Double–Resonance Factor Measurements
by Quantum Jump Spectroscopy
With the advent of high-precision frequency combs that can bridge large frequency intervals, new possibilities have opened up for the laser spectroscopy of atomic transitions. Here, we show that laser spectroscopic techniques can also be used to determine the ground-state factor of a bound electron: Our proposal is based on a double-resonance experiment, where the spin state of a ground-state electron is constantly being read out by laser excitation to the atomic L shell, while the spin flip transitions are being induced simultaneously by a resonant microwave field, leading to a detection of the quantum jumps between the ground-state Zeeman sublevels. The magnetic moments of electrons in light hydrogen-like ions could thus be measured with advanced laser technology. Corresponding theoretical predictions are also presented.
Recently, there has been a dramatic progress in the precision laser spectroscopy of atomic transitions, with uncertainties on the order of for two-photon transitions in hydrogen  and even for ultra-violet (UV) electric-quadrupole transitions in the mercury ion  (statistical effects led to a limitation on the order of for the evaluation of the latter measurement). By contrast, microwave measurements of the bound-electron factor in hydrogen-like ions [3, 4, 5] have been restricted to a comparatively low level of accuracy, namely in the range of , where the most accurate values have been obtained for bound electrons in hydrogen-like carbon and oxygen (for an introductory reviews on bound-electron factors and various related experimental as well as theoretical techniques, see [6, 7, 8]). It is tempting to ask if the accuracy gap between the two categories of measurements might leave room for improvement of the factor determination, as both measurements investigate the properties of bound electrons. More specifically, the question arises if the additional “channels” provided by laser excitation among the discrete states of the bound system, and the additional possibilites for the laser cooling of ions (following the original ideas formulated in Refs. [9, 10]), can be used as auxiliary devices to improve the accuracy of the factor determination via quantum jump spectroscopy. We also note that double-resonance techniques for stored ions have already been shown to open up attractive experimental possibilities with respect to hyperfine transitions as well as electronic and nuclear factors [11, 12, 13].
The bound-electron (Landé) factor for an electron bound in an ion with a spinless nucleus is the proportionality constant relating the Zeeman energy in the magnetic field (directed along the axis) and the Larmor precession frequency to the magnetic spin projection onto that same axis. In natural units (), we have
where is the Bohr magneton, expressed in terms of the electron charge and the electron mass . Deviations from the Dirac–Breit  prediction are due to quantum electrodynamic (QED), nuclear and other effects.
The purpose of this note is to answer the following question: “Is it possible to apply ultra-high precision atomic laser spectroscopy to bound-electron factor measurements?” Our answer will be affirmative.
In contrast to the continuous Stern–Gerlach effect , and complementing a recent proposal for a high-precision measurement of the factor in a highly charged ion , the current proposal is based on a Penning trap and will be studied here in conjunction with the hydrogen-like helium ion , which seems to be well suited for an experimental realization in the near future. The magnetic field strength in the Penning trap can be calibrated via a measurement of the cyclotron frequency of the trapped ion, and the Landé -factor for the bound electron is determined by the relation
where the electron-ion mass ratio is an external input parameter and is the nuclear charge number.
In a Penning trap, a single ion is confined by a strong homogeneous magnetic field in the plane perpendicular to the magnetic field lines and by a harmonic electrostatic potential in the direction parallel to the field lines . The three eigenmotions of a stored ion are the trap-modified cyclotron motion (frequency ), the axial motion (frequency ), and the magnetron motion (frequency ). The free-space cyclotron frequency of an ion with charge can be determined from the three eigenfrequencies by 
Experimentally, the eigenfrequencies of the stored ion can be measured by non-destructive detection of the image currents which are induced in the trap electrodes by the ion motion. Measurements on the level of are achieved [19, 20, 21] by careful anharmonicity compensation of the electrostatic trapping potential, optimizing the homogeneity and temporal stability of the magnetic field close to the Penning trap’s center, and cooling the motional amplitudes of the single trapped ion to low temperatures. Further optimization of experimental techniques should make it possible to reach an accuracy of (better than) . In our proposed factor measurement, advantage could be taken of the fact that two frequencies of the same particle are measured simultaneously (cyclotron vs. spin-flip), whereas in a mass measurement, the cyclotron frequencies of two different particles have to be determined.
In the following, we concentrate on the system, where the total angular momentum is equal to the total electron angular momentum . In the presence of the magnetic field in the Penning trap, the Zeeman splitting of the electronic ground state of the ion is given by Eq. (Double–Resonance Factor Measurements by Quantum Jump Spectroscopy). Correspondingly, the excited state is split into four Zeeman sublevels , with . The Landé factor can be obtained easily according to a modified Dirac equation which forms a basis for bound-state analysis 
where we take into account the leading QED and relativistic contributions and use for the relativistic term of order . Suppose now that one single ion in the Penning trap is prepared in the Zeeman sublevel of the electronic ground state . Narrow-band ultraviolet (UV) electromagnetic radiation with polarization and angular frequency drives the Lyman- transition , see Fig. 1. This is a closed cycle because decay by emission of a fluorescence photon is only possible to the initial state (if one ignores one-photon ionization into the continuum). Due to the short lifetime of the upper state , the fluorescence intensity of under saturation conditions makes it possible to detect a single trapped ion with high sensitivity . The Rabi frequency of the UV transition is given by , where is measured in units of .
Building a continuous-wave (cw) laser that operates at the Lyman- transition of , with a wavelength of , is certainly not a trivial task. However, a cw laser operating at the corresponding Lyman- transition for atomic hydrogen, with , has already been demonstrated . A possible pathway is higher-harmonic generation which has recently been described in Ref.  and leads to a pulsed UV excitation with a high repetition rate and a potentially discontinuous probing of the Zeeman ground-state sublevel. Note that a discontinuous probing of the ground-state sublevels does not inhibit the quantum jump detection scheme as outlined below. Groundwork for a detailed analysis of the dynamics of a pulsed excitation scheme in a very much analogous atomic system has recently been laid in Ref. ; in principle, one only has to ensure that the light intensity of the Lyman- source during a single laser pulse is sufficient to discern the presence or absence of fluorescence.
During excitation of the transition and detection of the corresponding fluorescence photons, a microwave field with frequency in resonance with the spinflip transition in the electronic ground state is irradiated on the single trapped ion (Fig. 1). Successful excitation of the spinflip transition results in an instantaneous stop of the fluorescence intensity, because the lower Zeeman level is not excited by the narrow-band Lyman- radiation. A quantum jump is thus directly observed with essentially 100% detection efficiency . A second spinflip restores the fluorescence intensity. A plot of the quantum jump rate versus excitation microwave frequency at yields the resonance spectrum of the Larmor precession frequency. The cyclotron frequency , which also enters Eq. (Double–Resonance Factor Measurements by Quantum Jump Spectroscopy), is measured simultaneously by non-destructive electronic detection of the image currents induced in the trap electrodes .
|Nuclear charge radius||frequency|
|factor contribution ()|
|Finite nuclear size|
While the UV laser light drives the transition , the absorption of an additional photon can take place, resulting in ionization through the channel , where are electronic continuum states (see Fig. 2). For this process, we obtain an ionization cross section of . This leads to an ionization rate of , where is the laser intensity measured in units of , corresponding to a depletion of the state with a time-dependent exponential . Here, we use the notational conventions of Ref. . In principle, since the ionization rate is proportional to the laser intensity, whereas the Rabi frequency is only proportional to its square root, it might be preferable to work at reduced laser intensities in order to increase the lifetime of the hydrogen-like charge state of the ion. However, at an incident typical laser intensity of , the lifetime of is against ionization, and this has to be compared to a Rabi frequency of . The ion has about Rabi cycles before it is ionized, and so the ionization channel does not limit the feasibility of the measurement at all.
Finally, we take notice of the ac Stark shift of the – transition due to non-resonant levels, which is , with given in . The ac Stark shift of the UV transition affects the two ground-state Zeeman levels slightly differently, but the relative shift of the spinflip transition frequency between them is a fourth-order effect and is suppressed with respect to the ac Stark shift by a factor of and thus negligible on the level in units of the microwave frequency, at a typical laser intensity of . Also, experimental procedures for previous factor measurements [4, 5] have included an extrapolation to zero intensity of the microwave fields, and the same can be done with the driving UV laser field in the proposed measurement scheme. Alternatively, one can perform the excitation of the Lyman- and the spinflip transitions in a time sequence, thus eliminating any systematic uncertainties of the factor determination related to the intensity of the UV laser light.
In order to lay a theoretical ground for the evaluation of the measurement, we present theoretical predictions for the transition frequencies and for the factor. According to the recent compilations [31, 32, 33], the ground state Lamb shift values are for the “old” value  of the nuclear charge radius and for the “new” value of the charge radius , which is . (The uncertainty estimate for the “old” value has given rise to discussions, see Ref. .) For the states, the Lamb shift is independent of the current uncertainty in the nuclear radius on the level of one kHz and reads (see Tables 3 and 4 of Ref. ). Using a proper definition of the Lamb shift as given, e.g., in Eq. (10) of Ref. , we then obtain the transition frequencies as given in Table 1.
The ground-state factor can be described naturally in an intertwined expansion in the QED loop expansion parameter and the electron-nucleus interaction strength . We follow the conventions of Ref.  and take into account all corrections that are relevant at the level of accuracy (see Table 2). The entry for the “ one-loop QED” is just the Schwinger term and it carries the largest theoretical uncertainty, because of the uncertainty in the fine-structure constant itself [36, 37].
In this note, we attempt to formulate a proposal by which ultra-accurate factor measurements in hydrogen-like systems with low nuclear charge number might be accessible to laser spectroscopic techniques. Within the next decade, it is realistic to assume that the necessary requirements for experiments will be provided that fully profit from both the electric coupling of the electron (via optical electric-dipole allowed Lyman- transitions) and from the magnetic coupling of the electron (via spin-flip transitions among the Zeeman sublevels of the ground state, see Fig. 1). The accuracy of the measurement of the free-electron factor has recently been increased to a level of [38, 29]. Within our proposed setup, an accuracy on the level of seems to be entirely realistic for bound-electron factors in hydrogen-like ions with a low nuclear charge number. It might be very beneficial if the extremely impressive, ultra-precise new measurement of the free-electron factor  could be supplemented by a potentially equally accurate measurement of the bound-electron factor in the near future, as an alternative determination of the fine-structure constant is urgently needed in conjunction with an improved determination of the electron mass.
The authors acknowledge helpful discussions with V. A. Yerokhin and K. Pachucki, and support from DFG (Heisenberg program) as well as from GSI (contract HD–JENT).
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