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Journal of Mathematical Physics

Quantum chaos in the group-theoretical picture

Leonid E. Kon’kov, Sergey V. Prants

Published Mar 1, 1996• 30 citations
Publisher

Abstract

The dynamical-group approach is developed and applied to investigate the problems of controllability and quantum chaos in two fundamental models of the matter–radiation interaction. It provides a new insight into the dynamics of nonstationary quantum process of the interaction between two-level atoms and a single-mode radiation field without and with the feedback. A sequence of transitions from the quasiperiodicity to chaos has been numerically observed for two-level atoms interacting with a self-consistently generated radiation field. The unitary irreducible representations of the SU(2) group of dynamical symmetry in a noncanonical parametrization is constructed, allowing one to use the results for describing the time evolution of any driven quantum system with the underlying SU(2) symmetry.

DOI
10.1063/1.531439

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The dynamical-group approach is developed and applied to investigate the problems of controllability and quantum chaos in two fundamental models of the matter–radiation interaction. It provides a new insight into the dynamics of nonstationary quantum process of the interaction between two-level atoms and a single-mode radiation field without and with the feedback. A sequence of transitions from the quasiperiodicity to chaos has been numerically observed for two-level atoms interacting with a self-consistently generated radiation field. The unitary irreducible representations of the SU(2) group of dynamical symmetry in a noncanonical parametrization is constructed, allowing one to use the results for describing the time evolution of any driven quantum system with the underlying SU(2) symmetry.
We apply the dynamical symmetry method for treating quantum atomic dynamics both in a prescribed laser field and in a self- consistent radiation field. In the framework of such an approach a new reasonable criterion for 'quantum chaos' that is connected with the behavior of the dynamical-group parameter in the pseudophase plane can be introduced in addition to the known criteria. For the simplest SU(2) model (an ensemble of two-level atoms), it is shown that the atomic dynamics essentially depend on a field polarization. The motion of SU(2) parameter g in the plane Re g - Im g is quite regular for a circular polarization and is ergodic for a linear one.
Dynamics of two-level atoms interacting with their own radiation field in a single-mode high-quality resonator is considered. The dynamical system consists of two second-order differential equations, one for the atomic SU(2) dynamical-group parameter and another for the field strength. With the help of the maximal Lyapunov exponent for this set, we numerically investigate transitions from regularity to deterministic quantum chaos in such a simple model. Increasing the collective coupling constant b is identical with 8(pi)N(sub 0)(d(exp 2))/hw, we observed for initially unexcited atoms a usual sharp transition to chaos at b(sub c) approx. equal to 1. If we take the dimensionless individual Rabi frequency a = Omega/2w as a control parameter, then a sequence of order-to-chaos transitions has been observed starting with the critical value a(sub c) approx. equal to 0.25 at the same initial conditions.
An analytic symmetry approach is proposed and applied to nonlinear Hamiltonian dynamics of the self-consistent interaction of a two-level atom in a high-Q cavity with its own radiation field. A numerical investigation is reported of a transition from quasiperiodic to deterministically chaotic oscillations in a system of this kind which has classical and quantum degrees of freedom. This occurs when the control parameter Ω, which is the constant of the atom—field interaction, is increased. The largest Lyapunov exponent (which is a measure of the degree of chaos of a system) is used to demonstrate an alternation of the order—chaos transitions with increase in Ω. The power spectra and the phase portraits obtained for different ranges of Ω confirm the predicted path of the appearance of dynamic chaos in the fundamental model of quantum electronics.
The coherent state representations of the group G equals W1 (direct product) G0 [where G0 equals SU(2) or SU(1,1)] are used in computer simulations of the dynamic of single two- level atom [G0 equals SU(2)] interacting with a quantized photon cavity mode [Jaynes-Cummings Model (JCM) without the rotating wave approximation] and, in general, nonlinear in photon variables. The second case [hyperbolic Jaynes - Cummings Model (HJCM), G0 equals SU(1,1)] corresponds to the quantum dynamics of quadratic nonlinear coupled oscillators (the parametric resonance on double field frequency and a three-wave parametric processes of nonlinear optics). Quasiclassical dynamical equations for parameters of approximately factorizable coherent states for these models are derived and regimes of motion for 'atom' and field variables are analyzed.
With the help of Lyapunov exponents different routes to quantum chaos are demonstrated numerically in two semiclassical models of the interaction of N two-level atoms with their own radiation field in a resonant cavity. The usual transition to chaos is observed when varying the external control parameter b which describes the interaction of the atomic ensemble with the field. After reaching its critical value b approximately equal to 1 the system becomes more and more chaotic as b increases. As the second control parameter a (the dimensionless individual Rabi frequency) increases the system undergoes transitions to chaos by a completely different way. We observe a complex sequence of order-to-chaos transitions that resembles those for the classical Lorenz attractor.
The dynamics of an ensemble of two-level atoms moving through a single-mode lossless cavity is investigated in the semiclassical and rotating-wave approximations. The dynamical system for the expectation values of the atomic and field observables is considered as a perturbation to one of the following integrable versions: (i) a model with atoms moving through a spatially inhomogeneous resonant field, and (ii) a model with atoms interacting with a nonresonant eigenmode which is assumed to be homogeneous on the cavity size. We find the general exact solutions for both the models and show that they contain special solutions describing a coherent effect of population and radiation trapping. Using the Melnikov method, we prove analytically transverse intersections of stable and unstable manifolds of a hyperbolic fixed point under a small modulation of the vacuum Rabi frequency. These intersections are believed to provide the Smale horseshoe mechanism of Hamiltonian chaos. The analytical results are accompanied with direct computation of topographical maps of maximal Lyapunov exponents that give a representative image of regularity and chaos in the atom-field system in different ranges of its control parameters--the frequency detuning, the number, and the velocity of atoms.