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physica status solidi (b)

Quantum Kinetic Equations with Two Particle Correlations Applied to an Exactly Solved Model of Relaxing Electrons

C. Wöhler, Kurt Schönhammer

Published Nov 1, 1997• 2 citations
Publisher

Abstract

Quantum kinetic equations (QKE) are used for the description of highly excited electrons in the conduction band of semiconductors in the short time regime. The relaxation process is described with QKE in Born approximation and with QKE including two particle correlations. The results of the approximations are compared for different particle densities with the exact solution of a one-dimensional model of N electrons with linear dispersion interacting with optical phonons. Especially the discussion of the processes contributing to the gain peak allows a better understanding of the approximations.

DOI
10.1002/1521-3951(199711)204:1<362::aid-pssb362>3.0.co;2-z

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Quantum kinetic equations (QKE) are used for the description of highly excited electrons in the conduction band of semiconductors in the short time regime. The relaxation process is described with QKE in Born approximation and with QKE including two particle correlations. The results of the approximations are compared for different particle densities with the exact solution of a one-dimensional model of N electrons with linear dispersion interacting with optical phonons. Especially the discussion of the processes contributing to the gain peak allows a better understanding of the approximations.
We present a one-dimensional microscopic model of interacting electrons coupled to phonons that allows the exact calculation of the time evolution of the momentum distribution functions starting from a nonequilibrium initial state. The model shows relaxation features known from approximate calculations for more realistic three-dimensional models. The exact results are compared with approximate ones resulting from the use of quantum kinetic equations which avoid the assumption of completed successive scattering processes implicit in simple Boltzmann equations. To renormalize the electronic but not the phonon propagation in the non-Markovian equations yields no improvement uniform in time compared to simpler approximations. Due to the presence of the Fermi sea only the additional renormalization of the phonon propagation leads to a drastic improvement in comparison with the exact results.
The exact nonequilibrium time evolution of the momentum distribution for a finite many particle system in one dimension with a linear energy dispersion coupled to optical phonons is presented. For distinguishable particles the influence function of the phonon bath can be evaluated also for a finite particle density in the thermodynamic limit. In the case of fermions the exact fulfillment of the Pauli principle involves a sum over permutations of the electrons and the numerical evaluation is restricted to a finite number of electrons. In the dynamics the antisymmetry of the wavefunction shows up in the obvious Pauli blocking of momentum states as well as more subtle interference effects. The model shows the expected physical features known from approximate treatments of more realistic models for the relaxation in the energy regime far from the bottom of the conduction band and provides an excellent testing ground for quantum kinetic equations.
The exact nonequilibrium time evolution of the momentum distribution for a finite many-particle system in one dimension with a linear energy dispersion coupled to optical phonons is presented. For distinguishable particles the influence function of the phonon bath can be evaluated also for a finite particle density in the thermodynamic limit. In the case of fermions the exact fulfillment of the Pauli principle involves a sum over permutations of the electrons and the numerical evaluation is restricted to a finite number of electrons. In the dynamics the antisymmetry of the wave function shows up in the obvious Pauli blocking of momentum states as well as more subtle interference effects. The model shows the expected physical features known from approximate treatments of more realistic models for the relaxation in the energy regime far from the bottom of the conduction band and provides an excellent testing ground for quantum kinetic equations.
We report direct investigations of the coherence of four-particle (exciton-exciton) correlations by ultrafast four-wave mixing experiments. We determine the sensitivity of the coherence of the four-particle correlations to temperature and excitation density, comparing it to the coherence of the two-particle correlations. We discuss our results in terms of numerical solutions of the general kinetic equations, treating the four-particle correlations in the second order Born approximation. We introduce a simple model deduced from these kinetic equations that provides an intuitive interpretation of our results.
We study the dynamics of relaxation and thermalization in an exactly solvable model of a quantum particle interacting with a thermal bath of oscillators with the goal of understanding the effects of off-shell processes. The focus is to compare the exact evolution of the distribution function with different approximations to the relaxational dynamics: Boltzmann, non-Markovian and Markovian quantum kinetics. The time evolution of the distribution function is evaluated exactly using different approaches where each method provides different insights. There are two different cases that are studied in detail: (i) no stable particle states below threshold of the bath and a quasiparticle resonance above it and (ii) a stable discrete exact ``particle'' state below threshold. The exact solution for the evolution allows us to investigate the concept of the formation time of a quasiparticle and to study the difference between the relaxation of the distribution of particles and quasiparticles. For the case of quasiparticles in the continuum (resonances) the exact quasiparticle distribution asymptotically tends to a statistical equilibrium distribution that differs from a simple Bose-Einstein form as a result of off-shell processes such as the strength of the quasiparticle poles, the width of the unstable particle and proximity to thresholds. In case (ii), the distribution of particles does not thermalize with the bath. We study the kinetics of thermalization and relaxation by deriving a non-Markovian quantum kinetic equation which resums the perturbative series and includes off-shell effects. A Markovian approximation that includes off-shell contributions and the usual Boltzmann equation are obtained from the quantum kinetic equation in the limit of a wide separation of time scales upon different coarse-graining assumptions. The relaxational dynamics predicted by the non-Markovian, Markovian and Boltzmann approximations are compared to the exact result of the model. The Boltzmann approach is seen to fail in the case of wide resonances and when threshold and renormalization effects are important. Implications for thermalization in field theory models are discussed.
We study the dynamics of relaxation and thermalization in an exactly solvable model of a particle interacting with a harmonic oscillator bath. Our goal is to understand the effects of non-Markovian processes on the relaxational dynamics and to compare the exact evolution of the distribution function with approximate Markovian and non-Markovian quantum kinetics. There are two different cases that are studied in detail: (i) a quasiparticle (resonance) when the renormalized frequency of the particle is above the frequency threshold of the bath and (ii) a stable renormalized "particle" state below this threshold. The time evolution of the occupation number for the particle is evaluated exactly using different approaches that yield to complementary insights. The exact solution allows us to investigate the concept of the formation time of a quasiparticle and to study the difference between the relaxation of the distribution of bare particles and that of quasiparticles. For the case of quasiparticles, the exact occupation number asymptotically tends to a statistical equilibrium distribution that differs from a simple Bose-Einstein form as a result of off-shell processes whereas in the stable particle case, the distribution of particles does not thermalize with the bath. We derive a non-Markovian quantum kinetic equation which resums the perturbative series and includes off-shell effects. A Markovian approximation that includes off-shell contributions and the usual Boltzmann equation (energy conserving) are obtained from the quantum kinetic equation in the limit of wide separation of time scales upon different coarse-graining assumptions. The relaxational dynamics predicted by the non-Markovian, Markovian, and Boltzmann approximations are compared to the exact result. The Boltzmann approach is seen to fail in the case of wide resonances and when threshold and renormalization effects are important.