Two derivations of the master equation of quantum Brownian motion

Abstract

Central to many discussion of decoherence is a master equation for the reduced density matrix of a massive particle experiencing scattering from its surrounding environment, such as that of Joos and Zeh. Such master equations enjoy a close relationship with spontaneous localization models, like the GRW model. This aim of this paper is to present two derivations of the master equation.The first derivation is a pedagogical model designed to illustrate the origins of the master equation as simply as possible, focusing on physical principles and without the complications of S-matrix theory. This derivation may serve as a useful tutorial example for students attempting to learn this subject area. The second is the opposite: a very general derivation using non-relativistic many body field theory. It reduces to the equation of the type given by Joos and Zeh in the one-particle sector, but correcting certain numerical factors which have recently become significant in connection with experimental tests of decoherence. This master equation also emphasizes the role of local number density as the "preferred basis" for decoherence in this model.

Decoherence, discord, and the quantum master equation for cosmological perturbations

“…Whilst Hermiticity and normalisation follow trivially, the most important property -that ρ(η) be positive definite -remains to be established. Indeed, even the Caldeira-Leggett model can lead to a small violation of positivity [10,34]. In fact, we shall see the operator form of the master equation and the oscillatory behaviour of the coefficients mean that the master equation manifestly violates positivity on sub-horizon scales.…”

“…A useful toy model which has many parallels with the present investigation is the much-discussed subject of quantum Brownian motion [10,33,34,40]. The system consists of a particle, which, in the simplest setup, moves in one dimension, interacting with an environment.…”

Adjoint master equation for quantum Brownian motion

“…which can be solved directly without having to solve a more complicated system of differential equations, as it is necessary when the solution is in the Schrödinder picture [2], as well as for the case of the Wigner function approach [41,51,52]. Once the interaction with the bath, i.e.…”

“…Once the functional dependence of the physical quantities on the initial values is computed, we direct obtain their time dependence for different initial states simply by inserting the initial expectation values. On the other hand, when working in the Schrödinger picture, as typically done in the literature [2,33], or with the Wigner formalism [41,51,52], one has to find the explicit time evolution of the initial state, which changes depending on the initial state.…”

Collisional decoherence of a tracer particle moving in one dimension

“…We wish to mention that a Wigner function description of quantum Brownian motion was already put forward previously in a very heuristic derivation [18], as well as in a more precise approach, but limited to states of the tracer particle which are close to a thermal state [19]. Both articles are concerned with the general form of a partial differential equation for the Wigner function, and do not study decoherence.…”