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Molecular crystals and liquid crystals science and technology. Section B, Nonlinear optics/Molecular crystals and liquid crystals scienceand technology. Section B, Nonlinear optics/Nonlinear optics

MICROSCOPIC THEORY OF FOUR-WAVE-MIXING PROCESSES WITH AN EXCITON-BOSONIZATION METHOD

SATORU OKUMURA, Tetsuo Ogawa

Published Dec 1, 2002• 2 citations
Publisher

Abstract

We theoretically investigate how deformation of an 1 s type exciton wavefunction affects four-wave-mixing signals near 1 s -exciton resonance in a semiconductor quantum well. For this we derive boson-boson interaction and boson-photon nonlinear coupling with the exciton-bosonization method, by taking into account the deformation of excitons.

DOI
10.1080/1058726021000045324

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We theoretically investigate how deformation of an 1 s type exciton wavefunction affects four-wave-mixing signals near 1 s -exciton resonance in a semiconductor quantum well. For this we derive boson-boson interaction and boson-photon nonlinear coupling with the exciton-bosonization method, by taking into account the deformation of excitons.
We derive a bosonized Hamiltonian for a semiconductor quantum well with a light field by a new bosonization method taking exactly into account the composite-particle effects of excitons. Then we apply the bosonized Hamiltonian to the four-wave-mixing (FWM) processes. The obtained Hamiltonian describes two-boson states corresponding to bound and unbound two-exciton states and guarantees exclusion of unphysical states, which cannot exist in the fermionic space due to the Pauli exclusion principle. Calculated results of the FWM signals are in good agreement with the experimental ones, and therefore, the bosonized Hamiltonian turns out to be able to describe two-exciton correlations effectively.
A theory of degenerate four-wave mixing (DFWM) in the low density excitation limit at the semiconductor band edge is developed on the basis of the bosonic representation of the electron-hole Hamiltonian. By accounting for the exciton-exciton interactions of the lowest order we arrive at the practical few-level density matrix description of the third-order excitonic nonlinearity. The developed theory accounts phenomenologically for the phase space filling, and excitation induced dephasing effects and well reproduces the results of the recent DFWM experiments in high- and low-fineness microcavity. We show that polarization-sensitive four-wave mixing measurements enable to discriminate the contributions to the DFWM signal from the exciton-exciton interactions of different symmetry.
The combined influence of biexcitons and the two-exciton scattering continuum is analyzed. The analysis is based on a density matrix description of a two-band semiconductor accounting for mean-field contributions, bound biexcitons and the correlated parts of the two-exciton scattering continuum on a common footing. Consistent with the microscopic formulation, the influence of biexcitons as well as that of the two-exciton continuum is represented by a correlation kernel which is obtained from an integral equation. For a one-dimensional model four-wave mixing signals are calculated solving numerically the resulting coupled nonlinear equations for excitation conditions in the vicinity of the 1s resonance. A strong compensation between the exciton–exciton continuum and the mean-field contribution is found for all excitation conditions studied. Surprisingly, biexcitonic signatures turn out to be enhanced when continua are included in the description.
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.
A theory of degenerate four-wave mixing in a semiconductor optical cavity with quantum wells is constructed. The nonlinear response of a microcavity can be four to five orders of magnitude stronger than that of an isolated quantum well. For P 2 E -and P 3 -type nonlinearities the damped diffracted signal oscillates with a period determined by the Rabi splitting. For a biexcitonic mechanism of nonlinearity, the signal contains damped overtones of the Rabi splitting and the biexciton binding energy.
We derive a bosonized Hamiltonian describing two-exciton correlation of semiconductor-photon coupled systems with a bosonization method, which takes into full account an effect of deviation of the excitons from ideal bosons. This deviation effect stems from the fact that excitons are composite particles, whose character appears clearly in the case where the excitons overlap each other. We call this effect a composite-particle effect (CPE). To our knowledges this effect was not considered completely in previous theoretical works on exciton-exciton interaction. The Hamiltonian introduced in this paper includes the results of the previous works as low-order terms of the CPE. After the introduction of a general theory of the bosonization method for arbitrary dimension and electron-hole mass ratio, we also demonstrate an application to a semiconductor bulk system coupled with a photon field in the heavy-hole limit. The bosonized Hamiltonian shows that the CPE brings about an enhancement of the exciton-exciton scattering strength and a qualitative change of the photo transition amplitude. It is also shown that the Hamiltonian describes two-exciton bound and scattering states.