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Physical Review A

Effects of correlation between inhomogeneously broadened transitions on quantum beats in transient four-wave mixing

Steven T. Cundiff

Published Apr 1, 1994• 67 citations
Publisher

Abstract

The transient four-wave-mixing signal from a three-level system is calculated in the short-pulse limit including a general distribution function for the energies of the levels between which beating occurs. This allows an examination of the effects of correlation between the energetic splitting of the levels. As expected, the results demonstrate that the degree of correlation can significantly alter the four-wave-mixing signal. However, the results also show that in the case where the level energies are completely correlated, in a rigorous sense, but that the correlation is not a simple offset, a strong modification of the beat amplitude occurs. This case describes the type of correlation expected in semiconductor quantum wells. Examination of several representative cases demonstrates that these effects can easily be incorrectly ascribed to dephasing.

DOI
10.1103/physreva.49.3114

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The transient four-wave-mixing signal from a three-level system is calculated in the short-pulse limit including a general distribution function for the energies of the levels between which beating occurs. This allows an examination of the effects of correlation between the energetic splitting of the levels. As expected, the results demonstrate that the degree of correlation can significantly alter the four-wave-mixing signal. However, the results also show that in the case where the level energies are completely correlated, in a rigorous sense, but that the correlation is not a simple offset, a strong modification of the beat amplitude occurs. This case describes the type of correlation expected in semiconductor quantum wells. Examination of several representative cases demonstrates that these effects can easily be incorrectly ascribed to dephasing.
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.
The transient four-wave mixing signal from a three-level system coupled to a bath of harmonic oscillators is calculated in the short-pulse limit. This allows an examination of the non-Markovian effects on the relaxation processes in the excited levels between which beating occurs. In particular, it is possible to distinguish between oscillations which are due to oscillatory dephasing, a consequence of the memory effects, and oscillations due to the quantum beats. This also allows one to analyze how these two kinds of oscillations are mixed together. We show that we not only have an amplitude modulation of the beats as in the Markovian case but also deformations of the beats for short delays between the exciting light pulses. These deformations are a consequence of the nonequilibrium motion of the surroundings of the system under optical excitation. \textcopyright{} 1996 The American Physical Society.
Oscillatory dephasing is a characteristic signature of non-Markovian relaxation phenomena. From its analysis one can obtain valuable information about the dynamics of the surroundings of a given system. In this paper we calculate the transient four-wave mixing signal in the short pulse limit from two-level systems coupled to a bath of harmonic oscillators for arbitrary coupling strengths and in the presence of inhomogeneous broadening. The strong and weak coupling cases having already been analyzed, we focus on the study of the intermediate case where oscillatory dephasing is observed. We specifically show that even in the presence of inhomogeneous broadening, information about the surroundings can still be extracted in the intermediate coupling regime. The formation and the evolution of the photon echo in such experiments are also analyzed. \textcopyright{} 1996 The American Physical Society.
Coherent transient spectroscopy of semiconductor nanostructures has become an active field of research in recent years. Most such experiments are analyzed in terms of a simple two-level model in which the effects of interaction and induced fields are ignored. We discuss two very different cases where inclusion of these generally-neglected effects leads to large and important modification in the temporal response of the system. In the first case, interactions between excitons strongly modify the temporal response of four-wave-mixing (FWM) signals from quantum wells. This is the temporal analog of well-known effects in the spectral domain, such as bleaching of the exciton absorption and bandgap renormalization. In the second case, the shape of a low-intensity pulse is severely distorted after transmission through a typical quantum well sample. Interferometric measurements show that reradiation of induced polarization is responsible for the distortion. The distortion depends strongly on the dephasing time and has important implications for all time-resolved measurements. 1.
Current issues in the study of light-matter interactions are coherence and quantum interference. With ultrafast nonlinear spectroscopy such as transient four-wave mixing (TFWM) the imposed coherence on the optical excitations in e.g. GaAs multiple quantum wells has been studied. In this contribution, we study the influence of inhomogeneous broadening on quantum beats observed by spectrally resolved TFWM. In artificially grown semiconductor structures as e.g. quantum wells, inhomogeneities due to interface-disorder has, up till now, been inevitable. In the experiments, the light-hole exciton (lhx) and the heavy-hole exciton (hhx) in a 100 Å GaAs multiple quantum well are excited simultaneously with 150 fs pulses from a self mode-locked Ti:sapphire laser. The spectral profile is shown in the inset of Fig. 1.
In recent years, transient four-wave mixing (FWM) techniques have been used to investigate the dephasing times of non-equilibrium carriers in semiconductor heterostructures by monitoring the decay of the macroscopic polarization. Initially, this decay was measured by temporally-integrating the scattered FWM signal as a function of time delay between two (or more) pump pulses. More recently, the macroscopic polarization decay has been measured by time-resolving the FWM signal by cross correlating it with an ultrashort laser pulse via frequency up-conversion in a nonlinear crystal. When excitons with slightly different energies are excited, oscillations (or beats) have been observed both in the time-integrated FWM signal and in the time-resolved FWM signal. Such beats have been observed, for example, between light- and heavy-hole excitons and between excitons in wells of different widths. Both the spectral behavior1 of the time-integrated FWM signal and the temporal behavior2 of the time-resolved signal have been shown to allow the distinction between the polarization interference associated with two independent oscillators and the quantum beats associated with two coupled oscillators which share a common level. In addition, it has been demonstrated3 that the quantum beats produced by incident pulses with parallel polarizations are exactly out of phase (by π) with the beats produced by orthogonally polarized pulses. In each case1-3, the period and phase of the beats and the polarization of FWM signal were explained by using a six-level model, which excluded many-body effects, to describe the J=1/2 conduction and J=3/2 valence states. Within the last few months, however, it has been shown that a complete description of the polarization selection rules for the FWM signal requires excitation-induced dephasing4 or disorder-induced coupling of the σ + and σ – excitonic transitions5.