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Physical Review A

Generalized perturbation theory of coherent optical emission

Masaharu Mitsunaga, Richard G. Brewer

Published Sep 1, 1985• 59 citations
Publisher

Abstract

A generalized perturbation theory is developed for an inhomogeneously broadened two-level quantum system interacting with an optical field of arbitrary temporal shape. Solutions of the Liouville equation carried out to nth order in the optical field strength exhibit a rephasing behavior and therefore are interpreted as an n-pulse echo where each pulse corresponds to one power of the field strength. We conclude that at low fields any coherence effect can be interpreted as a generalized stimulated echo phenomenon. Several well-understood coherence effects are explained and a few new ones are predicted, including one example where the light source is incoherent.

DOI
10.1103/physreva.32.1605

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arXiv:cond-mat/9411118•Physical review. B, Condensed matter•Oct 15, 1994•7 citations
A generalized perturbation theory is developed for an inhomogeneously broadened two-level quantum system interacting with an optical field of arbitrary temporal shape. Solutions of the Liouville equation carried out to nth order in the optical field strength exhibit a rephasing behavior and therefore are interpreted as an n-pulse echo where each pulse corresponds to one power of the field strength. We conclude that at low fields any coherence effect can be interpreted as a generalized stimulated echo phenomenon. Several well-understood coherence effects are explained and a few new ones are predicted, including one example where the light source is incoherent.
A time-dependent perturbational approach is used to study the Raman echo generated by an inhomogeneous ensemble of three-level atomic systems in the Λ configuration. The sixth-order perturbative solution for coherence between the two lower states contains eight echo terms that are visualized with double Feynman diagrams. The echo is interpreted as a response of the ensemble to six short single-frequency optical pulses whose timing satisfies a special condition. Nonperturbative numerical simulation confirms the predictions of the perturbation analysis about the main characteristics of the echo. When the excitation pulses vary adiabatically with respect to fast optical coherence decay the echo amplitude depends on interference of the eight terms. A computational scheme developed for the higher-order perturbative analysis is also discussed.
An analytical study of the photon echoes associated with the three-level system interacting resonantly with optical pulses possessing two different frequencies is performed. It is predicted that photon echoes arise at anomalous times corresponding to the correlation between the inhomogeneous broadenings for the different spectral lines as well as at the normal time expected in a simple two-level system. This new effect is caused by the coupling of the two resonant transitions through the common energy level. Furthermore, if the particle does not have inversion symmetry, the echoes with the sum or difference frequency of the exciting optical pulses can be observed in such a system. The quantitative analysis is also given for intensities, direction of propagation and polarizations of the anomalous echoes which depend upon the exciting pulses with different frequency.
A quasi-two-level system is a generalization of a two-level system where each of the single levels is replaced by a nearly degenerate multiplet. We develop the theory of the photon echo, the stimulated echo, and fluorescence emission for such systems when they are subjected to pulses of electromagnetic radiation that are short compared to the reciprocal (in frequency units) of the widths of the multiplets. The observable in each of these experiments is related to equilibrium time-correlation functions of the system in the absence of the radiation field. The correlation functions associated with these experiments involve more than two dynamical operators and more than two times, unlike the correlation functions for such processes as low-power optical absorption and neutron scattering. We introduce an exactly solvable factorization model for the transition moments, which includes as special cases the situation where either of the multiplets is a single state. We also consider the unitary model, which was introduced by Mims, and which leads to considerable simplification of the correlation functions. Several examples of these models are discussed to investigate the features of time-dependent fluorescence experiments and coherent optical transient experiments. In general, there is no simple connection between the time dependence of the decay of a photon echo and the Fourier transform of the low-power homogeneous absorption spectrum.
Photon echoes, observed recently in ruby and in NH2{\mathrm{NH}}_{2}NH2​D gas by two-pulse laser excitation, exhibit a modulation behavior as the pulse delay time is varied. The effect, which arises in a three-level quantum system, is due to the interference of two optically coupled transitions that share a common level. In this article, the three-level echo problem is treated exactly. Wave-function solutions are incorporated that are handled conveniently by a Laplace-transform technique. We, therefore, extend the earlier theory of echo modulation by Lambert et al., by removing their restriction of on-resonance pulse excitation. This procedure allows us to examine the effect of inhomogeneous broadening on echo amplitude explicitly, and we treat the problem by numerical and also by analytical means for the case of a Doppler-broadened gas. Analytic expressions are derived either for a high or a low laser power regime corresponding to excitation of either a large or a small fraction of the Doppler linewidth. The former case is analogous to pulsed nuclear magnetic resonance and is well known, whereas the latter result is new and applies to current photon echo measurements.
Using a perturbation theory based on the density-matrix formulation, we study the nonlinear optical responses of a noninteracting three-level model system to consecutive coherent pulsed excitations, as realized in several recent experiments involving excitonic transitions in quantum wells. The terahertz emission, which is a second-order response, and the four-wave mixing signal, a third-order response, are calculated within the rotating-wave approximation, in the presence of detuning, dephasing, and inhomogeneous broadening. We study the quantum beats in the photon echo from transient four-wave mixing experiments of a three-level system. We find that the temporal profile of the photon echo signal in the four-wave mixing experiments depends very strongly on the amount of inhomogeneous broadening of the energy levels involved. Both the position and the intensity of the peaks exhibit a smooth evolution from strong quantum beats to a conventional photon echo as the inhomogeneous broadening increases. These features from our noninteracting model system are compared with a recent experiment, and found to account for a number of experimental observations.