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Annalen der Physik

Equilibration of a dissipative quantum oscillator

Vinay Ambegaokar

Published Jan 19, 2007• 0 citations
Publisher

Abstract

Abstract An explicit demonstration is given of a harmonic oscillator in equilibrium approaching the equilibrium of a corresponding interacting system by coupling it to a thermal bath consisting of a continuum of harmonic oscillators.
DOI
10.1002/andp.200751905-601

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We discuss how various parts of a quantum many-body system exchange energies at thermal equilibrium. To show this, we assume a quantum system is coupled to a many-body environment (at thermal equilibrium with a bigger environment) consisting of a large number of independent and non-interacting quantum harmonic oscillators above a stable ground state. Once the coupling to a large environment is switched on, the system dissipates its energy continuously to the environment until it reaches equilibrium with the latter. We use the Quantum Langevin equation to show such energy exchange at equilibrium. We conclude that different parts of a physical system can exchange energies even at absolute zero temperature.
The motion of a free quantum particle in a thermal environment is usually described by the quantum Langevin equation, where the effect of the bath is encoded through a dissipative and a noise term, related to each other via the fluctuation dissipation theorem. The quantum Langevin equation can be derived starting from a microscopic model of the thermal bath as an infinite collection of harmonic oscillators prepared in an initial equilibrium state. The spectral properties of the bath oscillators and their coupling to the particle determine the specific form of the dissipation and noise. Here we investigate in detail the well-known Rubin bath model, which consists of a one-dimensional harmonic chain with the boundary bath particle coupled to the Brownian particle. We show how in the limit of infinite bath bandwidth, we get the Drude model, and a second limit of infinite system-bath coupling gives the Ohmic model. A detailed analysis of relevant equilibrium correlation functions, such as the mean squared displacement, velocity autocorrelation functions, and response function are presented, with the aim of understanding the various temporal regimes. In particular, we discuss the quantum-to-classical crossover time scales where the mean square displacement changes from a ∼lnt to a ∼t dependence. We relate our study to recent work using linear response theory to understand quantum Brownian motion.