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Journal of the Optical Society of America B

Two-photon free-induction decay

Margaret F. Marshman, Peter M. Farrell, W. R. MacGillivray, M. C. Standage

Published Apr 1, 1986• 2 citations
Publisher

Abstract

Density matrix formalism is used to model two-photon free-induction decay from a ladder-type, three-level scheme with a resonant intermediate state. Counterpropagating light beams are assumed. Only the upper transition is frequency shifted out of resonance with the injected field. Results indicate departure from single-photon behavior; in particular, there is a π phase shift in the oscillations when the Rabi frequencies associated with each transition are comparable in magnitude. The theory is applied to the 32S1/2 (F = 1) − 32P1/2 − 42D3/2 excitation scheme of sodium.

DOI
10.1364/josab.3.000607

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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.

•Jan 1, 1986•0 citations
Density matrix formalism is used to model two-photon free-induction decay from a ladder-type, three-level scheme with a resonant intermediate state. Counterpropagating light beams are assumed. Only the upper transition is frequency shifted out of resonance with the injected field. Results indicate departure from single-photon behavior; in particular, there is a π phase shift in the oscillations when the Rabi frequencies associated with each transition are comparable in magnitude. The theory is applied to the 32S1/2 (F = 1) − 32P1/2 − 42D3/2 excitation scheme of sodium.
A theoretical study is made of the steady-state intensity and squeezing properties of the fluorescent light from a three-level-atom system in a ladder configuration, which is subject to spontaneous-emission decay to the electromagnetic-field vacuum. Two cases are examined: in the equispaced (ES) level case, the two atomic transition frequencies are nearly equal and two photon absorption processes occur from a single coherent laser field that couples both transitions and has a small detuning from each; in the non- equispaced (NES) level case, the atomic transition frequencies are rather different and two photon absorption processes occur from two coherent laser fields, each coupled to a single transition and in near resonance with it. In both cases, a situation of small two-photon detuning occurs. Optical Bloch equations for the atomic density matrix in rotating frames are given and matrix expressions for determining the steady-state populations and coherences are obtained. Analytic expressions are also given for special cases. The fluorescent intensity is obtained from the populations of the intermediate and upper states, with the normally ordered variance (NOV) for quadrature components of the fluorescent field involving, in addition, the atomic coherences. The fluorescent intensities are shown graphically as a function of two-photon detuning for a variety of one-photon detunings and Rabi frequencies for the ES and NES cases.In both cases, the fluorescent intensities show the well known resonances from the upper state at zero two-photon detuning and from the intermediate state at zero one-photon detuning for the lower transition. However, further resonances are also found. The intermediate-state intensity displays a resonance at zero two-photon detuning. Also, the upper-state intensity shows a resonance at zero one-photon detuning for the upper transition, but only for the NES case; evidently, the transfer rate from upper to lower transition coherences destroys this resonance for the ES case. The time-averaged NOV is also shown graphically as a function of two-photon detuning for a variety of one-photon detunings and Rabi frequencies for the ES and NES cases. For the ES case, the quadrature frequency is chosen as the single laser frequency and for the NES case, it is chosen as either the average of the two laser frequencies or the lower transition laser frequency. Squeezing occurs near zero two-photon detuning for both ES and NES cases, though only for the quadrature frequency equal to the average laser frequency in the latter case. It also occurs near zero detuning for the lower transition for both cases, though only for the quadrature frequency equal to the lower transition laser frequency in the NES case. The squeezing minima show a splitting effect for large Rabi frequencies corresponding to two-photon or one-photon Rabi splitting of the dressed atom levels for the situation near two-photon or one-photon resonance, respectively. The large squeezing near two-photon resonance for moderate Rabi frequencies and large one-photon detuning corresponds to essentially pure three-level squeezing, while that near one-photon resonance for weak Rabi frequencies involves two-level squeezing.
Rabi oscillations reflect the existence of a coherent superposition of two molecular eigenstates during the interaction of molecules with a strong, resonant electromagnetic field. This phenomenon is not restricted to two-level systems. We demonstrate its occurrence in a three-level system where the radiation induces a two-photon absorption. The intermediate level is (slightly) detuned from resonance. For a theoretical description the dressed-state picture is adopted. The two-photon excitation is performed either by using one laser, yielding a one-color two-photon process, or by using two counterpropagating, spatially overlapping laser beams, yielding a two-color two-photon process. Rabi oscillations are produced by varying the fluence, i.e., by changing either the laser power or the molecule-laser interaction time. For the one-color case the two-photon transition dipole moment for an SF6{\mathrm{SF}}_{6}SF6​ transition has been determined from the relation between laser fluence and number of Rabi oscillations. For the two-color cases the detuning with the intermediate level is varied by tuning the respective laser frequencies while keeping the sum frequency constant. A dramatic change in the degree of excitation as a function of the intermediate detuning has been observed. \textcopyright{} 1996 The American Physical Society.
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.
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.