We propose to study the L 2 -norm distance between classical and quantum phase space distributions, where for the latter we choose the Wigner function, as a global phase space indicator of quantum–classical correspondence. For example, this quantity should be the key to understanding the correspondence between quantum and classical Loschmidt echoes. We concentrate on fully chaotic systems with compact (finite) classical phase space. By means of numerical simulations and heuristic arguments we find that the quantum–classical fidelity stays at one up to the Ehrenfest-type time scale, which is proportional to the logarithm of effective Planck constant, and decays exponentially with a maximal classical Lyapunov exponent, after that time.
We propose to study the L 2 -norm distance between classical and quantum phase space distributions, where for the latter we choose the Wigner function, as a global phase space indicator of quantum–classical correspondence. For example, this quantity should be the key to understanding the correspondence between quantum and classical Loschmidt echoes. We concentrate on fully chaotic systems with compact (finite) classical phase space. By means of numerical simulations and heuristic arguments we find that the quantum–classical fidelity stays at one up to the Ehrenfest-type time scale, which is proportional to the logarithm of effective Planck constant, and decays exponentially with a maximal classical Lyapunov exponent, after that time.
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.
We discuss the connection between the out-of-time-ordered correlator and the number of harmonics of the phase-space Wigner distribution function.In particular, we show that both quantities grow exponentially for chaotic dynamics, with a rate determined by the largest Lyapunov exponent of the underlying classical dynamics, and algebraically-linearly or quadratically-for integrable dynamics.It is then possible to use such quantities to detect in the time domain the integrability-to-chaos crossover in many-body quantum systems.
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.
We investigate how decoherence affects the short‐time separation between quantum and classical dynamics for classically chaotic systems, within the framework of a specific model, which corresponds to an ion trapped in a harmonic trap and submitted to non‐resonant laser pulses. For a wide range of parameters, the distance between the classical and quantum (Wigner function) phase‐space distributions depends on a single parameter χ that relates an effective Planck constant ℏeff (given by the square of the Lamb‐Dicke parameter), the Lyapunov coeffficient, and the diffusion constant. This distance peaks at a time that depends logarithmically on ℏeff, in agreement with previous estimations of the separation time for Hamiltonian systems. Due to the logarithmic dependence, this time may be very small, even for macroscopis sytems, which would seem to imply that quantum behavior should become manifest after a short time scale. We show however that, for χ1, the separation remains small, going down with ℏeff2. In this limit, the concept of separation time does not make sense anymore, and the classical behavior of the system is recovered.
The Loschmidt echo (LE) is a magnitude that measures the sensitivity of quantum dynamics to perturbations in the Hamiltonian. For a certain regime of the parameters, the LE decays exponentially with a rate given by the Lyapunov exponent of the underlying classically chaotic system. We develop a semiclassical theory, supported by numerical results in a Lorentz gas model, which allows us to establish and characterize the universality of this Lyapunov regime. In particular, the universality is evidenced by the semiclassical limit of the de Broglie wavelength going to zero, the behavior for times longer than Ehrenfest time, the insensitivity with respect to the form of the perturbation, and the behavior of individual (nonaveraged) initial conditions. Finally, by elaborating a semiclassical approximation to the Wigner function, we are able to distinguish between classical and quantum origin for the different terms of the LE. This approach renders an understanding for the persistence of the Lyapunov regime after the Ehrenfest time, as well as a reinterpretation of our results in terms of the quantum-classical transition.
We investigate how decoherence affects the short-time separation between quantum and classical dynamics for classically chaotic systems, within the framework of a specific model. For a wide range of parameters, the distance between the corresponding phase-space distributions depends on a single parameter $\ensuremath{\chi}$ that relates an effective Planck constant ${\ensuremath{\hbar}}_{\mathrm{eff}}$, the Lyapunov coefficient, and the diffusion constant. This distance peaks at a time that depends logarithmically on ${\ensuremath{\hbar}}_{\mathrm{eff}}$, in agreement with previous estimations of the separation time for Hamiltonian systems. However, for $\ensuremath{\chi}\ensuremath{\lesssim}1$, the separation remains small, going down with ${\ensuremath{\hbar}}_{\mathrm{eff}}^{2}$, so the concept of separation time loses its meaning.