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arXiv:2606.10665

Equilibrating continuous-variable open quantum systems using stochastic classical trajectories in path-integral space

William H. D. Moore, Stuart C. Althorpe

quant-phphysics.chem-phSubmitted Jun 9, 2026• Announced Jun 10, 2026• 0 citations
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Abstract

Beyond the weak-coupling limit, open quantum systems equilibrate to a highly entangled thermal state. For continuous-variable systems, this state can be written explicitly as an imaginary-time phase-space path integral, in which the positions are directly entangled with the bath, and the momenta are correlated with the positions through a phase term. Here, we ask to what extent this state can be reached by propagating stochastic classical trajectories in path-integral phase space. Surprisingly, we find that the trajectories equilibrate to the exact quantum equilibrium state, recovering the purely imaginary momentum-position correlation in the phase term. The trajectories are generated using a recently derived Matsubara generalized Langevin equation, which produces the imaginary correlations by evolving the stochastic variables into the complex plane. This makes the dynamics numerically unstable, but we are nonetheless able to demonstrate the equilibration of a quartic oscillator coupled to a white-noise bath. These unexpected findings could lead to new approximate methodologies for simulating continuous-variable open quantum systems.

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Beyond the weak-coupling limit, open quantum systems equilibrate to a highly entangled thermal state. For continuous-variable systems, this state can be written explicitly as an imaginary-time phase-space path integral, in which the positions are directly entangled with the bath, and the momenta are correlated with the positions through a phase term. Here, we ask to what extent this state can be reached by propagating stochastic classical trajectories in path-integral phase space. Surprisingly, we find that the trajectories equilibrate to the exact quantum equilibrium state, recovering the purely imaginary momentum-position correlation in the phase term. The trajectories are generated using a recently derived Matsubara generalized Langevin equation, which produces the imaginary correlations by evolving the stochastic variables into the complex plane. This makes the dynamics numerically unstable, but we are nonetheless able to demonstrate the equilibration of a quartic oscillator coupled to a white-noise bath. These unexpected findings could lead to new approximate methodologies for simulating continuous-variable open quantum systems.
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 consider the evolution of a quantum simple harmonic oscillator in a general Gaussian state under simultaneous time-continuous weak position and momentum measurements.We deduce the stochastic evolution equations for position and momentum expectation values and the covariance-matrix elements from the characteristic function of the system.By generalizing the Chantasri-Dressel-Jordan (CDJ) formalism (Chantasri et al., 2013(Chantasri et al., , 2015) ) to this continuous-variable system, we construct its stochastic Hamiltonian and action.Action extremization gives us the equations for the most likely readout paths and quantum trajectories.For steady states of the covariance-matrix elements, the analytical solutions for these most likely paths are obtained.Using the CDJ formalism, we calculate final-state probability densities exactly starting from any initial state.We also demonstrate the agreement between the optimal-path solutions and the averages of simulated clustered stochastic trajectories.Our results provide insights into the time dependence of the mechanical energy of the system during the measurement process, motivating their importance for quantum measurement-engine or refrigerator experiments.Although quantum mechanics fundamentally limits our ability to extract information via simultaneous measurements of noncommuting observables, insights into the dynamics of the system during such a process can lead to a deeper understanding of the physics of measurement.
We study the Caldeira-Leggett model where a quantum Brownian particle interacts with an environment or a bath consisting of a collection of harmonic oscillators in the path-integral formalism. Compared to the contours that the paths take in the conventional Schwinger-Keldysh formalism, the paths in our study are deformed in the complex time plane as suggested by the recent study by C. Aron, G. Biroli, and L. F. Cugliandolo [SciPost Phys. 4, 008 (2018)10.21468/SciPostPhys.4.1.008]. This is done to investigate the connection between the symmetry properties in the Schwinger-Keldysh action and the equilibrium or nonequilibrium nature of the dynamics in an open quantum system. We derive the influence functional explicitly in this setting, which captures the effect of the coupling to the bath. We show that in equilibrium the action and the influence functional are invariant under a set of transformations of path-integral variables. The fluctuation-dissipation relation is obtained as a consequence of this symmetry. When the system is driven by an external time-dependent protocol, the symmetry is broken. From the terms that break the symmetry, we derive a quantum Jarzynski-like equality for a quantum mechanical worklike quantity given as a function of fluctuating quantum trajectory. In the classical limit, the transformations becomes those used in the functional integral formalism of the classical stochastic thermodynamics to derive the classical fluctuation theorem.
We report the first numerical calculations in which converged Matsubara dynamics is compared directly with exact quantum dynamics with no artificial damping of the time-correlation functions (TCFs). The system treated is a Morse oscillator coupled to a harmonic bath. We show that, when the system-bath coupling is sufficiently strong, the Matsubara calculations can be converged by explicitly including up to M = 200 Matsubara modes, with the remaining modes included as a harmonic "tail" correction. The resulting Matsubara TCFs are in near-perfect agreement with the exact quantum TCFs, for non-linear as well as linear operators, at a temperature at which the TCFs are dominated by quantum thermal fluctuations. These results provide compelling evidence that incoherent classical dynamics can arise in the condensed phase at temperatures at which the statistics are dominated by quantum (Boltzmann) effects, as a result of smoothing of imaginary-time Feynman paths. The techniques developed here may also lead to efficient methods for benchmarking system-bath dynamics in the overdamped regime.
We establish a direct connection between the Feynman-Vernon path integral formalism for open quantum systems and the Wiener path integral used in classical stochastic dynamics. By considering a generalized influence functional in the strong decoherence limit, we demonstrate that integrating over the quantum coherence length leads to a derivation of stochastic Langevin dynamics. Specifically, we show that the quantum Feynman measure transforms into the stochastic Wiener measure. Applying this framework to the Wigner function representation, we show that the system follows a stochastic path interpretable via classical probability theory. Finally, we address the inverse problem: constructing an equivalent quantum influence functional from a given classical Langevin equation.
Consistent dynamics which couples classical and quantum degrees of freedom exists. This dynamics is linear in the hybrid state, completely positive and trace preserving. Starting from completely positive classical-quantum master equations, we derive a general path integral representation for such dynamics in terms of a classical-quantum action, which includes the necessary and sufficient conditions for complete positivity and trace preservation. The path integral we study is a generalization of the Feynman path integral for quantum systems, and the stochastic path integral used to study classical stochastic processes, allowing for interaction between the classical and quantum systems. When the classical-quantum Hamiltonian is at most quadratic in the momenta we are able to derive a configuration space path integral, providing a map between master equations and covariant classical-quantum path integrals.