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Journal of Physics B Atomic and Molecular Physics

Two-photon spectroscopy. III. General strong-field aspects

Rainer Salomaa

Published Oct 21, 1977• 53 citations
Publisher

Abstract

For pt.II see ibid., vol.9, p.1221 (1976). Discusses the steady-state response of a three-level atom in two-photon absorption. The applied laser fields can have arbitrary intensities (within the rotating-wave approximation (RWA)). A general expression is derived for the population excited on the uppermost energy level. Two special cases which allow a trivial Doppler averaging are treated in detail. For an off-resonant intermediate state the two-photon spectrum is a simple Lorentzian showing a power shift and modest cross power broadening. The same features are retained when several off-resonance intermediate levels are assumed. For very strong laser fields the Doppler shifts can be neglected in the first approximation. The corresponding spectra are analysed.

DOI
10.1088/0022-3700/10/15/012

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For pt.II see ibid., vol.9, p.1221 (1976). Discusses the steady-state response of a three-level atom in two-photon absorption. The applied laser fields can have arbitrary intensities (within the rotating-wave approximation (RWA)). A general expression is derived for the population excited on the uppermost energy level. Two special cases which allow a trivial Doppler averaging are treated in detail. For an off-resonant intermediate state the two-photon spectrum is a simple Lorentzian showing a power shift and modest cross power broadening. The same features are retained when several off-resonance intermediate levels are assumed. For very strong laser fields the Doppler shifts can be neglected in the first approximation. The corresponding spectra are analysed.
For pt.I see ibid., vol.8, p.1795 (1975) where the theory of Doppler-free two-photon transitions was discussed for the case of one single laser frequency. Here, the effects of two different frequencies are considered. When the residual Doppler width of the two-photon transition is less than the natural linewidth, the previous results are essentially regained. For large residual Doppler shifts, the absorption profile is a power-broadened Lorentzian if the frequency of the probe field is larger than the frequency of the strong field. In the opposite case, the absorption peak splits because of the AC Stark effect. The high-intensity distortions are minimized when equal frequencies can be used. Power-dependent shifts do not ordinarily occur with a resonantly tuned intermediate level. Some modifications of the spectra are introduced by phase-disturbing collisions.
Discusses the theory of Doppler-free two-photon transitions. The theory is worked out for the case of one arbitrarily strong field and one weak field. The Doppler limit is assumed. These simplifications allow the authors to discuss the experimental manifestations of various parameters. They consider power shifts and broadening the number of resonant peaks and their positions. The general case of an off-resonant intermediate state and the single-frequency case with a resonant intermediate state are discussed. The former is found to lead to a power shift, whereas the latter is unshifted in a certain region. In addition the resonant case gives an enhancement without an appreciable increase in width.
A complete and detailed treatment of a three-level atom interacting with two near-resonant monochromatic fields is presented. It is assumed that the only damping mechanism is radiative damping, and that the atoms all have the same resonance frequencies, as is the case in an atomic beam. Detailed analytic solutions as well as numerical examples are determined from quantum-electrodynamic equations of motion. In addition, approximation techniques are presented which allow one to get accurate quantitative predictions for strong applied fields from simple rate-equation-like arguments. Absorption and emission spectra are determined, as are the transient and steady-state response of the atom. Two of the more interesting predictions are an emission spectrum containing up to seven components and a prediction of steady-state populations larger than 0.5 in the second excited state.
A theory of two-photon ionisation of atomic systems is developed incorporating into the treatment details of the laser spectrum. A general expression for the time-integrated two-photon ionisation probability is obtained together with general expressions for the induced shifts and widths of the atomic levels. These expressions are valid near resonance, when in the ordinary perturbation treatment the energy denominators vanish. The general expressions are then evaluated for the special case of a Lorentzian laser spectrum, when simple analytic results are obtained. The treatment is not confined to the two-level atom or the rotating-wave approximation.
This dissertation examines aspects of the interaction of multiple coherent light fields for the two-photon two-level model. In this model the interacting energy levels are not connected by an atomic dipole and a two-photon transition between them is necessary. We employ the density matrix formalism allowing easy comparison between the one- and two-photon two-level models. Significant differences are found due to dynamic Stark shifts and conjugate scattering off the pump-induced two-photon coherence. Averages over Doppler broadening are performed and the new upper-level relaxation mechanisms of decay to an intermediate nonresonant level and ionization from the upper state are included. The new relaxation mechanisms, introduced to the theory to better model experiments, are similar except that ionization is intensity dependent. They cause the resulting probe absorption spectra to become more complex and in general asymmetric. Doppler broadening is also important in experiments using gases. We analytically average over a Lorentzian velocity distribution for both co- and counterpropagating pump and probe beams. For copropagating fields the results are similar to those for the one-photon case averaged over inhomogeneous broadening, whereas counterpropagating pump and probe fields yield the so-called Doppler-free configuration that is normally only modelled to third order in the pump amplitude. We consider the pump field amplitude to all orders and find that as long as the width of the Doppler velocity distribution is significantly larger than the two-photon Rabi frequency the results are Doppler-free. The final part of the dissertation treats the question of two-photon squeezed states. This requires quantized sidemodes. Squeezed states are minimum uncertainty states with unequal variances in the two quadratures of the electromagnetic field amplitude. One way to generate these states is via multiwave mixing and we present here the first calculation for nondegenerate two-photon multiwave mixing as it applies to squeezed states. We find that in general two-photon squeezed states require lower intensities and detuning than those predicted by the one-photon model.
Resonant two-photon absorption is discussed for the case in which the exciting field may be intense enough and near enough to resonance to cause appreciable saturation. A single pair of levels in an otherwise quite general atom is assumed to satisfy the appropriate resonance condition, and equations are derived governing the time evolution of the associated 2 \ifmmode\times\else\texttimes\fi{} 2 submatrix of the full atomic density matrix. Atomic relaxation is treated explicitly. The analysis, which treats both inversion-symmetric and inversion-nonsymmetric cases within the same general formalism, is developed systematically, the necessary approximations receiving careful discussion. The results are analogous to those for the corresponding one-photon process, but with a shift in the resonance frequency proportional to the intensity of the exciting field. This frequency shift is not identical to the one calculated by time-independent (second-order) perturbation theory, and becomes large when the energy of an intermediate state of the atom approaches the mean of the energies of the active states.