Ehrenfest's theorem beyond the Ehrenfest time

TL;DR

This paper establishes a quantitative bound on the environmental noise needed to maintain quantum-classical correspondence in open quantum systems, valid for times exponentially longer than the Ehrenfest time, and proposes an efficient classical simulation algorithm.

Contribution

It provides a rigorous condition relating diffusion strength to system parameters, extending Ehrenfest's theorem beyond traditional timescales in open quantum systems.

Findings

Quantum and classical evolutions are close when diffusion exceeds a specific threshold.

The bound applies for all observables and times exponentially longer than Ehrenfest time.

An efficient classical algorithm for simulating quantum Lindblad dynamics is proposed.

Abstract

In closed quantum systems, wavepackets can spread exponentially in time due to chaos, forming long-range superpositions in just seconds for ordinary macroscopic systems. A weakly coupled environment is conjectured to decohere the system and restore the quantum-classical correspondence while necessarily introducing diffusive noise -- but at what coupling strength, and under which conditions? For Markovian open systems with Hamiltonians of the form $H = p^2/2m+V(x)$ and Hermitian linear Lindblad operators, we prove the quantum and classical evolutions are close whenever the strength of the environment-induced diffusion satisfies $D \gg (\hbar/s_H)^{4/3} D_H$, where $s_H$ and $D_H$ are characteristic action and diffusion scales that we define precisely using the classical Hamiltonian $H$. The bound applies for all observables and for times exponentially longer than the Ehrenfest timescale, which is when the correspondence can break down in closed systems. The strength of the diffusive noise can vanish in the classical limit to give the appearance of reversible dynamics. The $4/3$ exponent may be optimal, suggested by heuristic arguments and prior numerical evidence. Based on our bound, we give an efficient classical algorithm for simulating quantum Lindblad dynamics, which becomes provably accurate when the strength of environmental coupling exceeds the above threshold.