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Limitations of the Markovian approximation in the harmonic oscillator

Michele Coppola, Zoubair Daouma, Malte Henkel

TL;DR

The paper investigates the limitations of Markovian dynamics in open quantum systems by comparing two constrained damped-oscillator models (friction and cavity) that share the same observable criteria but yield different dynamical properties.By deriving equivalent master equations and analyzing their Wigner-function dynamics and mean-field behavior, the authors show that Markovian truncation can produce non-CP generators, violate Ehrenfest-type relations, and lead to γ-dependent stationary states that fail to thermalise in a physically meaningful way when external fields are present.The work highlights that converting to a Markovian description can obscure non-Markovian memory effects captured by the fluctuation-dissipation theorem, underscoring the need for non-Markovian frameworks in accurate open-quantum-dynamics modelling and in understanding phase behavior in many-body systems.

Abstract

The quantum fluctuation-dissipation theorem is a central ingredient in the construction of quantum dynamics of Brownian motion which necessarily is non-Markovian. Yet, often Markovian approximations to quantum dynamics are studied in the literature. In this work, we investigate the limitations of the Markovian approximation within two paradigmatic models describing a single damped harmonic oscillator. These models are governed by distinct quantum Langevin equations, although both are constructed to satisfy the same set of phenomenological criteria: the canonical commutation relations between position and momentum, the Kubo response relation, the virial theorem, and the equilibrium quantum variance. The limitations of the Markovian approximation are underscored by the classical limit, violations of the Ehrenfest theorem, the breakdown of complete and simple positivity in the reconstructed master equations, and anomalies in thermalisation behaviour. Further phenomenological differences between the two models are illustrated through their quantum relaxation dynamics and phase diagrams, derived from their reinterpretation as mean-field approximations of a many-body interacting magnet. Our analysis explicitly reveals intrinsic inconsistencies introduced by the Markovian approximation, emphasising the need for non-Markovian frameworks for a consistent description of open quantum dynamics.

Limitations of the Markovian approximation in the harmonic oscillator

TL;DR

The paper investigates the limitations of Markovian dynamics in open quantum systems by comparing two constrained damped-oscillator models (friction and cavity) that share the same observable criteria but yield different dynamical properties.By deriving equivalent master equations and analyzing their Wigner-function dynamics and mean-field behavior, the authors show that Markovian truncation can produce non-CP generators, violate Ehrenfest-type relations, and lead to γ-dependent stationary states that fail to thermalise in a physically meaningful way when external fields are present.The work highlights that converting to a Markovian description can obscure non-Markovian memory effects captured by the fluctuation-dissipation theorem, underscoring the need for non-Markovian frameworks in accurate open-quantum-dynamics modelling and in understanding phase behavior in many-body systems.

Abstract

The quantum fluctuation-dissipation theorem is a central ingredient in the construction of quantum dynamics of Brownian motion which necessarily is non-Markovian. Yet, often Markovian approximations to quantum dynamics are studied in the literature. In this work, we investigate the limitations of the Markovian approximation within two paradigmatic models describing a single damped harmonic oscillator. These models are governed by distinct quantum Langevin equations, although both are constructed to satisfy the same set of phenomenological criteria: the canonical commutation relations between position and momentum, the Kubo response relation, the virial theorem, and the equilibrium quantum variance. The limitations of the Markovian approximation are underscored by the classical limit, violations of the Ehrenfest theorem, the breakdown of complete and simple positivity in the reconstructed master equations, and anomalies in thermalisation behaviour. Further phenomenological differences between the two models are illustrated through their quantum relaxation dynamics and phase diagrams, derived from their reinterpretation as mean-field approximations of a many-body interacting magnet. Our analysis explicitly reveals intrinsic inconsistencies introduced by the Markovian approximation, emphasising the need for non-Markovian frameworks for a consistent description of open quantum dynamics.
Paper Structure (17 sections, 112 equations, 2 figures, 2 tables)

This paper contains 17 sections, 112 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Quantum Langevin equations
  3. Friction model
  4. Cavity model
  5. Equivalent master equations
  6. Friction model
  7. Cavity model
  8. Wigner function dynamics
  9. Effective mean-field theories
  10. Friction model
  11. Cavity model
  12. Discussion and conclusion
  13. Noise correlators for the cavity model
  14. Phase-space formulation of quantum mechanics
  15. Wigner function dynamics
  16. ...and 2 more sections

Figures (2)

  • Figure 1: Mean-field phase diagrammes of the quantum spherical model at $T = 0$, for the cavity model (left) and the friction model (right).
  • Figure 2: Region in the parameter space $(\Gamma_1,\,\Gamma_2,\,\Gamma_3)$ of the friction model that satisfies Eqs. (\ref{['cond1']}), (\ref{['cond2']}) and (\ref{['cond3']}) simultaneously, leading to positive semi-definite density matrices to first order in time. Here we set $n_\omega=g=\omega=\gamma=\hbar=1$.