Solutions to Master Equations of Quantum Brownian Motion in a General Environment with External Force

TL;DR

This paper derives a closed-form solution to the master equation for a quantum Brownian oscillator under external force in general environments, enabling analysis of quantum limits and state evolution at arbitrary temperatures.

Contribution

It provides a new, general solution to the HPZ master equation for various spectral densities and temperatures, extending previous results to broader conditions.

Findings

Gaussian states are the long-term equilibrium states influenced by the thermal reservoir.

The solution reduces to a classical parametric oscillator problem for complex spectra.

Late-time uncertainty functions can be fully determined, aiding quantum measurement analysis.

Abstract

We revisit the model of a system made up of a Brownian quantum oscillator under the influence of an external classical force and linearly coupled to an environment made up of many quantum oscillators at zero or finite temperature. We show that the HPZ master equation for the reduced density matrix derived earlier [B.L. Hu, J.P. Paz, Y. Zhang, Phys. Rev. D 45, 2843 (1992)] with coefficients obtained from solutions of integro-differential equations can assume closed functional forms for a fairly general class of spectral densities of the environment at arbitrary temperature and coupling strength. As an illustration of these new results we solve the corresponding master equation and calculate, among other physical quantities, the uncertainty function whose late time behavior can be obtained fully. This produces a formula for investigating the standard quantum limit which is central to addressing many theoretical issues in macroscopic quantum phenomena and experimental concerns related to low temperature precision measurements. We find that any initial state always settles down to a Gaussian density matrix whose covariance is determined by the thermal reservoir and whose mean is determined by the external force. For more general spectra we show that the solution of the master equation can be reduced to solving for the motion of a classical parametric oscillator with parametric frequency determined by the unsolved for master equation coefficients. States in these systems experience evolution that is parametrically similar to the simpler evolution explicitly determined for in the case of Laurent-series spectra.