Statistical correlations in the oscillator model of quantum dissipative systems

TL;DR

This paper investigates the initial conditions in the oscillator model of quantum dissipative systems, proposing correlated initial states based on symmetry, which yield physically consistent results aligning quantum and classical Brownian motion.

Contribution

It introduces a new form of correlated initial conditions and a corresponding Lagrangian for quantum dissipative systems, improving upon unphysical results from previous models.

Findings

Correlated initial conditions produce physically consistent quantum Brownian motion results.

Uncorrelated or factorized initial conditions lead to non-physical outcomes.

Symmetry considerations uniquely determine the form of the Lagrangian.

Abstract

The problem of the initial conditions for the oscillator model of quantum dissipative systems is studied. It is argued that, even in the classical case, the hypothesis that the environment is in thermal equilibrium implies a statistical correlation between environment oscillators and central system. A simple form of initial conditions for the quantum problem, taking into account such a correlation in analogy with the classical ones, is derived on the base of symmetry considerations. The same symmetries also determine unambiguously the form of the Lagrangian. As a check of the new form of correlated initial conditions (and of that of the Lagrangian), the problem of a forced Brownian particle under the action of arbitrary colored noise is studied: it is shown that one obtains an average position of a quantum wave packet equal to that of the corresponding classical Brownian particle. Instead, starting from uncorrelated initial conditions based on the factorization hypothesis or from a different form of Lagrangian, non-physical results are obtained. Similar considerations apply also to the mean square displacement.