Mode coupling for precise measurements of the electronic g-factor of hydrogen-like ions in Penning traps
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Abstract
The g-factor is a constant which connects the magnetic moment $\vec{\mu}$ of a
charged particle, of charge q and mass m, with its angular momentum $\vec{J}$. Thus, the
magnetic moment can be writen $ \vec{\mu}_J=g_J\frac{q}{2m}\vec{J}$. The g-factor for a
free particle of spin s=1/2 should take the value g=2. But due to quantum
electro-dynamical effects it deviates from this value by a small amount, the so called
g-factor anomaly $a_e$, which is of the order of $10^{-3}$ for the free electron. This
deviation is even bigger if the electron is exposed to high electric fields. Therefore
highly charged ions, where electric field strength gets values on the order of
$10^{13}-10^{16}$V/cm at the position of the bound electron, are an interesting field of
investigations to test QED-calculations. In previous experiments [H"aff00,Ver04] using a
single hydrogen-like ion confined in a Penning trap an accuracy of few parts in $10^{-9}$
was obtained.
In the present
work a new method for precise measurement of magnetic the electronic
g-factor of hydrogen-like ions is discussed. Due to the unavoidable magnetic field
inhomogeneity in a Penning trap, a very important contribution to the systematic
uncertainty in the previous measurements arose from the elevated energy of the ion
required for the measurement of its motional frequencies. Then it was necessary to
extrapolate the result to vanishing energies. In the new method the energy in the
cyclotron degree of freedom is reduced to the minimum attainable energy. This method
consist in measuring the reduced cyclotron frequency $\nu_{+}$ indirectly by coupling the
axial to the reduced cyclotron motion by irradiation of the radio frequency
$\nu_{coup}=\nu_{+}-\nu_{ax}+\delta$ where $\delta$ is, in principle, an unknown detuning
that can be obtained from the knowledge of the coupling process. Then the only unknown
parameter is the desired value of $\nu_+$. As a test, a measurement with,
for simplicity,
artificially increased axial energy was performed yielding the result
$g_{exp}=2.000~047~020~8(24)(44)$. This is in perfect agreement with both the theoretical
result $g_{theo}=2.000~047~020~2(6)$ and the previous experimental result
$g_{exp1}=2.000~047~025~4(15)(44).$
In the experimental results the second error-bar is due to the uncertainty in the
accepted value for the electron's mass. Thus, with the new method a higher accuracy in
the g-factor could lead by comparison to the theoretical value to an improved value of
the electron's mass.
[H"af00] H. H"affner et al., Phys. Rev. Lett. 85 (2000) 5308
[Ver04] J. Verd\'u et al., Phys. Rev. Lett. 92 (2004) 093002-1