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Journal of Mathematical Physics

The weak coupling limit for Fermions

Luigi Accardi, A. Frigerio, Yun Gang Lu

Published Jun 1, 1991• 19 citations
Publisher

Abstract

It will be shown that, in the weak coupling limit, the matrix elements in the collective number vectors of the evolution in the interacting picture of a quantum system coupled to a quasifree Fermion bath with a laser type interaction, converge to the matrix elements in some number vectors, of a unitary quantum stochastic process satisfying a quantum stochastic differential equation driven by a Fermion Brownian motion with covariance uniquely determined by the initial Fermion bath.

DOI
10.1063/1.529268

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It will be shown that, in the weak coupling limit, the matrix elements in the collective number vectors of the evolution in the interacting picture of a quantum system coupled to a quasifree Fermion bath with a laser type interaction, converge to the matrix elements in some number vectors, of a unitary quantum stochastic process satisfying a quantum stochastic differential equation driven by a Fermion Brownian motion with covariance uniquely determined by the initial Fermion bath.
We complete the program started in [4] by proving that, in the weak coupling limit, the matrix elements, in the collective coherent vectors, of the Heisenberg evolved of an observable of a system coupled to a quasi-free reservoir through a laser type interaction, converge to the matrix elements of a quantum stochastic process satisfying a quantum Langevin equation driven by a quantum Brownian motion. Our results apply to an arbitrary quasi-free reservoir so, in particular, the finite temperature case is included.
The weak coupling limit of a ‘‘system+reservoir’’ model with nonlinear (quadratic) interaction is discussed. In a previous paper [Accardi and Lu, Stoch. Process. Phys. Geom., 1–26], it has been shown that, if the matrix elements of the van Hove rescaled evolution are taken with respect to the same collective vectors used for linear interactions, then the limiting evolution is not unitary. In the present paper, we choose a new type of collective vectors and obtain, in the weak coupling limit, a unitary evolution satisfying a quantum stochastic differential equation. A remarkable feature of the limiting process is that the quadratic nature of the interaction shows up in a nontrivial choice of the Hilbert space where the limiting quantum noise takes its values, which is qualitatively new with respect to all the models considered up to now.
Dynamics of quantum systems which are stochastically perturbed by linear coupling to the reservoir can be studied in terms of quantum stochastic differential equations (for example, quantum stochastic Liouville equation and quantum Langevin equation).In order to work it out one needs to define the quantum Brownian motion.As far as only its boson version has been known until recently, in the present paper we present the definition which makes it possible to consider the fermion Brownian motion as well.