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Physical Review Letters

Schrödinger's Equation as a Model Approach to Short Time-Scale Quantum Kinetics in a Semiconductor

J. A. Kenrow, T. Kenneth Gustafson

Published Oct 21, 1996• 10 citations
Publisher

Abstract

We present a new approach to quantum kinetics based on a Schr\"odinger's equation formalism. As an example we apply this method to a coupled electron-LO phonon system in a semiconductor. We demonstrate that it is possible to compute the full many-body wave function of the electron-phonon system for short times. The time-dependent electron probability distribution, extracted from the many-body wave function, illustrates the non-Markovian nature of the early time kinetics. The retarded onset of dissipation and kinetic energy overshoots are explained through virtual, nonresonant transitions.

DOI
10.1103/physrevlett.77.3605

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We present a new approach to quantum kinetics based on a Schr"odinger's equation formalism. As an example we apply this method to a coupled electron-LO phonon system in a semiconductor. We demonstrate that it is possible to compute the full many-body wave function of the electron-phonon system for short times. The time-dependent electron probability distribution, extracted from the many-body wave function, illustrates the non-Markovian nature of the early time kinetics. The retarded onset of dissipation and kinetic energy overshoots are explained through virtual, nonresonant transitions.
A phase space description of Schrödinger dynamics is provided in terms of a quantum kinetic formalism relying on the introduction of an appropriate extension of the well-known Wigner transform, also accounting for time delocalizations. This “space-time Wigner distribution,” built up in the framework of two-time correlation functions, is shown to be governed by a non-Markovian, integro-differential equation of convolution type. Its utility in investigating long time dynamics of quantum systems is also discussed and illustrated with some examples.
An effective single-particle Schrodinger equation to include dissipation into quantum devices is presented. This effective equation is fully understood in the context of Bohmian mechanics, a theory of particles and waves, where it is possible to define unambiguously the wave function of a subsystem, the so-called conditional wave function. In particular the change in energy and momentum of an electron when interacting with a phonon is presented, both theoretically and numerically. This work is a first step to include dissipation into the fully-quantum simulator BITLLES.
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.
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.
Starting from the ab initio many-body theory of electrons and phonons, we go through a series of well defined simplifications to derive a set of coupled equations of motion for the electronic occupations and polarizations, nuclear displacements as well as phononic occupations and coherences. These are the semiconductor electron-phonon equations (SEPE), sharing the same scaling with system size and propagation time as the Boltzmann equations. At the core of the SEPE is the mirrored Generalized Kadanoff-Baym Ansatz (GKBA) for the Green’s functions, an alternative to the standard GKBA which we show to lead to unstable equilibrium states. The SEPE treat coherent and incoherent degrees of freedom on equal footing, widen the scope of the semiconductor Bloch equations and Boltzmann equations, and reduce to them under additional simplifications. The new features of the SEPE pave the way for first-principles studies of phonon squeezed states and coherence effects in time-resolved absorption and diffraction experiments.
Without access to the full quantum state, modeling dissipation in an open system requires approximations. The physical soundness of such approximations relies on using realistic microscopic models of dissipation that satisfy completely positive dynamical maps. Here we present an approach based on the use of the Bohmian conditional wave function that, by construction, ensures a completely positive dynamical map for either Markovian or non-Markovian scenarios while allowing the implementation of realistic dissipation sources. Our approach is applied to compute the current-voltage characteristic of a resonant tunneling device with a parabolic-band structure, including electron-lattice interactions. A stochastic Schr\"odinger equation is solved for the conditional wave function of each simulated electron. We also extend our approach to (graphenelike) materials with a linear band structure using Bohmian conditional spinors for a stochastic Dirac equation.