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Metrologia

Effect of a Non-Resonant Irradiation on Atomic Energy Levels–Application to Light-Shifts in Two-Photon Spectroscopy and to Perturbation of Rydberg States

Claude N. Cohen-Tannoudji

Published Jul 1, 1977• 44 citations
Publisher

Abstract

Atomic or molecular energy levels are shifted by a non-resonant electromagnetic irradiation. These so-called "light-shifts" must be carefully taken into account in high resolution spectroscopy experiments since they perturb the atomic Bohr frequencies which one wants to measure or to use as frequency standards. In this paper, we present a brief survey of various types of light-shifts corresponding to different physical situations. We use a dressed atom approach allowing simple physical discussions and simple calculations. Some new suggestions are made.

DOI
10.1088/0026-1394/13/3/016

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Effect of a Non Resonant Irradiation on Atomic Energy Levels- Application to Light Shifts in Two-Photon Spectroscopy and to Perturbation of Rydberg States

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Atomic or molecular energy levels are shifted by a non-resonant electromagnetic irradiation. These so-called "light-shifts" must be carefully taken into account in high resolution spectroscopy experiments since they perturb the atomic Bohr frequencies which one wants to measure or to use as frequency standards. In this paper, we present a brief survey of various types of light-shifts corresponding to different physical situations. We use a dressed atom approach allowing simple physical discussions and simple calculations. Some new suggestions are made.
An atomic electron, in the vacuum of photons, can emit and reabsorb photons of any frequency. Such processes give rise to well known radiative corrections (Lamb—shift, g — 2, natural width of excited states...). We review in this paper similar effects which can be induced by the interaction of an atomic electron with an incident electromagnetic wave, and we try to get some physical insight in radiative processes by comparing these two types of « spontaneous » and « induced » effects. Optical pumping methods have been very useful for studying how atoms are perturbed by optical or RF photons. The Zeeman sublevels of an atomic ground state are broadened and shifted by a resonant or quasiresonant optical irradiation. The corresponding light-shifts (or « Lamp-shifts » as A. Kastler called them) are due to virtual absorptions and reemissions of incident photons. Another example of an induced effect is the modification of the magnetic properties of an atomic level due to a high frequency non resonant RF irradiation. The Larmor precession is slowed down and becomes anisotropic. The two previous examples correspond to two situations leading to simple calculations. For a resonant or quasiresonant excitation, a single atomic transition needs to be considered. Non perturbative treatments can be given which include the effect of the atom field coupling to all orders. For example, the « dressed atom » approach allows a unified treatment of various effects such as light shifts, Autler-Townes doublets resonance fluorescence triplets... On the other hand, for a high frequency irradiation, all atomic transitions must be included. It is possible in this case to derive an effective hamiltonian describing, to lowest order in the fine structure constant α, how electronic energy levels are perturbed in the presence of N photons ω. The remaining part of the paper is devoted to the analysis of such a situation. The terms of the effective hamiltonian proportional to the number N of incident photons correspond to the induced effects. They can be fully interpreted in semiclassical terms and they describe how the slow motion of the electron in the external static fields is modified by the high frequency vibration in the incident monochromatic wave. For example, the vibrating electron averages the static potential over a finite length, the angular oscillation of the spin reduces the effective magnetic moment. The remaining terms, which are independent of N, exist even in the vacuum of photons (N = 0) and describe the contribution of the mode ω to spontaneous radiative corrections. It is possible to split them into two parts. The first one has the same structure as the N-dependent terms, with N replaced by 1/2. It describes the corrections due to the vibration of the electron in vacuum fluctuations which have a spectral power density ħω/2 per mode. The second part represents the effect of radiation reaction, i.e. the interaction of the electron with its self field. Such an interpretation is confirmed by a Heisenberg equations approach where the total force acting upon the electron is split into two physical (i.e. hermitian) parts representing respectively the effect of the vacuum « free » field and the effect of the « source » field produced by the electron itself. These considerations are finally applied to the problem of the positive sign of g - 2. The g-factor can be written as 2 ωL/ωc, where ω L is the Larmor frequency of the spin and ωc the cyclotron frequency of the charge for an electron in a weak static field B 0. In the absence of radiative corrections, g = 2, and ω L = ωc. To lowest order in 1/c (order zero), radiation reaction slows down ωc but not ωL (a charge is more coupled to its self field than a magnetic moment). The effect of vacuum fluctuations appears only at order 2 in 1/c and corresponds to relativistic and magnetic effects which slow down both ωL and ωc. Therefore, it appears that, in the non relativistic domain, the main effect is a reduction of ωc by radiation reaction and this explains why 2 ωL/ωc becomes larger than 2. A complete relativistic calculation (to all orders in 1/ c, but to lowest order in α) does not change this conclusion since the contribution of non relativistic modes (ħω < mc2) is predominant in the integral giving g — 2.
We present a non-perturbative numerical technique for calculating strong light shifts in atoms under the influence of multiple optical fields with arbitrary polarization. We confirm our technique experimentally by performing spectroscopy of a cloud of cold 87^{87}87Rb atoms subjected to ∼\sim∼ kW/cm2^22 intensities of light at 1560.492 nm simultaneous with 1529.269 nm or 1529.282 nm. In these conditions the excited state resonances at 1529.26 nm and 1529.36 nm induce strong level mixing and the shifts are highly nonlinear. By absorption spectroscopy, we observe that the induced shifts of the 5P3/2 hyperfine Zeeman sublevels agree well with our theoretical predictions.. We propose the application of our theory and experiment to accurate measurements of excited-state electric-dipole matrix elements.
Optical trapping of atoms employs high-intensity fields that necessarily alter atomic level structure. The calculation of light shifts by perturbation theory fails for scenarios that arise, for example, when the trapping light is near an excited-state transition or for polychromatic fields. We show here that non-perturbative methods based on Floquet’s theorem elegantly handle such scenarios. We compare our calculation to precision absorption spectroscopy on cold 87Rb atoms in a bichromatic optical dipole trap at 1560 + 1529 nm. Proximity to excited-state resonances induces highly nonlinear level shifts, providing a strong test of theory. The good theory-experiment agreement suggests a new method for accurate measurements of excited-state electric-dipole matrix elements and a precision tool for engineering custom atomic level structures.
The review describes atomic transitions where two or more photons are simultaneously absorbed by an atom or a molecule (without any intermediate resonant level). The authors show theoretically and experimentally the possibilities of these transitions for numerous high-resolution spectroscopic studies. The Doppler broadening is suppressed in two-photon transitions if the atoms are irradiated by two laser waves travelling in opposite directions, because the Doppler shift in one wave cancels that in the other. A generalization is made for multiphotonic transitions. A typical experimental set-up with a CW dye laser is described in detail, but the possibilities of other light sources are also discussed.
Atoms in weak broadband quasi-resonant fields - light shifts atoms in linear superpositions of Zeeman sublevels-Zeeman coherences atoms in strong radiofrequency fields - the dressed-atom approach in the RF domain atoms in intense resonant laser beams - the dressed-atom approach in the optical domain the radiative cascade of the dressed atom - photon correlations and quantum jumps atoms in high-frequency fields or in the vacuum field - simple physical pictures for radiative corrections atomic motion in laser light - laser cooling and trapping.