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.     

Quantum nonintegrability and the classical limit for usp(4) systems

We investigate the transition from integrability to chaos in a system built of usp(4) elements, both in the quantum case and in its classical limit, obtained using coherent states. This algebraic Hamiltonian consists in an integrable term plus a nonlinear perturbation, and we see that the level spacing distribution for the quantum system is well approximated by the Berry-Robnik-Brody distribution, and accordingly the classical limit displays mixed dynamics.     

Integrable and nonintegrable classical spin clusters

This study investigates the nonlinear dynamics of a pair of exchange-coupled spins with biaxial exchange and single-site anisotropy. It represents a Hamiltonian system with 2 degrees of freedom for which we have already established the (nontrivial) integrability criteria and constructed the integrals of the motion provided they exist. Here we present a comparative study of the phase-space trajectories for two specific models with the same symmetry properties, one of which (the XY model with exchange anisotropy) is integrable, and the other (the XY model with single-site anisotropy) nonintegrable. In the integrable model, the integrals of the motion (analytic invariants) can be reconstructed numerically by means of time averages of dynamical variables over all trajectories. In the nonintegrable model, such time averages over trajectories define nonanalytic invariants, where the nonanalyticities are associated with the presence of chaotic trajectories. A prominent feature in the nonintegrable model is the occurrence of very long time scales caused by the presence of low-flux cantori, which form sticky coats on the boundary between chaotic regions and regular islands or leaky walls between different chaotic regions. These cantori dominate the convergence properties of time averages and presumably determine the long-time asymptotic properties of dynamic correlation functions. Finally, we present a special class of integrable systems containing arbitrarily many spins coupled by general biaxial exchange anisotropy.     

Spatio-Temporal Evolution of Initial Coherent State Under the Action of Quantum Integrable and Nonintegrable Systems

According to the exactly formulated concepts of quantum integrability and nonintegrability,an initial coherent state |Φ⁰(γ)>evolving under the action of an integrable Hamiltonian having the same dynamical symmetry is expressed as an analytical function of γ and t. The expectation values of various dynamical variables should be expressed as correlated analytical functions of γ and t . Thus, the general characters of initial coherent state would be maintained during the spatio-temporal evolution under the action of such a system even if the wavepacket is slightly distorted. In contrast, when the integrability condition is destroyed, these features will be lost during the evolution. This theoretical prediction is illustrated with numerical results computed for the two-level Lipkin model with a dou ble-well potential and dynamical explanations are given. And the Eherenfest time is verified with this model.     

Integrability, stability, and adiabaticity in nonlinear stimulated Raman adiabatic passage

We study the dynamics of a two-color photoassociation of atoms into diatomic molecules via nonlinear stimulated Raman adiabatic passage process. The system has a famous counterpart in (linear) quantum mechanics, and has been discussed recently in the context of generalizing the quantum adiabatic theorem to nonlinear systems. Here we use another approach to study adiabaticity and stability in the system: we apply methods of classical Hamiltonian dynamics. We find nonlinear dynamical instabilities, cases of complete integrability, and improved conditions of adiabaticity.     

Integrability of the S-matrix versus integrability of the Hamiltonian

This report reviews the relations between the integrability properties of the S-matrix and of the Hamiltonian. Particular emphasis is put on the situation where the Hamiltonian has a conserved quantity which is not compatible with the asymptotics and where correspondingly the integrability does not transfer to the S-matrix. As questions of integrability are more readily handled in classical dynamics, all developments are first performed classically. Several examples are discussed to illustrate the main points. The quantum mechanical discussion reveals that the eigenphase statistics of the S-matrix depends principally on the chaoticity of the scattering map while basis dependent quantities such as the distribution of matrix elements tend to have random matrix behaviour only in the presence of topological chaos. The relevance of these considerations to the evaluation of scattering data is discussed.     

Two-state system coupled to a boson mode: Quantum dynamics and classical approximations

The relation between classical and quantum mechanical integrability is investigated for a boson mode coupled to a two-level system. Different semi-classical approximations of this system are considered which are obtained by (i) factorization of expectation values of the two-state variable and the boson, (ii) making a WKB-type approximation, (iii) replacing the boson by a classical field of constant amplitude and fixed frequency and (iv) putting the boson into a self-consistent coherent state. The results vary considerably and include cases of non-integrable and integrable classical dynamics. Quantum mechanically the system is found to satisfy a criterion of quantum mechanical integrability, which we formulate, but the separated Hamiltonian of the boson alone does not have a well-defined classical limit. Numerical results for the energy spectrum and expectation values are obtained, which show a high degree of regularity but also display overlapping avoided crossings usually associated with non-integrable Hamiltonians. The exact dynamics of the occupation probabilities of the two levels is also analysed numerically. The dependence of quantum mechanical recurrence effects (in quantum optics known as revivals) on coupling strength, frequency detuning and initial conditions is studied. The revivals are found to disappear in the case of strong coupling. The Fourier spectra of the dynamical expectation values are also calculated     

Chaotic delocalization of two interacting particles in the classical Harper model

We study the problem of two interacting particles in the classical Harper model in the regime when one-particle motion is absolutely bounded inside one cell of periodic potential. The interaction between particles breaks integrability of classical motion leading to emergence of Hamiltonian dynamical chaos. At moderate interactions and certain energies above the mobility edge this chaos leads to a chaotic propulsion of two particles with their diffusive spreading over the whole space both in one and two dimensions. At the same time the distance between particles remains bounded by one or two periodic cells demonstrating appearance of new composite quasi-particles called chaons. The effect of chaotic delocalization of chaons is shown to be rather general being present for Coulomb and short range interactions. It is argued that such delocalized chaons can be observed in experiments with cold atoms and ions in optical lattices.     

Quantum chaos in nuclear physics

A definition of classical and quantum chaos on the basis of the Liouville–Arnold theorem is proposed. According to this definition, a chaotic quantum system that has N degrees of freedom should have M < N independent first integrals of motion (good quantum numbers) that are determined by the symmetry of the Hamiltonian for the system being considered. Quantitative measures of quantum chaos are established. In the classical limit, they go over to the Lyapunov exponent or the classical stability parameter. The use of quantum-chaos parameters in nuclear physics is demonstrated.     

Integrable and nonintegrable classical spin clusters

The nonlinear dynamics is investigated for a system ofN classical spins. This represents a Hamiltonian system withN degrees of freedom. According to the Liouville theorem, the complete integrability of such a system requires the existence ofN independent integrals of the motion which are mutually in involution. As a basis for the investigation of regular and chaotic spin motions, we have examined in detail the problem of integrability of a two-spin system. It represents the simplest autonomous spin system for which the integrability problem is nontrivial. We have shown that a pair of spins coupled by an anisotropic exchange interaction represents a completely integrable system for any values of the coupling constants. The second integral of the motion (in addition to the Hamiltonian), which ensures the complete integrability, turns out to be quadratic in the spin variables. If, in addition to the exchange anisotropy also singlesite anisotropy terms are included in the two-spin Hamiltonian, a second integral of the motion quadratic in the spin variables exists and thus guarantees integrability, only if the model constants satisfy a certain condition. Our numerical calculations strongly suggest that the violation of this condition implies not only the nonexistence of a quadratic integral, but the nonexistence of a second independent integral of motion in general. Finally, as an example of a completely integrableN-spin system we present the Kittel-Shore model of uniformly interacting spins, for which we have constructed theN independent integrals in involution as well as the action-angle variables explicitly.     

Quantum NOT operation and integrability in two-level systems

We demonstrate the surprising integrability of the classical Hamiltonian associated to a spin 1/2 system under periodic external fields. The one-qubit rotations generated by the dynamical evolution is, on the one hand, close to that of the rotating wave approximation (RWA), on the other hand to two different “average” systems, according to whether a certain parameter is small or large. Of particular independent interest is the fact that both the RWA and the averaging theorem are seen to hold well beyond their expected region of validity. Finally, we determine conditions for the realization of the quantum NOT operation by means of classical stroboscopic maps.     

Regular and chaotic quantum dynamics in atom-diatom reactive collisions

A new microirreversible 3D theory of quantum multichannel scattering in the three-body system is developed. The quantum approach is constructed on the generating trajectory tubes which allow taking into account influence of classical nonintegrability of the dynamical quantum system. When the volume of classical chaos in phase space is larger than the quantum cell in the corresponding quantum system, quantum chaos is generated. The probability of quantum transitions is constructed for this case. The collinear collision of the Li + (FH) → (LiF) + H system is used for numerical illustration of a system generating quantum (wave) chaos. The text was submitted by the authors in English.     

Chaos, fractals, and atomic flights in cavities

A semiclassical study is carried out of the nonlinear interaction dynamics between two-level atoms and a standing-wave field in a high-finesse cavity. As a result of atomic movement or wave amplitude modulation, a dynamic local instability occurs in a strongly coupled atom-field system. The appearance of dynamical Hamiltonian chaos, fractals, and Lévy flights is demonstrated for the models of two experimental devices: a (micro)maser with thermal Rydberg atoms and a microlaser with cold atoms. Numerical simulation showed that the manifestations of classical chaos, atomic fractals, and flights can be observed in the appropriate real experiments. Attention is drawn to the prospects provided by work on the atom-field systems in the coupling-modulated high-finesse cavities for further investigation of the quantum-classical correspondence, quantum chaos, and decoherence.     

Quantum Invariants for Solving the Time-Dependent Schrödinger Equation in One Dimension

Extending the work of Lewis and Leach on classical invariants for solving the classical equation of motion in one-dimensional system, the quantum invariants in polynomial form of momentum are obtained. The involved Hamiltonian is time-dependent and quadratic in momentum.     

Quantum monodromy and molecular spectroscopy

Monodromy (or once round) is a classical property of integrable dynamical systems in two or more degrees of freedom, which imposes a characteristic pattern on the quantum mechanical eigenvalue distribution. This article explains the connection by showing how the presence of an isolated critical point of the Hamiltonian leads to a classical action function that is multi-valued with respect to energy and angular momentum. Consequently, by the Bohr correspondence principle between actions and quantum numbers, there can be no uniquely defined global system of quantum numbers. Implications for the interpretation of highly excited molecular spectra are brought out by reference to quasi-linear molecules, which transfer one degree of freedom from rotational to vibrational motion during the excitation process. Emphasis is placed on the simplest examples, while a brief resumé of the wide scope of the quantum monodromy phenomenon is given in the final section.     

Ehrenfest ‘phase-space’ trajectories and quantum chaos in He atom under strong,oscillating magnetic fields: an application of time-dependent quantum fluid density functional theory (TDQFDFT)

Using a dynamical signature proposed earlier from our laboratory, quantum chaos in He atom interacting with strong, oscillating magnetic fields has been studied through a comparison between the nonlinear divergence of two neighbouring Ehrenfest ‘phase-space’ (EPS) trajectories differing slightly in initial conditions and the Loschmidt echo. The dynamical EPS signature can detect quantum chaos independently of the Loschmidt echo and in agreement with the latter, even for low-lying states, in the same spirit as that of classical chaos. This time-dependent signature extends the concept of quantum chaos to systems which have no classical counterparts and brings the concept of quantum chaos closer to that of classical chaos.     

Quantum Lyapunov Exponents

We show that it is possible to associate univocally with each given solution of the time-dependent Schrödinger equation a particular phase flow (quantum flow) of a non-autonomous dynamical system. This fact allows us to introduce a definition of chaos in quantum dynamics (quantum chaos), which is based on the classical theory of chaos in dynamical systems. In such a way we can introduce quantities which may be appelled quantum Lyapunov exponents. Our approach applies to a non-relativistic quantum-mechanical system of n charged particles; in the present work numerical calculations are performed only for the hydrogen atom. In the computation of the trajectories we first neglect the spin contribution to chaos, then we consider the spin effects in quantum chaos. We show how the quantum Lyapunov exponents can be evaluated and give several numerical results which describe some properties found in the present approach. Although the system is very simple and the classical counterpart is regular, the most non-stationary solutions of the corresponding Schrödinger equation are chaotic according to our definition.     

de Broglie-Bohm formulation of quantum mechanics, quantum chaos and breaking of time-reversal invariance

A recently developed unified theory of classical and quantum chaos, based on the de Broglie-Bohm (Hamilton-Jacobi) formulation of quantum mechanics is presented and its consequences are discussed. The quantum dynamics is rigorously defined to be chaotic if the Lyapunov number, associated with the quantum trajectories in de Broglie-Bohm phase space, is positive definite. This definition of quantum chaos which under classical conditions goes over to the well-known definition of classical chaos in terms of positivity of Lyapunov numbers, provides a rigorous unified definition of chaos on the same footing for both the dynamics. A demonstration of the existence of positive Lyapunov numbers in a simple quantum system is given analytically, proving the existence of quantum chaos. Breaking of the time-reversal symmetry in the corresponding quantum dynamics under chaotic evolution is demonstrated. It is shown that the rigorous deterministic quantum chaos provides an intrinsic mechanism towards irreversibility of the Schrodinger evolution of the wave function, without invoking ‘wave function collapse’ or ‘measurements’