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Physical Review B

Four-wave-mixing theory for two-photon generation of excitons in Cu2O

Igor V. Beloussov, John B. Ketterson, Yinghui Sun

Published Dec 23, 2009• 1 citation
Publisher

Abstract

We propose and examine theoretically a four-wave-mixing experiment in which excitons in ${\text{Cu}}_{2}\text{O}$ are pumped directly into a state with wave vector $\mathbf{k}=\mathbf{0}$ by two counter-propagating cross-polarized laser pulses and which is subsequently probed by a third time-delayed pulse. Approximate analytical solutions of the time evolution equations are found which describe the time dependence of the exciton and photon densities. Most importantly, we obtain the dependence of the resulting phase-conjugated signal versus the delay time ${t}_{d}$ between the probe and pump pulses. It is shown that the total number of photons generated in the course of the experiment is proportional to the areas of all three incoming pulses. The dependence of the resulting time-integrated signal on the delay time ${t}_{d}$ is the same as the exciton density dependence on the time $t$. Hence, one may directly study the time evolution of the exciton condensate. The approximate analytical results are compared with a numerical solution of the evolution equations for a wide range of incident pulse intensities. It is noted that an exciton condensate arising by an alternate route can also be probed through a phase-conjugated two-photon process.

DOI
10.1103/physrevb.80.245213

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The experimental situation in which excitons in Cu2O are pumped directly into a state with wave vector = k0 by two counter-propagating laser pulses, and subsequently probed by a third, time-delayed, pulse has been theoretically investigated. It has been shown that additional electromagnetic waves are excited in the system. A specific configuration of two pump pulses and one probe pulse has been found to generate only one additional electromagnetic wave. The dependence of the time-integrated photon number in this wave on the delay time between pump and probe pulses for some values of the detuning has been investigated.
The experimental situation in which excitons in Cu 2 O are pumped directly into a state with wave vector k → = 0 → by two counter-propagating laser pulses , and subsequently probed by a third, time-delayed, pulse has been theoretically investigated. It has been shown that additional electromagnetic waves are excited in the system. A specific configuration of two pump pulses and one probe pulse has been found to generate only one additional electromagnetic wave. The dependence of the time-integrated photon number in this wave on the delay time between pump and probe pulses for some values of the detuning has been investigated.
We propose and theoretically investigate a two-photon four wave mixing experiment to probe for BEC of excitons in Cu2O thin films. A relatively simple set of equations describing the dynamics of the system is obtained for a particular configuration of the exciting beams, and numerical and approximate analytical solutions are found. The resulting phase-conjugated signal vs. the delay time between the pump and probe pulses yields a direct measure of the time evolution of the exciton condensate, and exhibit peaks when the film thickness is a multiple of a half wavelength of light.
The phase-conjugation in crystalline CuCl has been studied at 1.6 K. It has been found that the two-photon resonance of the Γ1\varGamma_{1}Γ1​-excitonic molecule and the one-photon resonance of the Γ5\varGamma_{5}Γ5​-exciton contribute to the generation of the phase-conjugated wave. Each contribution has been separated spectroscopically by adjusting the polarization configuration of the incident fields adequately and has been compared qualitatively with the theory. The efficiency of the phase-conjugated reflection has become maximum at around the two-photon resonance of the excitonic molecule, 3.186 eV and has been ∼1% at the maximum.
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.