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Physical Review Letters

Effects of coherent polarization interactions on time-resolved degenerate four-wave mixing

Karl Leo, Martin Wegener, J. Shah, D. S. Chemla, Ernst Otto Göbel, Theo C. Damen, Stefan Schmitt-Rink, W. Schäfer

Published Sep 1, 1990• 296 citations
Publisher

Abstract

We report the first observation of coherent polarization interactions in dense media in time-resolved degenerate four-wave mixing in semiconductors. For near-resonant ultrashort-optical-pulse excitation, these interactions lead to a diffracted signal at negative delay times, with a slope half that for positive delay times. At higher intensities, two temporal maxima appear in the diffracted signal, in agreement with many-body theory.

DOI
10.1103/physrevlett.65.1340

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We report the first observation of coherent polarization interactions in dense media in time-resolved degenerate four-wave mixing in semiconductors. For near-resonant ultrashort-optical-pulse excitation, these interactions lead to a diffracted signal at negative delay times, with a slope half that for positive delay times. At higher intensities, two temporal maxima appear in the diffracted signal, in agreement with many-body theory.
Time-resolved four-wave-mixing experiments are usually interpreted in terms of noninteracting two-level systems in order to obtain information on the polarization dephasing time T2{\mathit{T}}_{2}T2​. Recent experiments involving excitonic resonances in semiconductor quantum wells (including results presented in this paper) show striking qualitative deviations from this simple picture. In particular, an exponential tail is observed at low excitation for negative time delays. At high excitation, the four-wave-mixing signal is found to evolve into two distinct temporal maxima. We demonstrate that the microscopic origin of this time dependence can be understood in terms of coherent exciton-exciton interactions. We show in fact that this behavior is more general and should be seen in numerous dense media where strong nonlinear interactions of polarizations occur. In addition to presenting rigorous numerical results, we analyze two simple situations in which such interactions exist: dielectric media with local-field effects and the anharmonic oscillator. We derive analytical expressions for their time-dependent four-wave-mixing response and discuss the physical origin of these new nonlinear-optical effects.
We show that four-wave mixing in optically dense media is very different than in an optically thin medium. In time-resolved degenerate four-wave mixing we experimentally observe ``negative'' time delay response, fast decay rates at short delay times, and broad shoulders at long delays. The observations are explained very well in terms of pulse propagation effects, and a theoretical framework is presented for the analysis of the nonlinear interaction. We treat both the incident and generated fields self-consistently, and solve for the interaction of short pulses with a resonantly absorbing medium. The implications for the extraction of relaxation rates from four-wave-mixing experiments involving ultrashort pulses in optically dense matter are discussed.
The optical density of a resonant medium significantly affects the output signal in time resolved nonlinear optical interactions. The resonant interaction with a medium reshapes any propagating short pulse, and the mutual interaction of several such pulses, as in four-wave mixing, must be treated self-consistently. We present a theoretical framework for the proper handling of the propagation of all pulses, input as well as generated, in the small area limit. For optically thick resonant absorbers, negative time delay signals are observed, and the apparent decay rate of the induced polarization is faster than the rate observed for thin samples. We experimentally measure degenerate four-wave mixing in an atomic medium as an example, and demonstrate the quality of the theoretical model by the excellent fit to measured signals over several orders of magnitude. The improved understanding enables us to provide a simple, but surprisingly accurate, estimate for the apparent decay rate in homogeneously broadened optically thick media: If the absorption is given by \ensuremath{\alpha}, the propagation length is L, and the transverse relaxation time is T2{\mathit{T}}_{2}T2​, the apparent decay rate 2/Ta{\mathit{T}}_{\mathit{a}}Ta​ of a time resolved four-wave mixing signal is given by 2/Ta{\mathit{T}}_{\mathit{a}}Ta​=(2/T2{\mathit{T}}_{2}T2​)(1+\ensuremath{\alpha}L/2).
We present a time-resolved study of the diffracted pulse in degenerate four-wave mixing experiments with short-response-time materials. The first step is the validation of a theoretical model that takes into account the temporal characteristics of the energy level (relaxation time T1 and dephasing time T2) and of the incident pulse (duration and spectral phase) in a simple particular case. Two methods are used for characterizing the diffracted beam: the amplitude correlation with the reference pulse and the spectrum intensity measurements. We discuss the experimental results, which show good agreement with the theoretical model and evidence the strong influence of the incident pulse characteristics.