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arXiv:0708.0442 · Europhysics Letters (EPL)

Large optical gain from four-wave mixing instabilities in semiconductor quantum wells

Stefan Schumacher, N. H. Kwong, Rolf H. Binder

Published Dec 6, 2007• 8 citations
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Abstract

Based on a microscopic many-particle theory, we predict large optical gain in the probe and background-free four-wave mixing directions caused by excitonic instabilities in semiconductor quantum wells. For a single quantum well with radiative-decay limited dephasing in a typical pump-probe setup we discuss the microscopic driving mechanisms and polarization and frequency dependence of these instabilities.

DOI
10.1209/0295-5075/81/27003

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Journal of Experimental and Theoretical Physics•Feb 1, 2006•1 citation
Based on a microscopic many-particle theory, we predict large optical gain in the probe and background-free four-wave mixing directions caused by excitonic instabilities in semiconductor quantum wells. For a single quantum well with radiative-decay limited dephasing in a typical pump-probe setup we discuss the microscopic driving mechanisms and polarization and frequency dependence of these instabilities.
We demonstrate spectroscopically how optical gain is formed in an interacting system of excitons and biexcitons in a single ZnSe quantum well. The dynamics of gain formation are studied on a subpicosecond time scale in pump-probe experiments, revealing the influence of coherent effects and the impact of exciton-exciton and exciton-biexciton interactions. Spectrally resolved, polarization-dependent degenerate four-wave-mixing experiments further demonstrate how the excitonic many-body phenomena profoundly influence the nonlinear optical response in this II-VI semiconductor heterostructure. \textcopyright{} 1996 The American Physical Society.
Excitons are sometimes trapped at impurities in crystals and at defects on interfaces in the confined systems, such as quantum wells and microcrystallites embedded in a host matrix. We discuss these trapping effects on the third-order optical polarization under nearly resonant pumping of the exciton. First we predict an enhancement of ${\mathrm{\ensuremath{\chi}}}^{(3)}$(\ensuremath{\Omega};\ensuremath{\Omega},-\ensuremath{\Omega},\ensuremath{\Omega}) by a factor ${\mathrm{\ensuremath{\Gamma}}}_{\mathit{n}\ensuremath{\rightarrow}\mathit{T}}$/${\mathrm{\ensuremath{\Gamma}}}_{\mathit{T}\ensuremath{\rightarrow}\mathit{g}}$ multiplicatively on that enhanced by the large excitonic dipole moments. Here ${\mathrm{\ensuremath{\Gamma}}}_{\mathit{n}\ensuremath{\rightarrow}\mathit{T}}$ and ${\mathrm{\ensuremath{\Gamma}}}_{\mathit{T}\ensuremath{\rightarrow}\mathit{g}}$ are, respectively, the rate at which the exciton is trapped and the decay rate of the trapped electron into the ground state. In many cases, ${\mathrm{\ensuremath{\Gamma}}}_{\mathit{n}\ensuremath{\rightarrow}\mathit{T}}$ is much larger than ${\mathrm{\ensuremath{\Gamma}}}_{\mathit{T}\ensuremath{\rightarrow}\mathit{g}}$ so that a large enhancement of ${\mathrm{\ensuremath{\chi}}}^{(3)}$ is expected. Second, the four-wave mixing spectrum \ensuremath{\Vert}${\mathrm{\ensuremath{\chi}}}^{(3)}$(2${\mathrm{\ensuremath{\Omega}}}_{1}$-${\mathrm{\ensuremath{\Omega}}}_{2}$;${\mathrm{\ensuremath{\Omega}}}_{1}$,-${\mathrm{\ensuremath{\Omega}}}_{2}$,${\mathrm{\ensuremath{\Omega}}}_{1}$)\ensuremath{\Vert} under two beams ${\mathrm{\ensuremath{\Omega}}}_{1}$ and ${\mathrm{\ensuremath{\Omega}}}_{2}$ is shown to consist of three hierarchical structures as a function of ${\mathrm{\ensuremath{\Omega}}}_{1}$-${\mathrm{\ensuremath{\Omega}}}_{2}$, i.e., the sharpest spike around ${\mathrm{\ensuremath{\Omega}}}_{1}$-${\mathrm{\ensuremath{\Omega}}}_{2}$=0 with the linewidth \ensuremath{\Vert}${\mathrm{\ensuremath{\Omega}}}_{1}$-${\mathrm{\ensuremath{\Omega}}}_{2}$\ensuremath{\Vert}=${\mathrm{\ensuremath{\Gamma}}}_{\mathit{T}\ensuremath{\rightarrow}\mathit{g}}$, the steep shoulder with the center also at ${\mathrm{\ensuremath{\Omega}}}_{1}$-${\mathrm{\ensuremath{\Omega}}}_{2}$=0 and the width ${\mathrm{\ensuremath{\Gamma}}}_{\mathit{n}}$ (the exciton decay rate), and the wide structure with the width ${\mathrm{\ensuremath{\Gamma}}}_{\mathit{n}\mathit{g}}$ around ${\mathrm{\ensuremath{\Omega}}}_{1}$=${\mathrm{\ensuremath{\omega}}}_{1\mathit{s}}$ and ${\mathrm{\ensuremath{\Omega}}}_{2}$=${\mathrm{\ensuremath{\omega}}}_{1\mathit{s}}$. Here ${\mathrm{\ensuremath{\Gamma}}}_{\mathit{n}\mathit{g}}$ is the relaxation rate of the exciton polarization and \ensuremath{\Elzxh}${\mathrm{\ensuremath{\omega}}}_{1\mathit{s}}$ the lowest exciton energy. Third, the differential transmission spectrum is found to show the strong absorption saturation peak or the sharp induced absorption dip as a function of pump-probe detuning ${\mathrm{\ensuremath{\Omega}}}_{2}$-${\mathrm{\ensuremath{\Omega}}}_{1}$, depending upon the pump frequency ${\mathrm{\ensuremath{\Omega}}}_{1}$ relative to the exciton peak ${\mathrm{\ensuremath{\omega}}}_{1\mathit{s}}$.
Coherently controlled four-wave-mixing (FWM) measurements have turned out to be a very sensitive tool to investigate the microscopic origin of the nonlinear optical response of semiconductor quantum-well structures, in particular if excitons are excited [1-3]. The measured signal oscillates as a function of the phase difference of the two interferometric pulses and is modulated additionally by contributions that oscillate with integer multiples of the excitonic resonance frequency. The influence of these harmonics can be measured in background free directions, i.e., the first (FWM) or second (six-wave mixing, SWM) diffracted order in direction 2k2-k1 or 3k2-2k1, respectively. Their intensity is easily quantified by inspecting the corresponding Fourier transforms (FT). Here, we report on transient interferometric SWM measurements and the influence of biexcitons on the signal.
The paper presents a description of two-pair correlations in an electron-hole system in laser-excited direct-gap semiconductors in terms of the exciton-boson formalism. The dependence of the states of quasi-particles on their angular momentum orientation is properly taken into account following group theory rules. It is shown for the example of heavy hole excitons in GaAs quantum wells that Coulomb two-exciton correlations include effective repulsion between excitons with nonzero total momenta and effective attraction between excitons with zero total momenta. They generate exciton and biexciton nonlinearities, respectively, in the exciton-biexciton system. The Heisenberg equations of motion derived are applied to four-wave mixing in the coherent mode. Two components of Coulomb two-exciton correlations were found to generate coherent four-wave mixing either separately, one at the cocircular and the other at the perpendicular-linear polarization of applied pulses, or jointly at parallel-linear polarization. General equations are obtained for the amplitude of four-wave mixing signals with time resolution in these polarization configurations, which, in the limit of ultranarrow pulses, reduce to analytic functions of time and excitation and sample parameters. These equations are used to analyze signal characteristics in every polarization configuration. The results give a correct picture of biexciton effects and the polarization dependence of coherent four-wave mixing in GaAs quantum wells.