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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Manifestations of quantum chaos in simple nonstationary optical processes with dynamic symmetry

Leonid E. Kon’kov, Sergey V. Prants

Published May 3, 1994• 0 citations
Publisher

Abstract

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.

DOI
10.1117/12.175139

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arXiv:quant-ph/9702016•Feb 6, 1997•0 citations
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.
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.
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.
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.
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.
Abstract This paper is a review article on some aspects of quantum–classical correspondence in chaotic dynamics of cold atoms interacting with a standing-wave laser field forming an optical lattice. The problem is treated from both (semi)classical and quantum points of view. In both approaches, the interaction of an atomic electic dipole with the laser field is treated quantum mechanically. Translational motion is described, at first, classically (atoms are considered to be point-like objects) and then quantum mechanically as a propagation of matter waves. Semiclassical equations of motion are shown to be chaotic in the sense of classical dynamical chaos. Point-like atoms in an absolutely deterministic and rigid optical lattice can move in a random-like manner demonstrating a chaotic walking with typical features of classical chaos. This behavior is explained by random-like ‘jumps’ of one of the atomic internal variable when atoms cross nodes of the standing wave and occurs in a specific range of the atom-field detuning. When treating atoms as matter waves, we show that they can make nonadiabatic transitions when crossing the standing-wave nodes. The point is that atomic wave packets split at each node in the same range of the atom-field detuning where the classical chaos occurs. The key point is that the squared amplitude of those semiclassical ‘jumps’ equal to the quantum Landau–Zener parameter which defines the probability of nonadiabatic transitions at the nodes. Nonadiabatic atomic wave packets are much more complicated compared to adiabatic ones and may be called chaotic in this sense. A few possible experiments to observe some manifestations of classical and quantum chaos with cold atoms in horizontal and vertical optical lattices are proposed and discussed.
The coherent state representations of the group G=W1⊗G0G = W_1 \otimes G_0G=W1​⊗G0​ (where G0=SU(2),SU(1,1)G_0 = SU(2), SU(1,1)G0​=SU(2),SU(1,1)) are used in computer simulation of the dynamics of single two-level atom (G0=SU(2))(G_0 = SU(2))(G0​=SU(2)) interacting with a quantized photon cavity mode - the 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=SU(1,1))G_0 = SU(1,1))G0​=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.