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Physical Review A

Nonlinear interactions of multilevel atoms with a near-resonant standing wave

T. Okane, Robert E Scholten, Peter M. Farrell, M. R. Walkiewicz

Published Jun 1, 1999• 7 citations
Publisher

Abstract

Using a semiclassical density matrix formalism we have calculated the behavior of multilevel atoms interacting with a standing wave field, and show how complex nonlinear phenomena, including multiphoton effects, combine to produce saturation spectra as observed in experiments. We consider both 20-level sodium and 24-level rubidium models, contrasting these with a simple 2-level case. The influence of parameters such as atomic trajectory and the time the atom remains in the beam are shown to have a critical effect on the line shape of these resonances and the emission/absorption processes. Stable oscillations in the excited state populations for both the two-level and multilevel cases are shown to be limit cycles. These limit cycles undergo period doubling as the system evolves into chaos. Finally, using a Monte Carlo treatment, these processes average to produce saturated absorption spectra complete with power and Doppler broadening effects consistent with experiment.

DOI
10.1103/physreva.59.4485

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Using a semiclassical density matrix formalism we have calculated the behavior of multilevel atoms interacting with a standing wave field, and show how complex nonlinear phenomena, including multiphoton effects, combine to produce saturation spectra as observed in experiments. We consider both 20-level sodium and 24-level rubidium models, contrasting these with a simple 2-level case. The influence of parameters such as atomic trajectory and the time the atom remains in the beam are shown to have a critical effect on the line shape of these resonances and the emission/absorption processes. Stable oscillations in the excited state populations for both the two-level and multilevel cases are shown to be limit cycles. These limit cycles undergo period doubling as the system evolves into chaos. Finally, using a Monte Carlo treatment, these processes average to produce saturated absorption spectra complete with power and Doppler broadening effects consistent with experiment.
We have developed a model for calculating saturated absorption spectra for dipole transitions in multi-level atoms. Using a semiclassical density matrix formalism, we derive a set of coupled differential equations for the internal state of the atom in a standing wave light field. The equations are solved using standard integration techniques. The absorption at each laser detuning is found from an average of the absorption for a number of velocities along the laser field, thermally weighted. The method is relatively efficient computationally yet quantitatively predicts important details of saturated absorption spectra including saturation, crossover resonances, merging of absorption lines at high intensity and optical pumping between hyperfine levels. We have measured saturated absorption and fluorescence spectra of 85 Rb, and compare to our computational results for a 36-level model.
The effects of a strong standing wave field on the absorption behaviour of a weak probe field are studied. An absorptive medium is assumed to consist of stationary two-level atoms with arbitrary decay and pumping scheme. The spectrum with a resonant saturator does not exhibit emissive behaviour as in the travelling wave cases. With a detuned saturator the spectrum closely resembles the corresponding three-level spectrum. These features are explained from the standing wave induced spatial selectivity and inhomogeneity. The connections to Doppler-broadened spectra are discussed.
Using a three-level model and semiclassical density matrix formalism, we show that optical pumping influences both line positions and amplitudes in Doppler-broadened alkali-atom D1 and D2 spectra for intensities much lower than the two-level atom saturation intensity. The influence on the D2 spectrum is particularly interesting due to the presence of closed hyperfine transitions unaffected by optical pumping. This effect is of importance for example when lasers are frequency stabilized using linear absorption or when the alkali-atom vapour pressure is determined using absorption measurements.
We treat semiclassical nonlinear dynamics of a two-level atom in the strong standing-wave field created in a high-quality cavity and in a free space with the help of two counterpropagating laser beams. First of all, we consider the dynamics of the strongly coupled atom-field system in an ideal cavity without any losses and discuss briefly nonlinear dynamical effects of random walking of an atom in a fully deterministic standing wave and Hamiltonian dynamical fractals. Then we study how these effects change in a free space when we take into account relaxation and external pumping of the standing wave. We discuss briefly new dissipative effects of synchronization between electronic and mechanical degrees of freedom of atoms, limit cycles of different periods and their bifurcations, and strange atomic attractors. Finally, we treat spontaneous relaxation as a random events in the equations of motion and study what happens with atomic fractality and limit cycles. It is shown that a self-similar structure of the scattering function (at least, on a few first levels) is conserved in the presence of spontaneous jumps. As to limit cycles, we could not resolve limit cycles of different period, but the fact of synchronization of internal and external atomic degrees of freedom is strictly established in the system with spontaneous jumps.
The nonlinear absorption or gain characteristics of an optical-frequency, Doppler-broadened atomic resonance involving levels with closely spaced structure are analyzed. The level structures are assumed to be resolved with respect to their natural widths, but not necessarily with respect to the Doppler width at the optical transition. The radiation field consists of two closely spaced monochromatic frequencies lying within the Doppler width of the resonance. This type of radiation field may be obtained, for example, from a laser operating in two of its Fabry-Perot resonator modes. It is shown that, because of saturation of level populations and double-quantum Raman transitions between levels, appreciable nonlinear coupling takes place between the two fields. This coupling shows a resonance behavior when the frequency separation of the two applied fields becomes equal to the frequency splitting between two of the components which form either level structure. The width of this resonance is determined entirely by the natural widths of these two level components and not by the Doppler width of the optical transition or the natural width of other level components. When such a resonance occurs, the over-all gain or attenuation characteristics of the atomic resonance change drastically. In practice the frequency spacing of the fields may be kept constant while the level structures are tuned, e.g., magnetically. The effects are analyzed for cases of running-wave and standing-wave radiation fields. The use of this effect in precise determination of level structures as well as mode or transition coupling of a gas laser is discussed. Portions of the analysis are applicable to resonances with more general forms of inhomogeneous broadening.
The nonlinear interaction of two intense optical waves with three energy levels of an atomic system is analyzed theoretically. The frequencies of the applied fields are assumed to be on or near two of the resonances involving the three levels. Particular attention is given to the generation of power at the difference frequency of the two applied fields where their frequency separation lies in the far-infrared range of the spectrum. In the limit of large input fields where atomic transitions are driven to saturation, the generated difference-frequency power is no longer proportional to the product of the two input powers. It is shown that, in this limit, the total rate of emission of photons at the difference frequency may, in principle, approach a value given by rNT\frac{\mathrm{rN}}{T}TrN​, where NNN is the total number of atoms, TTT is the average time of coherent interaction of atoms and the radiation field, and rrr is a factor which in favorable cases approaches unity at low temperatures. Also, a number of additional effects are encountered due to coherent double-quantum transitions among various levels which may be ignored only for weak input powers. These effects are highly dependent on the modes of relaxations of atomic levels and the details of the line shape. The results are applied to several cases, in which the resonance condition yields large efficiency in nonlinear frequency mixing.