Fat fractals in quantum chaos

Semiclassical quantum wavefunctions in a nonintegrable exchange-coupled three-spin system are studied by using Fock representations. Their projected binary phase patterns show a fat fractal area-scaling property, whose exponent distinguishes the difference between the quantum analogs of classical regular orbits and of chaos.     

Type II quantum chaos

A classification of quantum systems into three categories, type I, II and III, is proposed. The classification is based on the degree of sensitivity upon initial conditions, and the appearance of chaos. The quantum dynamics of type I systems is quasi periodic displaying no exponential sensitivity. They arise, e.g., as the quantized versions of classical chaotic systems. Type II systems are obtained when classical and quantum degrees of freedom are coupled. Such systems arise naturally in a dynamic extension of the first step of the Born-Oppenheimer approximation, and are of particular importance to molecular and solid state physics. Type II systems can show exponential sensitivity in the quantum subsystem. Type III systems are fully quantized systems which show exponential sensitivity in the quantum dynamics. No example of a type III system is currently established. This paper presents a detailed discussion of a type II quantum chaotic system which models a coupled electronic-vibronic system. It is argued that type II systems are of importance for any field systems (not necessarily quantum) that couple to classical degrees of freedom.     

Optical chaos in multiple degenerate four-wave mixing

Spatial evolution has been studied of an ensemble of equal-frequency light waves involved in several four-wave mixing processes under linear photoabsorption. It is shown that this system may exhibit spatial chaos caused by competition of optical mixing processes with strong energy exchange between the waves. For pulsed radiation, the temporal envelopes of interacting waves undergo modulations which may also be of chaotic character. Also discussed are possible variants of identifying chaos in systems of interacting light waves experimentally and the role of accompanying photoprocesses (photoabsorption, self-action of radiation).     

Disappearance of quantum chaos in coupled chaotic quantum dots

Statistical properties of the single electron levels confined in the semiconductor (InAs/GaAs, Si/SiO₂) double quantum dots (DQDs) are considered. We demonstrate that in the electronically coupled chaotic quantum dots the chaos with its level repulsion disappears and the nearest neighbor level statistics becomes Poissonian. This result is discussed in the light of the recently predicted “huge conductance peak” by R.S. Whitney et al. [Phys. Rev. Lett. 102 (2009) 186802] in the mirror symmetric DQDs.     

Transitions to chaos in intracavity parametric wave mixing

The complex dynamics of intracavity three-wave mixing are discussed. Detailed results obtained by numerical simulation of routes to chaos are presented for a wide range of parameters. As the pump intensity increases, a complex sequence of alternating regular and chaotic self-modulation regimes is observed. The relationship between these regimes and soliton formation and propagation is analyzed.     

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.     

Temporal chaos versus spatial mixing in reaction-advection-diffusion systems

We develop a theory describing the transition to a spatially homogeneous regime in a mixing flow with a chaotic in time reaction. The transverse Lyapunov exponent governing the stability of the homogeneous state can be represented as a combination of Lyapunov exponents for spatial mixing and temporal chaos. This representation, being exact for time-independent flows and equal Pe clet numbers of different components, is demonstrated to work accurately for time-dependent flows and different Pe clet numbers.     

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)     

Perspectives on relativistic quantum chaos

Quantum Chaos has been investigated for about a half century.It is an old yet vigorous interdisciplinary field with new concepts and interesting topics emerging constantly.Recent years have witnessed a growing interest in quantum chaos in relativistic quantum systems,leading to the still developing field of relativistic quantum chaos.The purpose of this paper is not to provide a thorough review of this area,but rather to outline the basics and introduce the key concepts and methods in a concise way.A few representative topics are discussed,which may help the readers to quickly grasp the essentials of relativistic quantum chaos.A brief overview of the general topics in quantum chaos has also been provided with rich references.     

Fundamental concepts of quantum chaos

We review the fundamental concepts of quantum chaos in Hamiltonian systems. The quantum evolution of bound systems does not possess the sensitive dependence on initial conditions, and thus no chaotic behaviour occurs, whereas the study of the stationary solutions of the Schrödinger equation in the quantum phase space (Wigner functions) reveals precise analogy of the structure of the classical phase portrait. We analyze the regular eigenstates associated with invariant tori in the classical phase space, and the chaotic eigenstates associated with the classically chaotic regions, and the corresponding energy spectra. The effects of quantum localization of the chaotic eigenstates are treated phenomenologically, resulting in Brody-like level statistics, which can be found also at very high-lying levels, while the coupling between the regular and the irregular eigenstates due to tunneling, and of the corresponding levels, manifests itself only in low-lying levels.     

Quantum instantons and quantum chaos

We suggest a closed form expression for the path integral of quantum transition amplitudes to construct a quantum action. Based on this we propose rigorous definitions of both, quantum instantons and quantum chaos. As an example we compute the quantum instanton of the double well potential.     

Topological aspects of quantum chaos

Quantized classically chaotic maps on a toroidal two-dimensional phase space are studied. A discrete, topological criterion for phase-space localization is presented. To each eigenfunction is associated an integer, analogous to a quantized Hall conductivity, which tests the way the eigenfunction explores the phase space as some boundary conditions are changed. The correspondence between delocalization and chaotic classical dynamics is discussed, as well as the role of degeneracies of the eigenspectrum in the transition from localized to delocalized states. The general results are illustrated with a particular model.     

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