We derive a ``classical-quantum'' approximation scheme for a broad class of bipartite quantum systems from fully quantum dynamics. In this approximation, one subsystem evolves via classical equations of motion with quantum corrections, and the other subsystem evolves quantum mechanically with equations of motion informed by the evolving classical degrees of freedom. Using perturbation theory, we derive an estimate for the growth rate of entanglement of the subsystems and deduce a ``scrambling time''---the time required for the subsystems to become significantly entangled from an initial product state. We argue that a necessary condition for the validity of the classical-quantum approximation is consistency of initial data with the generalized Bohr correspondence principle. We illustrate the general formalism by numerically studying the fully quantum, fully classical, and classical-quantum dynamics of a system of two oscillators with nonlinear coupling. This system exhibits parametric resonance, and we show that quantum effects quench parametric resonance at late times. Lastly, we present a curious late-time scaling relation between the average value of the von Neumann entanglement of the interacting oscillator system and its total energy: $S\ensuremath{\sim}2/3\mathrm{ln}E$.
We derive a classical-quantum'' approximation scheme for a broad class of bipartite quantum systems from fully quantum dynamics. In this approximation, one subsystem evolves via classical equations of motion with quantum corrections, and the other subsystem evolves quantum mechanically with equations of motion informed by the evolving classical degrees of freedom. Using perturbation theory, we derive an estimate for the growth rate of entanglement of the subsystems and deduce a scrambling time''---the time required for the subsystems to become significantly entangled from an initial product state. We argue that a necessary condition for the validity of the classical-quantum approximation is consistency of initial data with the generalized Bohr correspondence principle. We illustrate the general formalism by numerically studying the fully quantum, fully classical, and classical-quantum dynamics of a system of two oscillators with nonlinear coupling. This system exhibits parametric resonance, and we show that quantum effects quench parametric resonance at late times. Lastly, we present a curious late-time scaling relation between the average value of the von Neumann entanglement of the interacting oscillator system and its total energy: S\ensuremath∼2/3lnE.
We study a system of two quadratically-coupled harmonic oscillators, prepared respectively in the ground state and in a coherent state. This toy model can describe for instance the dynamics of one mode of a quantum field evolving in a harmonically-oscillating classical background. In such situations, it is well-known that, depending on the background oscillation amplitude and coupling strength, the system may exhibit parametric resonance. Using the Feynman-Vernon influence functional, we analyze the interplay between parametric resonance and quantum entanglement by perturbatively computing the purity of the mixed state resulting from tracing over one of the oscillators. We find a runaway behavior of the quantum entanglement between the two oscillators, as well as a rapid breakdown of the validity of perturbation theory, in regions of parametric instability. This provides a quantitative confirmation of various statements in the literature and is highly relevant to questions pertaining to the validity of the semiclassical approximation in coupled quantum-classical systems.
In this review we summarize and amplify recent investigations of coupled quantum dynamical systems with few degrees of freedom in the short-wavelength, semiclassical limit. Focusing on the correspondence between quantum and classical physics, we mathematically formulate and attempt to answer three fundamental questions. (i) How can one drive a small dynamical quantum system to behave classically? (ii) What determines the rate at which two single-particle quantum-mechanical subsystems become entangled when they interact? (iii) How does irreversibility occur in quantum systems with few degrees of freedom? These three questions are posed in the context of the quantum-classical correspondence for dynamical systems with few degrees of freedom, and we accordingly rely on two short-wavelength approximations to quantum mechanics to answer them: the trajectory-based semiclassical approach on the one hand, and random matrix theory on the other hand. We construct novel investigative procedures towards decoherence and the emergence of classicality out of quantumness in dynamical systems coupled to external degrees of freedom. In particular, we show how dynamical properties of chaotic classical systems, such as local exponential instability in phase space, also affects their quantum counterparts. For instance, it is often the case that the fidelity with which a quantum state is reconstructed after an imperfect time-reversal operation decays with the Lyapunov exponent of the corresponding classical dynamics. For related reasons, but perhaps more surprisingly, the rate at which two interacting quantum subsystems become entangled can also be governed by the subsystem's Lyapunov exponents. Our method allows us to differentiate quantum coherent effects (those related to phase interferences) from classical ones (those related to the necessarily extended envelope of quantal wavefunctions) at each stage in our investigations. This makes it clear that all occurrences of Lyapunov exponents we witness have a classical origin, although they require rather strong decoherence effects to be observed. We extensively rely on numerical experiments to illustrate our findings and briefly comment on possible extensions to more complex problems involving environments with many interacting dynamical systems, going beyond the uncoupled harmonic oscillators model of Caldeira and Leggett.
We study the back-reaction of quantum systems onto classical ones. Taking the starting point that semi-classical physics should be described at all times by a point in classical phase space and a quantum state in Hilbert space, we consider an unravelling approach, describing the system in terms of a classical-quantum trajectory. We derive the general form of the dynamics under the assumptions that the classical trajectories are continuous and the evolution is autonomous, and the requirement that the dynamics is linear and completely positive in the combined classical-quantum state. This requirement is necessary in order to consistently describe probabilities, and forces the dynamics to be stochastic when the back-reaction is non-zero. The resulting equations of motion are natural generalisations of the standard semi-classical equations of motion, but since the resulting dynamics is linear in the combined classical-quantum state, it does not lead to the pathologies which usually follow from evolution laws based on expectation values. In particular, the evolution laws we present account for correlations between the classical and quantum system, which resolves issues associated with other semi-classical approaches. In addition, despite a breakdown of predictability in the classical degrees of freedom, the quantum state evolves deterministically conditioned on the classical trajectory, provided a trade-off between decoherence and diffusion is saturated. As a result, the quantum state remains pure when conditioned on the classical trajectory. To illustrate these points, we numerically simulate a number of semi-classical toy models, including one of vacuum fluctuations as a source driving the expansion of the universe. Finally, we discuss the application of these results to semi-classical gravity, and the black-hole information problem.
Two particles, initially in a product state, become entangled when they come together and start to interact. Using semiclassical methods, we calculate the time evolution of the corresponding reduced density matrix rho(1), obtained by integrating out the degrees of freedom of one of the particles. We find that entanglement generation sensitively depends (i) on the interaction potential, especially on its strength and range, and (ii) on the nature of the underlying classical dynamics. Under general statistical assumptions, and for short-ranged interaction potentials, we find that P(t) decays exponentially fast in a chaotic environment, whereas it decays only algebraically in a regular system. In the chaotic case, the decay rate is given by the golden rule spreading of one-particle states due to the two-particle coupling, but cannot exceed the system's Lyapunov exponent.
We study a general bipartite quantum system consisting of a spin interacting with a bosonic field, with the initial state prepared as the product of a spin coherent state and a canonical coherent state. Our goal is to develop a semiclassical framework to describe the entanglement dynamics between these two subsystems. Using appropriate approximations, we derive a semiclassical expression for the entanglement entropy that depends exclusively on the trajectories of the underlying classical description. By analytically extending the classical phase space into the complex domain, we identify additional complex trajectories that significantly improve the accuracy of the semiclassical description. The inclusion of these complex trajectories allows us to capture the entanglement dynamics with remarkable precision, even well beyond the Ehrenfest time. The approach is illustrated with a representative example, where the role of real and complex trajectories in reproducing the quantum entanglement entropy is explicitly demonstrated.
The dynamical generation of entanglement in closed bipartite systems is investigated in the semiclassical regime. We consider a model of two particles, initially prepared in a product of coherent states, evolving in time according to a generic Hamiltonian, and derive a formula for the linear entropy of the reduced density matrix using the semiclassical propagator in the coherent-state representation. The formula is explicitly written in terms of quantities that define the stability of classical trajectories of the underlying classical system. The formalism is then applied to the problem of two nonlinearly coupled harmonic oscillators, and the result is shown to be in remarkable agreement with the exact quantum measure of entanglement in the short-time regime. An important by-product of our approach is a unified semiclassical formula, which contemplates both the coherent-state propagator and its complex conjugate.