ] [ ] [ ]
cos(β2 ) −sin(
β ) 0 −sinΩRt/2
ψ(t) = 2β = , (2.15)sin( 2 ) cos(
β
2 ) 1 cosΩRt/2
where the rotation angle around ŷ is β = ΩRt. The probability to find the spin in the 1/2 state is:
= 1− cosΩRtP1/2 2 , (2.16)
and in the -1/2 state:
= 1 + cosΩRtP−1/2 2 . (2.17)
These probabilities are oscillating as a function of time with a period equal to the inverse of the Rabi
frequency.
We can generalize this problem for any two-level system under the influence of a periodic perturba-
tion. In the case that interests us, where light couples to a two-level system, the Rabi frequency is
ΩR = dE0/ℏ, where E is the electric field and d is the electric-dipole matrix element.
2.3.2 Stimulated emission
Rabi oscillations cause the system’s population to transition periodically between the two states. The
atoms will be in the energetically lower state initially, but the Rabi oscillations will cause them to
periodically go to the excited state, gradually increasing the probability of the system being found in
the excited state. This happens through the mechanism of absorption: the system absorbs the energy
of a photon and undergoes a transition from a lower energy state to a higher energy state. Under
Rabi oscillations, the probability of the quantum system being in the excited state gets increased
28 Nonlinear light-atom interactions
Figure 2.6: The probability of finding the spin in the 1/2 or -1/2 state oscillates with time.
and absorption occurs less and less with time. There are two important processes that can happen
next: spontaneous emission and stimulated emission.
Figure 2.7: Absorption, spontaneous and stimulated emission from left to right.
Spontaneous emission is an important relaxation mechanism in which an atom goes from a higher
energy state to a lower energy state, and in this process it emits a photon that corresponds to the
difference of the two energy levels. For this to happen there is no external light-field required, but the
excited states are unstable and tend to decay to lower energy states with a certain decay rate{I}. The
time it takes for an excited atom to go from the excited to the ground state is an intrinsic physical
property related to Heisenberg’s energy-time uncertainty and is called the lifetime τ of the transition.
{I}Excited states with a longer lifetime are called metastable
Atomic transitions 29
This results always in broadening of a spectral line that we call natural linewidth of a transition
Γ0 ∝ 1/τ and it is not related to other sources of broadening like collisions or the Doppler effect.
In the process of stimulated emission, same as in spontaneous emission, an atom goes from a higher
energy state to a lower energy state, but this time this effect is “stimulated” by an electromagnetic
field with a frequency that matches the energy difference between the excited state and a lower
energy state. If our system is in the Rabi cycle in the upper state (not taking into account stimulated
emission) then it will emit photons from the upper (excited) state to the (lower) ground state in the
same direction and with the same phase as the field through the process of stimulated emission. The
resulting light is coherent{I}. Lasers are the prime example of light coherence, which is one of the
properties responsible for their unique properties.
2.3.3 Optical Bloch equations
Section 1.4.2 introduced the Liouville equation as the rate of change of the density matrix. The full
equation of motion for the density matrix, including the Hamiltonian part due to the interaction
with the external field and the spontaneous emission, is [41]:
[ ]
−Γρ Γ
( mm
ρmn
∂/∂t)ρ = (1/iℏ)[H , ρ] + 2Γ . (2.18)
2 ρnm Γρnn
Keep in mind that the diagonal elements of the density matrix ρ correspond to populations and the
non-diagonal to coherences of the corresponding levels. The diagonal elements of the last term show
the time evolution of the populations with a decay rate Γ and the non-diagonal elements show the
effect of the spontaneous emission on the coherences. We can view n as the ground state and m and
the excited state in a two-level system.
The above matrix equation can be written in the form of four (4) differential equations known as
the optical Bloch equations. The equations describe the time evolution of the populations and
coherences in a two-level system. In addition to these equations, the elements of the density matrix
must obey two other constraints: the sum of the populations must be equal to one (1), and the off-
diagonal matrix elements must be complex conjugates [41]. These conditions can be written as three
(3) independent components {u, v, w} of a vector called the Bloch vector and are proportional to:
u: the in-phase component of the atomic dipole moment,
v: the quadrature component of the atomic dipole moment,
w: the populations of the ground and excited states .
The reason we use this geometrical language is, once again, for visual comprehension: The three (3)
vector components of the Bloch vector are used as three dimensional coordinate system that defines a
sphere called the Bloch sphere. The Bloch sphere is a tool that allows us to see the evolution of the
{I}The phase relationship between different parts of the wave remains constant over time and space i.e. the wave has a
well-defined and stable phase
30 Nonlinear light-atom interactions
Figure 2.8: An illustration of the Bloch sphere.
system as a trajectory and represent a state with a vector from the origin to a point on that sphere.
The radius of the Bloch sphere has a length of one (1)– if we don’t consider spontaneous emission [41].
The diametrically opposite points of the north and south pole of the sphere correspond to the ground
and excited states of our system (w axis). Whenever the vector rests on the equator of the Bloch
sphere that corresponds to an equal superposition of the ground and excited state (50-50 probability
for the system to be observed on either the ground or excited state).
2.4 Optical pumping
You might have noticed by now that we still have not introduced how one polarizes an atom and what
happens with an ensemble of an enormous amount of atoms when light gets transmitted though it.
Pumping is a crucial nonlinear light-atom interaction [30] that allows us to manipulate atomic pop-
ulations and coherences and create what is the basis for a huge range of applications and studies:
magnetization of an ensemble. Through resonant excitation we can polarize enough atoms in an
ensemble to create polarization that manifests as (macroscopic) magnetization.
2.4.1 Selection rules
As everything in Quantum Mechanics, the possible transitions of a system from one state to another
are constrained, in this case by the angular-momentum selection rules. Light interacts with atoms
that are in states called bright states by transferring to these atoms part of the angular momentum
carried by the light beam [45]. The states that don’t interact with the light are called dark states.
Different states have different absorption rates based on the selection rules.
From the Equation 2.6 we know that the Rabi frequency of a driven transition is proportional to
the electric-dipole operator d and the latter is, in turn, proportional to the matrix element of the
Optical pumping 31
position vector r of the electron. Therefore, the probability of an electric-dipole transition to be
induced depends on the dipole moment of the transition, the amplitude of the electric field and the
direction of the light field relative to the atomic dipole moment [30].
For the purposes of this thesis we will not go into the formalism of how to derive the selection
rules or calculate the strength of the transitions. To establish these rules, one must study the non-
diagonal matrix elements (n ̸= n′) of the various atomic operators such as the electric and magnetic
dipoles [33]. The selection rules have their origin in the Wigner-Eckart theorem which states
that the matrix elements of a vector operator are a product of two factors: the first are called the
reduced matrix elements and do not depend on the angular momentum projections m; the second
are called the Clebsch-Gordan coefficients, are orientation dependent and show which states are
coupled and with what probabilities. These coefficients will vanish, unless J +1 ≥ J ′ ≥ J − 1, where
J is the angular momentum of the given state. The above condition is the reason there is only one
(1) unit of angular momentum added to the system when coupling vector operators [46]. A thorough
quantitative approach can be found in the literature [47], [34].
The selection rules in an electric-dipole transition between initial and final states for a single-electron
atom are:
∆S = 0 , (2.19a)
∆L = ±1 , (2.19b)
∆J = 0,±1 , (2.19c)
∆F = 0,±1 , (2.19d)
∆mJ = 0,±1 . (2.19e)
Let us qualitatively explain these rules one by one:
• ∆J = 0,±1 is a strict rule that arises from angular momentum conservation. Since the photon
spin is one (1), the case ∆J = 0 is not allowed for single-photon transitions.
• ∆S = 0 obeys the rule that the electron spin in an optical transition remains constant, meaning
that the spin wave function is preserved– that case is true only for LS coupling.
• ∆L = ±1 The electric dipole operator does not depend on the spin component S. These matrix
elements (see 2.10) vanish unless ∆L = ±1.
• ∆F = 0,±1 is similar to the J rule when the interaction involves an atom with nonzero nuclear
spin I. As long as J is conserved in the system, the selection rule ∆J = 0,±1 applies as well.
• So far, the rules were related to the magnitude of the angular momenta, but they also concern
their projections on the quantization axis which are related to the magnetic quantum numbers
m. The selection rules in this case depend on the polarization of the light. As a convention
32 Nonlinear light-atom interactions
in QED, the polarization is defined with respect to the quantization axis: right-circular (left-
circular) polarization is a property of the light whose polarization vector rotates clockwise
(counter-clockwise) as we look from the source towards the light propagation [30]. Later we will
describe polarized light with respect to a fixed axis (the axis of our magnetic field, usually also
the quantization axis). We call light that is linearly polarized along this axis π-polarized and
light that rotates clockwise (counterclockwise) as viewed from source σ+ polarized (σ−) [48].
Polarized light can be decomposed into the above three polarizations. Now, photons that are
right-circularly polarized (left-circularly polarized) have a projection of angular momentum of
+1 (−1) on the quantization axis. The selection rules are {I}:
For σ+ (right-polarized): ∆mJ = +1 and for σ− (left-polarized): ∆mJ = −1 .
Keep in mind here that in optics when we talk about right and left-circular polarization we
usually define it in the opposite way to the above convention.
Besides the angular-momentum selection rules, there is another important rule related to parity.
Remember that in atoms odd parity is defined when (−1)L = −1 and the opposite is true for even
parity. The electric-dipole moment has odd parity, which means that the initial and final states must
have different parity in order for the matrix elements to not give a zero value. This means that
electric-dipole transitions between states with the same parity are forbidden. We call this the parity-
selection rule. In some of our experiments we are interested in magnetic-dipole transitions, because
these are the ones that occur. In this case, some of the angular-momentum selection rules stay
the same and some change. For example, regarding the parity-selection rule: the magnetic-dipole
transitions couple states with the same parity.
2.4.2 Saturation parameters
Another way to distinguish between linear and nonlinear effects is the decay rate of the ground
and excite states. Nonlinear effects start appearing when the light is resonant or strong compared
to a characteristic saturation intensity. If, with the increase of the light intensity, the properties
of the medium depend on the light, we talk about a nonlinear process. It is important to realize
that the degree of nonlinearity strongly depends on atomic saturation by the light field as well as
the relaxation rate of atomic polarization [40]. There is a gradual transition from the linear to the
nonlinear that happens when the rate by which we excite atoms gets larger than the rate by which
atoms spontaneously return to the ground state i.e the relaxation rate (see equation 1.26):
= excitation rateκ . (2.20)
relaxation rate
To be more exact: The saturation parameter is the ratio of the rates of coherent light-atom in-
teractions (responsible for Rabi oscillations) and incoherent relaxation processes (e.g., spontaneous
decay) [40].
{I}We talk about J for an atom with zero nuclear spin I. If I is nonzero then these rules are the same for the total angular
momentum F .
Optical pumping 33
For a two-level system in which the upper state decays back to the lower state, when the light is
resonant, the saturation parameter for the upper state is:
2 2
κ1 =
d E0
2 2 , (2.21)ℏ γ0
where d is the dipole moment and γ0 the relaxation rate of the upper state, which would make
κ = d2E21 0/ℏ2γ0 the excitation rate. Remember from Section 2.3.1 that for a two-level system the
Rabi frequency is ωR = dE0/2. That would make κ1:
( )
ω 2∝ Rκ1 . (2.22)
γ0
We see that when κ1 ≪ 1 that means that the relaxation rate related to the spontaneous emission is
dominating and atoms are in the lower state. When κ1 ≫ 1 the Rabi oscillation term dominates and
the average population of the upper and lower state is 50–50; that is the nonlinear regime.
There is another case we need to study that is when the upper state decays predominantly to states
other than the lower state of the transition. The light will keep exciting atoms to the upper state,
while the excited state relaxes to the other states eventually leading all the atoms to these non-
interacting states, called dark states. There are processes, such as collisions, that can redistribute
atoms between the ground states at a rate γ.
Figure 2.9: An illustration of a dark state |3⟩ that does neither absorb nor emit photons. This is
not a two-level system, but since it contains three levels it is called a three-level system. It interacts
with the ground state |1⟩ only indirectly through γ. For more details see 3.3.3.
In this case the saturation parameter is:
d2E2
κ 02 = 2 . (2.23)ℏ γ0γ
34 Nonlinear light-atom interactions
If γ is smaller than γ0, which is the case sometimes in our experiments, then the denominator is small
and the fraction gets much bigger, which means that in this system we can have a nonlinear effect
with lower light intensities than in the case of κ1. The decay rate of the upper state stays the same,
but the population gets now redistributed back to the bright state at a slow rate determining the
behaviour of the whole system.
If we have a dark state in the ground levels of our system we are in the case where the saturation
parameter κ2 is relevant. This is the case when for example one pumps with circularly polarized
light or if the right- and left-circularly polarized light have different transition frequencies due to an
external magnetic field. As we will see next, these cases are relevant for this thesis as our experiments
are operating under similar principles.
Optical pumping that creates ground-state population imbalance such as is the case in many atomic
magnetometers, creates also atomic polarization by definition. And any process that depends on
atomic polarization is a nonlinear process since atomic polarization changes the properties of a me-
dium. The saturation parameter κ2 describes the generation of this polarization and the nonlinear-
ities arising from it [30].
2.4.3 Mirrorless lasers
In this context, we can describe an effect that we will comprehensively discuss in Chapter 6 and
7 called degenerate mirrorless lasing. Both saturation parameters play a role in this effect.
The principal idea is that optical pumping redistributes the population among the ground-state
Zeeman sublevels – κ2 is the relevant saturation parameter here. At sufficiently high light powers,
as κ1 reaches 1, the population is partially transferred to the excited state, which, in the case of an
F → F + 1 transition may result in population inversions between certain sublevels of the ground
and excited states. This inversion may, under right circumstances, result in directional emission from
the sample.
In the case of degenerate mirrorless lasing, amplified spontaneous emission is produced along or op-
posite to the pump-beam direction, with polarization orthogonal to that of the pump. The pump
field interacts with the Zeeman-sublevels (mF ) of a specific atomic transition and population inver-
sion is produced between certain magnetic sublevels.
Amplified Spontaneous Emission (ASE)
We already talked about spontaneous emission, a quantum process in which an atom in an excited
state spontaneously de-excites into a lower energy state by emitting a photon (the photon being
emitted into one of the modes of the vacuum). One could also understand this process as stimulated
emission by vacuum noise fluctuations.
Amplified spontaneous emission (ASE) is a phenomenon that occurs often in lasers and optical amp-
lifiers as a result of stimulated emission processes that amplify the spontaneous radiation field that
propagates in a medium with gain [49]. When the contribution from optical gain processes is greater
Optical pumping 35
than that of absorption and radiation that scatters away in a gain medium, we reach the threshold
for ASE.
In atomic gas systems, the spatial coherence and beam divergence of ASE are affected by the geo-
metry of the laser setup (the optically active part of the system) and the internal degrees of freedom
(atomic populations, coherences and spectral lineshape) allowing for higher ASE spacial coherence.
When one also assumes a pencil-like geometry for the pump laser (only a narrow cone of wavevectors
contributes to the ASE modes), large enough atomic density and after calculating the rate of radi-
ative transfer of the light field one can model the dynamics of an ASE field in a cold random atomic
gas, under the approximation that inter-atomic scattering plays little to no effect. For more details
read also [49].
We will delve experimentally into this fascinating topic in Chapters 6 and 7.
36 Nonlinear light-atom interactions
I write these words in steel,
for anything not set in metal cannot be trusted.
Brandon Sanderson, The Well of Ascension
3
Atomic magnetometers
3.1 Simple model of an optical magnetometer
Figure 3.1: Illustration of a basic setup of an atomic magnetometer.
This Chapter is all about atomic magnetometers: what they are, how they work, their types and
properties. After having discussed about the Rabi frequency, relaxation mechanisms, the time evolu-
tion of a two-level system, what optical pumping is and how nonlinear effects appear, we reached the
point in which we can illustrate the basics of an optical magnetometer or OPM (optically pumped
magnetometer).
37
38 Atomic magnetometers
3.1.1 Atomic structure
Many optical magnetometers are based on alkali atoms. A typical structure of the ground state
S and the first excited states of such an atom– in this case 87Rb –can be seen in Figure 3.2. The
transitions from S to the two excited states P1/2 and P3/2 form a fine-structure doublet, and each of
these transitions additionally have hyperfine structure [50].
Figure 3.2: Transition structure of an alkali atom with nuclear spin I = 3/2 such as 23Na, 41K or
87Rb [19].
We call the transitions between the magnetic (Zeeman) sublevels ∆mF , that belong to the same
hyperfine level F , Zeeman transitions. The energy difference between them corresponds to fre-
quencies in the radio-frequency range, while the hyperfine-structure transitions (∆F = 0,±1) are in
the microwave range. The transitions from S to the two excited states P1/2 and P3/2 are optical and
are used for pumping [19]. This is why later we refer our laser lights as D1 and D2, where often D1 is
used for pumping and D2 for probing.
Note here that, for some sublevels, the mF = −1 has a higher energy the mF = 1. Remember that
we calculate the Zeeman energy shift for the hyperfine F states from: ∆EmF = gFµBBmF . The
Lande factors g {I}F for the two hyperfine components of the ground state S1/2 play a role in these
energy shifts. Considering that F = I ± J and because the Lande factor depends on J and F ,
when electronic angular momentum J points in the same direction as the total angular momentum
{I}Lande factor is the g-factor for an electron
Simple model of an optical magnetometer 39
F , gF is positive, and when they are directed in opposite directions gF is negative. This creates this
asymmetry of the signs between various mF sublevels.
3.1.2 Vapor Cells
When we talk about atom-light interactions we know that we use lasers for the light-part of the
interaction (see more in Section 3.5.3). How do we use atoms? We have a piece of alkali-metal that
is placed inside a sealed glass cell, round or cylindrical. Inside the cell a small amount of the solid
alkali evaporates into the gas phase, so that the cell is filled with alkali-vapor. We need to polarize
the atoms/alkali-vapor which in turn should stay polarized for as long as possible.
In order to keep the atoms polarized for a longer time many cells use another gas called buffer gas
inside the cell to prevent the atoms from diffusing away from the pump beam and towards the walls.
Some cells have paraffin coating on the glass walls, so that, when the polarized atoms bounce at the
walls they don’t lose their polarization [51, 52]. In the magnetometer experiments of this thesis we
used paraffin-coated cells.
Figure 3.3: Vapor cell illustration: atoms are not to scale.
3.1.3 Pump
Let’s now present the basics of an optical magnetometer. We have alkali atoms in a two-level system
in a alkali vapor gas, as seen in Figure 3.4, with total angular momentum F = 1/2 for the ground
state and F ′ = 1/2 in the excited state. The ground state, according to the angular-momentum rules
1.11, has two mF sublevels 1/2 and −1/2. In Figure 3.4 in ⃝1 we see the system and its population
before we pump.
Next, we shine light to these atoms which is right-circularly or σ+ polarized, according to our con-
vention. The pump light is propagating along x and its frequency is near-resonant to the F → F ′
transition. When we apply the pump we drive Rabi oscillations, as detailed in Section 2.3.1. Based
on the selection rules 2.19e if we pump with σ+ light, we can only pump between sublevels that differ
by +1. As a result, and as we see in ⃝2 , we have a dark state in our Zeeman ground sublevels which
makes our system similar to the three-level system in Section 2.4.2 and so the saturation parameter
κ2 applies [40]. For now the magnetic field B is zero. We choose our quantization axis along the light
propagation x and we see that, because of the way we can pump, with time, the probability of an
40 Atomic magnetometers
Figure 3.4: 1. Before pumping 2. Light depletes population at B=0 3. Optical detection when
B ̸= 0.
atom to be found in the state 1/2 is increasing since the 1/2 sublevel cannot interact with our light
due to the selection rules and hence its population cannot be excited to higher levels. This creates a
net angular momentum along x [19].
3.1.4 Probe
After we pumped, we suddenly add a uniform magnetic field Bẑ and we reduce the intensity of the
light so that now the light beam pumps slowly. This is the case of ⃝3 in Figure 3.4 and now the
Zeeman sublevels are not degenerate anymore, due to Bẑ. If we choose our quantization axis now
along z we get reminded of the concept of quantum beats. What seemed abstract in Section 1.4.4 is
now tangible; the states |1/2⟩x and |−1/2⟩x are now rotated to be written in the z coordinate system
as:
|1/2⟩ |−⟩zx = √
+ |+⟩z , (3.1a)
2
|−1/2⟩x =
|−⟩z√− |+⟩z , (3.1b)
2
with mz = 1/2 (mz = −1/2) undergoing an energy shift of ℏΩL/2 (−ℏΩL/2), in other words the
superposition of the mF states along the axis z of the magnetic field is oscillating at a frequency
ΩL. The probability of finding the state in mx = 1/2 which is the state which can interact with the
light is also a sinusoidal function of ΩL. The light that reaches now the detector interacts with the
Simple model of an optical magnetometer 41
medium in a sinusoidal manner as well, since it gets absorbed more when the light state has more
atoms. This change is what we detect and is what we call Larmor precession in a magnetic field
B.
We can find magnetometers in a variety of configurations, some quite different to the example we
described above. However, at its heart optical magnetometry remains the same: we induce a change
in the opto-magnetic properties of the ensemble through pumping, which we then detect through
a change in the light absorption or its polarization rotation corresponding to a Larmor-precession
frequency.
Optical detection
The moment comes now to talk about detection in optical magnetometers. It is useful, once again,
to start with the linear case of the Macaluso-Corbino effect as seen in Section 2.1.2, where linearly
polarized light, composed of σ+ and σ− interacts with an atom. The frequencies of σ− and σ+
between the ground sublevels and the excited state are the same if the system is in zero field. When
an external magnetic field is added, the ground-sublevels energy is shifted and σ− and σ+ now
interact differently with the excited state. To explain this better we need to get back to the complex
refractive index of the atomic medium η introduced in 2.2.2:
η = n+ iκ . (3.2)
In the above relationship, n is the real index of refraction, that has a dispersive behaviour as a func-
tion of the angular frequency of the light ω and κ is the imaginary part related to absorption. A
change in the refractive index would result in a change in absorption or dispersion of light passing
through the medium. Absorption can be detected through transmission of light, while dispersion
is related to optical rotation. The change in the behaviour we detect, has its cause to the added
magnetic field which creates birefringence {I} and/or dichroism {II}: the polarization compon-
ents σ− and σ+ interact now differently with the medium and, as a result, they have a different
refractive index η− and η+ whose difference creates the measured rotation. The rotation angle ϕ of
the polarization of the light, in an atomic sample of length l, is given by:
ωl
ϕ = 2 Re(η+ − η−) . (3.3)c
For one refractive index value, the curve we detect is close to Lorentzian as seen in Figure 3.5. Gener-
ally in magneto-optical effects, and without getting into details about the formalism, optical rotation
is, in the resonant case, dependent on the magnetic field in a way that resembles a dispersive Lorent-
zian. This is intuitively understood by considering that the relaxation of the atomic coherences in
{I}Refraction of light depends on its polarization
{II}Absorption of light depends on its polarization
42 Atomic magnetometers
time is an exponential decay and the Fourier transformation of an exponential decay is a Lorentzian
function.{I}
Figure 3.5: Lorentzian shape of the refractive-index change of light due to the Macaluso-Corbino
effect.
The transitions driven by the light induce absorption with a Lorentzian shape and a phase shift with
a dispersive shape [30]. These curves have a characteristic width Γ, while the curve of the absorption
is an even function{II} with maximum at zero (0) light-detuning from resonance 2∆/Γ (or else has a
maximum on resonance). The curve of dispersion is an odd function{III} that crosses zero (0) at no
detuning/on-resonance.
Figure 3.6: The real parts of the refractive indices are creating dispersive curves that are shifted
in frequency due to the external magnetic field. Their difference is proportional to the linear optical
rotation ϕ of the light passing through the medium.
{I}This is not a very accurate model, because the light beam interacts with the atoms in a finite time. However, to a first
approximation the shape is qualitatively similar to a dispersive Lorentzian.
{II}f(−x) = f(x)
{III}f(−x) = −f(x)
Simple model of an optical magnetometer 43
We are in the linear case still, and here the optical rotation signal in a system with circular birefrin-
gence is a phase shift between σ− and σ+ when linear light propagates through an atomic medium.
Based on Equation 3.3 , the angle is given by the difference between the dispersive (phase-shift)
curves as seen in Figure 3.6.
The above description helps understanding linear effects, but nonlinear effects make the system much
more sensitive to changes of the magnetic field. In linear effects we have shifts in the sublevels due
to a magnetic field, but nonlinear effects are related to the evolution of created atomic polarization
and Larmor precession. We will take a closer look at throughout the next Chapters, and especially
in Section 3.3.3.
3.1.5 Shielding
In order to place the atoms inside zero (0) field, one needs to find a way to zero the external magnetic
fields first. Sometimes, as is the case for many experiments in this thesis, we might want to work
on the Earth’s field (about 0.5 G) and not in zero field. One might think that this would solve our
shielding problem, but, alas, the lab has an unexpected amount of items that can produce magnetic
fields, from screwdrivers and waveplates to the moving elevator in the corridor above. Thankfully for
us, we can work inside a shield and control this way magnetic fields. One can roughly divide the way
to shield the atoms in two categories: active and passive shielding.
In active shielding, pairs of coils are used to cancel out the external fields. Biot–Savart’s law states
that constant electric current creates a magnetic field and hence by adding current through these
coils we create opposite magnetic fields to the ones we want to eventually cancel.
Figure 3.7: Magnetic shield with high permeability.
In passive shielding, we take advantage of materials with high magnetic permeability{I} to reduce the
ambient magnetic noise. In our experiments we used four-layer magnetic shields from the company
Twinleaf made out of a nickel–iron alloy called mu metal. This material does not allow the magnetic
lines to pass through to its center where are cell is located, but rather it diverts the magnetic flux
{I}The magnetic permeability of a material is equal to the magnetic flux density B in a material created by a magnetic
field, divided by the magnetic field strength H of this field. We can consider it to be a measure of how much magnetic flux
can be concentrated inside a material.
44 Atomic magnetometers
and the magnetic field lines are now inside the material. Figure 3.7 shows a representation of the
divergence of the path of the magnetic field lines due to a material with high permeability.
One might ask themselves why we use multiple thin layers for a better shielding effect and not
one very thick piece. Our shielding ratio T (which needs to be as small as possible) would in that
case scale proportionally to the increase of thickness. For example, if we add a 3x thicker shield,
we get a shielding ratio of T/3x. If, however, we add different layers/shells separated by air in
between, shielding becomes more efficient. By taking an approach for which the layers are magnetic
circuits, one can calculate that for separate shells, T is proportional to the product of the thickness,
meaning that, if we add a 3x thicker shield, we get a shielding ratio of T/(x3). The shell shape
and the gap between the layers play a role as well. Adding to the fact that from time to time the
residual magnetization inside the shielding layers can be increased, it becomes quickly clear that
good magnetic shielding is not a trivial task.
3.1.6 Noise and Sensitivity
The concept of magnetic-field sensitivity is central to atomic magnetometry and it is often√used to
qualitatively compare different types of magnetometers. The units of sensitivity are often T/ Hz in
SI and it shows what the minimum detectable magnetic field is– i.e. a field that can be distinguished
from noise. This noise depends on the frequency bandwidth and this is why we need to normalize
sensitivity by dividing by frequency.
What we practically do in order to measure sensitivity is to measure the response of the magneto-
meter for some time t. Since the noise changes for different frequencies, we are interested in looking
at this response in the frequency domain, so we perform a Fourier transform (FFT). In order to
characterize noise, power spectral densities (PSD) are used instead of an FFT. To calculate a PSD
each frequency bin in an FFT is multiplied by its complex conjugate and this results in an amplitude
value (in our case B2 normalized to the frequency bin width to get the units of T 2/Hz. This way
our signal does not depend on bin width. To make comparisons easier we then take the root-mean-
square (RMS) of the T 2√ /Hz and we calculate the amplitude spectral density. This is how we report
the sensitivity in T/ Hz .
Another way to calculate sensitivity is by using the fundamental limits set by quantum mechanics.
There are two fundamentals limits: projection noise (PN) and photon shot noise (PSN).
Projection noise (PN)
Let us think of an atom that is polarized along a particular direction. From the uncertainty principle
we know that if we measure the angular-momentum projection on an orthogonal direction we get a
random result [18]. In this case, the sensitivity is:
δBPN ∝ √
1
, (3.4)
Nt2T
Simple model of an optical magnetometer 45
where t2 is the spin-coherence time related to transverse spin relaxation and N is the number of
atoms in the ensemble. This equation is derived from the fact that one atom has an uncertainty of
1 rad in the Larmor precession angle for a measurement of time t2. If we consider N times this atom
and repeating T times we reach the above formula.{I}
Photon shot noise (PSN)
The other limitation to sensitivity is the photon shot noise. This is related again to Heisenberg’s
uncertainty principle and the detection of light. When light is detected by a photodetector, the
number of photons arriving at any given time is random, and so the electrons produced by the
photodetector– and hence our signal– is also fluctuating in time. In this case, sensitivity scales as:
1
δBPSN ∝ √ , (3.5)ΦT
where Φ is the probe-photon flux (photons/second).
From the above we can conclude that in order to improve the sensitivity of a magnetometer (to make
it smaller) we need to either improve the spin-relaxation time or increase the number of atoms that
participate in the interactions. The first can be done by, for example, using coating and the latter by
increasing the density inside the cell through higher temperatures.
3.1.7 Polarimetry
Figure 3.8: Balanced polarimetry setup.
There are several methods that we can use to detect optical rotation ϕ. We will here introduce the
one used mostly in our experiments, since we detect at high-frequencies. This is called the balanced
{I}A way to overcome this limit is through spin squeezing
46 Atomic magnetometers
polarimetry technique which, as seen in Figure 3.8, consists of a polarizing beamsplitter which is
set at 45o to the probe’s polarization axis. This configurations splits the beam’s intensity I0 in two
parts with intensities I1 and I2:
= 2( − πI1 I0sin ϕ 4 ) , (3.6a)
I2 =
π
I0cos
2(ϕ− 4 ) , (3.6b)
where I0 = I1 + I2. The two intensities are equal, or in our case balanced, when ϕ = 0 (no rotation).
When the probe beam rotates, its optical rotation ϕ is given by:
= I1 − I2ϕ 2(I1 + )
, (3.7)
I2
and is valid for a small angle ϕ << 1.
3.1.8 Spin relaxation
Spin-exchange collisions
In our efforts to reduce noise through heating, we come across a new obstacle: if we increase the
temperature too much, collisions between the atoms are dominating the spin-relaxation time. In
this case, the total spin is preserved, but the direction of the spin can change– see Figure 3.9. The
nuclear spin I is not affected, but the atoms might change hyperfine states and get redistributed
among other unobserved hyperfine Zeeman sublevels. The change in mF states results in a change in
the angle of the spin and hence its precession, leading to loss of atomic polarization in the ensemble
(decoherence).
Figure 3.9: Spin-exchange mechanism. Red spheres are atoms that collide and after the collision
precess in different directions as indicated by the purple arrows.
Polarization moments 47
Note here that the total angular momentum is conserved: the vector sum of the spins after the
collision should be the same as that before the collision. The spin-exchange collisions contribute to
t2 time and as a result can become important for a magnetometer’s sensitivity.
Magnetic–field gradients
Many projects that are included in this thesis have been conducted in a magnetic field of the order of
the Earth’s field. Earth-field magnetometry can be very useful, since it does not require heavy and
expensive shields or complicated instrumentation to cancel out magnetic fields. To test our theories
in a controlled environ√ment we have used shields which can attenuate DC fields below nT and have a
sensitivity of few fT/ Hz . Unfortunately this is not the end of the story. Our cells are quite long,
with a radius of a few cm up to a length of 14 cm for the cylindrical cells. Since we zero the field at
a small area in the centre of the cell (and the shield), this means that the atoms experience different
magnetic fields along the cell. In a paraffin-coated cell the atoms average the magnetic field to a large
extent, but in cells with buffer gas the atoms diffuse slowly and the differences are much bigger. This
magnetic-field gradient along the cell is broadening our linewidths and reduces our sensitivity.
3.2 Polarization moments
It has become quite obvious so far that the polarization of an atom and its symmetry plays a crucial
role in atomic magnetometry and it can be useful to classify the symmetries that appear in a given
atomic state. To do that, we decompose the density matrix into components that have the sym-
metries of the spherical harmonics, because these symmetries help us describe the polarization in a
visual language. We therefore expand the density matrix over a set of irreducible tensor operators
T κq , with the rotational symmetries of the spherical harmonics. We call them polarization operators.
The density matrix ρ can now be written as:
∑2F ∑κ
ρ = ρκqT κq . (3.8)
κ=0 q=−κ
The polarization moments (PM) ρκq are characterizing the anisotropy of a state with total angular
momentum F . These are the coefficients in the expansion of the density matrix into the irreducible
tensor operators of rank κ = 0,...,2F and projection q = κ,...κ [53]. The covariant coefficients ρκq for
a state with total angular momentum F in the |m⟩, ⟨m′| basis are given by:
∑F
ρκq = (−1)F−m
′⟨F,m,F,−m′|κq⟩ρm′m , (3.9)
m,m′=−F
where ⟨...|...⟩ indicate the Clebsch-Gordan coefficients [54].
The names we give to the polarization moments depending on their κ–rank sound familiar to a
conversant in physics: they are borrowed from the multipole moment expansion of a static electric
48 Atomic magnetometers
field. As such, ρ0 is monopole moment or population, ρ1 is dipole moment or orientation, ρ2 is
quadrupole moment or alignment etc.
Figure 3.10: Polarization moments for a total angular momentum of F = 1. For spin →−s = 1
we have three (3) projections 0, 1, and -1 which results in a 3x3 density matrix. On the bottom
left are the nonphysical and on their right side are the physical density matrices representing the
PM’s [30]. Mathematically both are correct, however physically a diagonal with negative numbers
or all zero (0) has no meaning, since the diagonal elements represent the probabilities of finding the
the populations in a state. This is why we transform the density matrix to one where the diagonal
makes physically sense and this new density matrix might also consist of linear combinations of Tκq .
For each polarization moment the angular-momentum probability surface that corresponds to a
physical density matrix is illustrated. The values indicated by color are: the middle value of purple
is zero, dark purple are negative values and white and lighter purple correspond to positive values.
In the case of κ = 2, for example, we see that κ defines the type of polarization (alignment) and q
describes the axis of polarization. For κ = 2 we have three (3) possible axes for q > 0.
Let us give some examples to understand the properties of the PM. For κ > 0 we describe an ensemble
that has any moment. The polarization moments ρκ0 (with q = 0) represent a polarization along the
quantization axis{I} which corresponds only to populations and, accordingly, ρκq with q ̸= 0 describes
transverse polarization to the quantization axis and coherences between the sublevels. An example
of the polarization moments that could exist in the F=1 state is illustrated in Figure 3.10.
{I}In quantum mechanics we cannot know all the information of an operator. The maximum amount of information we
could have is a projection on an axis. For the electron’s angular momentum this projection is called the quantization axis
and equals to lz = mℏ. We often use the magnetic field direction as reference so the m is called the magnetic quantum
number. We choose the quantization axis according to what is convenient for our experiment.
Important types of atomic magnetometers 49
For a specific κ we get -κ...κ components that oscillate with q times the Larmor frequency ΩL. The
measured signals are proportional to the real-valued sum and difference of these components.
Important note: We use a very similar notation– ρ(n) –to distinguish between linear and nonlinear
effects in a medium (in Section 2.2.2). In this Section we use ρκq to classify the polarization moments
that can be created in an atom and their symmetries. These two things are not the same.
3.3 Important types of atomic magnetometers
3.3.1 Mx and Mz magnetometers
The Mx and Mz magnetometers are based on magnetic-resonance phenomena. Typically, we have
our atoms placed inside a bias (static) magnetic field B0. The atomic transitions are excited by a
resonant oscillating (RF–radio frequency) magnetic field B1 with frequency ω. Optical pumping
occurs with a resonant circularly-polarized light beam that creates magnetization of the ensemble of
atoms. Macroscopically, the pumping of the medium creates atomic magnetization M which evolves
around B0 due to Larmor precession.
To visualize the movement of M, we can use the Bloch sphere– see Section 2.8 –to visualize the
time evolution of the atomic magnetization including relaxation parameters.
Figure 3.11: The Block sphere visualizes the possible solutions of the Bloch equations. The thin
purple dashed line shows a possible path of the magnetization M .
Let us assume that for t = 0 the initial magnetization is M0 and is along w. Both magnetic fields
act on it and now magnetization M starts precessing around the effective field in a similar manner as
explained in Section 2.3.1, where we talked about how an oscillating magnetic field makes our spin
system evolve in time under Rabi frequency (or Larmor frequency in the resonant case, where ω is
close to the resonant frequency ω0).
50 Atomic magnetometers
Relaxation has to be considered as well: when the system returns to equilibrium, the longitudinal
and transverse magnetization relax differently. There are two important time constants to consider
here: t1 and t2. The longitudinal component returns to M0 value (the value at t=0) with t1 time
and the transverse component becomes zero (there is no transverse component initially) with t2.
Generally, Mz and Mx magnetometers are distinguished by the method we use to detect/probe the
signal [19]:
For Mz: we detect a component proportional to the longitudinal component of the magnetic moment
and so t1 time plays a role. In particular, t1 limits the temporal response of the magnetometer to
changes in the magnetic field.
For Mx: we detect the phase of the transverse component of the magnetic moment and the t1 and
t2 times are not limiting how fast the magnetometer responds to changes in the magnetic field. T2
limits does limit however the sensitivity of the magnetometer.
Mz magnetometer
Figure 3.12: Mz magnetometer illustration.
In this magnetometer, we employ circularly polarized light in order to optically pump the atoms
and create spin polarization in the ensemble and as a result magnetization. The bias magnetic field
B0 is along the pump direction, let’s say z. Let’s now apply an oscillating (RF) magnetic field B1
transverse to B0 with frequency close to the magnetic-resonance frequency ω0. This frequency is
related to the energy difference of the magnetic sublevels due to B0, in other words it is resonant
to their Zeeman splitting. In classical-physics terms, when applying the RF field the magnetization
vector gets tilted and now we do not only have longitudinal magnetization which is smaller than
before, but also a transverse magnetization that is rotating. Longitudinal magnetization is related
to a population difference created by the pump light along z and since now the projection of the
magnetization vector along z is smaller than the original magnetization, this means that the B1
field reduced the created population difference. By modulating the frequency of the B1 field we can
detect changes in the absorption of the pump light depending on how much population is left to be
Important types of atomic magnetometers 51
absorbed [19]. In other words, there is a periodic change of population on the magnetic sublevels
along z due to the RF field that affects the absorption of the pump light which we detect.
Mx magnetometer
Figure 3.13: Mx magnetometer illustration.
In this magnetometer we use, as in the Mz case, circularly polarized light to pump. The bias field
B0 is in this case oriented at 45◦ with respect to the pump light [55]. Equivalently, one can use two
beams, a pump beam along B0 and a probe beam perpendicular to it. Here again, we have a change
of coordinate system and the concept of coherences (see 1.4.4) apply: while the rate of change of the
populations is limited by t1 and t2 times, the oscillation frequency of the coherences reacts almost
immediately to the changes induced by the field and hence Mx magnetometers are faster than other
types. [19]. Because Mx magnetometers respond fast to magnetic-field changes, they are usually used
when one needs a movable device that responds fast, is sensitive and robust. They can also be good
vector magnetometers, meaning they can measure the magnetic-field vector component. For some
applications, Mz magnetometers are preferred, especially when one needs high accuracy.
3.3.2 Bell-Bloom magnetometer
Figure 3.14: Bell-Bloom magnetometer illustration.
52 Atomic magnetometers
In the example we gave about the basics of the operation of a optical magnetometer in Section 3.1.3,
we pumped under zero magnetic field and we probed under a bias magnetic field B0. In order to
simplify the process or to be able to employ a magnetometer in the earth’s-field– without a shield –it
would be useful to be able to enable pumping and probing at the same time. This is possible in the
Bell-Bloom magnetometric scheme.
In this type of magnetometer, the setup is the same as in Section 3.1.3 except for the fact that
the bias field B0 stays on for the whole process and the intensity of the pump-light is modulated
at a frequency Ωm. Modulation of light, (or a laser beam in our case) is a process by which the
output of the light, may it be phase, intensity, frequency, or polarization, is changing in time. In
our experiments we use external modulation, which means that we take the laser-light output and
we pass the light through a modulator, for example an Acousto-Optic Modulator or AOM, that
outputs light with our desired modulation (see also 3.5).
Figure 3.15: Magnetic Resonance. The X signal is the amplitude of the signal in-phase with ΩL
and Y is the signal amplitude out-of-phase with ΩL.
If Ωm = ΩL then pumping occurs at the same time as Larmor precession. The pump light acts here
as a driving force in a driven harmonic oscillator at time t− 2pi/ΩL, t− 4pi/ΩL, ... [19]. We can look
at it also from a different perspective: Since the intensity goes from 0 (zero) to maximum, we can say
that only at the maximum intensity the spins are pumped along x and the spin-state is stationary in
the rotating frame that rotates with frequency ΩL. In order to keep Ωm in resonance, we use often
a voltage-controlled oscillator or function generator– a device that produces a waveform. Often, the
self-oscillating mode is used, where the spin-precession signal is producing the frequency for the
pump modulation.
What we detect is a characteristic resonance when we scan either the modulation frequency Ωm or
the magnetic field strength– see Figure 3.15. In the “stroboscopic” picture, if the spin-state that is
stationary in the rotating frame is a dark state, the light does not interact with it and hence we see
a peak in light transmission. If it is a bright state, we see a peak in absorption. Based on these
measurements we can determine the magnitude of the field B0.
Important types of atomic magnetometers 53
3.3.3 Nonlinear magneto-optical rotation (NMOR)
Figure 3.16: NMOR-based magnetometer illustration.
Nonlinear magneto-optical rotation or nonlinear Faraday rotation was first discovered in 1974 [56]
and has since been extensively studied. Nonlinear magneto-optical rotation happens when linearly
polarized light propagates through a medium placed inside a magnetic field B0 and its polarization
plane rotates, with the rotation being dependent on the light-intensity. This effect produces a non-
linear rotation that is– for small magnetic fields– about 104 times larger than the rotation produced
by linear effects in a buffer-gas vapor cell [30]. As we will discover later, we can produce even bigger
rotations by using paraffin-coated cells, making our magnetometers super sensitive, that is, able to
detect extremely small changes in magnetic fields.
Three-level Λ-type system
Sometimes the two-level system is not sufficient to describe a system, for example in the case where
we have linearly polarized light. We are specifically interested in the Three-level Λ-type system which
will be briefly discussed in Section 2.4.2 and Figure 2.9 in order to be able to discuss about NMOR.
Figure 3.17: Three-level Λ-type system. Linearly polarized light couples the ground states to the
excited state.
54 Atomic magnetometers
An approximation of such a system is the F = 1 → F ′ = 0 with the ground sublevels m = 1 and
m = −1 coupling to the excited state m′ = 0 through the σ+ and σ− components of the linearly
polarized light, as was the case in Figure 2.2 of the Macaluso-Corbino effect [38] reproduced in
Figure 3.17. This system cannot be viewed as two independent two-level systems and coherences of
the ground levels need to be taken into account.
Since we have a magnetic field along the light, the field B0 splits the Zeeman sublevels and now the
coherences on the ground sublevels depend on B0.
Ground-state-coherence effects
An important feature, in NMOR experiments, are the narrow lines observed in paraffin-coated cells
which are related to the lifetime of the ground-state atomic polarization– since our linearly polarized
light creates coherences between those states{I}. Longer ground-state polarization times lead to
higher sensitivity.
In Section 2.2.2, we talked about the pertubative expansion of the density matrix in orders ρ(n) and
their use in classifying linear and nonlinear effects. Odd orders of ρ(n) (ρ(1), ρ(3),... etc) describe op-
tical coherences, while even orders of ρ(n) (ρ(2), ρ(4),... etc) describe density-matrix elements between
states with the same parity{II}. For transition between F levels, these density-matrix elements are
level populations and Zeeman coherences.
In the case of NMOR, we could use two beams instead of one{III}: the first to pump and the second
to probe. It helps also to look at the process of NMOR as a three-step process: In the first step, the
linearly-polarized light creates ground-state polarization through pumping within the ground Zee-
man sublevels, which is a ρ(2) order effect. This creates an aligned state– along the light-polarization
axis– which is dark for one polarization of the pump– let’s assume for y-polarized light– and it is
bright for the other, in this case, x-polarization{IV}. Now linear dichroism was created, meaning
that the absorption of the light that consists of orthogonal linear polarizations depends on these
polarizations: the atomic medium absorbs one polarization and transmits the other.
Now comes the second step of the NMOR process. Alignment has been created along the light-
polarization axis that rotates in the bias magnetic field B0 rotating also the axis of dichroism. This
can be visualized with the peanut shape as seen in Figure 3.10.
For the final stage, the probe beam produces optical coherences and these are influenced by the
ground-state polarization at first order ρ(1). This means that the probe beam interacts with the
{I}The effects in which the nonlinear rotation depends on the relaxation rate of the excited state are called Bennett-
structure effects. In this case the light interacts only with a specific velocity subgroup and in the velocity distribution
“peaks” and “holes” appear due to optical pumping
{II}As we found out in Section 2.4.1 these are magnetic-dipole transitions. In contrast, optical coherences are coherences
between states of the opposite parity
{III}One-beam experiments work similarly
{IV}It is not always the case that the states are dark for one of the pump polarizations. For example, when we pump with
F → F + 1, the pumping happens in a bright state, but it is brighter for one polarization than the other
Important types of atomic magnetometers 55
dichroic medium in a linear way{I}. In this case we can also use, instead of a second probe beam, just
the pump beam in a one-beam configuration.
Let us describe mathematically the rotation of the polarization by using the Λ system (see Sec-
tion 3.3.3) to illustrate this process. If we use y-polarized light to polarize the atoms, they get
polarized into the state:
|y⟩ = (|+1⟩ − |−1⟩) . (3.10)
By solving the time-dependent Schrödinger equation we find that the state |y⟩ evolves in time inside
a magnetic field as:
|ϕ(t)⟩ = −√i (|+1⟩ eiΩLt − |−1⟩)e−iΩLt , (3.11)
2
where ΩL is, as usual, the Larmor frequency and the initial state |y⟩ rotates with ΩL. The system
ϕ(t) reproduces itself every half of the Larmor period, so |ϕ(φ+ π/ΩL)⟩ = |ϕ(φ)⟩, where φ is the
relative phase. The axis of the alignment also rotates. When the polarization of the probe light and
the atomic-alignment axis are parallel, or at 90◦ we have no rotation. At any other angle there is
rotation. As a result, the probe light gets modulated at 2ΩL [38]. We will talk in more detail about
this in Section 3.3.4 and in Chapter 5.
In reality, the above three processes, which can be thought of as a model of a rotating polarizer,
happen all at the same time and the NMOR corresponds to a ρ(3)- or χ(3)-order effect. The CW
(continuous-wave) pump continuously creates polarization and the relaxation processes destroy it at
1/γ time, where γ is the transit rate. This is the rate at which atoms interact with the light beam
and is a source of ground-state relaxation. The lineshape produced in this process– if we calculate the
expectation value of the optical polarization and make assumptions about the relaxation rates γ and
Γ– is a dispersive Lorentzian antisymmetric around zero field, as we saw in previous Sections [30].
High light power
The above model is working for low light power, but for higher-enough power we need to take AC-
Stark shifts into account. AC-Stark shifts are energy shifts of atomic energy levels due to an
oscillating light field. AC-Stark shifts together with a magnetic field can lead to optical-alignment
that oscillates and gets converted to orientation, perpendicular to both the alignment and electric-
field axis, in a process called alignment-to-orientation conversion (AOC) [57]. This effect needs
to be taken into account, since it can also create optical rotation due to the orientation creating
circular birefringence in the material. For an AC-Stark shift to produce AOC, it has to have a
non-zero detuning between the optical frequency and the resonance [30] and so in a Doppler-free{II}
{I}This is true if the probe is not strong enough as we will see in the next Section
{II}Doppler broadening happens in all experiments, since atoms have a velocity along the light-propagation direction that
makes them “experience” different frequencies. We can create a setup in which Doppler broadening does not play a role, if
we manage to only “talk” to a group of atoms with a specific atomic velocity and we call it Doppler-free
56 Atomic magnetometers
experiment where the light is on resonance AOC is not relevant. Another effect related to the high
power of the light is power broadening, which is the effect of the magnetic resonance broadening
due to effective relaxation induced by pumping. This happens when the pumping rate Γp (Ω2R/Γ) is
bigger than the transit rate γ, or else, if Γp/γ = κ2 ≥ 1. In NMOR y-polarized light polarizes the
atoms along y which is now a dark state. Under a magnetic field, the aligned atoms rotate and fast
pumping would align atoms into y before a full rotation could happen, while an even faster pump
could completely destroy NMOR.
3.3.4 NMOR with modulated light
Figure 3.18: Setup of NMOR with modulated light. The linearly-polarized pump light gets mod-
ulated through a function generator. The modulated, detected, signal gets demodulated through a
lock-in amplifier.
NMOR in antirelaxation-paraffin-coated cells offered unprecedentedly narrow resonances, related to
ground-state atomic polarization that lives long, hence the name “antirelaxation”. As we will see
later, this fact plays an important role in how sensitive our magnetometers can be. The width of this
resonance is, in polarization-dependent effects such as NMOR, depending on the relaxation rate of
the atomic polarization. This practically means that these magnetometers can be sensitive, but only
in a small range of magnetic fields.
An experimental arrangement that can utilize the sensitive NMOR measurements with the efficient
Bell-Bloom pumping scheme, is a type of NMOR magnetometry that uses modulated light in order to
pump. One can modulate either the frequency (FM NMOR) [58] or the amplitude (AM NMOR) [59]
of the light. This simple modification of modulating our pump and detecting at this frequency can
allow us to get a magnetic resonance not only around zero field, but also at other values of magnetic
field, allowing us, by definition, to extend the dynamic range of our magnetometers. We will talk
here more about AM since this method is more often employed throughout the experiments of my
thesis.
Important types of atomic magnetometers 57
Amplitude modulation
Let us think of a linearly polarized pump beam that gets modulated at a frequency Ωm. This
modulation leads to higher harmonics of Ωm appearing in the spectrum of the light with which the
atoms can interact. What the response of the polarized atoms in a magnetic field looks like, is a
short-lived signal that depends on time. The signal looks like a decaying oscillation: when we pump
once we have the maximum optical rotation and then we see the signal relaxing towards zero.
What happens if we use more than one pulse to pump and polarize our atoms? If we pump with pulses
that repeat themselves at a random rate, these will create phase differences between the precession
of atoms pumped at different times this way destroying atomic polarization. There is a way, however
to enhance our pumping process through modulation, if we pump every time a precession is full, in
other words, when atomic polarization returns to its original state. Remember here, that linearly
polarized light creates alignment, a polarization moment of rank κ = 2, symmetrical along an axis,
as shown in Figure 3.19.
Figure 3.19: Angular-momentum polarization surface or alignment, produced by linearly polar-
ized light and its rotation along the z axis.
When point A rotates from the positive axis z and reaches point B on the negative axis z, the surface
has regained its symmetry. However a full rotation happens when point A reaches again the positive
axis. This means that for the polarization to return to its original state, half a Larmor period is
needed (or two times the Larmor frequency). For this reason, we synchronize the modulation of the
pump with two times the Larmor frequency and we get a resonance in the oscillating signal (See also
Figure 3.20.)
The length of the pulse of the modulation is defined by a quantity called duty cycle defined as:
W
D = 100% , (3.12)
T
where D is the duty cycle, W the pulse width and T the period of the signal. For a Doppler-broadened
transition, we achieve the best pumping with relatively short pulses of about 20% duty cycle [19].
58 Atomic magnetometers
Figure 3.20: Optical-rotation signal in AM NMOR. In 3. the modulation is already at twice the
Larmor frequency so the pulses are at the modulation frequency.
For probing we use CW light– which is the same as saying 100% duty cycle. This is one of the reasons
that a two-beam setup is better for optimization, we can separately optimize the pump and probe
beams: their duty cycle, their frequency and their power.
3.3.5 Spin-exchange-relaxation-free (SERF) magnetometer
Figure 3.21: SERF-magnetometer illustration. In the SERF regime the atomic spins are rapidly
kicked between the two ground-state hyperfine levels and this averages the oppositely signed pre-
cession.
For this overview to be complete, the spin-exchange-relaxation-free (SERF) magnetometer could not
be missing. Since it is out of the scope of this thesis I will only briefly introduce its characteristics.
As the name already hints, this is a magnetometer that is free of spin-exchange collisions. In Section
3.1.6 we outlined how spin-exchange collisions redistribute atoms among the Zeeman sublevels of
NLZ in Earth-Field magnetometry 59
the ground hyperfine states and this makes them precess in opposite directions [60]. These collisions
contribute a lot to the t2 relaxation thereby limiting our sensitivity. However, in 1973, Happer and
Tang discovered very narrow (about 200 Hz) magnetic resonances in alkali vapors at high-densities
and inside magnetic fields close to zero [61]. This is what we know today as the SERF regime: In
Liouville’s equation plus the relaxation terms (see Equation 1.4.2), when the term for spin-exchange
is much smaller than the hyperfine term, but larger than any other term, spin-exchange does not
contribute to spin relaxation. This means that when spin-exchange collisions happen faster than the
precession frequency, which is the case in high density and small magnetic fields, the atoms sample
all the Zeeman sublevels in both ground-state hyperfine levels fast enough for the collective, average
spin to experience an effective precession frequency. The spin-exchange broadening factor depends
on the how small the magnetic field is and how dense the atomic ensemble, this is why the SERF
regime is around zero field and heating of the cell is needed.
Figure 3.21 illustrates a classic SERF-magnetometer setup. The circularly polarized pump creates
orientation along its propagation direction. The magnetic field (perpendicular to both beams) ro-
tates the spins and we detect a rotation of the linearly polarized probe beam along the third axis.
The atomic densities required for SERF demand a cell temperature of the order of 200oC.
In order to prevent atoms from colliding with the glass walls of the cell and to prevent spontaneous
emission– both mechanisms would destroy polarization– a buffer gas is added, usually a noble gas
like He and a diatomic molecule like N2.
3.4 NLZ in Earth-Field magnetometry
3.4.1 Nonlinear Zeeman effect
Most of our experiments have been done with a vapor cell in a magnetic field that is of the order of
the Earth field and so it is important to understand how a field of this order of magnitude can affect
the states of our atoms differently than what we saw in Section ??. So far we saw the effects of small
magnetic fields, where the energy levels are changing in a linear way in the applied field. Now we
are going to take a look at how a sufficiently large magnetic field can affect the level splitting in a
nonlinear way.
The ground-state Hamiltonian for an atom in a magnetic field B for states with electronic angular
momentum J = 1/2{I}, is:
Ĥ = AJI · J+ gsµBS ·B− gIµNI ·B , (3.13)
where AJ is the hyperfine coupling constant, I is the nuclear spin, gs and gI are the electron-spin and
nuclear Landé factors of the atom, µB is the Bohr magneton, and µN is the nuclear magneton. The
{I}The multipole series that describes the hyperfine interaction consists of multiple terms from which the most important
are AJ and BJ . AJ characterizes the strength of the magnetic-dipole interaction and BJ the strength of the electric-
quadrupole interaction. However BJ is zero when I and J are both larger than 1/2, such as in the case presented here.
60 Atomic magnetometers
first term describes the hyperfine interaction and the second and third terms the interactions with
the magnetic field– i.e., Zeeman interactions.
In a atom that has one valence electron in an S(J = 1/2) level, the analytical solution for the
eigenvalues of the Hamiltonian is given by the Breit-Rabi formula 3.14. This gives us the energy
shifts Em of the magnetic sublevels |m⟩ for a state with a total angular momentum F in a magnetic
field of strength B [30, 62]:
( )1/2
Em =
∆hf ∆hf 4mξ
2(2 + 1) − gIµBmB ± 2 1 + 2 + 1 + ξ
2 , (3.14)
I I
where ξ = (gJ + gI)µBB/∆hf , ∆hf is the hyperfine-structure interval and the ± sign refers to the
F = I ± 1/2 hyperfine levels, gJ ≈ 2, while we neglect the Zeeman energy of the nuclear spin (last
term in Eq. (3.13)= since µN ≪ µB). The first two terms are linear to B and the third is nonlinear.
The nonlinear term can be neglected if we operate the magnetometer in low-field, but it becomes
important on the order of the Earth’s magnetic field. If we plot these energies as a function of
applied field, we get what we call Breit-Rabi diagrams, as illustrated in Figure 3.22. We chose
cesium (Cs) as an example, since it will be relevant to our first experiment presented in the next
Chapter.
Figure 3.22: Breit-Rabi diagram of Cs. At the bottom of the illustration one can see that on
Earth field, although 0.5G is small in magntiude, the energy levels in this regime are behaving
nonlinearly as a function of magnetic field.
If we expand the eigenenergies as a series in B around zero, the transition frequencies corresponding
to ∆m = 1 for the Cs 62S1/2 F = 4 system are
µBB (µBB)2
Em+1 − Em ≈ 4 + 16∆ (2m− 1) . (3.15)hf
The quantity ω 2rev = (µBB) /(16ℏ∆hf ) is the previously mentioned quantum-beat revival frequency.
If we are on Earth field (50 µT or 0.5 G), the revival frequency after calculations is ωrev = 2π·3.3Hz∝
20 Hz [30]. Remember, from Section 1.4.4, that “quantum beats” is the general term used to describe
the time evolution of a coherent superposition of non-degenerate energy eigenstates at a frequency
Experimental devices 61
determined by their energy splittings [35], for example, the splittings happening in the NLZ effect.
It is exactly this frequency that we have calculated here and that is comparable to the magnetic
resonance linewidth which makes the system strongly affected by the nonlinear Zeeman effect. The
Cs magnetic resonance is split into eight peaks and as a result it is broadening the linewidth and the
magnetometer sensitivity.
3.4.2 Heading error
A noise source that plays a role in scalar magnetometers, such as the ones studies in this thesis, is the
so-called heading error. This describes the error of a measurement due to its dependence on the
orientation of the sensor. There are a few applications, usually involving airborne or marine system,
where the sensor is not at a constant angle and, as a result, heading error can affect the measurement.
Sources of heading error involve magnetic materials inside the sensor, ferromagnetic electronics, but
also reasons related to the physics of the sensor, such as different gyromagnetic ratios of hyperfine
states and the nonlinear Zeeman effect. In Chapter 5, we will address heading errors pertaining to
the second category.
3.5 Experimental devices
For the biggest part of my PhD journey, I worked hands-on in the lab with various optical and
electronic devices. Since these are a predominant part of an experimentalist’s life, I would like to
briefly explain how some of them work.
3.5.1 Beam modulation
Modulation our light beam is vital for many of our experimental setups, as was demonstrated in
Chapter 3. In our experiments we use EOM and AOM to externally modulate our beams.
Electro Optic Modulator – EOM
Figure 3.23: Simple example of an EOM’s function, with voltage on and off.
An electro-optic modulator (EOM) works as an electrically-actuated polarising beamsplitter and
waveplate, meaning it changes the birefringence of a nonlinear crystal through applying a Voltage
62 Atomic magnetometers
and distorting the electrons in its internal structure. It can reach modulation speeds up to about
20MHz{I} and requires high driving voltages and special amplifiers to reach the hundreds of Volts
needed for affecting the crystal.
As an example, when one does not apply a Voltage, the light passes through the crystal and then a
polarizer at the output can allow full or zero transmission depending on how we aligned the polarizer.
If you apply voltage across the crystal, along one direction of it, it creates birefringence along this
axis and it creates a phase difference between the two polarizations of an incoming linearly polarized
light. Typically, an EOM relies on a linear electro-optic effect, which is known as the Pockels effect.
This effect explains the change of the refractive index of a nonlinear (non-centrosymmetric) crystal
by an electric field that is proportional to the strength of the field.
One can now control through the applied voltage the rotation of the polarization and the amount
of the transmitted light and, as a result, control the power, phase or polarization of a beam. If one
applies a varying voltage, the EOM frequency-shifts the light and so a very common use of EOMs is
to generate sidebands on the central frequency of the light.
Acousto Optic Modulator – AOM
An acousto-optic modulator (AOM) diffracts the light beam that passes through it into several
orders. It utilizes the acoustic waves causing an acousto-optic effect in which a material’s permittivity
changes due to mechanical strain, in this case using sound waves.
Figure 3.24: Simple example of an AOM’s function.
As in the EOM, the key element in this case is a crystal through which light passes. On the crystal,
there is a piezoelectric transducer attached which is connected to a source that provides to it an
oscillating electrical signal. The piezo creates a sound wave that produces an oscillating mechanical
strain through the material. The mechanical stress creates a medium density which is also periodic,
with regions of high and low density and so it induces a periodic change of the refractive index in the
crystal. The periodic refractive index in the crystal behaves like a grating creating constructive or
destructive interferences. Light passing through the AOM scatters off the created periodic grating
{I}We use less than 1MHz modulation in our experiments
Experimental devices 63
and experiences something like Bragg diffraction, as seen in Figure 3.24, operating under the Bragg
condition nλ = 2Λsinθ, where n = order of diffraction, λ = wavelangth of light, Λ = wavelength of
the acoustic wave and θ = angle of incidence.
When considering cost, the AOM is generally a more affordable option than the EOM and is usually
used for intensity modulation. In our case we were sometimes limited by the power output of the
AOM and used an EOM instead. On the other hand the AOM’s electronics are cheaper and it allows
high enough modulation, so that we used AOMs more often in our setups than EOMs. AOMs also
frequency shift the output beams, but these frequency shifts are generally smaller than for the case
of the EOMs.
3.5.2 Lock-in amplifier – LIA
For atomic-magnetometer experiments we need to be able to measure the Larmor frequency of our
system. However, this is not the only frequency present in our signal: in a lab we have a broad range
of white noise consisting of lower and higher frequencies than the one we want to detect. In order
to extract our signal from the noise, we use a device called the lock-in amplifier (LIA). A simple
explanation follows of how a LIA works in the time domain.
A LIA requires a frequency reference to detect the response of the experiment at this frequency. The
LIA amplifies the signal and multiplies it by the reference signal. After using a low pass filter and in
the case where the reference frequency equals to the signal frequency, the single-phase LIA output is
a signal proportional to the signal amplitude V0 and the phase φ of the reference:
Signal ∝ V0sin(ωt) , (3.16)
Reference ∝ V0sin(ωt+ φ) , (3.17)
1
LIAoutput = 2V0cos(φ) . (3.18)
From the above equations, we can easily see that, since the cosine component can be either negative
or positive, the output can also be either negative or positive. For φ = 0, the LIAoutput = 12V0 and
usually the calibrated output is LIAoutput = √12 V0. The peak-to-peak amplitude is Vpp = 2V0, so the
final LIA output is:
= V√ppLIAoutput . (3.19)2 2
The lock-in amplifiers that I refer to in this thesis are dual-phase LIA, which means that we get two
outputs X and Y with the reference oscillator shifted by 90◦. By calculating the magnitude R of the
vector (X,Y) one gets rid of the phase dependence as follows:
64 Atomic magnetometers
1
X = √ V0cos(φ) , (3.20)2
= √1Y V0sin(φ) , (3.21)2
1
R = (X2 + Y 2) 2 = V0 . (3.22)
We can see that the R, contrary to the output of the LIA, is always positive and does not depend on
the phase between the signal and lock-in reference.
3.5.3 Laser
The word laser has become part of everyday language over the past few decades. It is however
an acronym which stands for Light Amplification by Stimulated Emission of Radiation. The light
emitted from a laser is monochromatic{I}, collimated{II} and coherent{III}.
In order to create a laser usually three processes are needed:
1. Inversion of population: More atoms or molecules exist in a high-energy level than in lower-energy
levels, as is usually the case in nature. Inversion happens by transferring energy from external sources
to the system (pumping).
2. Optical amplification based on stimulated emission: This process creates photons of the same
frequency and phase. When an atom is in an excited state, it can be stimulated by a passing photon
with a frequency ν∆E that corresponds to the energy gap ∆E of the transition from the excited state
to the ground state. The excited atom returns to the ground state and emits a second photon with
the same frequency ν resulting in two photons with the same frequency. This process creates photons
of the same frequency and phase.
3. An optical resonator{IV}: two mirrors are positioned on the two ends of a cavity, so that the
photons created by the stimulated emission are reflected between the mirrors repeatedly, while a
small part of the laser light gets emitted from one side.
Diode Lasers
Although the work of this thesis belongs to the field of atomic physics, we have solid-state physics to
thank for a fundamental part of our setups: diode lasers are affordable, robust and compact.
In order to start talking about semi-conductors and diodes we need to start by describing the prob-
ability of electrons occupying different energy levels. In solid-state physics a band structure is often
{I}Light of a narrow wavelength, close to a single wavelength
{II}The light’s waves are parallel to each other, and hence the beam is not diverging or converging for a long distance
{III}The light’s waves are in phase with each other in time and space
{IV}Or is it? See Chapter 6 and 7
Experimental devices 65
used to describe the range of electron energy levels within the bands, and the probability of electrons
filling the bands is determined by the Fermi-Dirac distribution:
f(E) = 1 , (3.23)
1 + e(E−Ef )/kT
where T is the absolute temperature and k is Boltzmann’s constant. The Fermi level with Fermi
energy Ef is the highest filled energy level at 0◦ K. The valence electron band is formed by the lower
energy orbitals (which are always filled), and the conduction band is formed by the higher energy
bands. There is a gap in energy between the valence band and the conduction band, in other words,
there aren’t energy levels there in the gap: the smaller this gap is, the less energy is needed for
electrons to transfer between the two bands and the closer the material is to a conductor.
Figure 3.25: Photon emitted when electrons and holes are recombined in a direct-band-gap struc-
ture. One can determine what the conductivity of a material is by looking at the material’s band
structure. In semi-conductors the conduction and the valence bands are not overlapping, but the
gap is also small enough for an electron to gain energy and make the transition.
The population at the Fermi level is 50% and so it lies between the two bands and its distance from
the conduction band determines the material’s conduction properties. It can also change if one heats
the material or adds impurities to it, such as by introducing donor or acceptor atoms (n-type or
p-type). A laser diode is a p–n diode semi-conductor, which is a semi-conductor consisting of a p-
type semi-conductor and an n-type semi-conductor. When free holes are added in a semiconductor
(doping) this one is then called p-type, and when it is doped with free electrons, it is called n-type.
The population inversion is happening along the p-n junction. To have stimulated emission, the
electron population in the conduction band must be higher than in the valence band and the current
required for lasing to happen is called the threshold current. Similar to electrons changing atomic
states, when holes and electrons are in the same region, an electron may transition from its area (the
conduction band) to the valence band and emit a photon, see Figure 3.25. This process is called
recombination. In order to initiate it, an external voltage source is used causing electrons to move
through the junction and fall into holes.
66 Atomic magnetometers
The cavity of the laser is formed by two high reflective mirrors, with one of them having less than
100% reflectivity so that the laser beam can exit through it. The laser light coming out of the laser
diodes is linearly polarized along the junction, but their polarization is varying and depending on
temperature or optical power.
ECDL – Extended-Cavity Diode Laser
In order to fine-tune and stabilize parameters of a diode laser, such as power and wavelength, often
an optical and electronics system is used after the output of the diode-laser beam.
Figure 3.26: ECDL Littrow configuration.
One of these systems is the extended-cavity diode laser (ECDL) which consists of a laser diode inside
a larger cavity and other optical elements. A common configuration for the optical system of an
ECDL is the Littrow configuration: this consists of the external cavity defined between the 100%
rear reflective mirror of the diode and the grating few cm away, a collimating lens, and a diffraction
grating attached to a turning lever and a piezo element [63], [64]. Often a mirror is attached after
the grating so that, when the grating is moved to adjust the wavelength, the output beam does not
change direction. The external cavity’s length can also be controlled by changing the temperature
and hence stabilizing this temperature is crucial. The broad tuning range of the laser is determined
by rotation or translation of the grating, while smaller changes in the laser’s wavelength are made by
using a piezo actuator.
Experimental devices 67

You spin me right ’round, baby, right ’round
Like a record, baby, right ’round, ’round, ’round
Dead or Alive
4
All-optical spin locking in alkali-vapor
magnetometers
The text presented in this section is part of a paper that has been published as [65]. For the work
presented in this Chapter, I build upon the idea conceived by Dmitry Budker. Together with Guzhi
Bao we constructed the setup needed for the experiment. I took part of the data and analyzed part
of them. Together with the rest of the co-authors, I co-wrote and edited the manuscript.
4.1 The idea
In the first Chapters of my thesis, the theoretical basis was set for understanding nonlinear magneto-
optical effects and atomic magnetometry. By the end of Chapter 3, in Section 3.4.1 we took a look
at the non-linear Zeeman (NLZ) effect and how it affects our resonances in atomic magnetometry.
The NLZ effect is the biggest limitation when aiming for the high sensitivities achieved by OPMs
operating in Earth’s magnetic field. It causes two important problems in accurate measurements:
first, the splitting and, as a result, the broadening of the magnetic resonance which causes in turn
loss in sensitivity [30]. The second issue, is the measurement errors related to the orientation between
the lasers and the magnetic field, the so-called heading-error [66].
There have been several techniques proposed for canceling out the NLZ effect: tensor light shift
effects [67], double-modulated synchronous optical pumping [68], polarization modulation [69] and
frequency modulation [70] and more recently spin locking (SL) [71].
In the latter scheme, an oscillating magnetic field (RF field) or short magnetic-field pulses applied in
the laboratory frame results in an effective static magnetic field along the atomic magnetization in
69
70 All-optical spin locking in alkali-vapor magnetometers
the rotating frame (in the rotating frame the oscillating field looks like a static magnetic field). The
atomic spin state then precesses about this static field, rather than evolving into a different state,
as it would under the action of the nonlinear Zeeman effect alone. As a result, the spin-locking field
prevents splitting, shifts, and line-shape asymmetries from occurring.
Figure 4.1: NLZ effect in Cs. (From left to right) Figure 1: The Zeeman effect allows us to polarize
our atoms and get a magnetic resonance. Figure 2: The Nonlinear Zeeman effect is splitting the
resonance depending on the number of m levels. Figure 3: When the magnetic field and the beams
are not perpendicular to each other (for example, here they are at 20◦), the already split resonance
get an asymmetry.
A potential drawback to this approach is that applied magnetic fields may lead to crosstalk between
closely located sensors (for example in a gradiometric configuration). This would limit the applic-
ability of this technique to sensor networks, which are important in biomedical applications, such as
imaging the human heart or mapping brain activity [72–76], as well as magnetic field monitoring in
experiments searching for a permanent electric dipole moment of the neutron [77, 78]. Additionally,
in remote magnetometry applications, spin-locking magnetic fields cannot be directly applied to the
atomic sample [79–81]. The all-optical spin locking technique applied in this work is used to bypass
such difficulties.
4.2 Spin locking
Spin locking, as demonstrated in [71], was generated by the use of a small magnetic field applied in
the rotating frame along the main magnetization of the spins of Cs atoms. Spin locking could in
theory be induced by using a fictitious magnetic field– in the form of a light field– rather than a real
magnetic field. In the presence of light, the energies of Zeeman sublevels are subject to AC Stark
shifts, or “light shifts” [82–85].
Spin locking 71
Linearly polarized light causes scalar and tensor shifts, while circularly polarized light can cause a
vector shift. The scalar component shifts the Zeeman sublevels all together [86]. The effect of the
vector light shift (VLS) is analogous to a fictitious magnetic field: it acts like the Zeeman effect on the
atomic structure [83,87]. VLS have been studied in the context of all-optical magnetometry [18], [85];
in particular, light was used as a substitute for RF (radio frequency) fields [86,88] in optically pumped
RF-driven magnetometers.
Here, we demonstrate all-optical compensation of the nonlinear Zeeman shift in a magnetometer
using spin locking by replacing the RF field with an intensity- and polarization-modulated laser
beam. With this method we can build a highly sensitive multi-sensor magnetometer array capable
of working in the magnetic-field range of the Earth.
To describe the physics of spin locking, we start with a total angular momentum F = 1 system
interacting with a leading magnetic field along ẑ and an oscillating magnetic field along x̂. The
atomic spins are initially prepared in the mF = 1 state along the x̂ direction by a circularly polarized
pump field, and we assume that the probe-light power is sufficiently low to be neglected for the
dynamics. With the quantization axis along ẑ and in the rotating frame (applied field is co-rotating
with the angular-momentum vector) the initial state is:
 
 1ψ(0) = 12 √ 2 . (4.1)
1
The Hamiltonian for the system under the rotating-wave approximation (RWA){I} for the RF field
is [71]
  ωrev −
Ω√RF
2 2 0 
Ĥ = ℏ−Ω√RF 0 −Ω√RFRWA 2 2 2 2  , (4.2)
0 −Ω√RF2 2 ωrev
where ωrev is the “revival” frequency characterizing the strength of the NLZ effect (See 3.4.1) and
ΩRF is the Larmor frequency induced by the oscillating field and is proportional to its amplitude.
We write the state ψ(t) of F as a superposition of energy eigenstates Ψ with eigenvalues E :
∑ k k
ψ(t) = Ψ e−iEkt/ℏk . (4.3)
k
{I}An oscillating field with linear polarization is a superposition of rotating fields: one rotating in a clockwise sense and
the other rotating in a counter-clockwise sense. In the rotating frame on resonance, the co-rotating field is static and the
counter-rotating field appears to rotate at twice the frequency. The terms of the Hamiltonian that are rapidly oscillating
are then neglected and this approximation is called the rotating-wave approximation (RWA). When the driving field is
strong enough, the counter-rotating field could produce a shift in the resonance frequency of the system: the Bloch-Siegert
shift.
72 All-optical spin locking in alkali-vapor magnetometers
The probability P (t, 0) for an atom to be found in the initial state, |⟨ψ(t)|ψ(0)⟩|2, can be written as
( 0) = ω
2 2 2
P t, lock
+ΩRF + ωrev cos(ωlockt)
2 , (4.4)2ωlock
where ωlock = (ω2 2 1/2rev + ΩRF ) is the spin-locking oscillation frequency. With an increase in the
spin-locking field amplitude, the oscillating component of P (t, 0) decreases in amplitude and the
frequency of the oscillation increases (see Figure 4.2). In this simplified model, it appears that spin
locking improves with the amplitude of the applied field. However, when the RF field is large enough
so that the rotating-wave approximation is not valid anymore, the presence of the spin-locking field
leads to power broadening of the magnetic resonance. Under this condition the optimal amplitude of
the field is when ΩRF is comparable to ωrev.
1.2 Ωrf >> ωrev
Ω r f = 2ωrev
1.0 Ωrf << ωrev
0.8
0.6
0.4
0.2
0
0 0.25 0.5 0.75 1
time （2�/ωr e v )
Figure 4.2: Probability P (t, 0) for an atom to be found in the initial state. For small amplitudes of
the spin-locking field (Ωrf ≪ ωrev), the probability undergoes quantum beating. If the amplitude of
the spin-locking field is much larger than the NLZ parameter (Ωrf ≫ ωrev), the atoms remain in the
initial state. If the amplitude of the spin-locking field is equal to the NLZ parameter (Ωrf = 2ωrev),
the atoms are partially locked in the initial state and the populations undergo oscillation with
frequency ωlock.
4.2.1 Spin Locking with AC Stark shift
The vector component of the light shift induced by circularly polarized light can be interpreted as a
fictitious magnetic field along the light propagation direction, with magnitude [89]
−c(∆)Iϵ
Bfict = , (4.5)ℏγCs
Spin locking 73
where c(∆) is a proportionality constant that depends on atomic parameters and the frequency
detuning ∆ from an atomic resonance, I is the light intensity, γ = 2.2 · 1010Cs rad/(s · T) is the
Cesium gyromagnetic ratio and ϵ = [I(σ+)− I(σ−)] /I is the Stokes parameter specifying the degree
of circular light polarization (ϵ = +1 for σ+-polarized light, ϵ = −1 for σ−-polarized light and ϵ = 0
for linear polarization).
Consider the same system as in the previous section– an F = 1 system with a leading magnetic field
along ẑ and a circularly polarized pump beam propagating along x̂. Now, rather than an RF field, we
apply a modulated, circularly polarized light beam propagating in the same direction as the pump,
near-resonant with a transition to an F = 0 state, in order to induce vector light shifts. Next, let’s
calculate what the effective Stark-shift Hamiltonian is.
Effective Hamiltonian of Stark Shift
We will, here, calculate the light shift due to a circularly polarized field that is near-resonant with
an J = 1 → J ′ = 0 atomic transition. The detuning from the atomic resonance is called ∆ and
we assume the light is σ+ polarized and propagating along the x-direction. If we use the x-basis
{|1,−1⟩x, |1, 0⟩x, |1,+1⟩x, |0′, 0′⟩x}, the light is only affecting one transition |1,−1⟩x ↔ |0′, 0′⟩x due
to the selection rule for σ+ polarized light (see Figure 4.3). The dipole interaction H = −d · E(t)
can be used to describe the interaction between the atom and the light. The Hamiltonian in rotating
wave approximation (RWA) with rotating frequency ω0 +∆ is
 
0 0 0 0

0 0 0 0 
H = ℏ0 0 0 − Ω√R2 3  , (4.6)
0 0 − Ω√R2 3 −∆ x
where ΩR is the Rabi frequency and |Ω 2R| is proportional to the intensity of the light. The effect of
oscillating fields with frequency 2ω0+∆ is ignored by the RWA. The energy eigenvalue EJ,m of thisJ
system is:
ℏ √ 2
E1,−1 = 6(−3∆ + 9∆
2 + 3Ω2 ) ≈ ℏΩRR 12∆ ,
E1,0 = 0 ,
(4.7)
E1,1 = 0 ,
ℏ √ 2
E0′,0′ = 6(−3∆− 9∆
2 + 3Ω2 ) ≈ −ℏΩRR 12∆ .
Since the light is detuned from atomic resonances, it is possible to derive an effective Hamiltonian for
the ground states only. In the x-basis for the ground states {|1,−1⟩x, |1, 0⟩x, |1,+1⟩x}, this Hamilto-
nian reads
74 All-optical spin locking in alkali-vapor magnetometers
 
ℏΩ2
H Reff = 1 0 0 12∆ 0 0 0 . (4.8)
0 0 0
x
Figure 4.3: (a) Schematics for atomic levels J = 1 and J ′ = 0 coupled by circularly polarized
light propagating in the x-direction. Only two levels are coupled in the x-basis. (b) All levels are
coupled if one used the z-basis.
We are now interested in finding the Hamiltonian in the z-basis. In order to do so, we notice that a
π/2 rotation about the y-axis rotates the x-axis into the z-axis. This rotation matrix can be written
as
[ ( ) ]   1 −√1 12π 2 2 
R( ŷ, π/2) = exp −i Ĵy = √12 0 −√12 2  , (4.9)
1
2 √
1 1
2 2 x
We can now find the effective Hamiltonian in the z-basis as
Ĥ+eff = R(ŷ, π/2)HeffR(ŷ, π/2)† 
1 √1 1
ℏΩ2 = R 12∆ 
4 −2 2 4 
− √1 12 − √
1 
2 2 2 2  . (4.10)
1 − √1 14 2 2 4 z
Spin locking 75
For σ− polarized light, the effective Hamiltonian in the z-basis is:
 14 √1 1ℏΩ2  2 2 4 Ĥ− = R  √1 1 1 eff 12∆ 2 2 2 √2 2  . (4.11)
1 1 1
4 √2 2 4 z
With this we arrive at the effective Hamiltonian in the form of Eq. (4.12). The effective Stark-shift
Hamiltonian for σ+ and σ− beam can be represented as:
+ 2
Ĥ+ = (Ωℏ LS)eff ∆ (K1 −K2),
(Ω− 2
Ĥ− = ℏ LS)eff ∆ (K1 +K2),
1 
 
K1 = 1 0 10 2 0 , (4.12)48
 1 0 1
1 
√ 
0 2 0
√ √ 
K2 = 48 2 0 2

√  ,
0 2 0
where Ω±LS is the Rabi frequency of σ±- polarized light, |Ω
± 2
LS | is proportional to the intensity of
the light I(σ±), and ∆ is the detuning. The light-shift beam can be intensity- and/or polarization-
modulated as we will see in the details in the experimental section below. The intensity of σ±-
polarized light in Figure 4.6(c) is:
I( ) = cos (ωRF t) + | cos (ωRF t)|σ− I0 2 , (4.13)
I( ) = − cos (ωRF t) + | cos (ωRF t)|σ+ I0 2 ,
where ωRF is the light-shift-field modulation frequency. By substituting Eq. (4.13) into Eq. (4.12),
we get the total Stark-shift Hamiltonian:
= ℏ(Ω
m 2
Ĥ LS
)
LS ∆ [K1| cos (ωRF t)|+K2 cos (ωRF t)] , (4.14)
where ΩmLS is the amplitude of the modulated Rabi frequency. The first term is an unwanted per-
turbation caused by the light shift beam. The second term describes a pure fictitious RF field
Bfict ∝ I0 cos(ωRF t) which is used to perform all-optical spin locking.
76 All-optical spin locking in alkali-vapor magnetometers
(a) (b) (c)
1.0
0.0
-1.0
2.0
1.5
1.0 2ωrev
0.5
0.0
0.975 1 1.025 0.975 1 1.025 0.975 1 1.025
Ω L /Ωm Ω L /Ωm Ω L /Ωm
Figure 4.4: Theoretical calculations of signal. Theoretical calculated in-phase (top row) and
quadrature (bottom row) first-harmonic amplitudes of optical rotation signal. The NLZ effect
splits the magnetic resonance into two peaks: (a) without the the spin-locking field (b) with RF
field (c)with intensity modulated light-shift field. For these plots, the parameters Ωpr/γ = 100,
δpr/γ = 106, Ωp/γ = 104, δ /γ = 103p , ∆m/γ = 103, ω mrev/γ = 1.5, ΩRF /γ = 0.02, ΩLS/γ = 0.02,
∆/γ = 105 are chosen.
4.2.2 Optical Rotation Signal
Let’s assume that probe light that is linearly polarized along x̂ with detuning δpr and Rabi frequency
Ωpr, is used to measure the atomic state of Cs during its evolution. The propagation direction ŷ of
the probe is perpendicular to both the propagation direction of the pump light x̂ and the direction
of leading field ẑ. The pump field with central detuning δp and Rabi frequency Ωp is periodically
modulated with frequency Ωm. To simplify the calculations, we assume that the frequency of the
pump is sinusoidally modulated, with modulation amplitude ∆m.
When Ωm = ωRF (when the pump modulation frequency is equal to the light-shift-field modulation
frequency), we can solve the time-periodic evolution equation using Floquet theory [90,91]. Results of
numerical calculations with the AtomicDensityMatrix (ADM) package [30] are shown in Figure 4.4 to
illustrate the in-phase and quadrature first-harmonic amplitudes of the optical-rotation signal. Here,
we consider each sublevel to undergo relaxation (for example, due to spin-depolarizing collisions) at
a rate γ. In addition, the upper state undergoes spontaneous decay at a rate Γ = 106γ. Without the
light-shift beam [4.4(a)], the magnetic resonance is split due to the NLZ effect. Figures 4.4(b) and (c)
show the magnetic resonance with RF field and amplitude-modulated light-shift field, respectively.
Quadrature signal (a.u.) in-phase signal (a.u.)
Experimental setup 77
The NLZ splitting apparent in 4.4(a) is no longer visible in 4.4(b) or 4.4(c) as in both cases spin-
locking is achieved, either through an RF field or through the modulated light shift, demonstrating
that theoretically both approaches achieve the desired effect.
Up to now, we show the theoretical simulation of all-optical spin locking in F = 1 to F ′ = 0 system.
For a real atomic system, there are many hyperfine levels that need to be taken into account its
modeling, something that would increase the computational complexity. Hence, for this case, we
present only experimental results.
4.3 Experimental setup
Figure 4.5: Experimental setup for all-optical spin locking: AOM: acousto-optic modulator;
EOM: electro-optic modulator; HWP: half-wave plate; QWP: Quarter-wave plate; PBS: polarizing
beam splitter; PD: balanced photodetector; LIA: lock-in amplifier; LO: local oscillator. A partial
view of the magnetic shield is shown in the figure. Atoms are contained in a glass cell positioned in
the center of the magnetic shield and are pumped (along −x̂) and probed (along ŷ) by laser beams
under a static magnetic field (along ẑ). The intensity of the light-shift laser beam is sinusoidally
modulated with an AOM at a frequency Ωm, while its polarization is switched between the σ+
and σ− states every π/Ωm, using an EOM. Inset shows the monitor setup for polarization and
amplitude of the light-shift beam.
Figure 4.5 shows the experimental apparatus. A paraffin-coated cylindrical cell with a length of
5 cm and a diameter of 4 cm containing 133Cs at room temperature, is enclosed within a four-layer
mu-metal magnetic shield.
78 All-optical spin locking in alkali-vapor magnetometers
The atoms are pumped in the stretched state along the −x̂ direction with a circularly polarized, −x̂-
directed laser beam. The pump-laser frequency is locked to the Cs D2 62S1/2 F = 3→ 62P F ′3/2 =
4 transition at 852 nm with a dichroic atomic vapor laser lock (DAVLL) [92]. The beam is pulsed
(3% duty cycle) with an acousto-optic modulator (AOM). The time-averaged light power– while the
AOM is on –is 50 µW{I}.
The polarization rotation of a 10 µW, ŷ-directed probe beam induced by the polarized atomic
vapor is measured with a balanced polarimeter after its transmission through the cell. The beam
is linearly polarized along the x axis and detuned by about 4 GHz towards higher frequencies (blue
detuned) from the D2F = 4→ F ′ = 5 transition.
A circularly polarized light-shift beam produced with a Ti:sapphire laser propagates parallel to
the pump beam. The intensity of the beam is modulated with an AOM and its polarization is
modulated with an EOM in order to provide a time-varying light shift. The setup for monitoring
the modulation of the light-shift beam is shown in the inset of Figure 4.5. A quarter-wave plate is
used to convert the circular σ+ and σ− components into orthogonal linear polarization components
the intensities of which are monitored with two photodetectors. The waist of the collimated laser
beam in the vapor cell is 1.5 mm. The frequency of this laser can be widely tuned and is, for most
of the experiments presented here, detuned by 10 GHz from the 62S 2 ′1/2 F = 4 → 6 P3/2 F = 5
D2 transition towards lower frequencies (red detuned). Its frequency is stabilized to the internal
reference cavity of the laser. A detuning of 10 GHz was chosen to minimize optical pumping by
the light-shift beam, while maintaining a sufficient fictitious-magnetic-field amplitude (∼14 nT for
250 mW power) for spin locking.
To measure the magnetic resonance, we fix the modulation frequency Ωm of both the pump and
light-shift beams at a particular value (corresponding to the Larmor frequency for a magnetic field
of up to 100 µT). We scan the leading ẑ-directed magnetic field and observe the polarization of
the probe beam. The signal from the balanced polarimeter is fed into a lock-in amplifier and is
then demodulated at the modulation frequency. The magnetic resonance can be observed in the
polarization rotation amplitude and phase of the probe beam.
4.4 Results
We employed three different methods to modulate the light-shift beam and achieve spin locking.
Figure 4.6 shows the amplitude of the lock-in output as a function of the leading magnetic field
around 60 µT and with the pump-laser modulation frequency fixed at 216,620 Hz. The magnetic
resonance spectra are shown without and with application of the light-shift beam (black and red
curves, respectively), as well as without the pump beam (blue curve).
In the method depicted in Figure 4.6(a,b), either the intensity or the polarization of the light-shift
beam is modulated, to provide a sine-modulated light shift, as in Refs. [85,86]. In the polarization-
modulation scheme, the fictitious magnetic field is oscillating around zero. However, in this
{I}Our power meter takes a specific amount of samples per second and averages the values in this range
Results 79
（a） （b） （c）
pump
pump pump
intensity
ϭ
polarization intensity +polarization ϭ -
effective rf field 0 effective rf field effective rf field 0
0
3.2
without light shift
with light shift
2.8 without pump
2.4
2
1.6 FWHM: 25 Hz
1.2
0.8
0.4
0
216500 216550 216600 216650 216700 216550 216600 216650 216700 216550 216600 216650 216700
Larmor frequency (Hz) Larmor frequency (Hz) Larmor frequency (Hz)
Figure 4.6: (Experimental data) Magnetic-resonance line shape for a modulation frequency of
216,620 Hz as a function of the leading magnetic field along the z axis with applied light-shift field
and pump (red line), without light-shift field and pump (black line), and without pump field (blue
line). The amplitude of magnetic resonance without light shift is normalized to unity. The power in
(a) and maximum power in (b,c) of the light-shift beam is 200 mW. The inset shows the polarization
modulation (a), intensity modulation (b) and both modulated (c) schemes for the pump and light-
shift field.
scheme, the light is only purely circularly polarized when ϵ = ±1 (see formula 4.5); the presence
of the other polarizations causes tensor-light shifts that broaden the transition. In the amplitude-
modulation scheme, the VLS produces a fictitious magnetic field of magnitude Bfict ∝ [1 +
cos(Ωmt)]. The oscillating term of the fictitious field locks the spins. The static term of the fictitious
field plays no role in spin locking, but the constant light leads to broadening of the linewidth due
to residual optical pumping and heading error. In the absence of the light-shift beam, the magnetic
resonance is split into eight partially-resolved Lorentzian peaks, due to the NLZ effect. When we
apply the modulated light-shift beam we get a narrower full-width-half-maximum (FWHM) central
peak and an amplitude increase.
In Figure 4.6(c), the direction of the fictitious magnetic field is modulated by switching the polar-
ization of the light-shift beam from σ+ to σ− using an EOM. In addition, the intensity I of the
light-shift beam is modulated with an AOM as I ∝ | cos(Ωmt)|. In this modulation scheme, neglect-
ing the counter-rotating component, the fictitious RF field co-rotates with the precessing spins in the
laboratory frame. Since the polarization and propagation direction of the light-shift beam and the
Normalized magnetic resonance (R)
80 All-optical spin locking in alkali-vapor magnetometers
(a)
10 1.0
5 0.5
0 0.0
-0.5
-5
-1.0
-10
-0.05 0.00 0.05 0.10 0.15
(b) Time (s)
90
360 80
70
270
60
50
180
40
30
90
0 90 180 270
phase of polarization modulation ϕ2 (degree)
Figure 4.7: (a) Observed monitor signal and input intensity modulation signal. Here ϕ1 = −π/2
and ϕ2 = −π/2. The distortion of fictitious RF field is mainly caused by nonlinear response of the
AOM. (b) Phase scanning of ϕ1 and ϕ2.
pump beam are the same, the light-shift beam also pumps the atoms. The blue lines in Figure 4.6
show the magnetic resonance with the light-shift beam on but without a pump beam. The stochastic
nature of the optical pumping leads to a decrease in atomic-coherence and results in broadening of
the magnetic resonance.
The phase of the intensity and polarization modulation needs to be chosen carefully to ensure that
the spin-locking field points along the direction of the precessing spins [71]. To create a fictitious
phase of amplitude modulation ϕ1 (degree) monitor signal (V)
intensity modulation signal (V)
linewidth (Hz)
Results 81
RF field that changes smoothly, the polarization should be switched when the intensity is modulated
to zero. Additionally, the fictitious RF field has to be in-phase with the pump pulse. We show the
monitor signal (produced by subtraction of the σ+ and σ− recorded powers) for ϕ1 = −π/2 and
ϕ2 = −π/2 in Figure 4.7(a), as a fictitious RF field. Here ϕ1 is the phase of the intensity modulation
and ϕ2 is the phase of the polarization modulation. Figure 4.7(b) displays the magnetic-resonance
line width for different ϕ1 and ϕ2. Some of the best results are achieved around ϕ1 = 3π/2 and
ϕ2 = π/2.
Figure 4.8 shows the effective linewidth of the magnetic resonance in Earth-field range ( 60 µT) as
a function of the applied light-shift beam power and its detuning. When the light-shift beam has
low power and is detuned far off resonance, there is no spin locking and the effective line width is
≈100 Hz. When the light-shift beam is near resonant with the atomic transition, the effect of optical
pumping is much stronger than that of the VLS. As a result, the line width of the magnetic resonance
is even broader than that observed in the absence of the light-shift beam. When the light is far-off
resonant from the optical transition, the optical pumping is negligible and the VLS dominates the
interaction. We observe a minimum of the line width for a 220 mW light-shift beam, 10 GHz detuned
below the D2 62S1/2 F = 4 → 62P ′3/2 F = 5 transition. Note, however, that spin locking works well
also for the opposite sign of detuning, corresponding to a sign reversal of the effective RF field. We
could not apply higher powers because of the available laser.
180
15
160
10
140
5
120
0
100
-5
80
-10
60
-15
40
-20
50 100 150 200 250
power (mW)
Figure 4.8: Map of the magnetic-resonance linewidth as a function of the applied light-shift-field
power and detuning. The minimum linewidth is 25.25(6) Hz.
detunning (GHz)
linewidth (Hz)
82 All-optical spin locking in alkali-vapor magnetometers
4.5 Conclusions
In Chapter 3.1.6 the sensitivity of a magnetometer is explained and studied and it becomes clear
that the smaller the linewidth and the bigger the amplitude the higher the sensitivity of our mag-
netometer. Here we have demonstrated an all-optical method to suppress the NLZ effect in the
range of the Earth’s magnetic field by using spin locking, which effectively manifests by reducing the
linewidth or our magnetic resonance.
A polarization and intensity-modulated light-shift beam is applied which suppresses NLZ-related
broadening of the magnetic resonance. This method works with either application of intensity mod-
ulation or polarization modulation, but the combination of both yields the best results.
In contrast to other techniques, this method does not cause crosstalk in sensor networks and does
not interact with samples close to the sensor. The Larmor frequency induced by the optimally
effective spin-locking field in the rotating frame is one order of magnitude larger than the spin-
revival frequency; the phases (ϕ1 and ϕ2) are chosen such that the corotating part of the fictitious
RF magnetic field is colinear with the precessing spins.
We note that the method improves the sensitivity of Earth-field-range magnetometers for two reas-
ons: the increase in the magnetic-resonance signal amplitude and the reduction in the
effective linewidth.
Intuitively, the area under the black and red curve in Figure 4.6 should be of equal size since spin
locking would combine the 8 peaks to one. We observe however a larger under-the-curve area for spin
locking compared to no spin locking. That could be due to residual optical pumping by the light-shift
beam, which could be improved if we detuned the beam even further and used higher powers.
Another interesting finding is the resonance lines look more noisy from left to right as we look at
Figure 4.6. This is speculated to be a result of magnetic noise, which we could find out if we calculate
the sensitivity. Since the slope of our resonance increases, the quantum limit of the sensitivity is
improved, but there is also magnetic noise in our shield because of the noisy current source we are
using. If the quantum noise is bigger than the magnetic noise we won’t notice the latter, but when the
sensitivity improves the magnetic noise emerges. In better words: a more sensitive magnetometer
is also responding more to magnetic noise, hence the difference in the noise between our different
resonances.
Conclusions 83

The real voyage of discovery consists,
not in seeking new landscapes, but in having new eyes.
Marcel Proust
5
Rb magnetometry free from NLZ-effect-related
heading error in Earth-field range
The text presented in this section is part of a paper that has been published as Ref. [93]. For the
work presented in this Chapter, I build upon the idea conceived by Rui Zhang and we constructed
the setup needed for the experiment together. I took a big part of the data and analyzed part of
them. Together with the rest of the co-authors, I co-wrote and edited the manuscript.
5.1 The idea
Very sensitive and low-cost magnetometers that work on Earth-field, in an unshielded environment
have been the holy grail of optical magnetometry for many decades. We discussed in Chapter 3 that
such devices would be of great importance for the production of inexpensive and sensitive sensors or
imaging devices that can operate in many environments. This appealing perspective has been the
motivation for a collective effort of thinking and testing ways to tackle the biggest limiting factor for
scalar Earth-field magnetometers, namely the heading error.
In Section 3.4.2 one can find a more detailed description of the effect, but to freshen up the reader’s
memory, heading error is the dependence of the scalar magnetometer’s reading on the direction of
the magnetic field [19]. There are mainly three physical sources of heading error: the nonlinear
Zeeman effect (NLZ) due to the coupling between electron spin and nuclear spin [94–96], the dif-
ferent gyromagnetic ratios of two ground hyperfine states due to the linear nuclear Zeeman
effect (NuZ) [96,97], and the magnetic-field-direction-dependent light shift (LS) [95]. The first
two effects lead to the orientation-dependent asymmetry of the magnetic resonance curve, and the
85
86 Rb magnetometry free from NLZ-effect-related heading error in Earth-field range
third leads to the orientation-dependent shift of the magnetic resonance frequency. These changes
of our measured magnetic resonance lead to a loss of sensitivity (see Chapter 3). Since atomic mag-
netometry is famous for its unmatched sensitivity the heading error is posing a serious problem in
the field. We will in this Chapter occupy ourselves with a strategy to suppress NLZ-related heading
error. To paraphrase John F Kennedy, “Ask not what your magnetometer can do for you, ask what
you can do about its heading error”.
5.1.1 Strategies to suppress the NLZ-related heading error
Traditionally there are three (3) strategies to deal with NLZ-related heading error:
1. Physical or theoretical compensation, including compensation with tensor light shift [98],
compensation with spatially separated pumping beams of opposite circular polarizations [95], and
using a high-power pump and correcting with theoretical predictions [96]{I}.
2. Using modulation schemes to address atomic spins in states less sensitive to direction-
related magnetic resonance asymmetry, including synchronous optical pumping with double mod-
ulation [68], push-pull pump [99], and spin-locking with synchronous optical pumping and radio-
frequency (RF) [94], such as our project covered in Chapter 4 or modulated optical [65] field.
3. Using transitions intrinsically free from NLZ-related splitting, for example, excitation
of high-order atomic polarization moments [53]. Compared with the first two strategies whose per-
formance relies on careful adjustment of experimental parameters, theoretically, the latter strategy
should give a better heading-error suppression, as it is intrinsically free from the NLZ effect, and it
does not require special hardware or additional modulation, which is especially welcome for mobile
or wearable systems. However, these approaches suffer from dramatic decrease of signal amplitude
in the geomagnetic field range, which precludes their applications in high-sensitivity magnetometry.
Our idea belongs to the last category since we tried to take a more fundamental approach to the
problem by asking ourselves the question: What makes the magnetic resonance split in the first
place? And, what if, we could only have one resonance and conceptually avoid the asymmetry
that would arise from multiple peaks? In the following pages I will shed some light into how we
dealt with these questions and how we came up with the idea of a simple heading-error-free atomic
magnetometer.
5.2 The Physics behind the idea
This section introduces briefly the physics needed to understand how we built our magnetometer and
why it works. For a more fundamental understanding one should start by reading Chapters 1 and 2
of this thesis.
To prompt the reader’s memory, the density matrix describes polarized states, as well as unpolarized,
since not all atoms in a physical system can be described by the same wave function. The diagonal
{I}In some cases, heading errors due to NuZ [96] or LS [95] effect are suppressed together with the NLZ-related effect at
the same time.
Experiment 87
elements of the density matrix represent the probability of finding the atom in a state |m⟩ or else,
their population. The off-diagonal elements are called coherences and they are at the heart of
many nonlinear and quantum optical phenomena including our very own nonlinear magneto-optical
rotation (NMOR) [100].
Let us remember polarized atoms—atoms described by the same wave function. A visually easy but
precise way to imagine polarized atoms, is the visualization of the probability of finding the max-
imum projection of the angular momentum of a polarized ensemble along a direction. The angular-
momentum probability surfaces, as they are called, reveal the symmetry of the polarization
state. This anisotropy {I} of the atomic states is caused by population differences and coherences
between Zeeman sublevels. [53].
At this point we need to bring into the discussion the quantum beats and realize that here lies
the essence of our idea: in the quantum-beats picture, the superposition of the Zeeman mF states
along the magnetic field is oscillating at a frequency ΩL that we can directly measure. We would
ideally build a magnetometer in a way that we have only one coherence oscillating at one frequency,
because if we have more than one these will be different because of the NLZ effect. If an atom
is prepared in the superposition of two mF states and if these two states have different energies,
then the superposition will evolve at a frequency corresponding to the energy difference between
these two states divided by ℏ. We will then have only a single alignment magnetic resonance in
a ground hyperfine state (in F = 1), corresponding to the transition between these mF states. A
magnetometer based on this resonance would be free from the NLZ-related splitting and asymmetry
of the magnetic resonance curve, and thus would be intrinsically free from NLZ-related heading error.
5.3 Experiment
Let us make the basic idea behind our magnetometer concrete. The energies of the Zeeman sublevels
are nonlinearly dependent on the magnetic field strength and so the frequency intervals between
the Zeeman sublevels are unequal and the differences become bigger as the field increases—up to a
certain field (see Section 5.1 for more).
As we mentioned before, if we pump with circularly polarized light we create orientation which is an
asymmetric polarization surface on one half of the polarization axis and, if we pump with linearly
polarized light, we create alignment, which creates a symmetric surface along the polarization axis
(see Figure 3.10). For an orientation experiment, the coherences or populations will evolve between
∆m = 1 Zeeman sublevels and for an experiment based on alignment they evolve between ∆m = 2
Zeeman sublevels; remember here, that the sublevels are separated by unequal intervals due to the
NLZ effect.
For the orientation experiment, in the F = 2 (F = 1) hyperfine state, there are four (two) different
∆m = 1 magnetic resonance frequencies, which lead to asymmetric broadening or splitting of the
{I}Anisotropy is the quality of exhibiting properties with different values when measured along axes in different direc-
tions. [101]
88 Rb magnetometry free from NLZ-effect-related heading error in Earth-field range
Figure 5.1: Schematic of NLZ-heading-error-free magnetometer. Nonlinear Zeeman effect of 87Rb
ground state 52S1/2 and corresponding magnetic resonance frequencies. In the F = 2 (F = 1)
hyperfine state, there are four (two) different ∆m = 1 magnetic resonance frequencies, labelled
with ω1 to ω4 (ω5 to ω6), respectively, and 3 (1) different ∆m = 2 magnetic resonance frequencies,
labelled ω′ to ω′1 3 (ω
′
4), respectively.
magnetic resonance curve when the background field is in the Earth-field range [94–96]. The pop-
ulations and the matrix elements of the different Zeeman sublevels change with the magnetic field
direction. This means that the amplitudes of the different components of the magnetic resonance
change also with the magnetic field direction, which leads to the magnetic-field-direction-dependent
asymmetry of the magnetic resonance and gives rise to heading errors [94–96]. In contrast to the
∆m = 1 magnetic resonance in orientation, there is only a single ∆m = 2 resonance in the F = 1
hyperfine state for alignment. This would mean that our resonance will have only one peak and is
then by default free from heading error; Figure 5.1 explains visually this process.
5.3.1 Pumping and probing
An important parameter for the success of our experiment was the pump-probe scheme (illustrated in
Figure 5.2) used to generate and respectively detect atomic polarization in the F = 1 hyperfine state.
We pump with a 795 nm laser with a frequency resonant with the 87Rb D1 transition (52S1/2F =
2→ 52P ′1/2F = 2) and we probe with a 780 nm laser light, red shifted about 0.5 GHz from the 87Rb
D2 transition 52S1/2F = 1→ 52P F ′3/2 = 0 transition.
In Figure 5.3 we show the spectrum of our pump and in Figure 5.4 the spectrum of our probe and
it is noted where we pump and probe accordingly. The spectra show Bennett structures related to
Doppler-free saturated absorption as well as Doppler-broadened resonances. Important note: these
Experiment 89
Figure 5.2: Pump and probe scheme of alignment magnetometer. Energy levels of 87Rb atoms
and excitation scheme (not to scale). Pump beam is a 795 nm laser beam exciting the 52S1/2F = 2
to 52P1/2F ′ = 2 transition, which generates atomic spin polarization in the 52S1/2F = 1 state via
repopulation pumping; the probe beam is a 780 nm laser tuned to the low frequency side relative to
the 52S 2 ′′1/2F = 1 to 5 P3/2F = 0 transition.
spectra are produced by a cell we call reference cell and it is different than the main cell we use for
our experiments which produces the magnetic resonance and that contains only 87Rb. We use the
Doppler-free saturated absorption spectrum of the reference cells containing both 87Rb and 85Rb for
stabilizing our laser frequencies.
Both lasers propagate along the leading magnetic field and quantization axis z and are linearly
polarized along y, the same way as we explained in the example in 1.4.4. The pump beam generates
atomic polarization in the F = 1 state via repopulation pumping [102] {I} (Figure 5.2) and the
polarization is monitored by detecting optical rotation of the probe light.
In the Earth-field range, the difference between the F = 1 and F = 2 resonance frequencies is in the
kHz range, see Figure 5.9. This frequency difference is much larger than the relaxation rate of the
ground-state polarization and, as a result, the F = 2 state will not be trapped in the dark state and
the pumping process is efficient. The pump and probe are resonant with transitions that start from
{I}In repopulation pumping a large fraction of the atoms is prepared in a coherent superposition of ground states non-
coupled with the applied laser radiation (“dark state”) [103]
90 Rb magnetometry free from NLZ-effect-related heading error in Earth-field range
Figure 5.3: D1-line spectroscopy of rubidium. Frequency decreases from left to right. In the
alignment-based magnetometer experiment the pump light is tuned to the line marked with a
dashed box.
Figure 5.4: D2-line spectroscopy of rubidium. The dashed line indicates the probe tuning in the
magnetometry experiment.
Experiment 91
different ground hyperfine states, and as a result this technique constitutes indirect pumping, which
does not cause power broadening of the magnetic resonance [104,105].
In an NMOR magnetometer that uses circularly polarized light to pump, we modulate the pump
light with the Larmor frequency (See Chapter 3). In our case, to produce a magnetic resonance, the
pump beam is modulated at two times the Larmor frequency of the F = 1 hyperfine state. This
interesting requirement can be explained by taking a look at the polarization surfaces.
5.3.2 q-fold symmetry
As already mentioned, by choosing the quantization axis along z, we are producing coherences,
which correspond to non-diagonal density matrix elements ρn,m, where n−m = q and the medium
has transverse polarization. Remember, that each PM has the symmetry of a particular spherical
harmonic. The q-th PM component has a q-fold symmetry, which means that when the angular-
momentum probability surface is rotated by 2π/q around the polarization axis, it coincides with itself
(see also density matrix rotation in [30]). When we pump with linear light, we get polarization with
2-fold symmetry around the light-polarization direction. We can easily understand this by looking at
the polarization surfaces. In our case we would have a peanut shaped surface (quadrupole) polarized
transverse to the magnetic field (and quantization axis). What kind of rotation around the field
would make the surface turn back to its original state? That would be a rotation by 2π/q = π, which
corresponds to q = 2. It would therefore make sense to pump with light that is modulated at 2ΩL.
The signal that we detect will be the optical rotation of the probe beam oscillating at this frequency
and as a result we will need to demodulate it at 2ΩL.
5.3.3 Apparatus
Our experimental arrangement is built in such a way that we can switch between an orientation
and an alignment magnetometer. This way, we can use the exact same experimental conditions to
compare the two resonances with regards to the NLZ effect and heading error.
The experimental arrangement for measuring the alignment magnetic resonance is shown in Fig-
ure 5.5 and for the orientation magnetic resonance in Figure 5.6.
The cylindrical anti-relaxation-coated atomic vapor cell has a diameter of about 4 cm and length of
about 5 cm, filled with isotope-enriched 87Rb.
For the alignment magnetometer both the pump and the probe beam are linearly polarized and
propagate through the atomic vapor cell along the z direction. The pump beam is square-wave
modulated with an AOM at 2ΩL with a duty cycle of 20%.
For the orientation magnetometer The pump (probe) beam is circularly (linearly) polarized and
propagates through the atomic vapor cell along the y (x) direction. In this case, the pump beam is
modulated at the Larmor frequency.
92 Rb magnetometry free from NLZ-effect-related heading error in Earth-field range
Figure 5.5: Alignment resonance setup. Pump, a linearly polarized 795 nm laser beam used to
generate the atomic alignment polarization; Probe, a linearly polarized 780 nm laser beam used
to detect the Larmor precession of the atomic alignment polarization via optical rotation; AOM,
acousto-optic modulator used to pulse the pump beam; PBS, polarizing beam splitter; BS, beam
splitter; λ/2, half-wave plate; λ/4, quater-wave plate; BPD, balanced photodiode; BPF, bandpass
filter with central wavelength of 780 nm, which is used to prevent pump from entering the BPD;
BD: beam damp; LIA, lock-in amplifier; PC, personal computer.
In order to maintain the consistency of experimental conditions, the two setups share the same
vapor cell, shield, electronics and the pump and probe lasers. During the alignment or orientation
experiment, only the corresponding pump and probe beams are allowed to pass through the vapor
cell, while the other two beams are blocked.
The leading magnetic field is produced by applying current to the internal coil within the innermost
magnetic shield. We used a Thorlabs current source with an accuracy of 0.1 mA. We detected reson-
ances by scanning a small range around the modulation frequency and demodulating that signal at
the modulation frequency by using a lock-in amplifier.
5.4 Comparison to an orientation magnetometer
5.4.1 Resonances
A comparison of the alignment and orientation magnetic resonances is shown in Figure 5.7 in which
the background magnetic field is set along the z direction, with the field strength ranging from 20µT
to 50µT (data are shifted vertically for clarity).
Comparison to an orientation magnetometer 93
Figure 5.6: Orientation resonance setup. Pump, a circularly polarized 795 nm laser beam used to
generate the atomic orientation polarization; Probe, a linearly polarized 780 nm laser beam used
to detect the Larmor precession of the atomic orientation polarization via optical rotation; AOM,
acousto-optic modulator used to pulse the pump beam; PBS, polarizing beam splitter; BS, beam
splitter; λ/2, half-wave plate; λ/4, quater-wave plate; BPD, balanced photodiode; BPF, bandpass
filter with central wavelength of 780 nm, which is used to prevent pump from entering the BPD;
BD: beam damp; LIA, lock-in amplifier; PC, personal computer.
The time-averaged optical power of the pump and probe beam in the alignment (orientation) ex-
periment is 100µW (50µW) and 300µW (15µW), respectively. These parameters were chosen to
produce relatively strong signals with minimal power broadening of the magnetic resonances.
The shield is equipped with a ∂Bz/∂z gradient coil which was used to compensate the magnetic field
gradient by connecting it to a Throlabs current source with an accuracy of 0.01 mA. The gradient
was large enough to affect the resonance lines, but compensating only this first-order gradient was
sufficient to make it low enough as to not affect the sensitivity.
5.4.2 Heading error
The alignment signal is always a single Lorentzian peak. Its central frequency at 2ωL, corresponds
to the ∆m = 2 magnetic resonance between the F = 1,mF = −1 and F = 1,mF = 1 state. The
orientation signal has always two peaks, corresponding to the ∆m = 1 magnetic resonances between
F = 1,mF = −1 and F = 1,mF = 0 and F = 1,mF = 1 states and is split as the magnetic
field increases. This result shows that our expectations of a NLZ-free alignment magnetic resonance
94 Rb magnetometry free from NLZ-effect-related heading error in Earth-field range
(a) Alignment Resonance (b) Orientation Resonance
4 28 50 µT
50 µT
3 21
40 µT
40 µT
2 14
30 µT 30 µT
1 7
20 µT 20 µT
0 0
-80 -40 0 40 80 -40 -20 0 20 40
Modulation Detuning(Hz) Modulation Detuning(Hz)
Figure 5.7: Magnetic resonances of alignment and orientation polarization with background mag-
netic field of different strengths. (a) Alignment resonance. (b) Orientation resonance. Background
magnetic field is set along z direction, with strength ranging from 20µT to 50µT. Different mag-
netic resonances are shifted vertically for clarity. Since the alignment-resonance frequency is about
twice that of orientation, the scale of detuning in (a) is also twice that of (b). The amplitude of the
alignment signal is smaller than that of the orientation signal for the reasons explained in [25].
on Earth-field are confirmed. Both the alignment and the orientation resonances are broadened at
stronger magnetic fields due to residual magnetic gradients.
In the geomagnetic field range, the estimated photon-sh√ot-noise-limited sensitivities [106–110] of
the a√lignment-based magnetometry are in the tens of fT/ Hz range. Our best sensitivity is about
9 fT/ Hz (See Section 5.4.3). When the background field gets larger, there is a degradation of the
sensitivity. One possible reason for this is the increased magnetic field gradient, since image currents
in the shields may not be perfectly symmetric and thus there are induced gradients from the coils
and shields that scale with the applied magnetic field.
The magnetic-field-direction dependence of the alignment and orientation resonances is shown in
Figure 5.8.
The magnetic field strength is kept at 30µT and the angle between the field direction and the z axis
is scanned in a 120 degrees range, during which we keep Bx = By or Bx = −By, respectively. In that
process, the magnetic field behaves as a vector rotated in a plane defined by the equation x = y or
x = −y, which contains the z axis and is 45 degrees from the x-z plane and the y-z plane, as shown
in Figures 5.5 and 5.6 as light purple parallelograms nearing the coordinate system. When the field
angle is 0 degree, the magnetic field is along the z axis.
Amplitude (mrad)
Amplitude (mrad)
Comparison to an orientation magnetometer 95
(a) Alignment Resonance with Bx = By (c) Orientation Resonance with Bx = By60 60 1
40 40
20 20
0 0 0.8
-20 -20
-40 -40
-60 -60
-80 -40 0 40 80 -40 -20 0 20 40 0.6
Detuning Frequency (Hz) Detuning Frequency (Hz)
(b) Alignment Resonance with Bx = −By (d) Orientation Resonance with Bx = −By 0.460 60
40 40
20 20
0 0 0.2
-20 -20
-40 -40
-60 -60 0
-80 -40 0 40 80 -40 -20 0 20 40
Detuning Frequency (Hz) Detuning Frequency (Hz)
Figure 5.8: Magnetic resonances of alignment and orientation polarization with background mag-
netic field along different directions. (a) Alignment resonance with Bx = By. (b) Alignment
resonance with Bx = −By. (c) Orientation resonance with Bx = By. (d) Orientation resonance
with Bx = −By. The strength of the background magnetic field was kept at 30µT. The angle
between the background magnetic field and the z direction is represented as the vertical axis. The
broadening of the magnetic resonances at larger field angle is due to the increased magnetic field
gradients. Since this broadening does not affect our conclusion about resonance symmetry, and
that the compensation of such gradients along different directions is complex, these gradients are
not compensated during this experiment. Gradients are not compensated.
The relative heights of the two Lorentz peaks in the orientation magnetic resonance are dependent
on the magnetic field direction, which leads to an asymmetry and a central-frequency shift of the
magnetic resonance when the field is not along the z direction. This gives rise to the NLZ-related
heading error. In contrast, the alignment resonance is always symmetric. This means that mag-
netometry based on this (alignment) resonance is free from the NLZ-related heading error. The
broadening of both the orientation and the alignment magnetic resonance at a larger field angle
is due to the increased magnetic gradient, as the x and y-directed magnetic field produced by the
magnetic coil is less uniform than the z-directed magnetic field.
Another source of heading error comes from the different Larmor frequencies of the F = 1 and F = 2
ground hyperfine states (their difference is around 27.9 Hz/µT and lies in the kHz range in the geo-
magnetic field). The tails of the nearby F = 2 magnetic resonance change in size depending on the
magnetic field direction and that introduces magnetic-field-direction-related error to the measure-
ment of the F = 1 Larmor frequency. If we take a magnetic resonance at 20µT for example, as
shown in Figure 5.9, the F = 1 Larmor frequency can be determined from either the peak frequency
of the demodulated X signal or the zero-crossing frequency of the demodulated Y signal. These fre-
quencies, however, are shifted by the F = 2 resonance, because the X contribution from the F = 2
resonance leads to a sloping background and the Y contribution from the F = 2 resonance leads to a
residual background at around the F = 1 resonance frequency. Since the efficiency of producing and
detecting atomic spin polarization is dependent on the orientation of the pump/probe laser beam
Field Field
Angle (◦) Angle (◦)
Field Field
Angle (◦) Angle (◦)
R (Arb. Unit.)
96 Rb magnetometry free from NLZ-effect-related heading error in Earth-field range
1
X
0.75
0.01 Y
0.5 0
0.25 -0.01
-1200 -1000
0
-0.25
F=2 F=1
-0.5
-1500 -1000 -500 0
Detuning Frequency (Hz)
Figure 5.9: Typical F = 1 and F = 2 alignment magnetic resonance. Background magnetic field
is set to 20µT along the z direction. The insertion shows details about the F = 2 resonance. This
magnetic resonance is averaged from four scans.
relative to the magnetic field, the amplitude of the F = 2 resonance depends on the field direction
as well and thus leads to heading error. What we can do is reduce the relative amplitude of the
F = 2 resonance because this way its tails will be also reduced around the F = 1 resonance and as
a result the heading error. Due to the fully resolved ground hyperfine states in the buffer-gas-free
antirelaxation-coated cell, the probe beam is far detuned from the transitions involving the undesired
52S1/2 F = 2 hyperfine state, suppressing the amplitude of the F = 2 resonance to about 100 times
more than that of the F = 1 resonance, as shown in Figure 5.9. Considering the large frequency dif-
ference between these two resonances, the influence from the F = 2 resonance on the F = 1 Larmor
frequency only leads to heading error on the order of tens of fT.
The light shift due to the probe beam is also a source of heading error. When the background
magnetic field is not aligned with the polarization of the probe beam, a linearly polarized (probe)
beam can generally be viewed as three linearly polarized components. The one is the π component, a
linearly polarized light whose polarization aligns with the magnetic field, given that the quantization
axis is set along the background magnetic field. The other two consist of again linearly polarized
light with its polarization perpendicular this time to the magnetic field, also viewed as a σ+ and σ−
combination with equal strength. Because these two optical components are of equal strength, and
their frequency detunings relative to the atomic transitions are almost the same, their light shifts on
the (F = 1) m = −1 and m = 1 states are almost identical. [111]. Since the alignment resonance
frequency corresponds to the energy difference between the (F = 1) m = −1 and m = 1 states the
Amplitude (Arb. Unit.)
Comparison to an orientation magnetometer 97
light shift of the alignment resonance is largely suppressed.
The frequency detuning of σ+ and σ− are slightly different, because of the Zeeman shift of the
atomic transitions– σ− and σ+ have the same optical frequency, but the Zeeman effect leads to
different atomic-transition frequencies, so they end up with a different detuning. As a result, there is
still residual light shift due to the Zeeman-effect-related unbalanced detuning of optical transitions
involving, respectively, the (F = 1) m = 1 and m = −1 states, which is on the order of several pT,
according to simulations based on the ADM package [112]. If the optical polarization of the probe
beam is actively rotated to make it always perpendicular to the magnetic field, then the light shift
will only lead to a constant bias, rather than a heading error. This method will also help to build a
dead-zone-free magnetometer [113].
Finally, there is another effect that can lead to a systematic error in our magnetometric system. We
use a 4-level system to pump and probe in which interference effects can arise: In such a system
a resonance between two states can be affected by another resonance between two different states
although the latter do not share common final or initial states with the first two states (See Figure
5.2) [114, 115]. In our case a coherence between the pump states will be transferred to the F = 1
ground state which eventually leads to a frequency shift of the F = 1 resonance.
The presence of such effects can be identified by measuring the heading error as a function of the
light power (particularly, the pump power) and completely eliminated using a “free-decay” protocol,
where atomic evolution occurs in the absence of applied light between pump and probe light pulses–
or simpler, we measure the Larmor frequency while the pump and probe are off.
5.4.3 Estimation of NuZ-related Heading Error
As shown in Figure 5.9, the F = 1 resonance frequency at 20µT can be determined from either the
peak frequency of the demodulated X signal or the zero-crossing frequency of the demodulated Y
signal. The peak frequency of X (zero-crossing frequency of Y ), however, is also influenced by the
F = 2 resonance, because the X (Y ) contribution from the F = 2 leads to a sloping background (re-
sidual background) at around the F = 1 resonance frequency. Since the F = 2 resonance amplitude
is influenced by the magnetic-field direction, this background is also magnetic-field-direction related
and thus is a source of heading error. Assuming the contributions of F = α resonance (α=1 or 2) to
the demodulated X and Y data have the Lorentzian form of
( A δν ) A δν2
Xα = Re α α = α α( ,i(να − νm) + δνα ) (να − νm)2 + δν2α (5.1)
= Im Aαδνα = Aαδνα(να − νm)Yα
i(να − νm) + δνα (
,
να − νm)2 + δν2α
98 Rb magnetometry free from NLZ-effect-related heading error in Earth-field range
where Aα, να and δνα are the amplitude, central frequency and the half width at half maximum
(HWHM) linewidth of the F = α resonance, while νm is the modulation frequency of the pump
beam. {I}
The presence of the X2 and Y2 terms lead to a shift of the effective central frequency of the magnetic
resonance around ν1.
Taking the X signal as an example, the effective peak frequency is the modulation frequency νm at
around ν1 which leads to a zero slope of the overall X signal, e.g.,(∂X ∂X )∣∣∣1 + 2 = 0 . (5.2)
∂νm ∂νm νm∼ν1
According to Eq. (5.1), the slopes of the X1 and X2 measured data close to the resonance, i.e.,
ν1 = νm, have the form of
∂X ∣1 ∣∣ ≈ 2A1(ν1 − νm) ,
∂νm ν 2m∼ν1 δν1 (5.3)
∂X ∣∣∣ 22 ≈ 2A2δν2 ,
∂νm νm∼ν1 (ν2 − ν1)3
so the shift of the peak frequency of X is
A δν2δν2∆ 2ν ≈ 1 21X (5.4)
A1(
.
ν2 − ν1)3
As for the Y signal, the effective zero-crossing frequency is the modulation frequency νm at around
ν1 which leads to a zero value of the overall Y signal, e.g.,
∣
(Y1 + Y2)∣ ∼ = 0 . (5.5)νm ν1
According to Eq. (5.1), the Y1 and Y2 data in the vicinity of the resonance, i.e., ν1 = νm, have the
form of ∣∣
Y1∣∣ ≈
A1(ν1 − νm)
,
∣νm∼ν1 δν1 (5.6)A2δν2
Y2∣ ≈
νm∼ν1 (ν1 − ν2)
,
so the shift of the zero-crossing frequency of Y is
∆ ≈ A2δν1δν2ν1Y .
A1(ν2 − )
(5.7)
ν1
{I}Note that the lineshape for the Y2 case is not perfectly dispersive in practice as seen in Figure 5.9; One possible reason
is that the measured Y2 is a combination of a dispersive curve and an absorptive curve. When the modulation frequency
is far detuned, the dispersive (absorptive) curve scales as 1/detuning (1/detuning2). Because the absorptive component
decreases much faster with larger detuning than the dispersive component, the far-detuned behavior of the Y2 will be
dominated by its dispersive component. Hence, it is safe to use inverse-detuning scaling to estimate the upper limits on
the F = 1 resonance shift, even though in practice Y2 is not ideally dispersive.
Sensitivity 99
5.5 Sensitivity
The sensitivity of an optically pumped magnetometer is limited by two fundamental quantum noises:
the spin-projection noise–in measuring the pointing angle of the atomic spins and the photon
shot noise–in measuring the polarization angle of the probe beam [19]. The spin-projection-noise-
limited sensitivity measured in noise spectral density is given by [108,116]
≈ √1 √ 1δB , (5.8)
γ1 2F Nτ2
where γ1 ≈ −7Hz/nT is the gyromagnetic ratio of the F = 1 ground state, F = 1 is the total
angular momentum of the system, N is the atoms in the vapor cell and τ2 is the lifetime of the
atomic polarization. Given that the atoms N are ≈ 9 × 1010 in room temperature in a vapor cell
with a radius of 4 cm an√d a length of 5 cm and τ2 about tens of ms, the spin-projection-noise-limited
sensitivity is in the fT/ Hz range, which is one or two orders of magnitude smaller than the photon-
shot-noise limited sensitivity of our current setup. As a result, we focus on the photon-shot-noise
limited sensitivity, as shown below.
The photon-shot-noise limited sensitivity of an optical-rotation-based magnetometer is given by [106,
107] ( ∂ϕ ∣∣∣ )−1δB ≈ δϕ , (5.9)
∂B νm=κγ1B
where ϕ is the optical rotation amplitude of the probe beam and is a function of B (magnetic field)
and νm (modulation frequency of the pump beam), κ is the rank of atomic polarization (2 for align-
ment), γ1 ≈ −7Hz/nT is the gyromagnetic ratio of the F = 1 ground state, ∂ϕ/∂B|νm=κγ1B is the
resonant slope of ϕ with respect to the magnetic field B, which could be read from the Y component
of the magnetic re√sonance, and δϕ is the sensitivity to the optical rotation angle of the probe beam,
measured in rad/ Hz .
In the present magnetic resonance scanning, we scan the modulation frequency of the pump beam
νm rather than the magnetic field B. As a result, ∂ϕ/∂νm|νm=κγ1B can be directly measured from
the magnetic-resonance data. Considering the relation ϕ(B, νm) = ϕ(B + ∆B, νm + κγ1∆B) when
∆B ≪ B, which indicates
∂ϕ = lim ϕ(B +∆B, νm)− ϕ(B, νm)
∂B ∆B→0 ∆B
= lim ϕ(B, νm − κγ1∆B)− ϕ(B, νm)κγ1 (5.10)
∆B→0 κγ1∆B
= − ∂ϕκγ1 ,
∂νm
we have
∂ϕ ∣∣∣ = − ∂ϕ ∣∣κγ1 ∣
∂B νm=κγ1B ∂νm νm=κγ1B (5.11)
= − Aκγ1 = 2πτ2A|γ1| ,
δν
100 Rb magnetometry free from NLZ-effect-related heading error in Earth-field range
where A is the optical rotation amplitude on resonance, δν is HWHM linewidth of the resonance,
and τ2 = κ/(2πδν) is the lifetime of the atomic polarization. Now, the sensitivity is given by
δB ≈ δϕ2 . (5.12)πτ2A|γ1|
√
The photon-shot-noise-limited angular sensitivity δϕ is estimated to be 1/ P [106], where P is the
photon flux entering the polarimeter. In the present experiment, the power of the 780 nm probe beam
entering the vapor cell is about 300µW, and about one third of it is gathered with the polarimeter.
Considering that the energy of a photon with wavelength of 780 nm is about 2.55×10−19 J, a 100µW
optical power gathered with the polarimeter means that there are about 3.9× 1014 photons en√tering
the polarimeter per second, or photon flux P ≈ 3.9× 1014 s−1. So δϕ is about 5.1× 10−8 rad/ Hz .
The estimated photon-shot-noise-limited sensitivities of the alignment magnetometry as a function
of magnetic field strengths are shown in Figure 5.10(a), with the field kept along the z direction.
(a)
140
120 before heating
100 after heating
80
60
40
20
0
0 10 20 30 40 50 60
Background Field (µT)
(b) 120
before heating
90
after heating
60
30
0
0 10 20 30 40 50 60
Background Field (µT)
(c)
1.2
0.9
0.6
before heating
0.3 after heating
0
0 10 20 30 40 50 60
Background Field (µT)
Figure 5.10: (a) Photon-shot-noise-limited sensitivity, (b) Life time and (c) optical rotation
amplitude of alignment-based magnetometry as a function of background magnetic field. The
background magnetic fields are kept along the z direction. The black squares (red circles) represent
data taken before (after) vapor cell heating.
Opt. Rotation Life Time Sensitivity
√
Amplitude (mrad) (ms) (fT/ Hz)
Conclusions 101
We present two sets of results, one of which is measured before the vapor cell heating, and the other
is measured after the heating, during which the vapor cell body was heated to about 60 ◦C for several
hours √and the cell stem was kept as the coldest place of the cell.√The sensitivities are in the tens
of fT/ Hz range, with the best sensitivity reaching about 9 fT/ Hz . Generally, the sensitivities
degrade with the increased background field. This is possibly due to the increased magnetic field
gradient, which in turn decreases the lifetime and amplitude of atomic polarization and thus the
optical rotation amplitude, as shown in Figure 5.10(b) and (c). The overall performance of the
magnetometer is improved after heating, which is due to the increased spin lifetime [117]. These
results suggest that such an alignment resonance is promising for highly sensitive magnetometry in
the geomagnetic field range.
To achieve shot-noise-limited sensitivity in practice, there are some technical issues to tackle. The
first one is magnetic field fluctuations, which may be much larger than the shot-noise-limited sensit-
ivity. In the geomagnetic field range (5×10−4T level), a sensitivity for 1 Hz bandwidth of about 10 fT
(10−14T) means that the relative noise of the magnetic field should be about 2×10−10, which is bey-
ond the stability of commercial current sources (we use current sources to drive magnetic coils and
generate magnetic field). It is possible to achieve such sensitivity in the geomagnetic field range in
differential measurements by subtracting the common-mode signal or by actively stabilizing the mag-
netic field with atomic magnetometers [118]. Besides, one should make sure that the contributions
from electronic noise of the detector and technical noise due to laser frequency/power fluctuations
to the measured optical rotation noise is smaller than that of shot noise. Careful optimization of
detectors, lasers, and experimental parameters including optical power and demodulation frequency
are needed for this purpose [109,110].
5.6 Conclusions
In conclusion, we have demonstrated a sensitive scalar magnetometer free from NLZ related heading
error in the Earth-field range. This magnetometer is based on the ∆m = 2 magnetic resonance in
the 87Rb F = 1 ground hyperfine state. In contrast to conventional alkali-metal magnetometry,
where the magnetic resonance curve is split and distorted in the Earth field, the magnetic resonance
demonstrated here is only a single Lorentzian peak, which is thus free from NLZ induced splitting
and√asymmetry. For our magnetometer, the photon-shot-noise limited sensitivity reaches tens of
fT/ Hz in the Earth field range, with the vapor cell at the room temperature. The sensitivity could
be further improved by heating the cell to increase the atomic vapor density [117]. This scheme
is also effective at suppressing the heading error due to the NuZ and LS effects. Due to the fully
resolved ground hyperfine states in the antirelaxation-coated cell, the residual signal from F = 2
is relatively small and thus its influence is at most at the tens of fT level, which is on the limit of
the sensitivity of this magnetometer. Moreover, as there is only linearly polarized light used in this
magnetometer, it is free from the vector light shift [111], which is another possible source of heading
error{I}. Considering that the remaining tensor light shift will not change the frequency difference
{I}Although, cells can have birefringent windows which induce elliptical polarization of the light and thereby accidentally
cause nominally linearly polarized light to generate vector light shifts [119]
102 Rb magnetometry free from NLZ-effect-related heading error in Earth-field range
between m = ±1 magnetic sublevels in the ground F = 1 system, i.e., the central frequency of the
desired ∆m = 2 magnetic resonance, this scheme is promising for more accurate magnetometry. It
should be noted that the magnetic resonance frequency itself still has cubic correction in the strength
of the magnetic field. Finally, a similar method can be implemented with other alkali metals that
have a F = 1 ground level, such as 39K, 41K and 23Na.
Conclusions 103

Instead of obtaining a mirror,
Obtain a person.
Look into him.
Use different people.
Old, young, fat, small, etc.
Mirror Piece, Yoko Ono
6
Degenerate mirrorless lasing–Forward beam
Some of the text presented in this section is part of a paper that has been published as [120] and [49].
For the work presented in this Chapter, I build upon the idea conceived by Prof. D. Budker and
Aram Papoyan in Ref. [120]. I constructed part of the setup needed for the experiment in Mainz
together with Guzhi Bao. I took part of the data and analyzed them and together with the rest of
the co-authors, I co-wrote and edited the manuscript.
6.1 The idea
Until now, the focus of our work has been on nonlinear magneto-optical rotation, or else, nonlinear
Faraday rotation. We have extensively talked about this phenomenon and studied it in Chapters 2,
4 and 5. In the last two Chapters of this thesis we will discuss about another nonlinear phenomenon
called mirrorless lasing, broadly defined here as directed monochromatic emission from an ensemble
of atoms or molecules excited with pump laser light.
In the cases studied throughout my thesis so far, the pump beam was never polarized parallel to the
transverse magnetic field. This is intuitively understood, because in the case where the laser field
E with wave vector k is polarized parallel to the transverse magnetic field (E ∥ B, B ⊥ k) [121],
there is no Larmor precession of polarized atoms around the magnetic field, since they would all be
polarized along the magnetic field. This seems, at first sight, to eliminate any possibility to generate
narrow lines and thus makes such a configuration unattractive for spectroscopic studies.
105
106 Degenerate mirrorless lasing–Forward beam
Nevertheless, narrow magnetic resonances with high signal-to-noise ratio may arise due to coherence
effects related to partial inversion of population. This can happen on transitions between mag-
netic sublevels belonging to the excited and ground-state levels under intense excitation by linearly
polarized light of cycling transitions of an alkali-atom D2 line (Fe > Fg) [122,123].
In this project, we observed narrow, high-contrast, magneto-optical resonances, when irradiating a
Rb cell with a single laser beam under specific conditions: it is a continuous wave single-mode linearly
polarized laser beam at relatively high intensity, with its frequency tuned to the 85Rb Fg=3 → Fe=4
or 87Rb Fg=2 → Fe=3 transitions of the D2 line.
The magnitude and width of the resonances exhibit a strong nonlinear dependence on the laser-light
intensity and on the magnetic field, as well as a sub-natural linewidth. We called this observation
mirrorless lasing since the detected beam exhibits characteristics similar to a laser beam, but with a
single beam-pass through the cell/medium and of course without any mirrors existing to add to the
stimulated emission.
The results of this work were the fruits of our collaboration with the Institute for Physical Research,
NAS of Armenia, Ashtarak. for a clear overview of the experiment I will present in Chapters 6 and 7
the Sections into parts based on the experiments conducted in Mainz and Ashtarak respectively.
6.2 The Mechanism
Section 2.4.3 introduced the basic theoretical framework for mirrorless lasers. There are a few papers
that deal with the phenomenon experimentally and motivated our own project. In [122, 123] it was
demonstrated that interaction of linearly polarized light with an Fe > Fg cycling D2 transition of
alkali atoms may result in amplified radiation between certain Zeeman sublevels, as shown in the
illustration in Figure 6.1. This amplification occurs at intensities of light above ∼ 20 mW/cm2 in a
steady-state regime, due to redistribution of population towards mF = 0 [124, 125], which results in
population inversion on |mFe | = n→ |mFg | = n+ 1, where n is a natural number.
The population inversion has a strongly nonlinear dependence on the light intensity: above a threshold,
the inversion increases reaching saturation followed by a decline. For B = 0 and linearly polarized
pump light, there appears optical gain for polarization orthogonal to that of the pump light in
the forward and backward directions [122]. When we apply an external magnetic field, this should
influence the redistribution of population and the onset of population inversion.
Since we want to detect magnetic-field-induced changes in transmitted light due to gain, we need an
environment that is very well magnetically shielded and a measurement which is not affected by other
magneto-optical effects. For this reason, we chose the external magnetic field that will be applied to
test our theory to be parallel to the laser field B ∥ E and perpendicular to its direction k. In this
case, there is no optical rotation or ellipticity associated with the Faraday or Voigt effects [121].
In order to confirm our expectation that such a system can indeed produce an inversion of population,
we used the Atomic Density Matrix (ADM) package [112]. We calculated, using a simple model, that
The Mechanism 107
Figure 6.1: Inversion of population between magnetic sublevels. Onset of population inversion
and amplified radiation corresponding to ∆m = ±1 transitions under excitation of 85F Rb D2 line
with an intense linearly polarized laser radiation for zero magnetic field (the quantization axis Z is
chosen to be along the laser light polarization). Bars: population of sublevels (not to scale). Red
corresponds to population of excited states and purple population of ground states.
a magnetic field applied along the linear polarization of a laser beam establishes population inversion.
We observe that the steady-state populations of particular Zeeman sublevels, when excited with
linearly polarized light, depend on the magnetic field that is applied (when B ∥ E). For that, we
calculated the steady-state solution of the Liouville equation for different transitions.
The results of the calculations of magnetic field dependencies of steady-state populations for different
ground and excited state sublevels are shown in Figure 7.2.
Magnetic field dependencies of steady-state populations for the ground state sublevels (mFg) and
excited state sublevels (mFe) of the “cycling” transitions Fg=3 → Fe=4 (85Rb) and Fg=2 → Fe=3
(87Rb) of the rubidium D2 line (5S1/2 → 5P3/2), under excitation with linearly-polarized radiation
with S = 100, are presented in Figure 7.2a and Figure 7.2b, respectively. S is the saturation para-
meter defined as:
= |Ω|
2
S Γ2 4 + 2 , (6.1)/ δ
where Ω is the effective Rabi frequency, δ is the detuning from resonance, and Γ is the radiative decay
rate.
108 Degenerate mirrorless lasing–Forward beam
Figure 6.2: Numerical simulations with the ADM package for inversion of populations: B ∥ E
magnetic-field dependence of normalized steady state population of ground state (g|mF >) andg
excited state (e|mFe>) Zeeman sublevels linked by ∆m 85F = 0 transitions Rb Fg=3 → Fe=4 (a)
and 87Rb Fg=2 → Fe=3 (b), and population inversion on corresponding mFe → mFg transitions
(c,d). The light power is sufficient to provide population inversion. Γ is the decay rate of excited
state, ΩL is the Larmor frequency. The width of the resonance is not affected by the ground state
relaxation rate included in the model, which is much smaller than Γ.
The two axes of the simulated graph are: the population inversion versus the magnetic field, which is
expressed in ΩL/Γ normalized Larmor frequency units—ΩL = gFµBB/ℏ, gF is a Landé factor, µB is
the Bohr magneton, and B is the magnetic field strength—is shown in Figure 7.2c,d. The population
inversion is practically calculated by the difference between the population of an excited mF state
and the population of a ground mF state.
We can see from the simulation that population inversion establishes at zero magnetic field; it is
stronger for the 87Rb Fg=2 → Fe=3 transition. Application of a magnetic field corresponding to
ΩL/Γ ≈ 5−10 eliminates the population inversion.
An important point to note here is that we calculated population inversion but not gain. The model
predicts a bigger population at the 0 excited level than in the -1 and +1 of the ground state which
Evidence for degenerate mirrorless lasing in alkali metal vapor: forward beam magneto-optical
experiment 109
would create y linearly polarized light (orthogonal to the x polarized incoming light). However, we
did not take into account the absorption of the ground-state populations to the excited states. This
means that we might have population inversion, but no gain and this is very important to actually
see amplification of our inputing light.
Below, we present experimental data which indicate that, in this system, a number of different
nonlinear-optical effects take place, including degenerate four-wave mixing and mirrorless lasing.
The latter is a particularly interesting case because, in contrast to most other works on cw mirrorless
lasing in alkali atoms (see [126,127] and references therein), in the our case lasing occurs at the same
wavelength as optical pumping. The only feature that is different between the pump beam and the
lasing beam is the polarization, that is why we call this effect entirely degenerate mirrorless
lasing.
6.3 Evidence for degenerate mirrorless lasing in alkali metal vapor:
forward beam magneto-optical experiment
6.3.1 Technique
This experiment, as simple as it might be from an experimental point of view, needs a lot of precision
and care. This is of course due to the fact that one needs to detect the mirrorless lasing beam (our
signal) on the same path as that of the beam that creates the inversion (our pump) and the two
beams differ only in polarization. It also requires a well controlled magnetic-field environment as
well as almost perfect blocking of reflections.
For these reason, we collaborated with our colleagues in Ashtarak. Experimental measurements were
initially carried out in Ashtarak using the setup shown schematically in Figure 6.3 (A). In order to
check the reproducibility of the results, carry out independent measurements with better control of
the applied magnetic field, and providing the capability of precise measurements of the output-light
spatial profile, another set of measurements was carried out in Mainz, with the setup seen in Figure
6.3 (B). In order to make sure that the reader understands the similarities and small, but—as will be
discussed in Chapter 7—important differences between the two setups, I made a distinction between
the Mainz and the Ashtarak experiments in the pages to follow.
6.3.2 Laser
In Ashtarak:
In the Ashtarak setup (Figure 6.3A), a≈ 1.5 mm-diameter beam from a single-frequency cw extended-
cavity diode laser (wavelength 780 nm, linewidth 1 MHz, input intensity varies) was used.
In Mainz:
In the Mainz setup (Figure 6.3B), a ≈ 1.5 mm-diameter beam from a single-frequency cw extended-
cavity diode laser (wavelength 780 nm, linewidth 100 kHz, input intensity varies) was used.
110 Degenerate mirrorless lasing–Forward beam
6.3.3 Setup
In Ashtarak: The beam was purified of the initial linear vertical polarization with a Glan-Thomson
polarizer and directed to a 135 mm-long sealed-off room-temperature (RT) glass cell with a side-arm
containing natural rubidium. The cell had no coating or buffer gas.
The vapor-cell temperature was kept at 30 ◦C, which corresponds to a number density of N =
1.16×1010 cm−3 of Rb atoms. It was verified that small changes in atomic density did not qual-
itatively influence the observed effects. The cell was surrounded with an assembly of mutually ortho-
gonal pairs of calibrated Helmholtz coils with the axes carefully aligned along the X,Y, Z directions.
The coils were used for canceling the magnetic field in two directions (X,Y ), monitored to ≈ 10 mG
with a Fluxmaster fluxgate-magnetometer, and scanning within up to ± 1 G in the third direction
(Z).
In Mainz:
The Mainz setup (Figure 6.3B), contrary to the one in Ashtarak, includes a magnetic shield. The
coils installed inside the Twinleaf MS-2 magnetic shield allowed to apply and scan the√magnetic field
in the X,Y, Z directions. The magnetic shield can zero the outside fields up to 10 fT/ Hz [128].
Separate measurements have been done to evaluate possible contribution from the inhomogeneity
of the applied magnetic field. The contribution was found to be insignificant. In particular, the
measured longitudinal (BX) component was less than 1.5% of the applied BZ field. Two linear
polarizers (LPVIS050) were used for purification of the input laser light polarization and selection of
the orthogonally-polarized component in the output beam. The spatial distribution of the forward
beam emerging from the cell was studied with a beam profiler DMK 41BU02.H from the company
Imaging Source.
6.3.4 Detection
In Ashtarak:
The light transmitted through the cell was directed onto a polarizing beam splitter (PBS) decom-
posing the incident polarization into vertical and horizontal components that were simultaneously
detected with two photodiodes. Diaphragms were placed in front of the photodiodes located 70 cm
away from the cell in order to cut the most of the fluorescence of the Rb atoms that is a potential
noise source. The two detection channels were adjusted to have the same sensitivity. A fraction of
the laser light passed through an auxiliary room-temperature rubidium reference cell to monitor the
spectrum.
A set of calibrated neutral-density filters was used to change the incident light intensity before the
cell (“AT” in Figure 6.3 A). The filters from the set, which were not used for attenuation, were placed
at the position of AT after the cell. With this configuration, the off-resonance signal did not change,
so that there was no need to adjust the photodiode gain.
In Mainz:
Evidence for degenerate mirrorless lasing in alkali metal vapor: forward beam magneto-optical
experiment 111
Figure 6.3: Experimental setup and the measurement configuration in Ashtarak (A) and Mainz
(B). PBS: polarizing beam splitter; BS: beam splitter; λ/2: half-wave plate; λ/4: quater-wave plate;
PD: photodiode; AT: attenuator; P: polarizer; G-T: Glan-Thompson Polarizer; Iris: I
The light transmitted through the cell was directed onto a polarizer and then through a filter (AT)
and an iris directed to either a photodetector or to a camera depending on the data we took.
6.3.5 Results
In Ashtarak:
The first measurement was done by scanning the BZ in the ± 0.12 G range. The frequency of the
incident linearly polarized (EZ) light was at the maximum of the Doppler-overlapped transition
groups 85Rb Fg=3 → F 87e=2,3,4 or Rb Fg=2 → Fe=1,2,3 of Rb D2 line, and was monitored by the
reference signal. The intensity of the output light (with orthogonal polarization EY ) was recorded
with Photodiode 1 (see Figure 6.3 A) for 12 values of incident laser intensity IL ranging from 1 to
1100 mW/cm2. This output orthogonal light is what is expected to be our lasing beam.
The output intensity normalized by the input intensity is presented in Figure 6.4a,b. Neg-
ligible output signal was recorded for low laser radiation intensity (IL < 1 mW/cm2). At IL ≈
1.5 mW/cm2 a (≈ 15 mG FWHM) resonance feature appears first on the 87Rb Fg=2 → Fe=3 trans-
ition, centered around B = 0, and then also on the 85Rb Fg=3 → Fe=4 transition (≈ 20 mG FWHM
at IL ≈ 3 mW/cm2). Further increase of intensity results in broadening and amplitude increase of
the resonance feature to a maximum normalized value recorded at I 2 87L ≈ 35 and 85 mW/cm for Rb
112 Degenerate mirrorless lasing–Forward beam
Figure 6.4: Dependences of normalized output intensity with orthogonal polarization (see text)
on magnetic field (B ∥ E) for 85Rb Fg=3 → Fe=4 (a,c) and 87Rb Fg=2 → Fe=3 (b,d) D2 trans-
itions, measured in Ashtarak (a,b) and Mainz (c,d).
Fg=2 → F =3 and 85e Rb Fg=3 → Fe=4, respectively. By increasing the intensity even more we get a
reduction of the transmitted intensity at B = 0 and further broadening of the resonance.
The scaled and absolute intensity values of B = 0 resonances, as well as their widths depend-
ing on the laser light intensity, derived from the Ashtarak measurements presented in Figure 6.4a,b
are shown in Figure 6.5. As seen in the figure, the absolute output intensity reaches 10 mW/cm2 (or
1% of the incident-light intensity). The input-intensity dependence of the output signal is strongly
nonlinear (note the log-log scale needed to present the data spanning several orders of magnitude).
The low-intensity range for the absolute intensity dependence is magnified in the inset of Figure
6.5a in log-log scale, exhibiting threshold behavior of the output light. The resonance linewidth
rises monotonically from ≈ 0.02 to 0.1 G when changing IL (Input light intensity) by three orders of
magnitude, from ≈ 1 to 1000 mW/cm2, with a steeper slope for 87Rb Fg=2 → Fe=3.
Measurements similar to those presented in Figure 6.4 were performed also for the output light with
initial laser polarization (EZ), recorded with the Photodiode on the right (see Figure 6.3a). A dip
Evidence for degenerate mirrorless lasing in alkali metal vapor: forward beam magneto-optical
experiment 113
Figure 6.5: Input-intensity dependences of (a) absolute (black symbols, left axis) and normal-
ized (red symbols, right axis) of the output intensity with orthogonal polarization for B = 0; (b)
FWHM width of B = 0 resonance for the output orthogonally-polarized light, retrieved from the
measurements presented in Figure 6.4a,b. Solid squares: 85Rb Fg=3 → Fe=4; open squares: 87Rb
Fg=2 → Fe=3. Solid lines are drawn as a guide to the eye.
centered at B = 0 complementary to the peak for the orthogonal polarization was recorded for the
same conditions and with nearly the same parameters (amplitude and width) as for the case of the
orthogonal polarization. Unlike zero-background peaks for EY , the EZ dip appears on top of a
background resulting in an inconspicuous, low-contrast signal. For this reason, we do not present
this graph here, but it will be explored below, when presenting the results of the double-scanning
measurements.
Circular polarization analysis of the output light was also performed with the Ashtarak setup.
Figure 6.6 presents the circular polarimeter output signal versus B ∥ E for the case of 85Rb Fg=3 →
Fe=4 transition measured at different values of IL corresponding to the conditions of Figure 6.4a,b.
One can clearly see that ellipticity in output light polarization appears around B = 0 for IL ≈
114 Degenerate mirrorless lasing–Forward beam
Figure 6.6: Ellipticity in output light: Magnetic-field dependence of the output signal for the 85Rb
Fg=3 → Fe=4 transition recorded with a circular polarimeter at different input-light intensities
(same as in Figure 6.4a,b). No normalization; the curves are shifted vertically by equal amounts.
10 − 500 mW/cm2, i.e. in the input-intensity range where the orthogonally polarized output light
is significant. This is a signature of non-random phase of the generated light with respect to the
excitation-light phase, which cannot be explained directly by mirrorless lasing that implies random
phase relation between the generated and the pump radiation. The observed phase correlation may
occur due to an accompanying wave-mixing process, which contributes to the detected signal because
of insufficient spatial resolution. The results of further experimental studies of the generated beam
profile performed with the Mainz setup indicating the presence of a concomitant four-wave mixing
are presented below.
The next series of measurements carried out in Ashtarak were aimed at detailed studies of the fre-
quency dependence of the resonances across the whole spectral range of the Rb D2
line. Simultaneous double linear scanning of the laser frequency and the B ∥ E magnetic field was
performed by applying triangular control pulses with different repetition frequencies.
Evidence for degenerate mirrorless lasing in alkali metal vapor: forward beam magneto-optical
experiment 115
Figure 6.7: Double-scanning measurement for the output light with polarization parallel (∥E)
and orthogonal (⊥E) to the input laser polarization for incident light intensity of 87.9 mW/cm2. A
single slow (compared to the frequency of the spectral scanning) B-field scan (B ∥ E) in the range
of ±0.12 G is combined with fast laser-frequency scans over 10 GHz covering the Rb D2 line. Upper
inset: zoomed output light spectrum in the region of B = 0; lower inset: zoomed signals of ⊥E
output light (lower trace), and ∥E output light in the region of the dip (upper trace).
The results of such measurements for the linear polarization components of the output light with
simultaneous slow scanning of the B-field (B ∥ E) and fast scanning of the laser frequency across
the D2 line are presented in Figure 6.7 for an incident intensity corresponding to a maximum output
signal with orthogonal polarization for the 85Rb Fg=3 → Fe=4 transition. It can be seen from the
upper zoomed-in inset that the ⊥ E polarized-light spectrum (lower trace) does not contain the
transition groups 85Rb F 87g=2 → Fe=1,2,3 and Rb Fg=1 → Fe=0,1,2 unlike the ∥ E polarized-
light spectrum (upper trace), where these transitions are well pronounced. Note that the maximum
amplitude of the resonance is observed when the laser frequency is tuned to the absorption maximum
of the cycling transitions. It is noteworthy that for the B = 0 peak in the orthogonally-polarized
output light there is a complementary dip in signal with unchanged input polarization. This is a
direct signature of parallel-to-orthogonal polarization conversion in the output light.
In Mainz:
Similar measurements about dependences of normalized output intensity with orthogonal
polarization on magnetic field were carried out in the Mainz setup. The cell length was the same,
temperature was similar (RT), and the input beam diameter was of similar size. The resulting
peaks exhibit ,qualitatively similar behavior, with somewhat higher threshold and maximum conver-
sion values of input light intensity (Figure 6.4c,d). Among other quantitative distinctions from the
116 Degenerate mirrorless lasing–Forward beam
Figure 6.8: Spatial-profile measurements of the orthogonally polarized component of the forward
beam at zero magnetic field. Upper curve: saturated-absorption spectrum for reference.
Ashtarak experiment are lower maximum conversion efficiency (≈ 0.25%), and a sharper shape of
the B = 0 peak.
We tried to study this quantitative discrepancy between the Ashtarak and Mainz results for the input
beam intensity values where the orthogonally-polarized output beam is efficiently generated, as well
as for the conversion efficiency (Figure 6.4). One could ascribe this to the different output-beam
detection geometries used on the two setups. The detection angle in Ashtarak was about 15 mrad,
while in Mainz it was 6 mrad. Another possible reason is the magnetic-field environment (different
precision of the B-field control). In order to clarify this issue, additional measurements were done in
Mainz using a beam profiler for spatial analysis of the output beam.
The spatial profile of the orthogonally polarized output beam versus frequency, covering the 85Rb
Fg=3 → Fe=2,3,4 and 87Rb Fg=2 → Fe=1,2,3 transitions groups, is shown in Figure 6.8 for different
values of input beam power. We are here at zero field. In these measurements, the extinction ratio for
the crossed polarizers was somewhat worsened by a slight tilt of the analyzer, so that when the laser
radiation frequency is off-resonance from the atomic transitions, some residual light remains in the
transmitted beam, with intensity increasing linearly with input power. When the laser frequency is
tuned closer to the atomic transition, the residual light gets absorbed. A bright beam emerges above
the threshold input power of ≈ 10 mW. In the region of 85Rb Fg=3 → Fe=4 transition, the emerging
narrow axial beam is surrounded by a ring structure, which remains, even becoming stronger, with
some blue frequency shift from the resonance. The threshold of the ring structure is somewhat
different from the one of the central beam. The ring structure is not observed around the 87Rb Fg=2
→ Fe=3 transition.
Discussion 117
Figure 6.9: Spatial-profile measurements of the orthogonal component of the forward beam for
different laser frequencies and different values of the magnetic field with a fixed input beam power
of 12 mW. Upper curve: saturated-absorption spectrum for reference.
The same technique was employed for the study of the output beam profile versus laser frequency
and B ∥ E magnetic field. For this measurement, the value of the input light power was kept at
12 mW (above the observed threshold for the 85Rb Fg=3 → Fe=4 transition). The results of this
measurement are shown in Figure 6.9. One can see that the increase of the magnetic field leads to
vanishing of the central bright beam, unlike the ring structure which becomes even more pronounced.
The results of these measurements indicate that the spatial profile of the beam and the detection
angle (aperture) are of importance for the studied process, which may explain the quantitative dis-
tinction of Ashtarak and Mainz results observed in Figure 6.4.
6.4 Discussion
We explored a variety of experimental methods to be able to observe the mechanism described in the
Introduction. These experiments provided a link between the observed features with the formation
of amplified orthogonally polarized light (mirrorless lasing) under excitation of an Fe > Fg atomic
cycling system with intense linearly polarized light.
We believe the obtained experimental results are consistent with the mechanism described in the
Introduction, sufficiently proving the link between the observed features with the formation of amp-
lified orthogonally polarized light (mirrorless lasing) under excitation of an Fe > Fg atomic cycling
118 Degenerate mirrorless lasing–Forward beam
system with intense linearly polarized light. A short list of our results together with our interpreta-
tion follows.
Linearly polarized light induces polarization of the Fe > Fg atomic system resulting in light-induced
alignment, redistributing the ground state population towards the mFg = 0 Zeeman sublevel. In the
weak-light-intensity limit, when we apply weak BZ (B ∥ E) magnetic field the transmitted radiation
is not influenced. However, when the excitation intensity of the laser beam is strong enough (above
saturation), light is generated, which is orthogonally polarized with respect to the incident radiation
and this gets influenced by magnetic fields. In particular the atomic polarization component that
is created can undergo Larmor precession around BZ , since it is orthogonal to that field. This
precession redistributes the ground-state population among Zeeman sublevels, and eventually affects
the conditions of population inversion. The characteristic strength of the magnetic field needed to
destroy the amplification is consistent with time-of-flight broadened ground state coherence time
(time-of-flight broadening for an atom traversing the laser beam γc/2π ≈ 100 kHz).
In the results in Mainz, this whole process is evidenced by sharp B = 0 peaks in Figure 6.4, blue
curves. Broader peak features observed in the Ashtarak experiment (Figure 6.4 a,b) could be caused
by the absence of shielding, which results in residual ac components of the laboratory magnetic field,
estimated to be about 10 mG.
Note that the coherence time can be substantially increased (by orders of magnitude) using antirelaxation-
coated cells, cells with buffer gas, or cold atoms [40], allowing a potential reduction of the magnetic
resonance width down to ∼ 0.1 mG.
There is yet another mechanism possibly responsible for effective broadening of the magnetic reson-
ance with IL. At high light intensity, the distribution of the ground-state Zeeman sublevel popula-
tions is dominated by light-induced polarization, thus diminishing the effect of Larmor coupling. As
a result, the alignment remains preserved for a larger range of the B-field. This causes a continuous
growth of the magnetic-resonance width as a function of light power (Figure 6.5b), as opposed to the
output light intensity, which undergoes saturation at high values of IL (Figure 6.5a). Comparison of
Figure 6.4 a and b shows that the B = 0 peak signal appears at lower values of I 87L for Rb Fg=2
→ Fe=3 as compared with 85Rb Fg=3 → Fe=4. This result is in qualitative agreement with the
calculations presented in [122], which give the values for inversion threshold intensity S ≈ 13 and ≈
25, respectively. The suppression of orthogonally polarized amplified radiation at B = 0 observed for
I > 100 mW/cm2L (Figure 6.5a) is in qualitative agreement with the experimental results obtained
in [122]. This decline may be attributed to saturation effects.
About the overall polarization of the output beam measured in Figure 6.6: Generation of the ortho-
gonally polarized output beam at B = 0 should cause ellipticity in the overall polarization of the
transmitted beam. This is consistent with the measurements done with a circular polarimeter con-
figuration (Figure 6.6) where the ellipticity is observed only in the conditions favorable for mirrorless
lasing. When we apply a BZ magnetic field, this tends to cease the generation of ellipticity, thus
recovering linear polarization of the output beam. On the contrary, around B = 0, where we also
expect mirrorless lasing to happen, a peak feature is observable.
Discussion 119
For non-cycling transitions (85Rb Fg=2 → Fe=2, Fg=2 → Fe=1 and Fg=1 → Fe=1 and 87Rb Fg=1
→ Fe=0 and Fg=1 → Fe=1) pumped with a linearly polarized laser light, practically no atoms
remain in the excited state (see, for example, [125]). Consequently, no population inversion can
develop. Indeed, no mirrorless lasing is observed on these transitions.
6.4.1 Ring structure
One point that we need to understand in this phenomenon is the ring structure observed in the
spatial profile of the output beam in the conditions close to the optimal conditions for mirrorless
lasing. This could be an indication for a four-wave mixing (FWM) process which can develop in
a system of magnetic sublevels of degenerate two-level atoms pumped with intense linearly polarized
light.
FWM is a nonlinear phenomenon related to phase-matching or simply put: momentum conservation.
In our case, because of the degeneracy, we deal with degenerate four-wave mixing, a configuration in
which three different light fields of same frequency are propagating in a nonlinear optical material to
and a fourth beam [129].
Figure 6.10: Schematic diagram of entirely-degenerate four-wave mixing process.
A schematic diagram of such a process is presented in Figure 6.10.
Excitation on transitions ⃝1 drives atoms from mFg = 0 sublevel to mFe = 0 and from mFg = 1
sublevel to mFe = 1, followed by transfer to mFg = ±1 by inversion-amplified radiation ⃝2 with
120 Degenerate mirrorless lasing–Forward beam
subsequent laser excitation to mFe = ±1, and finally the loop is closing by emission of induced light
⃝3 . To make it more clear, the phase relationship would be k⃗1 + k⃗1 = k⃗2 + k⃗3.
The figure shows only two symmetric FWM channels linked with mF = 0, but the process can sim-
ilarly occur also on neighboring magnetic sublevels. Again here, the distinction of this process from
other known schemes is that it is entirely degenerate (all the four waves have the same frequency,
they differ only by polarization).
Efficient generation of FWM requires phase-matching, the conditions for which are related to the dis-
persive properties of the nonlinear medium (in our case the Rb gas). Conical emission results from
phase matching and can be straightforwardly separated from the axial mirrorless lasing, the latter
having much lower divergence.We estimated the divergence in this experiment to be ≈ 10−3 rad. If
all fields are having the same frequency and are co-linear, a non-collinear direction for the FWM-
generated field could be only due to very different refractive index for the inversion-amplified radi-
ation ⃝2 .
Being assisted by the population inversion, the FWM process competes with the direct mirrorless
lasing, which results in different optimal conditions for their observation as is seen in Figs. 6.8 and
6.9. Four-wave mixing is related to Kerr nonlinearity effects: more about that in the next Chapter.
Summarizing, let us list the indications that support our claim that we observed degenerate mirror-
less lasing:
1) The absence of magnetic resonance for BZ (B ∥ E) below IL ≈ 1.5 mW/cm2 (87Rb Fg=2→ Fe=3)
and ≈ 3 mW/cm2 (85Rb Fg=3 → Fe=4); nearly exponential initial above-threshold growth of the B
= 0 feature until it saturates (Figs. 6.4, 6.5a). This is an indication of a nonlinear effect.
2)There is a narrow resonance in the magnetic-field dependence of the generated light: this is our
strongest evidence so far. In this configuration, there is no such resonance expected in this field
range for non-lasing processes, because of symmetry reasons (B-field is directed along the light po-
larization). The big question here is, if any possibility of misalignment or self-induced ellipticity or
birefringence of cell windows could break this symmetry.
3) Consistency of the laser-intensity range where the magnetic resonance is observed (Figure 6.4)
with the results of [122]. In this paper amplification due to partial population inversion on 85Rb
Fg=3 → Fe=4 and 87Rb Fg=2 → Fe=3 transitions was directly recorded in the backward direction.
4) Outright absence of any orthogonally polarized output light for 85Rb F =2 → F =1,2,3 and 87g e Rb
Fg=1 → Fe=0,1,2 (Fe < Fg) transitions, independently of the BZ value (Figure 6.7).
Conclusions 121
A comprehensive theoretical treatment and modeling of the problem is quite complex and would
ideally include many processes such as propagation effects with partial depletion of the pump beam.
The careful reader will have noticed that we are discussing about lasing only in the forward direction
although the directionality did not seem important in the theoretical description of the problem. The
absence of backward lasing will be discussed in the next Chapter.
6.5 Conclusions
In conclusion, we have experimentally observed effects consistent with degenerate mirrorless forward
lasing in rubidium vapor contained in uncoated buffer-gas-free vapor cells under excitation with
linearly polarized laser radiation tuned to the 85Rb Fg=3 → Fe=4 and 87Rb Fg=2 → Fe=3 D2
transitions.
The generated beam is orthogonally polarized to the incident light polarization, and has low diver-
gence (≈ 10−3 rad). The lasing occurs above a certain threshold value of the incident light intens-
ity (IL ∼ 3 mW/cm2), and grows nonlinearly at least up to ∼ 0.3% conversion efficiency at IL ≈
90 mW/cm2. Further increase of the pump power IL results in a decrease of the generated light in-
tensity, which almost completely vanishes at IL ∼ 1000 mW/cm2. The generated light is sensitive
to the application of a transverse (B ∥ E) magnetic field, forming a narrow (∼ 20 mG) resonance
centered at B = 0.
Based on the analysis of the experimental results, the observed effect is attributed to the population
inversion established on |mFe | = n → |mFg | = n + 1 transitions between the Zeeman sublevels
(where n is a non-negative integer), resulting in an orthogonally-polarized beam under excitation of
a D2 Fe > Fg transition with linearly-polarized laser light. However other nonlinear effects seem to
interfere with our forward lasing.
If we conduct the experiment with antirelaxation-coated or buffered vapor cells the obtained res-
ults may be of interest for optical magnetometry. For instance, mirrorless lasing in the backward
direction, could be useful for remote mesospheric magnetic sensing using the sodium layer as a
sensor [130].
122 Degenerate mirrorless lasing–Forward beam
Refutations have often been regarded as establishing the failure of
a scientist, or at least of his theory. It should be stressed that this
is an inductivist error. Every refutation should be regarded as a
great success. Even if a new theory should meet an early death, it
should not be forgotten; rather its beauty should be remembered,
and history should record our gratitude to it.
Karl Popper
7
Unsolved (lab) mysteries
7.1 Degenerate mirrorless lasing- Backward Beam
Some of the text presented in this section is part of a paper that has been published as [49]. For
the work presented in this Chapter, I worked for part 1 experimentally together with my colleague
Guzhi Bao to set up and test the experiment in Mainz and, for part 2, I rebuild the whole setup and
analysed the data by myself.
7.1.1 The Idea
In Chapter 6 we introduced the idea of degenerate mirrorless lasing, a phenomenon in which amp-
lified spontaneous emission is produced along the pump direction in a Rb vapor, with orthogonal
polarization to that of the pump, when the pump beam has linear polarization under zero magnetic
field. The pump field interacts with the Zeeman-sublevels (mF ) of a specific atomic transitions and
population inversion is produced between certain magnetic sublevels. Specifically, the transitions in
which we expected and observed the lasing are Fe > Fg cycling transitions in the D2 line of rubidium
and in particular 85Rb Fg=3 → Fe=4 and 87Rb Fg=2 → Fe=3.
What we have detected so far is nonlinear dependence between the incoming and outgoing beam as
well as a narrow magnetic resonance when we scan the magnetic field around zero and when this
field is parallel to our pump polarization. These are a strong indication of amplified emission in
the forward direction or else forward mirrorless lasing. While forward mirrorless lasing can be an
interesting phenomenon, backward mirrorless lasing can be of even more interest, since it can be
used for distant magnetometry, for example in laser guide stars [130]. In the simple model we have
123
124 Unsolved (lab) mysteries
so far it seems that there should not be a reason for not having backward amplified emission and
only forward. To make things easier, in the backward direction, four-wave mixing processes and Kerr
effect does not contribute. And yet, backward degenerate mirrorless lasing seemed to be a much
harder case than we expected.
It is known that there is a difference between backward and forward emission, since the velocity
distribution of atoms and light shifts can produce a forward-backward asymmetry [131]. Special
in our case was that strong backward emission was observed in Armenia, and this could not be
reproduced in Mainz, even with the same vapor cell and the same experimentalists.
I will give a side by side overview of the experiments made in Mainz and Ashtarak, although the
timeline is quite nonlinear.
7.1.2 In Ashtarak, part I
Our collaborators in Ashtarak used the same setup as the one for forward lasing, with an additional
part for backward detection. The cell, laser, the pumping/input frequency and the shielding coils
are all the same as in the forward-beam experiment in Chapter 6. The experiment was conducted at
room temperature (in Armenia during the summer).
Figure 7.1: Backward lasing setup in Ashtarak. AT: Attenuator for controlling laser power; PD:
Photodetector; PBS: Polarizing beam splitter; BS: Beam splitter; PD1 is used for detecting the
orthogonal polarization of the backward beam, PD2 detects fluorescence and PD3 is monitoring
the frequency of the laser to have a reference on which frequency we pump.
The 780 nm input laser beam has a 2 mm diameter and a linewidth of 15 MHz. Its frequency is tuned
to the cycling transitions where we expect inversion of population to happen [(]Fg=3→ F =4 (85e Rb)
and Fg=2 → F 87e=3 ( Rb) of the rubidium D2 line: 5S1/2 → 5P3/2].
The setup has three parts: the part on the right, monitored by PD3, is using spectroscopy on the
reference cell in order to monitor the frequency of the laser beam that we also call pump, that will
Degenerate mirrorless lasing- Backward Beam 125
create the inversion of population. The beam afterwards reaches the middle part of the setup, goes
through the cell and gets detected in the forward direction setup. For degenerate mirrorless lasing
we expect a backward beam with a polarization orthogonal to that of the pump beam. This is why
this part contains a beam blocker and a Glan-Thompson polarizer. The beam blocker acts also as
a pinhole for the backward lasing to go through, but is blocking all four reflections coming from
the four window surfaces of the cell (two from the faces of the window in the front and two of the
back window of the cell). The backward beam is detected with PD1 or an IR digital camera. As a
complementary measurement, the fluorescence from the cell was detected with PD2.
To monitor the nonlinear behavior of the backward beam, a magnetic field B was applied in the dir-
ection along z (Bz ∥ Ein)—where we do not expect nonlinear magneto-optical effects. The incoming
pump beam had a 23 mW power and its frequency was linearly scanned across all the D2 lines to get
the whole spectrum of the lasing beam and the fluorescence.
Results
Without revealing too much, since this paper has not been published yet, our collaborators detected
in the backward direction, by scanning the magnetic field Bz from -0.5 to 0.5 G, a beam with the
largest intensity at zero field.
Figure 7.2: Backward-direction in Ashtarak for different magnetic fields. The blue line is the
signal in the backward direction and the yellow line is the fluorescence. Left: Signal at 0.5 G,
Middle: Signal around 0.2 G, Right: Signal at 0 G.
The backward signal develops only in the region of the cycling transitions (Fe > F ) 85g Rb Fg=3 →
Fe=4 and 87Rb Fg=2 → Fe=3. Even after blocking the four main reflections of the cell, we see on
zero field some scattering signal from the incoming laser beam that is not blocked but propagates
freely until it is stopped by the wall a few meters away. The spectrum of the backward beam has
some sub-Doppler features at the crossover resonances CO32 87Rb and CO43 85Rb.
When applying a longitudinal magnetic field (Bx//k), there is positive correlation between the
magnetic field and the signal: at ±0.5G we have an increase of 30 x in the backward signal. No data
is provided for this observation in this thesis.
Discussion
Our collaborators in Armenia observed in the backward direction a beam that is appearing only at
the same cycling transitions as we observed for the forward direction and has orthogonal polarization
126 Unsolved (lab) mysteries
to the incoming beam. The beam appears around zero field (B parallel to the electric field of the
incoming beam) with its highest amplitude at 0. Some important things to notice:
• The beam appears- after a threshold in input laser power.
• The cell is 8 cm long, sapphire windows, no buffer gas and no coating. In order to observe
backward emission, the cell was being moved slowly by hand until we found a signal. The signal
was then immediately appearing in full amplitude or not at all. This effect was attributed to
residual birefringence of the cell.
• The smaller signal in the backward direction compared to the forward lasing we observed in
Chapter 6 is hypothesized to exist due to the the atomic velocity distribution: The backward-
emitted photons are absorbed by opposite-velocity atoms, so there is bleaching/saturation of
the medium for the velocity group of atoms corresponding to forward transmission.
7.1.3 Mainz, part I
Back to Mainz, we initially built a setup to monitor possible backward lasing and at the same time
we used another laser (probe) to detect whether there is gain in the backward direction.
Figure 7.3: Saturation spectroscopy of probe beam. The dark-red line is the pump, the light-red
line is the probe. We monitor both beams to see possible gain in the expected frequencies on the
probe beam, as illustrated at the top right.
The idea was that, if there is a population inversion established, then the incoming probe beam,
scanned through the frequencies at which we expect population inversion, will show some gain. This
would be a direct observation of amplification happening in the backward direction. We tried many
pump powers, ranging from a few µW to 22 mW. Our probe beam was ranging between 100−2µW
Degenerate mirrorless lasing- Backward Beam 127
and but no gain was seen, including at the 10 times lower probe power than the beam seen in
Ashtarak. The pump was equally strong as in Armenia (20 mW)and our beam size smaller, but we
failed to see any gain on the probe beam.
Figure 7.4: Setup for measuring probe gain in 85Rb.
Figure 7.5: Setup for measuring probe gain in 87Rb.
Since I did not see any signal in the backward direction, I decided to improve the forward-direction
setup and check all the things that could be different from the Ashtarak setup. A list with the setup
parameters that were studied follows.
Polarizers
Linear polarization of our incoming pump beam seems to be of importance to our experiment. I
checked the extinction ratio of our polarizers in order to be able to have an estimation of how well
the pump is polarized.
128 Unsolved (lab) mysteries
The extinction ratio of a polarizer is a measure of how well the light is polarized in a linear polariza-
tion mode. The two orthogonal linear polarization states related to transmission and reflection are S
and P. The polarization that gets transmitted through a polarizeri is called Pi and is parallel to the
incident surface, while the Si polarization is orthogonal to it and gets reflected. The extinction ratio
is defined as the power of Pi to Si, where S and P are always defined with respect to the plane of in-
cidence. The plane of incidence is defined by the vector of the beam’s direction and the vector of the
incidence surface. Ideally a reflective polarizer will transmit only one of the two linear polarizations
and reflect the other, reducing by half the incoming light intensity.
Figure 7.6: Example of S and P polarization of light. The incident light contains two types of
polarization: S and P . The P component is parallel to the plane of incidence and the S component
is perpendicular to it.
I tested the polarizers I was using with a good Glan-Thompson polarizer, with an extinction ratio
of 100000 : 1. Figure 7.7 shows the setup used to test out polarizers by measuring the incoming
and outcoming powers of a laser going through a good and already tested polarizer. The resulting
extinction ratio was around 300 : 1 and was depending on the position of the polarizer: on the edges
of the polarizing material there are more asymmetries than in the middle. This ratio was less than
expected from the polarizers’ specs and so we ordered new and better polarizers to replace the former
with.
Temperature
Another parameter we needed to check was the temperature at which the experiment took place. In
Ashtarak the work was conducted under RT. I started with RT and heated the cell with a heat gun
up to 50 ◦C.
Laser Power
Since there seems to be a threshold in detecting a forward beam and a range of optimal powers, I
tuned the power of our laser to the maximum I could which was around 22 mW before the cell. Lower
Degenerate mirrorless lasing- Backward Beam 129
Figure 7.7: Measuring extinction ratio of PBSi. We test how well linearly polarized our laser
with power I1. I2 is the power of the transmitted beam, I3 the power of the reflected beam and IA
power of the possibly absorbed beam due to the G-T polarizer. The G-T polarizer is not a cube,
but it is shown as such here for simplicity.
powers were also investigated in case we encountered a similar phenomenon as in Figure 6.4 where
the forward magnetic resonance declines with higher powers. At no point between a few µW and
22 mW did we get a backward signal. One could assume that the laser power might not be enough to
create all along the cell the conditions necessary for lasing. However the power after the cell is over
the threshold for forward lasing, meaning we should have population inversion all along the cell.
Beam size
The beam diameter D in the improved setup was about 100µm at the point of focus. The diffraction
angle θ is θ ∼ λD ∼ 10
−2. The cell was 14 cm long and we made sure that it focuses in the middle of
the cell, so at the output of the cell the beam size is 70mm · 10−2 = 0.7mm. This makes it about
half the size of the 1.5 mm beam size in Ashtarak.
Identify reflections
A very important background signal for backward detection are the reflections coming from the in-
coming pump beam getting reflected on windows of the Rb cell. The light reflects two times on the
front and two times on the back of the windows covering the cylindrical cell. We can detect this way
two strong and two weaker reflections on the backward direction which we must block. (The weaker
ones are from the external antireflection-coated surfaces). It is also important to block the beam
with a very black material after is passes the cell to avoid partial reflection back into the cell. As
we will reveal later, the only signal we ever had in the backward direction in Mainz was due to the
scattering of the pump beam on a wall 4 m away.
130 Unsolved (lab) mysteries
Cell birefringence
The initial guess about the discrepancy between Mainz and Ashtarak regarding the backward-
direction was the possible birefringence of the cells we used. Normally glass does not induce bi-
refringence, however stress caused by pressure or glue used to attach the windows to the cylindrical
part of the cell can locally induce changes in the glass and make it birefringent. These local inhomo-
geneities can change the refractive index associated with a specific polarization going through the
glass. If the first glass window alters the polarization of the pump beam it could be that the strict
conditions for population inversion are not met.
The problem with testing this theory is that we cannot measure inside the cell to compare the
possibly induced birefringence between the two setups. If we measure with crossed polarizers on the
other side of the cell, the risk exists that the back window of the cell changes the output polarization
as well and what we measure does not accurately reflect what the atoms see.
In order to detect backward signal in Ashtarak the cell was moved carefully and at a specific point;
there was full signal detected, in contrast to zero signal when the cell was moved off this point. In
Mainz it was much harder to move the cell carefully because it was inside a shield and we could move
it only with a non-magnetic stick or string attached to the cell. However, we tried to compensate
for birefringence by using either a quarter-wave plate and or a half-wave plate; this did not have any
effect.
Scanning rate and function
Since we wanted to measure the backward signal and the fluorescence and compare to reference
signals at any time, we scanned the laser frequency with a linear function from an external function
generator (triangular pulses) and slow enough –between 20−0.5 Hz– for the population inversion to
be established. The scanning rate was similar to the one in Ashtarak.
Magnetic field
A big difference between our two setups was the magnetic field compensation. The model we used
in the forward-direction experiment requires zero field for the population inversion to be established.
The compensation mechanism in Ashtarak however, were homemade Helmholtz coils with the dis-
tance of the inner smaller coils being almost equal to the length of their cell. That would create a big
magnetic gradient along the cell with the sides of the cell being under a nonzero magnetic field.
Laser linewidth
Another difference was the linewidth of our pump laser. In Ashtarak it was between 15−10 MHz
while our laser had sub-MHz linewidth. Later we tried to repeat the experiment with a broader laser
but we still did not see backward lasing.
Degenerate mirrorless lasing- Backward Beam 131
Forward Detection
After checking out and fixing some of the differences between our setups, I repeated parts of the
forward beam experiment to detect possible changes.
Figure 7.8: Forward beam experiment. We replaced all the polarizers with good ones and heated
the cell starting from RT and up to 45◦C. We detected either with a camera or with a photodetector
to compare the data. We operated at zero(0) magnetic field.
The forward detection signal looks a lot like the results we had before. I scanned the laser at 10 mW
over a 9 GHz range covering the D2 line and on the camera we can saw a bright spot appearing with
a black ring in the middle for some frequencies, which seem to be shifted compared to our previous
experiment. I wanted to study the ring effect a bit more. We had attributed this effect to FWM
which required linear polarization, so I added a quarter-wave plate to see what happens with circular
polarization. At low pump power (1.7 mW) the dark ring disappeared and re-appeared again at
high power. Then, I removed the second crossed polarizer which means the detected polarization
remained vertical and I could still see the ring. This phenomenon can be explained by the Kerr
effect.
The Kerr effect is a 3rd order nonlinear effect related to the degenerate FWM effect as described in
Chapter 6 only we do not need a specific polarization for it to happen. If we use a strong laser beam
interacting with an atomic vapor, the latter behaves as a Kerr medium, meaning it has an intensity-
dependent refractive index [132]. Our beam has a Gaussian intensity profile, meaning the intensity
of the beam changes according to the Gaussian function and hence we have radial dependence of the
index of refraction. The cross-section of the beam with the vapor causes nonlinearities. The high-
intensity laser changes the refractive index of our vapor and the vapor acts as a lens to the beam.
To better understand this process, I show Figure 7.9: the refractive index indicates how much light
132 Unsolved (lab) mysteries
bends in a material. In our case, between the maximum and the minimum refractive index, we have
maximum absorption, which creates the pattern of the black spot we see in the middle of the beam
when we scan around a resonance with frequency ω0.
Figure 7.9: Kerr effect/self-focusing. The index of refraction n and the absorption coefficient in
relation to frequency are illustrated for a better understanding of the Kerr/self-focusing effect. The
phenomenon arises when a high-intensity laser beam propagates through a Rb vapor while the laser
frequency is scanned around a resonant frequency ω0.
I filmed the effect of self-focusing on our beam: in Figure 7.10 one can see how the beam evolves from
1-8 when we detect it using a laser-viewing card for near infrared light. In pictures 1 and 7 there is
a small black ring in the middle of the beam, exactly where we have the beam maximally absorbed.
Again, the beam shows this inhomogeneous behavior because its intensity in not evenly distributed
and the refractive index depends on the intensity.
Backward detection
So far we have studied the forward beam in more detail. The next step was to build a setup for
backward detection now that our system was better controlled and studied. Figure 7.11 shows the
experimental setup.
At 50°C and at 500 mHz scanning rate across the first two resonances where we expect lasing to
happen, we get faint light in the backward direction. The beam in the forward direction after the cell
is at this point blocked with a black carton. The signal on the camera is two big (compared to the
central spot) faint rings in the background and a flashing light in the center.
It seems that this signal arises entirely from scattering: the beam structure we see on the camera
has a component that is frequency dependent (fluorescence from scattering on the carton) and one
weaker and larger in the back, frequency independent (incoherent scattering on carton). When we
block the beam in the forward direction with black aluminum foil, the backward signal completely
disappears.
Degenerate mirrorless lasing- Backward Beam 133
Figure 7.10: Kerr effect/self-focusing on a laser-viewing card. Pictures 1−8 show the time evolu-
tion of our detected beam without a crossed polarizer: size, shape and intensity are changing time.
In picture 1 and 7 one can see a small black dot in the middle of the beam indicating self-focusing.
The next step was to put limits on backward detection: I calibrated the intensity on the camera to
estimate how much power we detected. For attenuated pump, when we reflect it back with a mirror,
we get 6 pW on the camera. When we block it with black carton, but not perfectly, we get 0,5 nW.
When we block with good quality black paper the limit is set at 0,1 nW. Based on that and with a
30 mW beam, the estimated detected lasing efficiency is 3∗10−9, meaning we can detect a beam that
is more than 1 billion times weaker than our pump.
It is important to note here is that when I directed the beam 3 m away and focused it on the wall, I got
a lot of frequency dependent light back (see above), a strong indication that scattering is important
since it can create a false signal and should be checked thoroughly. The intensity we detect depends
on the distance from the camera and focus of the beam as well as angle of the surface we use to block
the beam with.
Discussion
The only signal in the backward direction that we were able to detect was due to scattering of the
forward beam. The flashes on the camera that were frequency dependent were detected with a
photodetector as well and coincided with the cross-over resonances.
This signal can arise from the system imperfections. The transmitted beam scatters on the impact
surface and some of it is reflected back to the vapor cell. Due to imperfections such as window
birefringence or residual magnetic field, the polarization of the back-scattered light can change so
134 Unsolved (lab) mysteries
Figure 7.11: Backward beam experiment. We replaced all the polarizers with good ones and
heated the cell starting from RT and up to 45◦C. We detected either with a camera or with a
photodetector to compare the data. We operated at zero(0) magnetic field. The beam in the
forward direction is blocked with various materials.
that part of it would appear in the polarization channel where we expect mirrorless-lasing light,
producing a Doppler-free signal. This might explain why the flashes on the camera coincide with the
CO lines. At that point the scattering of the forward beam seemed to be a possible explanation for
the signal seen in Ashtarak.
7.1.4 Ashtarak, part II
In order to better understand the disagreement of the results of the two setups, I took a trip to
Armenia in Spring of 2019. By then, the possibles theories for the the discrepancy of our results
were:
• The signal in Armenia is scattering from the wall.
• The laser in Ashtarak has a much broader linewidth of 16 MHz compared to the laser in Mainz
(<100 kHz); however, we do not know why the linewidth would have such a crucial impact on
the phenomenon and we can only speculate. Further experiments with a broader-linewidth
laser in Mainz did not yield results.
• Difference in magnetic field compensation due to lack of a shield in Ashtarak or a much bigger
gradient along the cell.
• The cell in Armenia inducing less birefringence because of the way it was made.
Degenerate mirrorless lasing- Backward Beam 135
• Something else or a combination of the above.
With me I brought the Rb cell used in Mainz as well as the very black aluminum sheet with which I
fully blocked the scattering of the forward beam.
After working on their existing setup and looking at the backward-detection signal, we replaced
their cell with ours and we blocked the scattering with the black sheet. We had to move the cell
from Mainz by hand a bit longer than Ashtarak’s cell but we managed to get a signal with it. The
black sheet suppressed the background, but not the entire signal. We also replaced the laser with an
Extended Cavity Diode Laser of 1 MHz linewidth resembling our conditions a lot more. The results
were the same as reported to us before: the signal appears in the region of the cycling transitions
(F > F ) 85Rb F =3 → F =4 and 87e g g e Rb Fg=2 → Fe=3. We can see scattering signal from the wall
which almost disappears when we later block it with the very dark sheet. There are also sub-Doppler
features at the crossover resonances CO32 87Rb and CO43 85Rb. The power of the detected beam is
about 104 times smaller than the input beam and the relation between input and output beam seems
quite linear, while the input vs output intensity for the forward beam was highly nonlinear.
Figure 7.12: Backward beam experiment in Ashtarak. Backward-direction detection at zero field
in Ashtarak. The blue line in the oscilloscope is the signal in the backward direction and the yellow
line is the fluorescence.
Discussion
At this point the question remains: is this signal backward lasing?
Indications that this could be backward lasing:
136 Unsolved (lab) mysteries
Figure 7.13: Data from Ashtarak, not yet published: Zero magnetic field input power dependence
of the absolute (left panel) and normalized (right panel) backward radiation power. Solid symbols:
amplitudes of Doppler-broadened contours of 87Rb (circles) and 85Rb (squares) transitions, Open
symbols: amplitudes of the narrow peaks located at corresponding crossover resonances CO32
(circles) and CO43 (squares). Lines are drawn to guide the eye.
• When a random back reflection from the cell windows goes in the beam path, we get no signal.
• There is a magnetic field dependence of the backward signal when none is expected without
mirrorless lasing, beam vanishes at ±0.5G.
• Peaks are only seen at the expected transitions Fe > Fg from our model in Chapter 6
Issues of concern:
• To get a backward signal there is always a back reflection from a window 1-3 beam diameters
from the pinhole.
• The nature of the Doppler-free structure is not clear. It is usually created by a counter-
propagating beam, so this enhances the idea that this might be a back reflection of some
Degenerate mirrorless lasing- Backward Beam 137
sort. You can also have these nonlinear ellipticity effects, induced by cell birefringence that
give quite similar signals even in zero field, see [133].
• The fact that the cell needs to be moved around in order to get a signal could indicate induced
birefringence. However, the always sudden appearance of the effect implies that the conditions
for the polarization have to be very strict, avoiding the tiniest ellipticity. It could also imply
that there is seeding of the effect with the right ellipticity in some backward scattering.
• Could uncompensated magnetic fields create other effects? Or could the gradient field that is
bigger in the Ashtarak experiment play a role in the creation of a backward-lasing beam? A
strange effect in this experiment was that when applying a Bx field (along k) the backward
signal grew from 0 to 0.5 .
7.1.5 Mainz, part II
The last part of the experiment took place in Mainz and some of our collaborators from Ashtarak
visited as well. The two final parameters that struck us as most important were the possible birefrin-
gence of the cell and the different magnetic environment or gradient along the cell.
In order to be able to move the cell inside the magnetic shield, our collaborators designed a 3D
printed plastic case which could be translated from outside. It was not as precise as moving the cell
by the hand, but it was still a good enough motion for testing different input spots on the windows.
The next idea was to apply a magnetic field along the longitudinal direction which showed an increase
in backward signal in Ashtarak. This did not give any signal back with the exception of adding a
mirror after the crossed polarizer: When we had a back-reflection, and when we added magnetic
field along x (parallel to k), we got a signal in the backward direction that resembles the one seen in
Ashtarak. However we could not get backward radiation in the way our collaborators seem to have
detected. We have postponed further experiments until we have a more solid model that calculates
the possible gain in this system. To this day, the mystery of backward degenerate lasing has not been
solved.
7.1.6 Conclusions
Degenerate mirrorless lasing is an interesting phenomenon that has been experimentally observed
in the forward direction. However, more experimental work needs to be done for backward directed
emission and the theory behind needs to be better understood.
138 Unsolved (lab) mysteries
7.2 Spin locking in Rb
In this final part of the Section of unsolved lab mysteries, I will shortly describe a project that never
came to an end due to discrepancies with previously published results. It serves more as a report, in
the case that a future researcher wants to look into the topic of spin-locking.
7.2.1 The idea
The idea of this project was to take the method of spin locking as published in [94] and apply it
on Rb in an unshielded environment. We worked on two levels in the case: We were building the
equipment for working eventually on earth field without a shield and in parallel we were testing out
how spin locking works in Rb inside a magnetic shield.
7.2.2 Experiment
We simulated the conditions of the spin locking paper in Cs [94]. We attempted to ask the following
questions in 87Rb:
• How does the resonance look at low fields and what is the linewidth?
• Are there residual fields in the shield?
• Is there a big gradient along the main magnetic field?
• How does the resonance change with higher fields?
• How does the resonance look at earth field and what is the width of each peak?
• Scan amplitude and phase of the RF field to find where and if there is a narrowing in the
linewidth through spin locking
The setup is shown in Figure 7.16 and the experimental parameters follow in detail.
Cell
I used a Rb cell containing both 87Rb and 85Rb. In comparison the project we tried to reproduce
was using Cs. Both the Rb and the Cs cell are 5 cm.
Pump-probe
The pump was locked to the 85Rb D1 line F = 3 and the probe was D2 87Rb F = 3 detuned towards
F = 3. The axes were the same as in [94]: The leading field along z (long axis of the shield and cell),
the pump along y (from the top of the cell) and the probe along x.
Pump-probe powers
The light power of the pump during the “on” part of the cycle is 1mW and the probe power is at
20µW . In [94]: the pump power was at 50µW and the probe at 10µW .
Spin locking in Rb 139
Figure 7.14: Spin-locking setup in 87Rb.
Duty circle
The duty circle was 5% of the pump modulation in our case and 3% in the Cs spin-locking paper.
Leading field
The leading field was 55000nT in our case and 98000nT in [94].
RF (spin-locking) field
In both projects, the spin locking (RF) field is applied along the pump axis, is a sine wave and its
frequency is equal to the modulation frequency. I tried out different RF-field strengths which we
report in mV. The Voltage corresponds to a magnetic field of 41.2 ∗V oltage. For example: 10mV →
0.412nT, 167mV → 6.8nT, etc.
7.2.3 Results
After setting up the experiment in Rb, as seen in Figure 7.14, I recorded the magnetic resonance
without a spin-locking field on Earth field (in this case 30µT). As seen in Figure 7.15 the resonance
splits in 4 peaks. The NLZ revival frequency for 87Rb for 30µT is calculated to be 6.45 Hz, hence
the split from one peak to the other should be about 12.9 Hz (2∗revival frequency). At a first glance,
the split seems to fit the calculation. To be more precise, I fit the the X, Y and R components of
the resonance with 4 different parameters for width and 4 for amplitude. The split the fit calculates
140 Unsolved (lab) mysteries
agrees to what we roughly assumed and is around 11.6 Hz, the amplitudes vary quite a lot. The split
we see comes from the NLZ effect, but there seem to be asymmetries in the shape and amplitude of
the peaks possibly due to heading error or gradient.
X
12
Y
12
12
R
A B
Apr-24 Dimitra Kanta                              1
Figure 7.15: Rb Resonance under Earth field, at 30µT. The resonance and an estimation of the
split is seen on the left and the right side shows the fitting of the X, Y and R components of the
resonance.
I measured the residual field in the middle of the shield and cell– when all other fields are off – and I
found along x: 16nT, along y: 16nT and along z: 30nT. Next, while having the field on along z at
18.4µT, I measured the magnetic field with a fluxgate (by hand) along the whole shield with a step
of 1 cm, starting at 1 cm away from the inner coil. The gradient seems to be very roughly 100nT
along the cell (5 cm).
Another parameter we tested was the phase of the RF field; I choose the “best phase” based on which
phase gives the maximum amplitude. This seems to change with different amplitudes of the RF field.
Figure 7.16 shows the Rb resonance at 55µT for different RF-field amplitudes.
7.2.4 Discussion
It seems that in our Rb setup, the RF field makes the amplitude bigger and narrows the whole
resonance, which is consistent to the results in Cs [94]. However, it does not decrease the signal’s
amplitude in the off-resonant case as seen in Figure 7.17 and marked by the dashed green lines. In
our case, the RF field increases the signal, but in [94] the spin locking field dramatically decreases
the signal.
Spin locking in Rb 141
Figure 7.16: Spin locking in Rb at 55µT for different RF-field amplitudes.
Figure 7.17: Comparison between spin locking in Rb (left) and Cs (right) and how it affects the
off-resonant case.
7.2.5 Conclusions
Although narrowing is observed in both cases, there are differences between the two and one would
need to make a direct “longitudinal” comparison between different signals in the two cases. There are
also more subtle things to discuss such as asking ourselves if both cases should be called spin locking
or if one of the two helps magnetometry more.
142 Unsolved (lab) mysteries
8
Conclusions
In this thesis, I describe the major projects that I worked on during my PhD research. Each chapter
in the thesis provides a separate outlook for each project.
In particular, an important part of the work was focused on Earth-Field magnetometry and how to
improve the sensitivity of the magnetometers working in the geomagnetic field range (Chapters 4,
5). Specifically, my first project was on all-optical spin locking, where we developed an all-optical
magnetometer and demonstrated a method to suppress the NLZ effect in the range of the Earth’s
magnetic field. This method improves the sensitivity of earth-field-range magnetometers for two
reasons: an increase in the magnetic resonance signal amplitude and a reduction in the effective
linewidth. On the same topic, we came up with the idea of making a magnetometer that is completely
free of heading error. Here, we presented a sensitive scal√ar magnetometer that is free from (NLZ)
related heading error and has a sensitivity of tens of fT/ Hz in the Earth’s magnetic field range.
Next steps in this field would include building these magnetometers without any kind of shielding
and testing them on Earth-Field and in an environment that is not magnetically noisy.
In parallel to the Earth-Field magnetometer projects I was working on mirrorless lasing (Chapters
7, 6). We observed degenerate mirrorless lasing in the forward direction, but we had different results
than our collaborators for the backward direction. More experimental work is needed on backward
directed emission, and the theory behind it must be better understood through the construction of
more accurate models.
Results presented in this thesis were first published in the following papers: [93], [65], [120], [49]
143

Bibliography
[1] M. W. McElhinny, R. Merrill, The earth’s magnetic field : its history, origin, and planetary
perspective (London ; New York : Academic Press, 1983).
[2] T. Gillian, North Pole, South Pole : the epic quest to solve the great mystery of Earth’s
magnetism (New York, NY: The Experiment, 2010).
[3] J. Osborne, J. Orton, O. Alem, V. Shah, Proc. of SPIE 10548, 105481G (2018).
[4] J. Kitching, Applied Physics Reviews 5, 031302 (2018).
[5] D. Sheng, A. R. Perry, S. P. Krzyzewski, J. K. S. Geller, S. Knappe, Applied Physics Letters
110, 031106 (2017).
[6] H. Dehmelt, Modulation of a light beam by precessing absorbing atoms, Phys. Rev. 105,
1924–1925 (1957).
[7] W. Bell, A. Bloom, Optical detection of magnetic resonance in alkali metal vapor, Phys. Rev.
107, 1559–1565 (1957).
[8] W. Bell, A. Bloom, Optically driven spin precession, Phys. Rev. Lett. 6, 280–281 (1961).
[9] I. K. Kominis, T. W. Kornack, J. C. Allred, M. V. Romalis, Nature 422, 596–599 (2003).
[10] J.-H. Storm, P. Hömmen, D. Drung, R. Körber, Appl. Phys. Lett. 110, 072603 (2017).
[11] H. B. Dang, A. C. Maloof, M. V. Romalis, Applied Physics Letters 97, 151110 (2017).
[12] K. Jensen, M. A. Skarsfeldt, H. Stærkind, J. Arnbak, M. V. Balabas, S.-P. Olesen, B. H.
Bentzen, E. S. Polzik, Scientific Reports 8 (2018).
[13] I. A. et al., Phys. Rev. Lett. 103 (2009).
[14] J. M. Higbie, E. Corsini, D. Budker, Review of Scientific Instruments 77 (2006).
[15] A. L. Bloom, Applied Optics 1 (1962).
[16] G. Bison, R. Wynands, A. Weis, Optics Express 11 (2009).
[17] M. D. et al., Space Sci. Rev. 114 (2004).
[18] D. Budker, M. Romalis, Nature Physics 3 (2007).
145
146 BIBLIOGRAPHY
[19] D. Budker, D. F. J. Kimball, Optical Magnetometry (Cambridge University Press, 2013).
[20] A. E. Siegman, Lasers (University Science Books, 1986).
[21] T. Dudok, Y. Nastishin, Optically pumped mirrorless lasing. a review. part i. random lasing,
Ukrainian Journal of Physical Optics 14, 146-170 (2013).
[22] M. S. Malcuit, J. J. Maki, D. J. Simkin, R. W. Boyd, Transition from superfluorescence to
amplified spontaneous emission, Phys. Rev. Lett. 59, 1189–1192 (1987).
[23] H. Cao, J. Xu, Y. Ling, A. Burin, E. Seeling, X. Liu, R. Chang, Random lasers with coherent
feedback, IEEE Journal of Selected Topics in Quantum Electronics 9, 111-119 (2003).
[24] B. V. Zhdanov, R. J. Knize, Review of alkali laser research and development, Optical Engin-
eering 52 (2012).
[25] M. Auzinsh, D. Budker, S. M. Rochester, Light-induced polarization effects in atoms with par-
tially resolved hyperfine structure and applications to absorption, fluorescence, and nonlinear
magneto-optical rotation, Phys. Rev. A 80, 053406 (2009).
[26] J. Weiner, F. Nunes, Light-Matter Interaction: Physics and Engineering at the Nanoscale,
Second Edition (Oxford University Press, 2017).
[27] E. Liarokapis, Notes in Atomic and molecular physics (School of applied mathematical and
physical sciences, NTUA, Athens, 2012).
[28] T. Henrich, Optisches Pumpen von 3He und Polarisationsanalyse, Basierend auf der wis-
senschaftlichen Prüfungsarbeit für die erste Staatsprüfung für das Lehramt an Gymnasien
(Johannes Gutenberg Universität, Mainz, 2008).
[29] E. Tiesinga, P. J. Mohr, D. B. Newell, B. N. Taylor, Codata recommended values of the funda-
mental physical constants: 2018, Rev. Mod. Phys. 93, 025010 (2021).
[30] M. Auzinsh, D. Budker, S. Rochester, Optically Polarized Atoms: Understanding Light-atom
Interactions (Oxford University Press, 2010).
[31] R. Coker, Physics 362l, https://web2.ph.utexas.edu/~coker2/index.files/
ch36pt2317L.htmaccessedon2022-11-24.
[32] P. Dirac, The principles of quantum mechanics (Oxford University Press, 1947).
[33] C. Cohen-Tannoudji, B. Diu, F. Laloe, Quantum Mechanics, Vol. 1 (Wiley-VCH, 1977).
[34] W. Happer, Optical pumping, Reviews of modern physics 44 (1972).
[35] E. B. Alexandrov, M. Auzinsh, D. Budker, D. F. Kimball, S. M. Rochester, V. V. Yashchuk,
Dynamic effects in nonlinear magneto-optics of atoms and molecules, Journal of the Optical
Society of America B 22 (2005).
BIBLIOGRAPHY 147
[36] T. Haider, A review of magneto-optic effects and its application, International Journal of
Electromagnetics and Applications 7, 17-24 (2017).
[37] D. Vojna, M. Duda, R. Yasuhara, O. Slezák, W. Schlichting, K. Stevens, H. Chen, A. Lu-
cianetti, , T. Mocek, Verdet constant of potassium terbium fluoride crystal as a function of
wavelength and temperature, Optics Letters 45 (2020).
[38] S. Pustelny, Nonlinear magneto-optical effects, Ph.D. thesis, Jagiellonian University Krakow
(2007).
[39] M. Mazilu, J. Baumgartl, S. Kosmeier, K. Dholakia, Optical eigenmodes; exploiting the quad-
ratic nature of the energy flux and of scattering interactions, Optics Express 19 (2011).
[40] D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, A. Weis, Resonant
nonlinear magneto-optical effects in atoms, Rev. Mod. Phys. 74 (2002).
[41] M. P. A. Jones, Atom-light interactions.
[42] F. X. Kärtner, O. D. Mücke, Nonlinear Optics (Center for Free-Electron Laser Science, DESY
Department of Physics, University of Hamburg, 2016).
[43] R. Pool, Molecular orientation at biological interfaces, water and lipids studied through
surface-specific vibrational spectroscopy, Ph.D. thesis, University of Amsterdam (2013).
[44] S. B. Mendes, P. Tandy, Advanced optics phys 545, https://www.physics.louisville.edu/
sbmendes/ (2010). Accessed on 2022-12-09.
[45] C. Cohen-Tannoudji, Manipulating atoms with photons (Nobel Lecture, 1997).
[46] J. A. McGovern, Advanced quantum mechanics i, lecture notes, https://theory.physics.
manchester.ac.uk/~judith/ (2012). Accessed: 2023–27-03.
[47] C. Cohen-Tannoudji, J. Dupont-Roc, G. Grynberg, Photons and Atoms : Introduction to
Quantum Electrodynamics (Wiley-VCH, 1989).
[48] A. Fabricant, I. Novikova, G. Bison, How to build a magnetometer with thermal atomic vapors:
A tutorial, New J. Phys. 25 (2023).
[49] A. Ramaswamy, J. Chathanathil, D. Kanta, E. Klinger, A. Papoyan, S. Shmavonyan, A. Khan-
bekyan, A. Wickenbrock, D. Budker, S. A. Malinovskaya, Mirrorless lasing: a theoretical per-
spective (2023).
[50] D. A. Steck, Rubidium 87 d line data, http://steck.us/alkalidata (2010). Revision 2.1.4,
23 December 2010).
[51] M. T. Graf, D. F. Kimball, S. M. Rochester, K. Kerner, C. Wong, D. Budker, E. B. Alexandrov,
M. V. Balabas, V. V. Yashchuk, Relaxation of atomic polarization in paraffin-coated cesium
vapor cells, Phys. Rev. A 72, 023401 (2005).
148 BIBLIOGRAPHY
[52] E. B. Alexandrov, M. V. Balabas, D. Budker, D. English, D. F. Kimball, C.-H. Li, V. V.
Yashchuk, Light-induced desorption of alkali-metal atoms from paraffin coating, Phys. Rev. A
66, 042903 (2002).
[53] V. M. Acosta, M. Auzinsh, W. Gawlik, P. Grisins, J. M. Higbie, D. F. J. Kimball, L. Krzemien,
M. P. Ledbetter, S. Pustelny, S. M. Rochester, V. V. Yashchuk, D. Budker, Production and
detection of atomic hexadecapole at earth’s magnetic field, Opt. Express 16, 11423–11430
(2008).
[54] V. V. Yashchuk, D. Budker, W. Gawlik, D. F. Kimball, Y. P. Malakyan, S. M. Rochester,
Selective addressing of high-rank atomic polarization moments, Phys. Rev. Lett. 90 (2003).
[55] S. Groeger, G. Bison, J.-L. Schenker, R. Wynands, A. Weis, A high-sensitivity laser-pumped
mx magnetometer, Eur. Phys. J. D 38, 239–247 (2006).
[56] W. Gawlik, J. Kowalski, R. Neumann, F. Träger, Strong narrowing of the na forward scattering
signals due to the interaction with an intense dye laser field, Phys. Rev. A 48.
[57] D. Budker, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, Nonlinear magneto-optical rota-
tion via alignment-to-orientation conversion, Phys. Rev. Lett. 85, 2088–2091 (2000).
[58] D. Budker, D. F. Kimball, V. V. Yashchuk, M. Zolotorev, Nonlinear magneto-optical rotation
with frequency-modulated light, Phys. Rev. A 65, 055403 (2002).
[59] W. Gawlik, L. Krzemień, S. Pustelny, D. Sangla, J. Zachorowski, M. Graf, A. O. Sushkov,
D. Budker, Nonlinear magneto-optical rotation with amplitude modulated light, Applied Phys-
ics Letters 88, 131108 (2006).
[60] S. J. Seltzer, Developments in alkali-metal atomic magnetometry, Ph.d. dissertation, Princeton
University, Princeton, NJ (2008).
[61] W. Happer, H. Tang, Spin-exchange shift and narrowing of magnetic resonance lines in optic-
ally pumped alkali vapors, Physical Review Letters 31, 273 (1973).
[62] G. Breit, I. I. Rabi, Measurement of nuclear spin, Physical Review 38, 2082 (1931).
[63] MOGLabs, External cavity diode laser manual, https://www.moglabs.com/products/
tunable-lasers/littrow-lasers (2014-2023).
[64] Vitawave, Technical description and instruction manual of an extended cavity diode laser ecdl-
7830r, http://vitawave.ru/eng/manuals/ (2014).
[65] G. Bao, D. Kanta, D. Antypas, S. Rochester, K. Jensen, W. Zhang, A. Wickenbrock,
D. Budker, All-optical spin locking in alkali-vapor magnetometers, Phys. Rev. A 105 (2022).
[66] G. Oelsner, V. Schultze, R. IJsselsteijn, F. Wittkämper, R. Stolz, Sources of heading errors
in optically pumped magnetometers operated in the earth’s magnetic field, Phys. Rev. A 99
(2019).
BIBLIOGRAPHY 149
[67] K. Jensen, V. M. Acosta, J. Higbie, M. P. Ledbetter, S. Rochester, D. Budker, Phys. Rev. A
79 (2009).
[68] S. J. Seltzer, P. J. Meares, M. V. Romalis, Phys. Rev. A 75 (2007).
[69] A. Ben-Kish, M. Romalis, Physical Review Letters 105 (2010).
[70] V. Acosta, M. P. Ledbetter, S. Rochester, D. Budker, D. F. J. Kimball, D. C. Hovde, W. Gaw-
lik, S. Pustelny, J. Zachorowski, V. V. Yashchuk, Phys. Rev. A 73 (2006).
[71] G. Bao, A. Wickenbrock, S. Rochester, W. Zhang, D. Budker, Physical Review Letters 120
(2018).
[72] G. Bison, N. Castagna, A.Hofer, P. Knowles, J.-L. Schenker, M. Kasprzak, H. Saudan, A. Weis,
A room temperature 19-channel magnetic field mapping device for cardiac signals, Appl. Phys.
Lett. 95, 173701 (2009).
[73] G. Lembke, S. N. Erné, H. Nowak, B. Menhorn, A. Pasquarelli, G. Bison, Optical multichannel
room temperature magnetic field imaging system for clinical application, Biomed. Opt. Express
5, 876–881 (2014).
[74] S. Groeger, G. Bison, A. Weis, Design and performance of laser-pumped cs-magnetometers
for the planned ucn edm experiment at psi, Journal of research of the National Institute of
Standards and Technology 110, 179 (2005).
[75] E. Aleksandrov, M. Balabas, S. Dmitriev, N. A. Dovator, A. I. Ivanov, M. I. Karuzin, V. N.
Kulyasov, A. S. Pazgalev, A. P. Serebrov, Quantum magnetometer for stabilization of the
neutron magnetic resonance, Tech. Phys. Lett. 32, 627–629 (2006).
[76] A. Borna, T. R. Carter, J. D. Goldberg, A. P. Colombo, Y.-Y. Jau, C. Berry, J. McKay,
J. Stephen, M. Weisend, P. D. D. Schwindt, A 20-channel magnetoencephalography system
based on optically pumped magnetometers, Phys. Med. Biol. 62, 8909 (2017).
[77] T. M. Ito, the nEdm Collaboration, Journal of Physics: Conference Series (IOP Publishing,
2007), vol. 69, p. 012037.
[78] A. Kraft, H.-C. Koch, M. Daum, W. Heil, T. Lauer, D. Neumann, A. Pazgalev, Y. Sobolev,
A. Weis, Development of a 3he magnetometer for a neutron electric dipole moment experiment,
EPJ Techniques and Instrumentation 1, 8 (2014).
[79] J. M. Higbie, , S. M. Rochester, B. Patton, R. Holzlöhner, D. B. Calia, D. Budker, Mag-
netometry with mesospheric sodium, Proceedings of the National Academy of Sciences 108,
3522–3525 (2011).
[80] B. Patton, O. O. Versolato, D. C. Hovde, E. Corsini, J. M. Higbie, D. Budker, A remotely
interrogated all-optical 87 rb magnetometer, Appl. Phys. Lett. 101, 083502 (2012).
150 BIBLIOGRAPHY
[81] F. P. Bustos, D. B. Calia, D. Budker, M. Centrone, J. Hellemeier, P. Hickson, R. Holzlöhner,
S. Rochester, Remote sensing of geomagnetic fields and atomic collisions in the mesosphere,
arXiv preprint arXiv:1802.04686 (2018).
[82] B. S. Mathur, H. Tang, W. Happer, Light shifts in the alkali atoms, Phys. Rev. 171, 11 (1968).
[83] C. Cohen-Tannoudji, J. Dupont-Roc, Experimental study of zeeman light shifts in weak mag-
netic fields, Phys. Rev. A 5, 968 (1972).
[84] F. L. Kien, P. Schneeweiss, A. Rauschenbeutel, Dynamical polarizability of atoms in arbitrary
light fields: general theory and application to cesium, The European Physical Journal D 67, 92
(2013).
[85] B. Patton, E. Zhivun, D. C. Hovde, D. Budker, All-optical vector atomic magnetometer, Phys.
Rev. Lett. 113, 013001 (2014).
[86] E. Zhivun, A. Wickenbrock, B. Patton, D. Budker, Alkali-vapor magnetic resonance driven by
fictitious radiofrequency fields, Appl. Phys. Lett. 105, 192406 (2014).
[87] E. Zhivun, A. Wickenbrock, J. Sudyka, B. Patton, S. Pustelny, D. Budker, Vector light shift
averaging in paraffin-coated alkali vapor cells, Opt. Express 24, 15383–15390 (2016).
[88] Z. Lin, X. Peng, W. Li, H. Wang, H. Guo, Magneto-optical double resonance driven by fictitious
fields, Opt. Express 25, 7668–7676 (2017).
[89] B. Julsgaard, Entanglement and quantum interactions with macroscopic gas samples, Ph.D.
thesis, Aarhus Universitetsforlag (2003).
[90] S. M. Rochester, Modeling nonlinear magneto-optical effects in atomic vapors, Ph.D. thesis,
UC Berkeley (2010).
[91] S. Coop, S. Palacios, P. Gomez, Y. N. M. de Escobar, T. Vanderbruggen, M. W. Mitchell,
Floquet theory for atomic light-shift engineering with near-resonant polychromatic fields, Opt.
Express 25, 32550–32559 (2017).
[92] V. V. Yashchuk, D. Budker, J. R. Davis, Laser frequency stabilization using linear magneto-
optics, Review of Scientific Instruments 71, 341–346 (2000).
[93] R. Zhang, D. Kanta, A. Wickenbrock, H. Guo, D. Budker, Heading-error-free optical atomic
magnetometry in the earth-field range, Phys. Rev. Lett. 130 (2023).
[94] G. Bao, A. Wickenbrock, S. Rochester, W. Zhang, D. Budker, Suppression of the nonlinear
zeeman effect and heading error in earth-field-range alkali-vapor magnetometers, Phys. Rev.
Lett. 120, 033202 (2018).
[95] G. Oelsner, V. Schultze, R. IJsselsteijn, F. Wittkämper, R. Stolz, Sources of heading errors
in optically pumped magnetometers operated in the earth’s magnetic field, Phys. Rev. A 99,
013420 (2019).
BIBLIOGRAPHY 151
[96] W. Lee, V. G. Lucivero, M. V. Romalis, M. E. Limes, E. L. Foley, T. W. Kornack, Heading
errors in all-optical alkali-metal-vapor magnetometers in geomagnetic fields, Phys. Rev. A 103,
063103 (2021).
[97] Y. Chang, Y.-H. Guo, S.-A. Wan, J. Qin, Asymmetric heading errors and its suppression in
atomic magnetometers (2021).
[98] K. Jensen, V. M. Acosta, J. M. Higbie, M. P. Ledbetter, S. M. Rochester, D. Budker, Cancel-
lation of nonlinear zeeman shifts with light shifts, Phys. Rev. A 79, 023406 (2009).
[99] A. Ben-Kish, M. V. Romalis, Dead-zone-free atomic magnetometry with simultaneous excita-
tion of orientation and alignment resonances, Phys. Rev. Lett. 105, 193601 (2010).
[100] S. Pustelny, M. Koczwara, L. Cincio, W. Gawlik, Tailoring quantum superpositions with lin-
early polarized amplitude-modulated light, Phys. Rev. A 83 (2011).
[101] E. G. The Editors of Encyclopaedia Britannica, anisotropy. URL: https://www.britannica.
com/science/anisotropy. Accessed on 2021-10-29.
[102] W. Happer, Optical pumping, Rev. Mod. Phys. 44, 169–249 (1972).
[103] F. Renzoni, E. Arimondo, Depopulation and repopulation pumping in coherent population
trapping, EPL 46 (1999).
[104] W. Chalupczak, R. M. Godun, S. Pustelny, W. Gawlik, Room temperature femtotesla radio-
frequency atomic magnetometer, Appl. Phys. Lett. 100, 242401 (2012).
[105] R. Gartman, W. Chalupczak, Amplitude-modulated indirect pumping of spin orientation in
low-density cesium vapor, Phys. Rev. A 91, 053419 (2015).
[106] D. Budker, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, M. Zolotorev, Sensitive magneto-
metry based on nonlinear magneto-optical rotation, Phys. Rev. A 62, 043403 (2000).
[107] V. Acosta, M. P. Ledbetter, S. M. Rochester, D. Budker, D. F. Jackson Kimball, D. C. Hovde,
W. Gawlik, S. Pustelny, J. Zachorowski, V. V. Yashchuk, Nonlinear magneto-optical rotation
with frequency-modulated light in the geophysical field range, Phys. Rev. A 73, 053404 (2006).
[108] E. Zhivun, Vector ac stark shift in 133Cs atomic magnetometers with antirelaxation coated
cells, Ph.D. thesis, University of California, Berkeley, Berkeley, CA, USA (2016).
[109] C. Troullinou, R. Jiménez-Martínez, J. Kong, V. G. Lucivero, M. W. Mitchell, Squeezed-light
enhancement and backaction evasion in a high sensitivity optically pumped magnetometer,
Phys. Rev. Lett. 127, 193601 (2021).
[110] V. G. Lucivero, P. Anielski, W. Gawlik, M. W. Mitchell, Shot-noise-limited magnetometer with
sub-picotesla sensitivity at room temperature, Review of Scientific Instruments 85, 113108
(2014).
152 BIBLIOGRAPHY
[111] F. Le Kien, P. Schneeweiss, A. Rauschenbeutel, Dynamical polarizability of atoms in arbitrary
light fields: general theory and application to cesium, Eur. Phys. J. D 67, 1–16 (2013).
[112] S. Rochester, Adm package, https://rochesterscientific.com/ADM.
[113] T. Wu, X. Peng, Z. Lin, H. Guo, A dead-zone free 4He atomic magnetometer with intensity-
modulated linearly polarized light and a liquid crystal polarization rotator, Rev. Sci. Instrum.
86, 103105 (2015).
[114] M. Horbatsch, E. A. Hessels, Shifts from a distant neighboring resonance, Phys. Rev. A 82,
052519 (2010).
[115] A. Marsman, M. Horbatsch, E. A. Hessels, Interference between two resonant transitions with
distinct initial and final states connected by radiative decay, Phys. Rev. A 96, 062111 (2017).
[116] D. Budker, M. G. Kozlov, Sensing: Equation one (2020).
[117] W. Li, M. Balabas, X. Peng, S. Pustelny, A. Wickenbrock, H. Guo, D. Budker, Characteriz-
ation of high-temperature performance of cesium vapor cells with anti-relaxation coating, J.
Appl. Phys. 121, 063104 (2017).
[118] K.-M. C. Fu, G. Z. Iwata, A. Wickenbrock, D. Budker, Sensitive magnetometry in challenging
environments, AVS Quantum Sci. 2, 044702 (2020).
[119] D. F. Jackson Kimball, J. Dudley, Y. Li, D. Patel, J. Valdez, Constraints on long-range spin-
gravity and monopole-dipole couplings of the proton, Phys. Rev. D 96, 075004 (2017).
[120] A. Papoyan, S. Shmavonyan, A. Khanbekyan, H. Azizbekyan, M. Movsisyan, G. Bao,
D. Kanta, A. Wickenbrock, D. Budker, Evidence for degenerate mirrorless lasing in alkali metal
vapor: forward beam magneto-optical experiment, Journal of Physics B: Atomic, Molecular
and Optical Physics 52 (2019).
[121] G. Nienhuis, F. Schuller, Magneto-optical effects of saturating light for arbitrary field direction,
Opt. Commun. 151, 40-45 (1998).
[122] M. Movsisyan, S. Shmavonyan, A. Papoyan, Amplification of radiation in atomic vapor in-
duced by a linearly polarized laser radiation, Cent. Eur. J. Phys. 9, 948-955 (2011).
[123] E. Gazazyan, G. Grigoryan, A. Papoyan, Amplification of a weak circularly polarized light
signal in a multilevel atomic medium, J. Contemp. Phys. (Arm. Acad. Sci.) 46, 145–149 (2011).
[124] A. Taichenachev, A. Tumaikin, V. Yudin, An atom in an elliptically polarized resonant field:
the exact steady-state solution for closed jg = j → je = j+1 transitions, J. Exp. Theor. Phys.
46 (2011).
[125] G. Nienhuis, A. Taichenachev, A. Tumaikin, V. Yudin, Steady state of atoms in a monochro-
matic elliptically polarized light field, Europhys. Lett. 44 (1998).
BIBLIOGRAPHY 153
[126] A. Sharma, N. D. Bhaskar, Y. Q. Lu, W. Happer, Continuous-wave mirrorless lasing in optic-
ally pumped atomic cs and rb vapors, Appl. Phys. Lett. 39 (1981).
[127] A. Akulshin, F. Bustos, D. Budker, Continuous-wave mirrorless lasing at 2.21 µm in sodium
vapors, Opt. Lett. 43, 5279-5282 (2018).
[128] https://twinleaf.com/shield/ms-2/ .
[129] B. Ghosh, P. Chakraborty, Reference Module in Materials Science and Materials Engineering
(Elsevier, 2022).
[130] F. Bustos, D. Calia, D. Budker, M. Centrone, J. Hellemeier, P. Hickson, R. Holzlöhner,
S.Rochester, Continuous-wave mirrorless lasing at 2.21 µm in sodium vapors, Nature Com-
mun. 9 (2018).
[131] S. Reynaud, M. Himbert, J. Dupont-Roc, H. H. Stroke, C. Cohen-Tannoudji, Experimental
evidence for compensation of doppler broadening by light shifts, Phys. Rev. Lett 42 (1979).
[132] W. S. Martins, H. L. D. de S. Cavalcante, T. P. de Silans, M. Oriá, M. Chevrollier, Two-beam
nonlinear kerr effect to stabilize laser frequency with sub-doppler resolution, Applied Optics 51
(2012).
[133] S. M. Rochester, D. S. Hsiung, D. Budker, R. Y. Chiao, D. F. Kimball, V. V. Yashchuk, Self-
rotation of resonant elliptically polarized light in collision-free rubidium vapor, Phys. Rev. A
63 (2001).