Matsubara Dynamics for Open Quantum Systems

Abstract

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.