Classical correspondence beyond the Ehrenfest time for open quantum systems with general Lindbladians

TL;DR

The paper proves a rigorous quantum-classical correspondence for open quantum systems under general Lindblad dynamics, extending beyond the Ehrenfest time. By constructing a time-dependent mixture of not-too-squeezed Gaussian states and evolving them with a local harmonic approximation, the authors bound the quantum-trace and classical-L1 deviations between the quantum Lindblad evolution and its classical Fokker-Planck counterpart. The bounds scale as $t oldsymbol{ ext{ħ}}^{1/2} g^{-3/2}$, with $g$ the relative diffusion strength and are modulated by classical and quantum anharmonicity factors, establishing a regime where diffusion strong enough to counteract phase-space squeezing ensures long-lived correspondence. A key technical tool is a detailed analysis of harmonic Gaussian evolution and causal Duhamel estimates, complemented by a novel Moyal-product bound to control higher-order quantum corrections. The results apply to general smooth Hamiltonians and Lindblad operators, with a companion paper providing explicit bounds for a common linear-Lindblad, quadratic-Hamiltonian case, illustrating applicability to physical systems.

Abstract

Quantum and classical systems evolving under the same formal Hamiltonian $H$ may dramatically differ after the Ehrenfest timescale $t_E \sim \log(\hbar^{-1})$, even as $\hbar \to 0$. Coupling the system to a Markovian environment results in a Lindblad equation for the quantum evolution. Its classical counterpart is given by the Fokker-Planck equation on phase space, which describes Hamiltonian flow with friction and diffusive noise. The quantum and classical evolutions may be compared via the Wigner-Weyl representation. Due to decoherence, they are conjectured to match closely for times far beyond the Ehrenfest timescale as $\hbar \to 0$. We prove a version of this correspondence, bounding the error between the quantum and classical evolutions for any sufficiently regular Hamiltonian $H(x,p)$ and Lindblad functions $L_k(x,p)$. The error is small when the strength of the diffusion $D$ associated to the Lindblad functions satisfies $D \gg \hbar^{4/3}$, which allows vanishing noise in the classical limit. Our method uses a time-dependent semiclassical mixture of variably squeezed Gaussian states. The states evolve according to a local harmonic approximation to the Lindblad dynamics. Both the exact quantum trajectory and its classical counterpart can be expressed as perturbations of this semiclassical mixture, with the errors bounded using Duhamel's principle. We present heuristic arguments suggesting the $4/3$ exponent is optimal and defines a boundary in the sense that asymptotically weaker diffusion permits a breakdown of quantum-classical correspondence at the Ehrenfest timescale. In a shorter companion paper, we treat the special case of Hamiltonians that decompose into kinetic and potential energy with linear Lindblad operators, with explicit bounds that can be applied directly to physical systems.

Figures (2)

• Figure 1: We illustrate the quantum-classical correspondence time (also know as the quantum breaking time) in different regimes of the coupling strength $\gamma$. Unless they are set to exactly zero (e.g. $L_{k}=0$), the classical diffusion $D$ and classical friction $G$ scale like $\gamma\hbar$ and $\gamma$ respectively. When $\hbar^0 \ll \gamma$, friction must be assumed exactly zero or else the classical dynamics will become will become singular (and likewise for diffusion when $\hbar^{-1} \ll \gamma$).
• Figure 2: (a) An initial pure quantum Gaussian state $\rho(t \mathord{=} 0)$ evolves in phase space. (b) At short times the dynamics admit a local harmonic (quadratic) approximation, broadening the distribution via diffusion (purple arrows) and possibly squeezing it via classical flow (red arrows). For diffusion strength $D$ and local Lyapunov exponent $\lambda_{\mathrm{L}}$ of the flow, the Gaussian state (ellipse) has a minimum thickness: the diffusion broadens the ellipse at speed $\dot{w} \sim D/w$, while the the Hamiltonian flow can shrink the width by at most $\dot{w} \sim -w /\lambda_{\mathrm{L}}$, with the competing effects balanced at $w \sim (D/\lambda_{\mathrm{L}})^{1/2}$. (c) After $\rho(t)$ becomes mixed due to diffusive broadening, it can be approximated by a mixture $\tilde{\rho}(t)$ of pure Gaussian states (ellipses) that are individually less squeezed. Each evolves by its own local harmonic dynamics while continuously being further decomposed. (d) As $\rho(t)$ spreads in phase space, our approximation $\tilde{\rho}(t)$ uses ellipses of fixed area $\hbar$ but varying amounts of squeezing. (e) The minimum thickness $w$ controls the error of the harmonic approximation: the dynamics are perturbed by the leading-order anharmonicity $\nabla^3 H$, which is strongest (relative to the center) at the tips of the ellipse lying on either end of the long axis $v \sim \hbar/w$. This changes the speed of the local flow by $s \lesssim v^2 \|\nabla^3 H\|$, so the discrepancy (red shaded area) between the true distribution (curved boomerang) and the ellipse grows at rate $\lesssim sv$. Compared to the ellipse's area $\hbar$, this gives an error rate $sv/\hbar \lesssim (\hbar^{4/3}/D)^{3/2} \lambda_{\mathrm{L}}^{3/2} \|\nabla^3 H\|$, which is small when $D \gg \hbar^{4/3}$.

Theorems & Definitions (38)

• Theorem 1.1: Main result, simplified
• Definition 3.1: Corresponding dynamics
• Definition 3.2: Relative diffusion strength
• Definition 3.3: Not-too-squeezed states
• Theorem 3.1: Main result
• proof : Proof of Theorem \ref{['thm:informal']} assuming Theorem \ref{['thm:mainResult']}
• Lemma 5.1: Trace formula for mixtures of Gaussians
• proof
• Lemma 5.2: Gaussian harmonic evolution
• proof