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

2023/07/07 by Felipe Hernández, Hernández, Felipe, Daniel Ranard +3 · 5 citations
Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Physics (quant-ph) #Spectroscopy and Quantum Chemical Studies #Statistical Mechanics and Entropy

paper · pdf · doi:10.48550/arxiv.2307.05326

openalex publication_date 2023/07/07 · openalex created_date 2023/07/13 · openalex updated_date 2026/07/28

Abstract

Quantum and classical systems evolving under the same formal Hamiltonian H may dramatically differ after the Ehrenfest timescale t_E ∼ log(ℏ^-1), even as ℏ → 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 ℏ → 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 ≫ ℏ^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.

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• Frozen Gaussian sampling algorithms for simulating Markovian open quantum systems in the semiclassical regime
• Quantitative Hypocoercivity and Lifting of Classical and Quantum Dynamics
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Related

• The threshold for quantum-classical correspondence is D ∼ ℏ^\frac43
• Ehrenfest's theorem beyond the Ehrenfest time
• Classical-Quantum correspondence in Lindblad evolution
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