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arXiv:quant-ph/0403202 · Physical Review A

Relativistic and radiative corrections to the Mollow spectrum

Jörg Evers, Ulrich D. Jentschura, Christoph H. Keitel

Published Dec 28, 2004• 18 citations
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Abstract

The incoherent inelastic part of the resonance fluorescence spectrum of a laser-driven atom is known as the Mollow spectrum [B. R. Mollow, Phys. Rev. 188, 1969 (1969)]. Starting from this level of description, we discuss theoretical foundations of high-precision spectroscopy using the resonance fluorescence light of strongly laser-driven atoms. Specifically, we evaluate the leading relativistic and radiative corrections to the Mollow spectrum, up to the relative orders of ${(Z\ensuremath{\alpha})}^{2}$ and $\ensuremath{\alpha}{(Z\ensuremath{\alpha})}^{2}$, respectively, and Bloch-Siegert shifts as well as stimulated radiative corrections involving off-resonant virtual states. Complete results are provided for the hydrogen $1S\text{\ensuremath{-}}2{P}_{1∕2}$ and $1S\text{\ensuremath{-}}2{P}_{3∕2}$ transitions; these include all relevant correction terms up to the specified order of approximation and could directly be compared to experimental data. As an application, the outcome of such experiments would allow for a sensitive test of the validity of the dressed-state basis as the natural description of the combined atom-laser system.

DOI
10.1103/physreva.70.062111

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Physical Review A•Oct 1, 1976•47 citations
The incoherent inelastic part of the resonance fluorescence spectrum of a laser-driven atom is known as the Mollow spectrum [B. R. Mollow, Phys. Rev. 188, 1969 (1969)]. Starting from this level of description, we discuss theoretical foundations of high-precision spectroscopy using the resonance fluorescence light of strongly laser-driven atoms. Specifically, we evaluate the leading relativistic and radiative corrections to the Mollow spectrum, up to the relative orders of (Z\ensuremathα)2{(Z\ensuremath{\alpha})}^{2}(Z\ensuremathα)2 and \ensuremathα(Z\ensuremathα)2\ensuremath{\alpha}{(Z\ensuremath{\alpha})}^{2}\ensuremathα(Z\ensuremathα)2, respectively, and Bloch-Siegert shifts as well as stimulated radiative corrections involving off-resonant virtual states. Complete results are provided for the hydrogen 1S\ensuremath-2P1∕21S\text{\ensuremath{-}}2{P}_{1∕2}1S\ensuremath-2P1∕2​ and 1S\ensuremath-2P3∕21S\text{\ensuremath{-}}2{P}_{3∕2}1S\ensuremath-2P3∕2​ transitions; these include all relevant correction terms up to the specified order of approximation and could directly be compared to experimental data. As an application, the outcome of such experiments would allow for a sensitive test of the validity of the dressed-state basis as the natural description of the combined atom-laser system.
This paper is concerned with two rather basic phenomena: the incoherent fluorescence spectrum of an atom driven by an intense laser field and the coupling of the atom to the (empty) modes of the radiation field. The sum of the many-photon processes gives rise to the inelastic part of the atomic fluorescence, which, for a two-level system, has a well-known characteristic three-peak structure known as the Mollow spectrum. From a theoretical point of view, the Mollow spectrum finds a natural interpretation in terms of transitions among laser-dressed states which are the energy eigenstates of a second-quantized two-level system strongly coupled to a driving laser field. As recently shown, the quasi-energies of the laser-dressed states receive radiative corrections which are nontrivially different from the results which one would expect from an investigation of the coupling of the bare states to the vacuum modes. In this article, we briefly review the basic elements required for the analysis of the dynamic radiative corrections, and we generalize the treatment of the radiative corrections to the incoherent part of the steady-state fluorescence to a three-level system consisting of 1S, 3P and 2S states.
We discuss radiative corrections to an atomic two-level system subject to an intense driving laser field. It is shown that the Lamb shift of the laser-dressed states, which are the natural state basis of the combined atom-laser system, cannot be explained in terms of the Lamb shift received by the atomic bare states which is usually observed in spectroscopic experiments. In the final part, we propose an experimental scheme to measure these corrections based on the incoherent resonance fluorescence spectrum of the driven atom.
We consider the resonance fluorescence of a two-level atom in a vacuum with a high intensity driving field, such that the Rabi frequency of the field is non-negligible up to frequencies comparable to the transition frequency of the atom. At these high intensities, the conventional rotating wave approximation cannot be applied to the atom-laser Hamiltonian and the modal densities of the vacuum at the dressed transition frequencies differ significantly from each other. On treating the counter-rotating part of the Hamiltonian as a perturbation and carrying out the Weisskopf-Wigner approximation in the dressed states basis, we find that the high driving intensity leads to significant deviations from the traditional Mollow resonance fluorescence spectrum. In particular we find unequally intense Mollow side-bands and that the generation of higher order triplets and harmonics is substantially enhanced by the high vacuum modal density at the corresponding high resonance frequencies.
An experiment was performed recently [Y. Zhu, A. Lezama, Q. Wu, and T. W. Mossberg (unpublished)] in which a two-level atom of transition frequency \ensuremath{\Omega} was placed in a bichromatic laser field with frequencies \ensuremath{\Omega}\ifmmode\pm\else\textpm\fi{}\ensuremath{\delta}, of sufficient intensity to produce dynamic Stark splitting. The observed fluorescence spectrum differed qualitatively in almost all its details from the well-known Mollow triplet [B. R. Mollow, Phys. Rev. 188, 1969 (1969)]. In this article, we use a dressed-atom model to explain the observed results.
We have investigated the resonance fluorescence spectrum from two identical atoms driven by a strong standing wave field. We found interesting modifications of the Mollow spectrum, caused by interferences between the two atoms, and in particular a succession of dark and bright states as one of the atoms is moved to successive nodes of the field. Large enough separation between the atoms leads to the disappearance of these interferences, and the atoms act then as independent radiators, the fluorescence spectrum becoming then independent of the interatomic distance. From the point of view advocated by E. Wolf,1 one can interpret these modifications as caused by source correlations. For instance, we obtain new sidebands which are altered as the separation between atoms is modified. When these sidebands eventually cross, the fluorescence spectrum displays interesting properties such as “dark” and “bright” states. These investigations lay the foundations for a discussion of the influence of particle statistics on spectral properties of atoms cooled and trapped in optical molasses. The next step in this direction will include the effects of translational motion on these spectra.2 This program will eventually lead to a full-fledged field theoretical description of the spectral and translational correlation properties of an ensemble of cooled and trapped atoms.
The calculation is given of the spectral distribution of resonance fluorescence from an atom that is driven near a j=12j=\frac{1}{2}j=21​ to j=12j=\frac{1}{2}j=21​ transition by a monochromatic, xxx-polarized electric field and that is subject to radiation damping only. In accordance with Mollow's results for a two-level system, the power spectrum of the scattered radiation corresponding to the xxx component of the dipole moment contains a coherent part for all driving-field intensities. The coherent part is absent in the fluorescence radiation corresponding to the yyy and zzz components of the dipole moment. For weak driving fields all power scattered by these latter components is present in a narrow Lorentzian. Consistent with energy conservation, the width of the Lorentzian corresponds to the probability of excitation of the atom out of its ground level due to the presence of the driving field. For strong driving fields the power spectra corresponding to the xxx component and to the yyy or zzz component mainly differ in the heights and widths of their central incoherent band and their Rabi-precession-frequency-shifted side bands.