Abstract Quantum and classical systems evolving under the same formal Hamiltonian H may exhibit dramatically different behavior after the Ehrenfest timescale
tE∼log(ℏ−1) t E ∼ log ( ħ - 1 ) , even as
ℏ→0 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 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
Lk(x,p) L k ( x , p ) . The error is small when the strength of the diffusion D associated to the Lindblad functions satisfies
D≫ℏ4/3 D ≫ ħ 4 / 3 , in particular allowing 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 constructed from a second-order Taylor expansion of the Lindbladian. 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. Our presentation aims to be comprehensive and accessible to both mathematicians and physicists. 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.
A semiclassical approximation is derived by using a family of wavepackets to map arbitrary wavefunctions into phase space. If the Hamiltonian can be approximated as linear over each individual wavepacket, as often done when presenting Ehrenfest's theorem, the resulting approximation is a linear first-order partial differential equation on phase space, which will be referred to as the Schrödinger-Ehrenfest or SE equation. This advectively transports wavefunctions along classical trajectories, so that as a trajectory is followed in time the amplitude remains constant while the phase changes by the action divided by
ℏ. The wavefunction's squared-magnitude is a plausible phase space density and obeys Liouville's equation for the classical time evolution. This is a derivation of the Koopman-von~Neumann (KvN) formulation of classical mechanics, which previously was postulated but not derived. With the time-independent SE equation, continuity of the wavefunction requires the change of phase around any closed path in the torus covered by a classical trajectory to be an integer multiple of
2π, giving a standing wave picture of old quantum mechanics. While this applies to any system, for separable systems it gives Bohr-Sommerfeld quantization.
Abstract We consider a frictionless system coupled to an external Markovian environment. The quantum and classical evolution of such systems are described by the Lindblad and the Fokker–Planck equation, respectively. We show that when such a system is given by an at most quadratically growing Hamiltonian and at most linearly growing real jump functions, the quantum and classical evolutions remain close on time scales much longer than Ehrenfest time. In particular, we show that the evolution of a density matrix by the Lindblad equation is close in trace norm to the quantization of the corresponding evolution by the Fokker–Planck equation. Such agreement improves upon recent results (Galokowski and Zworski in Classical quantum correspondence in Lindblad evolution, 2024. arXiv:2403.09345 ; Hernández et al. in Decoherence ensures classicality beyond the Ehrenfest time as
ℏ→0 ħ → 0 , 2023. arXiv:2306.13717 , Hernández et al. in The limit of open quantum systems with general Lindbladians: vanishing noise ensures classicality beyond the ehrenfest time, 2023. arXiv:2307.05326 ), which proved long-time agreement in weaker norms.
A long-standing challenge in mixed quantum-classical trajectory simulations is the treatment of entanglement between the classical and quantal degrees of freedom. We present a novel approach that describes the emergence of entangled states entirely in terms of independent and deterministic Ehrenfest-like classical trajectories. For a two-level quantum system in a classical environment, this is derived by mapping the quantum system onto a path-integral representation of a spin 1/2. We demonstrate that the method correctly accounts for coherence and decoherence and thus reproduces the splitting of a wave packet in a nonadiabatic scattering problem. This discovery opens up a new class of simulations as an alternative to stochastic surface-hopping, coupled-trajectory, or semiclassical approaches.
A long-standing challenge in mixed quantum-classical trajectory simulations is the treatment of entanglement between the classical and quantal degrees of freedom. We present a novel approach which describes the emergence of entangled states entirely in terms of independent and deterministic Ehrenfest-like classical trajectories. For a two-level quantum system in a classical environment, this is derived by mapping the quantum system onto a path-integral representation of a spin-1/2. We demonstrate that the method correctly accounts for coherence and decoherence and thus reproduces the splitting of a wavepacket in a nonadiabatic scattering problem. This discovery opens up a new class of simulations as an alternative to stochastic surface-hopping, coupled-trajectory or semiclassical approaches.
We derive a generalization of the Ehrenfest theorem valid for open quantum systems. From this result, we identify three contributions to the evolution of expected values: (i) the explicit time dependence of the observable, (ii) the incompatibility between the observable and an operator which plays the role of an effective Hamiltonian, and (iii) entropy changes. Considering the local Hamiltonian as the observable, and adopting a specific interpretation of the nature of thermal interactions, we obtain an alternative version of the first law of thermodynamics. Within this framework, we show that in some cases the power performed by the system can be considered as a quantum observable. As an application, the pure-dephasing process is reinterpreted from this perspective.