We show that for the Lindblad evolution defined using (at most) quadratically growing classical Hamiltonians and (at most) linearly growing classical jump functions (quantized into jump operators assumed to satisfy certain ellipticity conditions and modeling interaction with a larger system), the evolution of a quantum observable remains close to the classical Fokker–Planck evolution in the Hilbert–Schmidt norm for times vastly exceeding the Ehrenfest time (the limit of such agreement with no jump operators). The time scale is the same as in the recent papers by Hernández–Ranard–Riedel but the statement and methods are different. The appendix presents numerical experiments illustrating the classical/quantum correspondence in Lindblad evolution and comparing it to the mathematical results.
We show that for the Lindblad evolution defined using (at most) quadratically growing classical Hamiltonians and (at most) linearly growing classical jump functions (quantized into jump operators assumed to satisfy certain ellipticity conditions and modeling interaction with a larger system), the evolution of a quantum observable remains close to the classical Fokker–Planck evolution in the Hilbert–Schmidt norm for times vastly exceeding the Ehrenfest time (the limit of such agreement with no jump operators). The time scale is the same as in the recent papers by Hernández–Ranard–Riedel but the statement and methods are different. The appendix presents numerical experiments illustrating the classical/quantum correspondence in Lindblad evolution and comparing it to the mathematical results.
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.
The threshold for quantum-classical correspondence is D∼ℏ34
In chaotic quantum systems, an initially localized quantum state can deviate strongly from the corresponding classical phase-space distribution after the Ehrenfest time tE∼log(ℏ−1), even in the limit ℏ→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≫ℏ34, in units set by the classical Hamiltonian. At the same time, some heuristic arguments have suggested the weaker condition D≫ℏ2 always suffices. Here we construct an explicit Lindbladian that demonstrates that the scaling D∼ℏ34 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 ℏ-independent quantum-classical discrepancy at the Ehrenfest time whenever D≪ℏ34, even for ℏ-independent "macroscopic" smooth observables.
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.
Two approaches to quantum-classical correspondence are distinguished according to the classical dynamical theory with which quantum theory is compared. The first of these, Ehrenfest correspondence, defines a dynamical regime in which the quantum expectation values follow approximately a classical trajectory. The second of these, Liouville correspondence, applies when the quantum probability distributions remain well approximated by a density in the classical phase space. The former applies only for narrow states, whereas the latter may remain valid even for quantum states that have spread to the system size. A spin model is adopted for this correspondence study because the quantum state is discrete and finite- dimensional, and thus no artificial truncation of the Hilbert space is required. The quantum time-evolution is given by a discrete unitary mapping. The corresponding classical model is volume-preserving (non-dissipative) and the time-evolution is given by a symplectic map. In classically chaotic regimes, the widths of initially narrow quantum states grow, on average, exponentially with time, until saturation at the system size. This initial spreading rate is well approximated by the classical Lyapunov exponent when the accessible classical phase space is predominantly chaotic. Because of the exponential growth rate of the quantum variance, the Ehrenfest regime is delimited by a break-time that grows logarithmically with increasing quantum numbers. The small differences between quantum expectation values and corresponding Liouville averages of dynamical variables also grow exponentially, initially, if the classical behaviour is chaotic. This exponential rate is independent of h and consistently larger than the classical Lyapunov exponent by at least a factor of two. Interestingly, this exponential growth rate does not continue until these differences approach magnitudes of order the system size, but crosses over to power-law growth (or simply saturates), when the differences have reached magnitudes no larger than order ħ. Because of the exponential growth of these quantum-classical differences, the Liouville regime is also delimited by a break-time that grows logarithmically with increasing quantum numbers. However, due to the early saturation of the exponential growth of the differences, the Liouville log break-time applies only in a restricted domain. After spreading to the system size, the quantum states relax to an approximately time-independent state. This is characterised by equilibrium distributions that are subject to rapidly oscillating fluctuations about the coarse-grained classical steady-state. If the accessible classical phase space is predominantly chaotic, then the equilibrium quantum probability distributions are nearly microcanonical. Otherwise the steady-state quantum probability distributions accurately reflect the details of the KAM surfaces in the classical phase space. The equilibrium regime quantum fluctuations, which are quantum-classical differences, are shown to approach zero as a negative power of the quantum numbers. Since the differences between quantum expectation values and individual classical trajectories grow exponentially in time to a magnitude that scales with the system size, the Ehrenfest regime is delimited by an unrestricted log break-time. The log break-time may be short compared to experimental time-scales for some chaotic macroscopic bodies, and, as a result, an observable breakdown of Ehrenfest correspondence may arise for some macroscopic systems. Although the Liouville regime is also delimited by a log break-time, an observable breakdown of correspondence between quantum mechanics and Liouville mechanics for macroscopic bodies is not likely since their differences, expressed relative to the dimensions of the observable, approach zero in the limit of large quantum numbers.
In closed quantum systems, wavepackets can spread exponentially in time due to chaos, forming long-range superpositions in just seconds for ordinary macroscopic systems. A weakly coupled environment is conjectured to decohere the system and restore the quantum-classical correspondence while necessarily introducing diffusive noise -- but at what coupling strength, and under which conditions? For Markovian open systems with Hamiltonians of the form H=p2/2m+V(x) and Hermitian linear Lindblad operators, we prove the quantum and classical evolutions are close whenever the strength of the environment-induced diffusion satisfies D≫(ℏ/sH)4/3DH, where sH and DH are characteristic action and diffusion scales that we define precisely using the classical Hamiltonian H. The bound applies for all observables and for times exponentially longer than the Ehrenfest timescale, which is when the correspondence can break down in closed systems. The strength of the diffusive noise can vanish in the classical limit to give the appearance of reversible dynamics. The 4/3 exponent may be optimal, suggested by heuristic arguments and prior numerical evidence. Based on our bound, we give an efficient classical algorithm for simulating quantum Lindblad dynamics, which becomes provably accurate when the strength of environmental coupling exceeds the above threshold.
Abstract The dynamics of Gaussian states for open quantum systems described by Lindblad equations can be solved analytically for systems with quadratic Hamiltonians and linear Lindbladians, showing the familiar phenomena of dissipation and decoherence. It is well known that the Lindblad dynamics can be expressed as an ensemble average over stochastic pure-state dynamics, which can be interpreted as individual experimental implementations, where the form of the stochastic dynamics depends on the measurement setup. Here we consider quantum-jump and stochastic Schrödinger dynamics for initially Gaussian states. While both unravellings converge to the same Lindblad dynamics when averaged, the individual dynamics can differ qualitatively. For the stochastic Schrödinger equation, Gaussian states remain Gaussian during the evolution, with stochastic differential equations governing the evolution of the phase-space centre and a deterministic evolution of the covariance matrix. In contrast to this, individual pure-state dynamics arising from the quantum-jump evolution do not remain Gaussian in general. Applying results developed in the non-Hermitian context for Hagedorn wavepackets, we formulate a method to generate quantum-jump trajectories that is described entirely in terms of the evolution of an underlying Gaussian state. To illustrate the behaviours of the different unravellings in comparison to the Lindblad dynamics, we consider two examples in detail, which can be largely treated analytically, a harmonic oscillator subject to position measurement and a damped harmonic oscillator. In both cases, we highlight the differences as well as the similarities of the stochastic Schrödinger and the quantum-jump dynamics.