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Physical Review B

Time-resolved four-wave-mixing spectroscopy of excitons in a single quantum well

Jeffrey A. Davis, Jeremiah J. Wathen, Valérie Blanchet, R. T. Phillips

Published Jan 11, 2007• 12 citations
Publisher

Abstract

We have investigated, by fully-time-resolved degenerate four-wave mixing (FWM), excitonic coherence phenomena in single $\mathrm{Ga}\mathrm{As}∕\mathrm{Al}\mathrm{Ga}\mathrm{As}$ quantum wells which are known to exhibit localization. A magnetic field $(0--12\phantom{\rule{0.3em}{0ex}}\mathrm{T})$ was applied in Faraday geometry as a means of altering the nature of the excitonic transitions in a controlled manner. When the system is excited by a sequence of pulses tuned in resonance with the heavy-hole excitons, the predominant behavior observed at positive interpulse delays is beating in the FWM emission due to polarization interference between uncoupled excitons localized by monolayer islands. For negative interpulse delay the variation of beat phase with delay changes sign as the magnetic field is increased. In order to understand this feature, we modeled a coupled three-level system and included local field effects as a simple phenomenological approach to describing many-body terms. Comparison with experiments shows that with a larger magnetic field, the contribution to the signal from the coupled system reduces while that from two noninteracting two-level systems increases. We attribute this to the increased confinement of the excitons with field, leading to reduced wavefunction overlap, and hence reduced coupling.

DOI
10.1103/physrevb.75.035317

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We have investigated, by fully-time-resolved degenerate four-wave mixing (FWM), excitonic coherence phenomena in single GaAs∕AlGaAs\mathrm{Ga}\mathrm{As}∕\mathrm{Al}\mathrm{Ga}\mathrm{As}GaAs∕AlGaAs quantum wells which are known to exhibit localization. A magnetic field (0−−12T)(0--12\phantom{\rule{0.3em}{0ex}}\mathrm{T})(0−−12T) was applied in Faraday geometry as a means of altering the nature of the excitonic transitions in a controlled manner. When the system is excited by a sequence of pulses tuned in resonance with the heavy-hole excitons, the predominant behavior observed at positive interpulse delays is beating in the FWM emission due to polarization interference between uncoupled excitons localized by monolayer islands. For negative interpulse delay the variation of beat phase with delay changes sign as the magnetic field is increased. In order to understand this feature, we modeled a coupled three-level system and included local field effects as a simple phenomenological approach to describing many-body terms. Comparison with experiments shows that with a larger magnetic field, the contribution to the signal from the coupled system reduces while that from two noninteracting two-level systems increases. We attribute this to the increased confinement of the excitons with field, leading to reduced wavefunction overlap, and hence reduced coupling.
We report time-resolved degenerate four-wave-mixing experiments on a high-quality single GaAs quantum well. The signals for parallel and cross-polarized exciting pulses show pronounced quantum beats with different frequencies corresponding to the heavy/light-hole splitting and the biexciton binding energy, respectively. Comparison of the experimental data with solutions of the optical Bloch equations for a phenomenological model system shows that these features result from strong contributions of the biexcitonic state. The importance of biexcitons to the nonlinear coherent response of the two-dimension exciton is further confirmed by the frequency dependence of the time-integrated four-wave-mixing signal amplitudes measured for the two polarization geometries.
We investigate the properties of coherent exciton states in GaAs/Alx{\mathrm{Al}}_{\mathit{x}}Alx​${\mathrm{Ga}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$As quantum wells using the four-wave-mixing technique with subpicosecond time resolution. In the first part of this work, we show that many-body interactions in a solid manifest themselves in a line shape of the four-wave-mixing signal that is substantially different from the predictions of the generally used simple two-level model. In particular, the coherent interaction of the quasiparticles leads to a signal at negative delay times, in agreement with recent theoretical predictions. In the second part, we report in detail about experiments where energetically close exciton transitions are excited coherently with a short laser pulse. The quantum or polarization interference leads to the observation of quantum beats in the decay of the diffracted signal. These results demonstrate that quantum-beat spectroscopy can be applied to study intrinsic excitations in solids.
We show that time-resolved quantum beat spectroscopy in the four-wave mixing (FWM) geometry is a unique tool to investigate the electronic multi-level system associated with spectrally adjacent excitonic resonances. We have studied heavy- and light-hole excitons, excitons associated with different and spatially separated quantum wells, and excitons associated with spatially separated growth islands within the same quantum well. Though the linear spectra are qualitatively similar in all three cases, the respective beating of the time-resolved FWM-signal shows a very different dynamic behaviour in each case. We discuss appropriate electronic multi-level schemes and show that exciton transitions within different quantum wells can be described by independent twolevel systems. In contrast, excitonic transitions associated with different and spatially separated growth islands having different quantum well thickness are not independent but coupled.
We demonstrate the presence of strong exciton-exciton correlations in GaAs-AlGaAs multiple quantum wells by using the polarization selection rules for four-wave mixing and a test that was originally designed to distinguish between quantum beats and polarization interference. We show that, when the four-wave-mixing signal is produced by two pulses that have the same circular polarization, dynamic beating behavior is observed at the heavy-hole-light-hole frequency that provides evidence of Coulomb-induced correlations that go beyond those expected from the semiconductor Bloch equations in the Hartree-Fock approximation and beyond the local field corrections used in simple phenomenological models. Moreover, this behavior cannot be explained by invoking biexcitonic effects. We demonstrate that this dynamic excitonic coupling can be explained in terms of a competition between quantum-beat-like and polarization-interference-like behavior, and we show that such behavior can be used to place quantitative limits on the coupling strength. In addition, we use a simple phenomenological model to show that excitation-induced-dephasing can produce such correlations.
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.
Shifts in the relative phases of quantum beats in the time-resolved four-wave-mixing signals from a GaAs/Alx{\mathrm{Al}}_{\mathit{x}}Alx​${\mathrm{Ga}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$As multiple quantum well are observed. The relative phases are shown to depend strongly on the intensities of the incident pulses as well as their detuning from the heavy-hole exciton. A simple coupled six-level model shows that the phases of the quantum beats are directly determined by many-body effects such as density-dependent dephasing and the local-field correction.