Decoherence and the rate of entropy production in chaotic quantum systems

Behavior of Correlation Functions in the Dynamics of the Multiparticle Quantum Arnol'd Cat

The multi-particle Arnol'd cat is a generalization of the Hamiltonian system, both classical and quantum, whose period evolution operator is the renowned map that bears its name. It is obtained following the Joos-Zeh prescription for decoherence by adding a number of scattering particles in the configuration space of the cat. Quantization follows swiftly if the Hamiltonian approach, rather than the semiclassical approach, is adopted. The author has studied this system in a series of previous works, focusing on the problem of quantum-classical correspondence. In this paper, the dynamics of this system are tested by two related yet different indicators: the time autocorrelation function of the canonical position and the out-of-time correlator of position and momentum.

Quantum Entropies and Decoherence for the Multiparticle Quantum Arnol'd Cat

I study the scaling behavior in the physical parameters of dynamical entropies, classical and quantum, in a specifically devised model of collision-induced decoherence in a chaotic system. The treatment is fully canonical and no approximations are involved or infinite limits taken. I present this model in a detailed way, in order to clarify my views in the debate about the nature, definition, and relevance of quantum chaos.

Quantum coherence controls the nature of equilibration and thermalization in coupled chaotic systems

A bipartite system whose subsystems are fully quantum chaotic and coupled by a perturbative interaction with a tunable strength is a paradigmatic model for investigating how isolated quantum systems relax toward an equilibrium. It is found that quantum coherence of the initial product states in the energy eigenbasis of the subsystems-quantified by the off-diagonal elements of the subsystem density matrices-can be viewed as a resource for equilibration and thermalization as manifested by the entanglement generated. Results are given for four distinct perturbation strength regimes, the ultraweak, weak, intermediate, and strong regimes. For each, three types of tensor product states are considered for the initial state: uniform superpositions, random superpositions, and individual subsystem eigenstates. A universal timescale is identified involving the interaction strength parameter. In particular, maximally coherent initial product states (a form of uniform superpositions) thermalize under time evolution for any perturbation strength in spite of the fact that in the ultraweak perturbative regime the underlying eigenstates of the system have a tensor product structure and are not at all thermal-like; though the time taken to thermalize tends to infinity as the interaction vanishes. Moreover, it is shown that in the ultraweak regime the initial entanglement growth of the system whose initial states are maximally coherent is quadratic-in-time, in contrast to the widely observed linear behavior.

Many-Body Systems and Quantum Chaos: The Multiparticle Quantum Arnol’d Cat

A multi-particle extension of the Arnol’d cat Hamiltonian system is presented, which can serve as a fully dynamical model of decoherence. The behavior of the von Neumann entropy of the reduced density matrix is studied, in time and as a function of the physical parameters, with special regard to increasing the mass of the cat particle.

Classical Branch Structure from Spatial Redundancy in a Many-Body Wave Function

When the wave function of a large quantum system unitarily evolves away from a low-entropy initial state, there is strong circumstantial evidence it develops "branches": a decomposition into orthogonal components that is indistinguishable from the corresponding incoherent mixture with feasible observations. Is this decomposition unique? Must the number of branches increase with time? These questions are hard to answer because there is no formal definition of branches, and most intuition is based on toy models with arbitrarily preferred degrees of freedom. Here, assuming only the tensor structure associated with spatial locality, I show that branch decompositions are highly constrained just by the requirement that they exhibit redundant local records. The set of all redundantly recorded observables induces a preferred decomposition into simultaneous eigenstates unless their records are highly extended and delicately overlapping, as exemplified by the Shor error-correcting code. A maximum length scale for records is enough to guarantee uniqueness. Speculatively, objective branch decompositions may speed up numerical simulations of nonstationary many-body states, illuminate the thermalization of closed systems, and demote measurement from fundamental primitive in the quantum formalism.

Signature candidate of quantum chaos far from the semiclassical regime

We numerically investigated the entanglement product in the simplest coupled kicked top model with the spin j = 1. Different from the dynamical pattern of entanglement in the semiclassical regime, two similar initial states may have discordant entanglement oscillations. A candidate of the quantum signature of this classical chaotic system was proposed. The potential antimonotonic relation between the rank correlation coefficient qualifying the concordant of two entanglement evolutions and the stationary entanglement was preliminarily revealed.

Lyapunov decoherence rate in classically chaotic systems

We provide a path integral treatment of the decoherence process induced by a heat bath on a single particle whose dynamics is classically chaotic and show that the decoherence rate is given by the Lyapunov exponent. The loss of coherence is charaterized by the purity, which is calculated semiclassically within diagonal approximation, when the particle initial state is a single Gaussian wave packet. The calculation is performed for weak dissipation and in the high-temperature limit. This situation allows us to simplify the heat bath description to a single random potential. Although the dissipative term is neglected in such approach, the fluctuating one can be treated phenomenologically to fit with the above regime. Our results are therefore valid for times shorter than the inverse of the dissipation rate.

Relatively robust classical structures in dissipative quantum chaotic systems

We study the stability of classical structures in chaotic systems when a dissipative quantum evolution takes place. We consider a paradigmatic model, the quantum baker map in contact with a heat bath at finite temperature. We analyze the behavior of the purity, fidelity and Husimi distributions corresponding to initial states localized on short periodic orbits (scar functions) and map eigenstates. Scar functions, that have a fundamental role in the semiclassical description of chaotic systems, emerge as robust relative to other states (which are localized on classical structures) against environmental perturbations. Also, purity and fidelity show a complementary behavior as decoherence measures.

Scaling laws in the quantum-to-classical transition in chaotic systems

We study the quantum-to-classical transition in a chaotic system surrounded by a diffusive environment. First, we analyze the emergence of classicality when it is monitored by the Renyi entropy, a measure of the entanglement of a system with its environment. We show that the Renyi entropy has a transition from quantum to classical behavior that scales with heff2D, where heff is the effective Planck constant and D is the strength of the noise. However, it was recently shown that a different scaling law controls the quantum-to-classical transition when it is measured comparing the corresponding phase-space distributions. Then, we discuss the meaning of both scalings in the precise definition of a frontier between the classical and quantum behaviors. Finally, we show that there are quantum coherences that the Renyi entropy is unable to detect, which questions its use in studies of decoherence.

Decoherence induced by an interacting spin environment in the transition from integrability to chaos

We investigate the decoherence properties of a central system composed of two spins 12 in contact with a spin bath. The dynamical regime of the bath ranges from a fully integrable limit to complete chaoticity. We show that the dynamical regime of the bath determines the efficiency of the decoherence process. For perturbative regimes, the integrable limit provides stronger decoherence, while in the strong coupling regime the chaotic limit becomes more efficient. We also show that the decoherence time behaves in a similar way. On the contrary, the rate of decay of magnitudes like linear entropy or fidelity does not depend on the dynamical regime of the bath. We interpret the latter results as due to a comparable complexity of the Hamiltonian for both the integrable and the fully chaotic limits.

Quantum entropy dynamics for chaotic systems beyond the classical limit

The entropy production rate for an open quantum system with a classically chaotic limit has been previously argued to be independent of h and D, the parameter denoting coupling to the environment, and to be equal to the sum of generalized Lyapunov exponents, with these results applying in the near-classical regime. We present results for a specific system going well beyond earlier work, considering how these dynamics are altered for the Duffing problem by changing h,D and show that the entropy dynamics have a transition from classical to quantum behavior that scales, at least for a finite time, as a function of h2/D.

Lyapunov generation of entanglement and the correspondence principle

We show how a classically vanishing interaction generates entanglement between two initially nonentangled particles, without affecting their classical dynamics. For chaotic dynamics, the rate of entanglement is shown to saturate at the Lyapunov exponent of the classical dynamics as the interaction strength increases. In the saturation regime, the one-particle Wigner function follows classical dynamics better and better as one goes deeper and deeper in the semiclassical limit. This demonstrates that quantum-classical correspondence at the microscopic level does not require coupling to a large number of external degrees of freedom.

Decoherence, tunneling, and noise-induced activation in a double-potential well at high and zero temperature

We study the effects of the environment on tunneling in an open system described by a static double-well potential. We describe the evolution of a quantum state localized in one of the minima of the potential at t = 0, in both the limits of high and zero environment temperature. We show that the evolution of the system can be summarized in terms of three main physical phenomena--namely, decoherence, quantum tunneling, and noise-induced activation--and we obtain analytical estimates for the corresponding time scales. These analytical predictions are confirmed by large-scale numerical simulations, providing a detailed picture of the main stages of the evolution and of the relevant dynamical processes.

Spectral properties and classical decays in quantum open systems

We study the relationship between the spectral properties of diffusive open quantum maps and the classical spectrum of Ruelle-Pollicott resonances. The leading resonances determine the asymptotic time regime for several quantities of interest--the linear entropy, the Loschmidt echo, and the correlations of the initial state. A numerical method that allows an efficient calculation of the leading spectrum is developed using a truncated basis adapted to the dynamics.

Entanglement production in coupled chaotic systems: Case of the kicked tops

Entanglement production in coupled chaotic systems is studied with the help of kicked tops. Deriving the correct classical map, we have used the reduced Husimi function, the Husimi function of the reduced density matrix, to visualize the possible behaviors of a wave packet. We have studied a phase-space based measure of the complexity of a state and used random matrix theory (RMT) to model the strongly chaotic cases. Extensive numerical studies have been done for the entanglement production in coupled kicked tops corresponding to different underlying classical dynamics and different coupling strengths. An approximate formula, based on RMT, is derived for the entanglement production in coupled strongly chaotic systems. This formula, applicable for arbitrary coupling strengths and also valid for long time, complements and extends significantly recent perturbation theories for strongly chaotic weakly coupled systems.

Decoherence and the Loschmidt echo

Decoherence causes entropy increase that can be quantified using, e.g., the purity sigma=Trrho(2). When the Hamiltonian of a quantum system is perturbed, its sensitivity to such perturbation can be measured by the Loschmidt echo M(t). It is given by the squared overlap between the perturbed and unperturbed state. We describe the relation between the temporal behavior of sigma(t) and the average Mmacr;(t). In this way we show that the decay of the Loschmidt echo can be analyzed using tools developed in the study of decoherence. In particular, for systems with a classically chaotic Hamiltonian the decay of sigma and Mmacr; has a regime where it is dominated by the Lyapunov exponents.

Classical decays in decoherent quantum maps

The linear entropy and the Loschmidt echo have proved to be of interest recently in the context of quantum information and of the quantum to classical transitions. We study the asymptotic long-time behavior of these quantities for open quantum maps and relate the decays to the eigenvalues of a coarse-grained superoperator. In specific ranges of coarse graining, and for chaotic maps, these decay rates are given by the Ruelle-Pollicott resonances of the classical map.

When is quantum decoherence dynamics classical?

A direct classical analog of quantum decoherence is introduced. Similarities and differences between decoherence dynamics examined quantum mechanically and classically are exposed via a second-order perturbative treatment and via a strong decoherence theory, showing a strong dependence on the nature of the system-environment coupling. For example, for the traditionally assumed linear coupling, the classical and quantum results are shown to be in exact agreement.

Decoherence for classically chaotic quantum maps

We study the behavior of an open quantum system, with an N-dimensional space of states, whose density matrix evolves according to a nonunitary map defined in two steps: A unitary step, where the system evolves with an evolution operator obtained by quantizing a classically chaotic map (baker's map and Harper's map are the two examples we consider). A nonunitary step where the evolution operator for the density matrix mimics the effect of diffusion in the semiclassical (large N) limit. The process of decoherence and the transition from quantum to classical behavior are analyzed in detail by means of numerical and analytic tools. The existence of a regime where the entropy grows with a rate that is independent of the strength of the diffusion coefficient is demonstrated. The nature of the processes that determine the production of entropy is analyzed.

Quantum effects after decoherence in a quenched phase transition

We study a quantum mechanical toy model that mimics some features of a quenched phase transition. Both by virtue of a time-dependent Hamiltonian or by changing the temperature of the bath we are able to show that even after classicalization has been reached, the system may display quantum behavior again. We explain this behavior in terms of simple nonlinear analysis and estimate relevant time scales that match the results of numerical simulations of the master equation. This opens new possibilities both in the study of quantum effects in nonequilibrium phase transitions and in general time-dependent problems where quantum effects may be relevant even after decoherence has been completed.

Decoherence in a classically chaotic quantum system: entropy production and quantum-classical correspondence

We study the decoherence process for an open quantum system that is classically chaotic (a quartic double well with harmonic driving coupled to a sea of harmonic oscillators). We carefully analyze the time dependence of the rate of entropy production showing that it has two relevant regimes: For short times it is proportional to the diffusion coefficient (fixed by the system-environment coupling strength); for longer times (but before equilibration) it is fixed by dynamical properties of the system (and is related to the Lyapunov exponent). The nature of the transition time between both regimes is investigated and the issue of quantum to classical correspondence is addressed. Finally, the impact of the interaction with the environment on coherent tunneling is analyzed.