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The Journal of Chemical Physics

Extra resonances in four-wave mixing as a probe of exciton dynamics: The steady-state analog of the transient grating

Roger F. Loring, Shaul Mukamel

Published Feb 1, 1986• 19 citations
Publisher

Abstract

We present a microscopic theory of the nonlinear susceptibility χ(3) of a crystal composed of interacting two-level absorbers. The relationship between the signals in a steady-state four-wave mixing experiment and in a time-resolved transient grating experiment is established. In general, the steady-state signal is not simply the Fourier transform of the time-resolved signal, although such a relationship does hold under certain experimental conditions. We evaluate χ(3) for the Haken–Strobl model. χ(3) displays a dephasing-induced extra resonance, whose intensity and line shape are sensitive to the degree of exciton coherence. We also calculate χ(3) for a crystal of interacting three-level absorbers, under the assumption that population in the higher excited state relaxes rapidly to the lower excited state. We propose steady-state four-wave mixing as an alternative to the transient grating experiment in investigating the nature of exciton motion.

DOI
10.1063/1.450515

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Optics Letters•May 15, 1993•2 citations
We present a microscopic theory of the nonlinear susceptibility χ(3) of a crystal composed of interacting two-level absorbers. The relationship between the signals in a steady-state four-wave mixing experiment and in a time-resolved transient grating experiment is established. In general, the steady-state signal is not simply the Fourier transform of the time-resolved signal, although such a relationship does hold under certain experimental conditions. We evaluate χ(3) for the Haken–Strobl model. χ(3) displays a dephasing-induced extra resonance, whose intensity and line shape are sensitive to the degree of exciton coherence. We also calculate χ(3) for a crystal of interacting three-level absorbers, under the assumption that population in the higher excited state relaxes rapidly to the lower excited state. We propose steady-state four-wave mixing as an alternative to the transient grating experiment in investigating the nature of exciton motion.
We present the first fully microscopic theory of the generation of the transient grating signal from a crystal of interacting molecules. We derive a general expression for the signal, measured at a time immediately after a short probe pulse, in terms of the exciton Green function. This expression agrees with previous calculations of the grating signal, in which the experiment was treated as a diffraction process. It is also shown that the temporal profile of the signal pulse is, in general, different from that of the probe pulse. This result is inconsistent with the usual diffraction picture. Our treatment of this experiment as a transient four-wave mixing process rather than as a diffraction process allows us to derive an expression for the signal in terms of equilibrium correlation functions of the dipole operator. Using this formulation, we discuss the relationship of this experiment to other nonlinear spectroscopic techniques.
A picosecond time-resolved four-wave mixing technique is used to study the dephasing of nonlinear optical resonances in CdSe. At low temperature strong coherent, nonlinear signals at the free exciton resonance, at the biexciton resonance, and at the exciton-biexciton transition are observed. The observed correlation traces, i.e. the diffracted intensity versus the delay between the interfering light pulses, yield information on the dephasing process. Dephasing times for excitons as long as 40 ps, decreasing with increasing excitation density and temperature, are observed. A two-level model is applied to describe theoretically the dynamical behavior of the resonances, and to extract their homogeneous broadening, as well as their inhomogeneous broadening.
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 present an experimental and theoretical investigation of ultrafast transient four-wave mixing of GaAs/ Alx{\mathrm{Al}}_{\mathit{x}}Alx​${\mathrm{Ga}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$As quantum wells for coherent excitation of exciton and continuum states. The signal appears at the exciton resonance and is shown to consist of two contributions: an intense spectrally narrow part only present for short time delays and a weak exponentially decaying part reflecting the intrinsic coherence properties of the exciton states. The intense part of the signal is due to the excitation of the continuum states mediated by interaction-induced nonlinearities. \textcopyright{} 1996 The American Physical Society.
Spectrally resolved and time-integrated degenerate four-wave mixing (FWM) experiments with ultra-short pulses have been used to investigate the relaxation times and coherent interactions of excitons in the Boltzmann and in the quantum kinetic regime in ZnSe single quantum wells.Three-pulse FWM experiments using 100 fs pulses in a 10 nm thick ZnSe/ZnMgSe single quantum well reveal nearly equal zero-density dephasing rates for coherences between + σ and - σ heavy-and/or light-hole exciton states as compared to the dephasing rate of the interband coherence between these states and the ground state.The measurements also demonstrate that exciton-exciton scattering destroys the phase coherence between heavy-and light-hole excitons about three times more effectively than the coherence between + σ and + σ heavy-hole excitons.The different coherent nonlinear signals obtained in a three-beam FWM experiment are explained and reproduced by numerical calculations of the optical Bloch equations for a ten-level model including excitation induced dephasing (EID).Two-beam FWM experiments using 30 fs pulses on a 10 nm ZnSe quantum well reveal marked quantum beats and a pronounced non-exponential decay for delay times smaller ~500 fs.The pronounced peak at pulse overlap is attributed to rapid destructive interference of the created polarization and to spin dependent quantum kinetic effects caused by the transient formation of an electron-hole (eh) pair grating.The three-beam FWM trace shows a bi-exponential decay which is attributed to a nonlinear polarization that is caused by the interaction of polarization 3 k with both an exciton grating and an eh-pair grating.Model calculations using the OBEs of a two level system that includes ∑ -+
We show that extra resonances, such as those caused in frequency-domain nonlinear wave mixing by pure dephasing or laser fluctuations, can also be induced by operation in the time domain. These pulse-length-induced extra resonances arise in transient-grating experiments when the laser pulses are short enough that a steady state cannot be achieved during the excitation process. We show theoretically that these resonances increase in strength with decreasing excitation pulse length until the pulse length becomes shorter than the dephasing time of the medium and quote an experimental example to support this interpretation.