The threshold for quantum-classical correspondence is $D \sim \hbar^{\frac43}$

TL;DR

This paper establishes that in chaotic quantum systems, the quantum-classical correspondence persists beyond the Ehrenfest time only if the environmental diffusion strength D scales as  /3 with , providing a precise threshold for decoherence effects.

Contribution

The authors construct an explicit Lindbladian model demonstrating that the D  /3 scaling is the critical threshold for maintaining quantum-classical correspondence beyond the Ehrenfest time.

Findings

Quantum-classical correspondence breaks down if D  /3 .

Explicit Lindbladian example confirms the  /3 threshold.

Discrepancy persists at the Ehrenfest time when D  /3.

Abstract

In chaotic quantum systems, an initially localized quantum state can deviate strongly from the corresponding classical phase-space distribution after the Ehrenfest time $t_{\mathrm{E}} \sim \log(\hbar^{-1})$, even in the limit $\hbar \to 0$. Decoherence by the environment is often invoked to explain the persistence of the quantum-classical correspondence at longer timescales. Recent rigorous results for Lindblad dynamics with phase-space diffusion strength $D$ show that quantum and classical evolutions remain close for times that are exponentially longer than the Ehrenfest time whenever $D \gg \hbar^{\frac43}$, in units set by the classical Hamiltonian. At the same time, some heuristic arguments have suggested the weaker condition $D \gg \hbar^{2}$ always suffices. Here we construct an explicit Lindbladian that demonstrates that the scaling $D \sim \hbar^{\frac43}$ is indeed the threshold for quantum-classical correspondence beyond the Ehrenfest time. Our example uses a smooth time-dependent Hamiltonian and linear Lindblad operators generating homogeneous isotropic diffusion. It exhibits an $\hbar$-independent quantum-classical discrepancy at the Ehrenfest time whenever $D \ll \hbar^{\frac43}$, even for $\hbar$-independent "macroscopic" smooth observables.