Classical-Quantum correspondence in Lindblad evolution

TL;DR

This paper demonstrates that under specific conditions, quantum observables evolving via Lindblad equations stay close to classical Fokker-Planck dynamics for extended times, supported by theoretical analysis and numerical experiments.

Contribution

It establishes a novel connection between quantum Lindblad evolution and classical Fokker-Planck equations for long times, using different methods than previous studies.

Findings

Quantum observables remain close to classical evolution beyond Ehrenfest time.

The time scale of classical-quantum correspondence matches recent results.

Numerical experiments confirm the theoretical predictions.

Abstract

We show that for the Lindblad evolution defined using (at most) quadratically growing classical Hamiltonians and (at most) linearly growing classical jump functions (quantized into jump operators assumed to satisfy certain ellipticity conditions and modeling interaction with a larger system), the evolution of a quantum observable remains close to the classical Fokker--Planck evolution in the Hilbert--Schmidt norm for times vastly exceeding the Ehrenfest time (the limit of such agreement with no jump operators). The time scale is the same as in the recent papers by Hern\'andez--Ranard--Riedel but the statement and methods are different. The appendix presents numerical experiments illustrating the classical/quantum correspondence in Lindblad evolution and comparing it to the mathematical results.