Quantum Brownian Motion with Inhomogeneous Damping and Diffusion

TL;DR

This paper develops a detailed quantum master equation for inhomogeneous damping and diffusion in quantum Brownian motion, revealing Gaussian stationary states and potential squeezing effects, with implications for experimental realization in quantum gases.

Contribution

It introduces a systematic derivation of the quantum master equation with inhomogeneous damping, including novel terms for nonlinear couplings, and analyzes stationary states and decoherence in this context.

Findings

Gaussian stationary states can be squeezed and super-localized.

Derived conditions for the validity of the Markov approximation.

Predicted decoherence rates for quantum superpositions in position.

Abstract

We analyze the microscopic model of quantum Brownian motion, describing a Brownian particle interacting with a bosonic bath through a coupling which is linear in the creation and annihilation operators of the bath, but may be a nonlinear function of the position of the particle. Physically, this corresponds to a configuration in which damping and diffusion are spatially inhomogeneous. We derive systematically the quantum master equation for the Brownian particle in the Born-Markov approximation and we discuss the appearance of novel terms, for various polynomials forms of the coupling. We discuss the cases of linear and quadratic coupling in great detail and we derive, using Wigner function techniques, the stationary solutions of the master equation for a Brownian particle in a harmonic trapping potential. We predict quite generally Gaussian stationary states, and we compute the aspect ratio and the spread of the distributions. In particular, we find that these solutions may be squeezed (super-localized) with respect to the position of the Brownian particle. We analyze various restrictions to the validity of our theory posed by non-Markovian effects and by the Heisenberg principle. We further study the dynamical stability of the system, by applying a Gaussian approximation to the time dependent Wigner function, and we compute the decoherence rates of coherent quantum superpositions in position space. Finally, we propose a possible experimental realization of the physics discussed here, by considering an impurity particle embedded in a degenerate quantum gas.