Quantum Brownian Motion as a Classical Stochastic Process in Phase Space

TL;DR

This work shows that the exact quantum dynamics of a Brownian particle in the Caldeira-Leggett model can be mapped, at any temperature, to a classical non-Markovian stochastic process in phase space, provided the particle potential is quadratic. The mapping leverages a correlated thermal equilibrium state and encodes quantum environments through a quantum noise consistent with the quantum fluctuation-dissipation theorem, while arbitrary state preparations are handled via the Wigner representation. For non-quadratic potentials, a natural bath-controlled coherence length $\lambda$ provides a controlled small parameter $\lambda/L$, enabling a systematic, all-temperature approximation. The authors develop a stochastic Monte Carlo method to simulate these trajectories, validate it against known Gaussian results, and demonstrate its ability to capture fast decoherence and low-temperature dynamics that are missed by classical or high-temperature master equations. This framework offers a versatile, efficient tool for driven-dissipative quantum protocols across regimes from classical high temperature to deep quantum behavior.

Abstract

We establish that the exact quantum dynamics of a Brownian particle in the Caldeira-Leggett model can be mapped, at any temperature, onto a classical, non-Markovian stochastic process in phase space. Starting from a correlated thermal equilibrium state between the particle and bath, we prove that this correspondence is exact for quadratic potentials under arbitrary quantum state preparations of the particle itself. For more general, smooth potentials, we identify and exploit a natural small parameter: the density matrix becomes strongly quasidiagonal in the coordinate representation, with its off-diagonal width shrinking as the bath's spectral cutoff increases, providing a controlled parameter for accurate approximation. The framework is fully general: arbitrary initial quantum states-including highly non-classical superpositions-are incorporated via their Wigner functions, which serve as statistical weights for trajectory ensembles. Furthermore, the formalism naturally accommodates external manipulations and measurements modeled by preparation functions acting at arbitrary times, enabling the simulation of complex driven-dissipative quantum protocols.

Figures (3)

• Figure 1: Time evolution of the coordinate dispersion $\sigma^2(t)$ for a Gaussian preparation at $T=0$. Blue line: stochastic Monte Carlo simulation. Orange line: analytical result from Eq. \ref{['eq:analytical_sigma']}. The agreement validates the numerical method. Parameters: $\sigma_0=1$, $\gamma=\pi/2$, $\varepsilon=0.5$, $m=\hbar=1$.
• Figure 2: Decoherence of a Schrödinger cat state. The observable $\langle O \rangle$ (Eq. \ref{['eq:cat_observable']}) decays fastest for the exact quantum noise (blue line). The high-temperature master equation (orange) and white-noise process (green) give similar, slower decoherence, failing to capture the enhanced quantum fluctuations at low temperature. Parameters: $\gamma=\pi/2$, $\varepsilon=0.01$, $T=1$, $m=\hbar=1$.
• Figure 3: Momentum dispersion $\langle p^2(t) \rangle$ relaxation for Schrödinger cat state. The exact quantum result (blue and orange lines, analytical and stochastic) shows rapid initial growth on a timescale $\sim \varepsilon$, absent in the high-temperature white-noise model (green). This demonstrates the quantum bath's role in fast early-time dynamics. Parameters: $\gamma=\pi/2$, $\varepsilon=0.01$, $T=1$, $m=\hbar=1$.