Long time behavior in the quantized standard map with dissipation

A dissipative version of the quantized standard map is constructed by analytical means and iterated numerically to study the long time behavior in various regions of the damping rate. For weak dissipation, stochastic transitions induced by the heat bath disrupt the localization in the action variable, which suppresses chaotic motion in the conservative quantized standard map, and tend to restore diffusion of action. A steady state is reached on the time scale of classical relaxation. For strong dissipation, observable deviations from classical behavior both in the transients and in the statey state are due to quantum noise. They are reproduced by a classical stochastic map which is approached by the dissipative quantum map as its semi-classical limit.     

Quantization of the kicked rotator with dissipation

The effect of dissipation on a quantum system exhibiting chaos in its classical limit is studied by coupling the kicked quantum rotator to a reservoir with angular momentum exchange. A master equation is derived which maps the density matrix from one kick to the subsequent one. Several limiting cases are investigated. The limits of 0 and of vanishing dissipation serve as tests of consistency, in reproducing the maps of the classical kicked damped rotator and of the kicked quantum rotator, respectively. In the limit of strong dissipation the classical map reduces to a circle map. A quantum map corresponding to the circle map is therefore obtained in this limit. In the limit of infinite dissipation the density matrix becomes independent of the initial condition after a single application of the map, allowing for a simple analytical solution for the density matrix. In the semi-classical limit the quantum map reduces to a classical map with quantum mechanically determined classical noise terms, which are evaluated. For sufficiently small dissipation the physical character of the leading quantum corrections changes. Quantum mechanical interference effects then render the Wigner distribution negative in some parts of phase space and prevent its interpretation in classical terms. Numerical results will be presented in a subsequent paper.     

A Semiclassical Theory of a Dissipative Henon—Heiles System

A semiclassical theory of a dissipative Henon—Heiles system is proposed. Based on -scaling of an equation for the evolution of the Wigner quasiprobability distribution function in the presence of dissipation and thermal diffusion, we derive a semiclassical equation for quantum fluctuations, governed by the dissipation and the curvature of the classical potential. We show how the initial quantum noise gets amplified by classical chaotic diffusion, which is expressible in terms of a correlation of stochastic fluctuations of the curvature of the potential due to classical chaos, and ultimately settles down to equilibrium under the influence of dissipation. We also establish that there exists a critical limit to the expansion of phase space. The limit is set by chaotic diffusion and dissipation. Our semiclassical analysis is corroborated by numerical simulation of a quantum operator master equation.     

Dissipative quantum chaos: transition from wave packet collapse to explosion

Using the quantum trajectories approach, we study the quantum dynamics of a dissipative chaotic system described by the Zaslavsky map. For strong dissipation the quantum wave function in the phase space collapses onto a compact packet which follows classical chaotic dynamics and whose area is proportional to the Planck constant. At weak dissipation the exponential instability of quantum dynamics on the Ehrenfest time scale dominates and leads to wave packet explosion. The transition from collapse to explosion takes place when the dissipation time scale exceeds the Ehrenfest time. For integrable nonlinear dynamics the explosion practically disappears leaving place to collapse.     

Limit cycles in quantum theories

Renormalization group limit cycles and more chaotic behavior may be commonplace for quantum Hamiltonians requiring renormalization, in contrast to experience based on classical models with critical behavior, where fixed points are far more common. We discuss the simplest quantum model Hamiltonian identified so far that exhibits a renormalization group with both limit cycle and chaotic behavior. The model is a discrete Hermitian matrix with two coupling constants, both governed by a nonperturbative renormalization group equation that involves changes in only one of these couplings and is soluble analytically.     

The crossover from classical to quantum behavior in Duffing oscillator based on quantum state diffusion

The classical Duffing oscillator is a dissipative chaotic system, and so there is not a definite Hamiltonian. We quantize the Duffing oscillator on the basis of quantum state diffusion (QSD) which is a formulation for open quantum systems and a useful tool for analyzing nonlinear problems and classical limits. We can define a ‘Lyapunov exponent’, which corresponds to the classical one in the proper limit, and investigate the behavior of the system by varying the Planck constant effectively. We show that there exists a critical stage, where the behavior of the system crosses over from classical to quantum one.     

Spectral properties of dissipative chaotic quantum maps

I examine spectral properties of a dissipative chaotic quantum map with the help of a recently discovered semiclassical trace formula. I show that in the presence of a small amount of dissipation the traces of any finite power of the propagator of the reduced density matrix, and traces of its classical counterpart, the Frobenius-Perron operator, are identical in the limit of variant Planck's over 2pi -->0. Numerically I find that even for finite variant Planck's over 2pi the agreement can be very good. This holds in particular if the classical phase space contains a strange attractor, as long as one stays clear of bifurcations. Traces of the quantum propagator for iterations of the map agree well with the corresponding traces of the Frobenius-Perron operator if the classical dynamics is dominated by a strong point attractor. (c) 1999 American Institute of Physics.     

Dissipative chaotic quantum maps: Expectation values, correlation functions and the invariant state

I investigate the propagator of the Wigner function for a dissipative chaotic quantum map. I show that a small amount of dissipation reduces the propagator of sufficiently smooth Wigner functions to its classical counterpart, the Frobenius-Perron operator, if . Several consequences arise: the Wigner transform of the invariant density matrix is a smeared out version of the classical strange attractor; time dependent expectation values and correlation functions of observables can be evaluated via hybrid quantum-classical formulae in which the quantum character enters only via the initial Wigner function. If a classical phase-space distribution is chosen for the latter or if the map is iterated sufficiently many times the formulae become entirely classical, and powerful classical trace formulae apply. Received 7 October 1999     

Dissipative death of quantum coherences in a spin system

We investigate the influence of weak dissipation on the dynamics of a kicked spin system. Compared to its classical limit the quantum system is strongly affected by small dissipation. We present the attenuation of two different intrinsically quantum phenomena, recurrencies and tunneling. We find agreement of analytical perturbative estimates and numerical results. We further show that dissipation acts quite differently on the quantum evolution depending on whether we are in a classically chaotic or regular domain.     

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.     

Anomalous transport and quantum-classical correspondence

We present evidence that anomalous transport in the classical standard map results in strong enhancement of fluctuations in the localization length of quasienergy states in the corresponding quantum dynamics. This generic effect occurs even far from the semiclassical limit and reflects the interplay of local and global quantum suppression mechanisms of classically chaotic dynamics. Possible experimental scenarios are also discussed.     

Nonmonotonicity in the quantum-classical transition: chaos induced by quantum effects

The classical-quantum transition for chaotic systems is understood to be accompanied by the suppression of chaotic effects as the relative variant Planck's over 2pi is increased. We show evidence to the contrary in the behavior of the quantum trajectory dynamics of a dissipative quantum chaotic system, the double-well Duffing oscillator. The classical limit in the case considered has regular behavior, but as the effective variant Planck's over 2pi is increased we see chaotic behavior. This chaos then disappears deeper into the quantum regime, which means that the quantum-classical transition in this case is nonmonotonic in variant Planck's over 2pi.     

Time-reversal characteristics of quantum normal diffusion: time-continuous models

In quantum map systems exhibiting normal diffusion, time-reversal characteristics converge to a universal scaling behavior which implies a prototype of irreversible quantum process [H.S. Yamada, K.S. Ikeda, Eur. Phys. J. B 85, 41 (2012)]. In the present paper, we extend the investigation of time-reversal characteristic to time-continuous quantum systems which show normal diffusion. Typical four representative models are examined, which is either deterministic or stochastic, and either has or not has the classical counterpart. Extensive numerical examinations demonstrate that three of the four models have the time-reversal characteristics obeying the same universal limit as the quantum map systems. The only nontrivial counterexample is the critical Harper model, whose time-reversal characteristics significantly deviates from the universal curve. In the critical Harper model modulated by a weak noise that does not change the original diffusion constant, time-reversal characteristic recovers the universal behavior.     

Classical and quantum chaos for a kicked top

We discuss a top undergoing constant precession around a magnetic field and suffering a periodic sequence of impulsive nonlinear kicks. The squared angular momentum being a constant of the motion the quantum dynamics takes place in a finite dimensional Hilbert space. We find a distinction between regular and irregular behavior for times exceeding the quantum mechanical quasiperiod at which classical behavior, whether chaotic or regular, has died out in quantum means. The degree of level repulsion depends on whether or not the top is endowed with a generalized time reversal invariance.     

Quantization of Hénon's map with dissipation

Hénon's map with dissipation is suspended to the nonlinearly kicked damped harmonic oscillator and then quantized. The ensuing master equation between two subsequent kicks is solved exactly in the representation by the Wigner distribution, resulting in a quantized version of Hénon's dissipative map. The semi-classical limit of the map is studied. The leading quantum corrections are shown to be associated with dissipation and can be formulated as a classical map with classical stochastic perturbations. The next-to-leading quantum corrections, arising from the nonlinearity of the kicks, are similar as in the area conserving map and cannot be described within the framwork of classical statistics. The Wigner distribution in the steady state is investigated in the limit of strong dissipation, where Hénon's map is reduced to the logistic map. The insensitivity of the main results against details of the quantization procedure is demonstrated by comparing with the results of a different phenomenological quantization procedure.Dedicated to B. Mühlschlegel on the occasion of his 60th birthdayOn leave from and address after 1^st of February 1985: Institute for Theoretical Physics, Eötvös University, H-1088 Budapest, Hungary     

Squeezed states and quantum chaos

We examine the dynamics of a wave packet that initially corresponds to a coherent state in the model of a quantum rotator excited by a periodic sequence of kicks. This model is the main model of quantum chaos and allows for a transition from regular behavior to chaotic in the classical limit. By doing a numerical experiment we study the generation of squeezed states in quasiclassical conditions and in a time interval when quantum-classical correspondence is well-defined. We find that the degree of squeezing depends on the degree of local instability in the system and increases with the Chirikov classical stochasticity parameter. We also discuss the dependence of the degree of squeezing on the initial width of the packet, the problem of stability and observability of squeezed states in the transition to quantum chaos, and the dynamics of disintegration of wave packets in quantum chaos. Zh. éksp. Teor. Fiz. 113, 111–127 (January 1998)     

Stable quantum computation of unstable classical chaos

We show on the example of the Arnold cat map that classical chaotic systems can be simulated with exponential efficiency on a quantum computer. Although classical computer errors grow exponentially with time, the quantum algorithm with moderate imperfections is able to simulate accurately the unstable chaotic classical nonlinear dynamics for long times. The algorithm can be easily implemented on systems of a few qubits.     

Quantum signatures of chaos or quantum chaos?

A critical analysis of the present-day concept of chaos in quantum systems as nothing but a “quantum signature” of chaos in classical mechanics is given. In contrast to the existing semi-intuitive guesses, a definition of classical and quantum chaos is proposed on the basis of the Liouville–Arnold theorem: a quantum chaotic system featuring N degrees of freedom should have M < N independent first integrals of motion (good quantum numbers) specified by the symmetry of the Hamiltonian of the system. Quantitative measures of quantum chaos that, in the classical limit, go over to the Lyapunov exponent and the classical stability parameter are proposed. The proposed criteria of quantum chaos are applied to solving standard problems of modern dynamical chaos theory.     

Non-Markovian decoherent quantum walks

Quantum walks act in obviously different ways from their classical counterparts, but decoherence will lessen and close this gap between them. To understand this process, it is necessary to investigate the evolution of quantum walks under different decoherence situations. In this article, we study a non-Markovian decoherent quantum walk on a line. In a short time regime, the behavior of the walk deviates from both ideal quantum walks and classical random walks. The position variance as a measure of the quantum walk collapses and revives for a short time, and tends to have a linear relation with time. That is, the walker's behavior shows a diffusive spread over a long time limit, which is caused by non-Markovian dephasing affecting the quantum correlations between the quantum walker and his coin. We also study both quantum discord and measurement-induced disturbance as measures of the quantum correlations, and observe both collapse and revival in the short time regime, and the tendency to be zero in the long time limit. Therefore, quantum walks with non-Markovian decoherence tend to have diffusive spreading behavior over long time limits, while in the short time regime they oscillate between ballistic and diffusive spreading behavior, and the quantum correlation collapses and revives due to the memory effect.     

量子参数激励单摆的局域效应研究

分析了二能级原子在振幅调制主波场中动量扩散模型。这是一个量子参数激励单摆系统。这个量子系统在经典极限下表现混沌行为。在相同参数条件下,这个系统具有动力学特征。