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Open Systems & Information Dynamics

Low Density Limit and the Quantum Langevin Equation for the Heat Bath

Ameur Dhahri

Published Nov 30, 2009• 0 citations
Publisher

Abstract

We consider a repeated quantum interaction model describing a small system [Formula: see text] in interaction with each of the identical copies of the chain ⊗ℕ* ℂn+1, modeling a heat bath, one after another during the same short time intervals [0, h]. We suppose that the repeated quantum interaction Hamiltonian is split into two parts: a free part and an interaction part with time scale of order h. After giving the GNS representation, we establish the connection between the time scale h and the classical low density limit. We introduce a chemical potential µ related to the time h as follows: h2 = eβµ. We further prove that the solution of the associated discrete evolution equation converges strongly, when h tends to 0, to the unitary solution of a quantum Langevin equation directed by the Poisson processes.

DOI
10.1142/s1230161209000268

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We consider a repeated quantum interaction model describing a small system [Formula: see text] in interaction with each of the identical copies of the chain ⊗ℕ* ℂn+1, modeling a heat bath, one after another during the same short time intervals [0, h]. We suppose that the repeated quantum interaction Hamiltonian is split into two parts: a free part and an interaction part with time scale of order h. After giving the GNS representation, we establish the connection between the time scale h and the classical low density limit. We introduce a chemical potential µ related to the time h as follows: h2 = eβµ. We further prove that the solution of the associated discrete evolution equation converges strongly, when h tends to 0, to the unitary solution of a quantum Langevin equation directed by the Poisson processes.
We consider a repeated quantum interaction model describing a small system $\Hh_S$ in interaction with each one of the identical copies of the chain $\bigotimes_{\N^*}\C^{n+1}$, modeling a heat bath, one after another during the same short time intervals [0,h][0,h][0,h]. We suppose that the repeated quantum interaction Hamiltonian is split in two parts: a free part and an interaction part with time scale of order hhh. After giving the GNS representation, we establish the relation between the time scale hhh and the classical low density limit. We introduce a chemical potential μμμ related to the time hhh as follows: h2=eβμh^2=e^{βμ}h2=eβμ. We further prove that the solution of the associated discrete evolution equation converges strongly, when hhh tends to 0, to the unitary solution of a quantum Langevin equation directed by Poisson processes.
We consider the physical model of a classical mechanical system (called "small system") undergoing repeated interactions with a chain of identical small pieces (called "environment"). This physical setup constitutes an advantageous way of implementing dissipation for classical systems, it is at the same time Hamiltonian and Markovian. This kind of model has already been studied in the context of quantum mechanical systems, where it was shown to give rise to quantum Langevin equations in the limit of continuous time interactions ([2]), but it has never been considered for classical mechanical systems yet. The aim of this article is to compute the continuous limit of repeated interactions for classical systems and to prove that they give rise to particular stochastic differential equations in the limit. In particular we recover the usual Langevin equations associated to the action of heat baths. In order to obtain these results, we consider the discrete-time dynamical system induced by Hamilton's equations and the repeated interactions. We embed it into a continuous-time dynamical system and we compute the limit when the time step goes to 0. This way we obtain a discrete-time approximation of stochastic differential equation, considered as a deterministic dynamical system on the Wiener space, which is not exactly of the usual Euler scheme type. We prove the Lp and almost sure convergence of this scheme. We end up with applications to concrete physical exemples such as a charged particule in an uniform electric field or a harmonic interaction. We obtain the usual Langevin equation for the action of a heat bath when considering a damped harmonic oscillator as the small system.
The white noise approach to the investigation of the dynamics of a quantum particle interacting with a dilute and in general non-equilibrium gaseous environment in the low density limit is outlined. The low density limit is the kinetic Markovian regime when only pair collisions (i.e., collisions of the test particle with one particle of the gas at one time moment) contribute to the dynamics. In the white noise approach one first proves that the appropriate operators describing the gas converge in the sense of appropriate matrix elements to certain operators of quantum white noise. Then these white noise operators are used to derive quantum white noise and quantum stochastic equations describing the approximate dynamics of the total system consisting of the particle and the gas. The derivation is given ab initio, starting from the exact microscopic quantum dynamics. The limiting dynamics is described by a quantum stochastic equation driven by a quantum Poisson process. This equation then applied to the derivation of quantum Langevin equation and linear Boltzmann equation for the reduced density matrix of the test particle. The first part of the paper describes the approach which was developed by L. Accardi, I.V. Volovich and the author and uses the Fock-antiFock (or GNS) representation for the CCR algebra of the gas. The second part presents the approach to the derivation of the limiting equations directly in terms of the correlation functions, without use of the Fock-antiFock representation. This approach simplifies the derivation and allows to express the strength of the quantum number process directly in terms of the one-particle SSS-matrix.
Starting with a quantum Langevin equation describing in the Heisenberg representation a quantum system coupled to a quantum bath, the Markov approximation and, further, the closure approximation are applied to derive a semiclassical Langevin equation for the second-order quantized Hamilton dynamics (QHD) coupled to a classical bath. The expectation values of the system operators are decomposed into products of the first and second moments of the position and momentum operators that incorporate zero-point energy and moderate tunneling effects. The random force and friction as well as the system-bath coupling are decomposed to the lowest classical level. The resulting Langevin equation describing QHD-2 coupled to classical bath is analyzed and applied to free particle, harmonic oscillator, and the Morse potential representing the OH stretch of the SPC-flexible water model.
In this paper, we derive the stochastic master equations corresponding to the statistical model of a heat bath. These stochastic differential equations are obtained as continuous time limits of discrete models of quantum repeated measurements. Physically, they describe the evolution of a small system in contact with a heat bath undergoing continuous measurement. The equations obtained in the present work are qualitatively different from the ones derived in [6], where the Gibbs model of heat bath has been studied. It is shown that the statistical model of a heat bath has a clear physical interpretation in terms of emissions and absorptions of photons. Our approach yields models of random environment and unravelings of stochastic master equations. The equations are rigorously obtained as solutions of martingale problems using the convergence of Markov generators.
In this article, we derive the stochastic master equations corresponding to the sta-tistical model of a heat bath. These stochastic differential equations are obtained as continuous time limits of discrete models of quantum repeated measurements. Physically, they describe the evolution of a small system in contact with a heat bath undergoing continuous measurement. The equations obtained in the present work are qualitatively different from the ones derived in [6], where the Gibbs model of heat bath has been studied. It is shown that the statistical model of a heat bath provides clear physical interpretation in terms of emissions and absorptions of photons. Our approach yields models of random environment and unravelings of stochastic master equations. The equations are rigorously obtained as solutions of martingale problems using the convergence of Markov generators.