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The European Physical Journal Special Topics

Quantum Brownian motion for embedded system

Meng Xu, Joachim Ankerhold

Published Sep 19, 2025• 2 citations
Publisher

Abstract

Abstract Conventional derivations of the Wigner phase-space Fokker–Planck equation for open quantum systems often rely on heuristic coarse-graining of system-bath interactions. We present quantum dissipation in a minimally extended state space (QD-MESS), a unified framework that systematically models Gaussian bath dynamics in Wigner phase space with controllable numerical accuracy, yields a transparent quantum-classical picture for arbitrary noise spectra, and recovers the quantum-optical, Caldeira–Leggett, and quantum Smoluchowski limits.

DOI
10.1140/epjs/s11734-025-01925-0

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Quantum Brownian motion for embedded system

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Conventional derivations of the Wigner phase-space Fokker–Planck equation for open quantum systems often rely on heuristic coarse-graining of system-bath interactions. We present quantum dissipation in a minimally extended state space (QD-MESS), a unified framework that systematically models Gaussian bath dynamics in Wigner phase space with controllable numerical accuracy, yields a transparent quantum-classical picture for arbitrary noise spectra, and recovers the quantum-optical, Caldeira–Leggett, and quantum Smoluchowski limits.

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Abstract Conventional derivations of the Wigner phase-space Fokker–Planck equation for open quantum systems often rely on heuristic coarse-graining of system-bath interactions. We present quantum dissipation in a minimally extended state space (QD-MESS), a unified framework that systematically models Gaussian bath dynamics in Wigner phase space with controllable numerical accuracy, yields a transparent quantum-classical picture for arbitrary noise spectra, and recovers the quantum-optical, Caldeira–Leggett, and quantum Smoluchowski limits.
Conventional derivations of the Wigner phase-space Fokker–Planck equation for open quantum systems often rely on heuristic coarse-graining of system-bath interactions. We present quantum dissipation in a minimally extended state space (QD-MESS), a unified framework that systematically models Gaussian bath dynamics in Wigner phase space with controllable numerical accuracy, yields a transparent quantum-classical picture for arbitrary noise spectra, and recovers the quantum-optical, Caldeira–Leggett, and quantum Smoluchowski limits.
We develop an exact quantum thermostat that generates the quantum Boltzmann distribution in path integral phase space and so correctly captures the imaginary position–momentum correlations. This extends beyond the capability of existing quantum thermostats such as the PILE thermostat, which can only achieve the marginal of this distribution in position, with the momentum acting purely as a sampling tool and satisfying classical Maxwell–Boltzmann statistics. The dynamics correspond to the treatment of the Caldeira–Leggett system–bath model with Matsubara dynamics, a theory for approximating nuclear quantum dynamics by propagating smooth imaginary-time Feynman paths in real time. A Matsubara generalised Langevin equation that obeys certain quantum fluctuation–dissipation relations is mapped onto a Markovian stochastic process by introducing auxiliary variables; this allows the construction of a corresponding Fokker–Planck equation from which the stationary distribution in the extended variable space is determined. We show that this corresponds to the joint quantum Boltzmann distribution in position and momentum, a finding that holds for any bath spectral density and for anharmonic systems. The sole approximation in Matsubara dynamics is the neglect of real-time quantum coherence; consequently, provided there is sufficient quantum decoherence, the dynamics of the exact quantum thermostat also well approximates the exact quantum dynamics of the system–bath. In the limit of infinite system–bath coupling, the level of real-time coherence becomes negligible and so the Matsubara dynamics of the thermostat tends to the exact dynamics. The quantum thermostat can therefore be used to study this limit exactly and, in so doing, we demonstrate the existence of a quantum mechanical overdamped regime analogous to its classical counterpart that is characterised by a Matsubara Smoluchowski equation. The drawback of the quantum thermostat is that one of its associated quantum fluctuation–dissipation relations is imaginary. This results in complex noise and dynamical instabilities that often preclude the numerical implementation of the thermostat. Instead, it primarily serves as a theoretical framework that can be used as a starting point for introducing approximations that yield more practical methods. One such approximation that results in real noise is considered here and leads to a practical approach for calculating linear time-correlation functions of bound open quantum systems. These exhibit good agreement with exact quantum mechanical calculations, particularly at strong system–bath coupling, where the real-time quantum coherence is increasingly suppressed.
We establish that the exact quantum dynamics of a Brownian particle in the Caldeira-Leggett model, with at most quadratic external potential, 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 demonstrate that this correspondence is exact for quadratic potentials under arbitrary quantum state preparations of the particle itself. Our approach allows to consider arbitrary initial quantum states - including highly non-classical superpositions - which 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. For more general, smooth potentials, we identify 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, suggesting a controlled parameter for a possible approximation.
Open Quantum Brownian Motion (OQBM) is a new class of quantum Brownian motion in which the dynamics of the Brownian particle depend not only on interactions with a thermal environment but also on the state of its internal degrees of freedom. For an Ohmic bath spectral density with a Lorentz--Drude cutoff frequency at a high-temperature limit, we derive the Born--Markov master equation for the reduced density matrix of an open Brownian particle in a harmonic potential. The resulting master equation is written in phase-space representation using the Wigner function, and due to the separation of associated timescales in the high-damping limit, we perform adiabatic elimination of the momentum variable to obtain OQBM. We numerically solve the derived master equation for the reduced density matrix of the OQBM, using both Gaussian and non-Gaussian initial distributions. In each case, the OQBM dynamics converge to several Gaussian distributions. To gain physical insight into the studied system, we also plotted the dynamics of the off-diagonal element of the open quantum Brownian particle and found damped coherent oscillations. Finally, we investigated the time-dependent variance in the position of the OQBM walker and observed initial diffusive spreading, followed by a crossover to a finite steady-state value.
A model that generalizes the study of quantum Brownian motion (BM) is constructed. We consider disordered environment that may be either static (quenched), noisy or dynamical. The Zwanzig-Caldeira-Leggett BM model formally constitutes a special case where the disorder autocorrelation length is taken to be infinite. Alternatively, a localization problem is obtained if the noise autocorrelation time is taken to be infinite. Also the general case of weak nonlinear coupling to a thermal, possibly chaotic bath is handled by the same formalism. A general, Feynman-Vernon type path-integral expression for the propagator is introduced. A Wigner transformed version of this expression is utilized in order to facilitate comparison with the classical limit. It is demonstrated that nonstochastic genuine quantal manifestations are associated with the model. It is clarified that such effects are absent in the standard BM model, either the disorder or the chaotic nature of the bath are essential. Quantal correction to the classical diffusive behavior is found even in the limit of high temperatures. The suppression of interference due to dephasing is discussed, leading to the observation that due to the disorder the decay of coherence is exponential in time, and no longer depends on geometrical considerations. Fascinating non-Markovian effects due to time-correlated (colored) noise are explored. For this, a strategy is developed in order to handle the integration over paths. This strategy is extended in order to demonstrate how localization comes out from the path-integral expression.