Quantum Brownian Motion: Drude and Ohmic Baths as Continuum Limits of the Rubin Model

Abstract

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 correlation functions, such as the mean squared displacement, velocity auto-correlation functions, and the response function are presented, with the aim of understanding of the various temporal regimes. In particular, we discuss the quantum to classical crossover time scales where the mean square displacement changes from a $\sim \ln t$ to a $\sim t$ dependence. We relate our study to recent work using linear response theory to understand quantum Brownian motion.

arXiv comment11 pages, 10 figures

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