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

TL;DR

This paper establishes that the threshold $D\sim \hbar^{4/3}$ is the sharp boundary for preserving quantum-classical correspondence beyond the Ehrenfest time in open chaotic systems. By constructing a smooth, time-dependent Hamiltonian and isotropic diffusion, the authors produce an explicit Lindbladian counterexample showing an $\hbar$-independent discrepancy between quantum and classical evolutions at times $T=O(1+\log\hbar^{-1})$ when $D\lesssim \hbar^{4/3}$, even for macroscopic observables. The analysis combines a three-step protocol—state and observable squeezing, a nonlinear cubic kick, and reverse squeezing—with precise Duhamel-based bounds and exact asymptotics involving Airy and parabolic cylinder functions to demonstrate the breakdown. The work delineates macroscopic, mesoscopic, and quantum-classical regimes via the decoherence length $\ell_{\mathrm{dec}}\sim \hbar/\sqrt{D}$ and discusses implications for chaotic systems and potential extensions to Anosov flows, highlighting the tightness of previous bounds and guiding expectations for decoherence-assisted classicalization.

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.

Figures (4)

• Figure 1: The dynamics of our counterexample in phase space. Quantum and classical Gaussian states are uniquely represented by their covariance ellipse. We assume an initial pure unsqueezed Gaussian state (blue circle) with dimensions $\hbar^{\frac{1}{2}}\times \hbar^{\frac{1}{2}}$, i.e., covariance matrix $\Sigma_0 \sim \mathop{\mathrm{diag}}\nolimits(\hbar,\hbar)$. In Step 1 we squeeze with $H_1$ for time $\tau_1$ to produce a Gaussian state (blue ellipse) with dimensions $\hbar^{\frac{1}{3}}\times\hbar^{\frac{2}{3}}$. In Step 2 we apply the nonlinearity $H_2$ for unit time, producing a non-Gaussian state (pink banana) whose variance has changed only by order unity. The quantum (but not classical) momentum distribution now contains oscillations induced by interference between opposite ends of the banana, but confined to the mesoscopic scale. In Step 3 we squeeze in the opposite direction to produce the final state (squashed pink banana), bringing the $p$ distribution to the macroscopic scale (see Fig. \ref{['fig:p-distribution']}) with dimensions of order $\hbar^1\times \hbar^0$.
• Figure 2: The classical (blue) momentum distribution $k_3(p) = \int\! f_3(x,p)\, \mathop{}\!\mathrm{d} x$ vs. the quantum (orange) momentum distribution $m_3(p) = |\hat{\psi}_3 (p)|^2$ at the end of Step 3 of our protocol, plotted for the closed system, which approximates the open system when decoherence is weak, $D \ll \hbar^{\frac{4}{3}}$. We choose $\tau_1=\tfrac{1}{6}\log (2/\hbar)$ and $\tau_3=\tfrac{2}{3}\log (2/\hbar)$, ensuring the distributions are independent of $\hbar$, and compare $\tau_2 = 1$ (main) with $\tau_2 = 10$ (inset). The interference-induced oscillations in the quantum (but not classical) momentum distribution for positive $p$ are produced by interference between the two ends of the squashed banana in Fig. \ref{['fig:ellipse-to-banana']}. These oscillations already exist in the momentum distribution at the end of Step 2, but at a mesoscopic scale. Evidence of the interference is magnified to the macroscopic scale in Step 3 at the cost of allowing it to decohere.
• Figure 3: The three smooth bumps $\chi_{i=1,2,3}$ depicted respectively in red, green, and blue. They vanish outside the intervals $[t_{i-1},t_i]$ of width $\tau_i = t_i-t_{i-1}$ and satisfy $\int\! \chi_i(t) \, dt= \tau_i$. For the illustration we have chosen the specific shared bump shape described by \ref{['eq:chi-first']}--\ref{['eq:chi-last']}. We have also chosen the specific semiclassical parameter $h=\hbar/2=10^{-6}$ along with our standard choices $\tau_1=_\tau\frac{1}{6}\log h^{-1}$, $\tau_2=_\tau 1$, and $\tau_3=_\tau \frac{2}{3}\log h^{-1}$.
• Figure 4: The expectation values $\langle A_n\rangle_{\mathring{f}_3}$ and $\langle \hat{A}_n\rangle_{\psi_3}$ of the $n$-indexed bounded smooth observable $A_n := p^n \exp[-p^2]$ with respect to the classical (blue) closed-system momentum distribution $\mathring{k}_3(p)$ vs. the quantum (orange) closed-system momentum distribution $\mathring{m}_3(p) = |\hat{\psi}_3 (p)|^2$. We choose $\tau_1=\tfrac{1}{6}\log h^{-1}$ and $\tau_3=\tfrac{2}{3}\log h^{-1}$, ensuring the functions are independent of $h$, and compare $\tau_2 = 1$ (left) with $\tau_2 = 10$ (right). See Fig. \ref{['fig:p-distribution']} for the distributions themselves.

Theorems & Definitions (1)

• Theorem 1