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Classical-quantum oscillators as diffusive processes in phase space

Emanuele Panella

TL;DR

This work analyzes a solvable classical-quantum oscillator, showing that diffusion in the classical sector and decoherence in the quantum sector drive the system to a unique non-equilibrium steady state, computable via a mapping to classical OU diffusion. Starting from a master equation and a path-integral formalism, the authors derive exact and perturbative results for both purely classical and CQ cases, including steady-state covariances, unequal-time correlators, and a phase-space (Wigner) representation. They reveal that, for harmonic interactions, CQ dynamics are exactly equivalent to classical diffusion in phase space, and in the high-diffusion limit the hybrid state thermalizes to a canonical form with temperature set by the classical bath, $T_C=D/(2\alpha)$. The findings provide a controlled framework to study non-equilibrium thermodynamics in hybrid classical-quantum systems and suggest directions for deriving fluctuation relations and entropy production within CQ dynamics.

Abstract

The dynamics of hybrid systems -- i.e. ones in which classical and quantum degrees of freedom co-exist and interact -- feature both diffusion in the classical sector and decoherence in the quantum state. In this article, we will consider the simple setup of a classical damped oscillator interacting with its quantum counterpart and show that, for any initial state, the dynamics flows to a unique (non-equilibrium) steady state, which we compute explicitly. To do so, we make use of a useful mapping between hybrid and classical diffusive dynamics, which we characterise in detail in the master equation formalism.

Classical-quantum oscillators as diffusive processes in phase space

TL;DR

This work analyzes a solvable classical-quantum oscillator, showing that diffusion in the classical sector and decoherence in the quantum sector drive the system to a unique non-equilibrium steady state, computable via a mapping to classical OU diffusion. Starting from a master equation and a path-integral formalism, the authors derive exact and perturbative results for both purely classical and CQ cases, including steady-state covariances, unequal-time correlators, and a phase-space (Wigner) representation. They reveal that, for harmonic interactions, CQ dynamics are exactly equivalent to classical diffusion in phase space, and in the high-diffusion limit the hybrid state thermalizes to a canonical form with temperature set by the classical bath, . The findings provide a controlled framework to study non-equilibrium thermodynamics in hybrid classical-quantum systems and suggest directions for deriving fluctuation relations and entropy production within CQ dynamics.

Abstract

The dynamics of hybrid systems -- i.e. ones in which classical and quantum degrees of freedom co-exist and interact -- feature both diffusion in the classical sector and decoherence in the quantum state. In this article, we will consider the simple setup of a classical damped oscillator interacting with its quantum counterpart and show that, for any initial state, the dynamics flows to a unique (non-equilibrium) steady state, which we compute explicitly. To do so, we make use of a useful mapping between hybrid and classical diffusive dynamics, which we characterise in detail in the master equation formalism.

Paper Structure

This paper contains 17 sections, 92 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. CQ master equation
  4. CQ path integral
  5. Main results
  6. The classical case
  7. The steady-state
  8. MSR path integral
  9. The classical-quantum case
  10. Occupation number
  11. Thermal limit
  12. CQ in phase space
  13. Discussion
  14. Uncoupled oscillators
  15. Non-dissipative oscillator
  16. ...and 2 more sections

Figures (2)

  • Figure 1: The $\Omega_1$ pole and its reflections in the complex plane.
  • Figure 2: Sample trajectory for the couple oscillators with $m_*=1 \textrm{Kg}$, $\omega_*=\gamma_1=1 \textrm{s}^{-1}$ and $\lambda=0.05 \text{Nm}^{-1}$. Dashed lines are the theoretical predictions for the variance in the positions of the two oscillators. The numerical integration was performed with an Euler-Maruyama forward scheme with timestep $\Delta t=0.0005 \textrm{s}$.