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arXiv:math-ph/0206032 · Infinite Dimensional Analysis Quantum Probability and Related Topics

A STOCHASTIC GOLDEN RULE AND QUANTUM LANGEVIN EQUATION FOR THE LOW DENSITY LIMIT

Luigi Accardi, Alexander Nikolaevich Pechen, Igor Vasil'evich Volovich

Published Sep 1, 2003• 18 citations
arXivHTMLPDF

Abstract

A rigorous derivation of quantum Langevin equation from microscopic dynamics in the low density limit is given. We consider a quantum model of a microscopic system (test particle) coupled with a reservoir (gas of light Bose particles) via interaction of scattering type. We formulate a mathematical procedure (the so-called stochastic golden rule) which allows us to determine the quantum Langevin equation in the limit of large time and small density of particles of the reservoir. The quantum Langevin equation describes not only dynamics of the system but also the reservoir. We show that the generator of the corresponding master equation has the Lindblad form of most general generators of completely positive semigroups.

DOI
10.1142/s0219025703001304

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A rigorous derivation of quantum Langevin equation from microscopic dynamics in the low density limit is given. We consider a quantum model of a microscopic system (test particle) coupled with a reservoir (gas of light Bose particles) via interaction of scattering type. We formulate a mathematical procedure (the so-called stochastic golden rule) which allows us to determine the quantum Langevin equation in the limit of large time and small density of particles of the reservoir. The quantum Langevin equation describes not only dynamics of the system but also the reservoir. We show that the generator of the corresponding master equation has the Lindblad form of most general generators of completely positive semigroups.
We use the stochastic limit method to study long time quantum dynamics of a test particle interacting with a dilute Bose gas. The case of arbitrary form factors and an arbitrary, not necessarily equilibrium, quasifree low density state of the Bose gas is considered. Starting from microscopic dynamics we derive in the low density limit a quantum white noise equation for the evolution operator. This equation is equivalent to a quantum stochastic equation driven by a quantum Poisson process with intensity S−1, where S is a one-particle S matrix. The novelty of our approach is that the equations are derived directly in terms of correlators, without use of a Fock–anti-Fock (or Gel’fand–Naimark–Segal) representation. Advantages of our approach are the simplicity of derivation of the limiting equation and that the algebra of the master fields and the Ito table do not depend on the initial state of the Bose gas. The notion of a causal state is introduced. We construct master fields (white noise and number operators) describing the dynamics in the low density limit and prove the convergence of chronological (causal) correlators of the field operators to correlators of the master fields in the causal state.
A new derivation of the quantum stochastic differential equation for the evolution operator in the low density limit is presented. We use the distribution approach and derive a new algebra for quadratic master fields in the low density limit by using the energy representation. We formulate the stochastic golden rule in the low density limit case for a system coupling with a Bose field via quadratic interaction. In particular, the vacuum expectation value of the evolution operator is computed and its exponential decay is shown.
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.
We investigate the structure of quantum Markov generators that describe the reduced dynamics of a test particle interacting with a dilute Bose gas in the low density limit. These generators, called low density limit type generators (LDL), differ from the weak coupling limit type generators (WCL) studied in [4, 5] because of the presence of the T-operator that describes the change in momentum of the gas particle due to collisions with the test particle. We propose a general definition of Markov generator of stochastic limit type that includes (almost) all generators arising in the stochastic limit approach and we prove that the associated semigroups as well as their invariant states (or weights) have a very special structure which extends the explicit representation for the generic quantum Markov semigroups of weak coupling limit type, due to Accardi, Hachicha and Ouerdiane [7]. We also investigate the structure of invariant states and give conditions on the T - operator for the existence of a unique invariant state and of equilibrium states.
The authors conclude a research programme developed by various authors in several papers. The basic idea of this programme is to explain the irreversible behaviour of quantum systems as a limiting case (in a sense to be made precise) of usual quantum dynamics. One starts with a system interacting with a reservoir and, in the first attempts to deal with this problem, only limits of observables of the system and deduced master equations were considered. In their approach they study the limits of quantities related to the whole compound system. As a corollary they obtain an explanation of the physical origins of the quantum Brownian motion. They study the low-density limit of a system coupled to a quasi-free boson reservoir in the equilibrium state at inverse temperature beta and fugacity z, through an interaction of the scattering type, i.e. one which preserves the total number of particles of the reservoir.
The effects of decoherence for quantum system coupled with a bosonic field are investigated. An application of the stochastic golden rule shows that in the stochastic limit the dynamics of such a system is described by a quantum stochastic differential equation. The corresponding master equation describes convergence of a system to equilibrium. In particular it predicts exponential damping for off-diagonal matrix elements of the system density matrix, moreover these elements for a generic system will decay at least as \exp(-tN{kT\over\hbar}), where N is a number of particles in the system. As an application of the described technique a derivation from first principles (i.e. starting from a Hamiltonian description) of a quantum extension of the Glauber dynamics for systems of spins is given.