Uncertainty Relation for a Quantum Open System

TL;DR

This paper derives a generalized uncertainty relation for a quantum open system interacting with a thermal bath, revealing how environmental fluctuations influence quantum-to-classical transition and decoherence times.

Contribution

It provides analytic expressions for the uncertainty relation at high temperature and weak damping, including effects of initial squeezing, and links decoherence to the transition from quantum to classical behavior.

Findings

System evolves from quantum to thermal state at decoherence time

Analytic uncertainty relations derived for high temperature and weak damping

Squeezing parameter affects the generalized uncertainty relation

Abstract

We derive the uncertainty relation for a quantum open system comprised of a Brownian particle interacting with a bath of quantum oscillators at finite temperature. We examine how the quantum and thermal fluctuations of the environment contribute to the uncertainty in the canonical variables of the system. We show that upon contact with the bath (assumed ohmic in this paper) the system evolves from a quantum-dominated state to a thermal-dominated state in a time which is the same as the decoherence time in similar models in the discussion of quantum to classical transition. This offers some insight into the physical mechanisms involved in the environment-induced decoherence process. We obtain closed analytic expressions for this generalized uncertainty relation under the conditions of high temperature and weak damping separately. We also consider under these conditions an arbitrarily-squeezed initial state and show how the squeeze parameter enters in the generalized uncertainty relation. Using these results we examine the transition of the system from a quantum pure state to a nonequilibrium quantum statistical state and to an equilibrium quantum statistical state. The three stages are marked by the decoherence time and the relaxation time respectively. With these observations we explicate the physical conditions when the two basic postulates of quantum statistical mechanics become valid. We also comment on the inappropriateness in the usage of the word classicality in many decoherence studies of quantum to classical transition.