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

Chaotic dynamics for a system of two-level atoms interacting with photons

Alexander V. Gorokhov, E. V. Rogacheva, V. V. Ruchkov

Published Oct 24, 2001• 0 citations
Publisher

Abstract

Some quantum optical models possessing a transition from regular to chaotic dynamics and influence of 'quantum chaos' to photons statistics are investigated. Particular cases of semiclassical dynamics of a single atom and cooperative system of these atoms with (2j + 1) -- equidistant levels interacting with a quantized photon mode in an ideal cavity were considered. We take into account parametric pumping. Corresponding equations of motion are received without the rotation-wave approximation. It was demonstrated the scenario of transition to chaos, realized by doubling frequencies mechanism with growing of 'atom + field' coupling constant. Chaotic behavior and squeezing degree time dependence were obtained in computer simulations for coupling constant values of order of the atomic frequency, the calculation of the maximal Lyapunov coefficient for the dynamics of many two- level atoms interacting with photon cavity mode as function of 'atoms + field' interaction constant is performed. An influence of possible parametric instabilities of nonlinear media and effects of dissipation for atoms are discussed.

DOI
10.1117/12.445691

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Optics and Spectroscopy•May 1, 2002•3 citations
Some quantum optical models possessing a transition from regular to chaotic dynamics and influence of 'quantum chaos' to photons statistics are investigated. Particular cases of semiclassical dynamics of a single atom and cooperative system of these atoms with (2j + 1) -- equidistant levels interacting with a quantized photon mode in an ideal cavity were considered. We take into account parametric pumping. Corresponding equations of motion are received without the rotation-wave approximation. It was demonstrated the scenario of transition to chaos, realized by doubling frequencies mechanism with growing of 'atom + field' coupling constant. Chaotic behavior and squeezing degree time dependence were obtained in computer simulations for coupling constant values of order of the atomic frequency, the calculation of the maximal Lyapunov coefficient for the dynamics of many two- level atoms interacting with photon cavity mode as function of 'atoms + field' interaction constant is performed. An influence of possible parametric instabilities of nonlinear media and effects of dissipation for atoms are discussed.
Some quantum optical models possessing a transition from regular to chaotic dynamics and influence of `quantum chaos' to squeezing are investigated. Particular cases of semiclassical dynamics of a single atom and cooperative system of these atoms with (2j + 1)--equidistant levels interacting with a quantized photon mode in an ideal cavity were considered. Corresponding equations of motion are received without the rotation-wave approximation. Chaotic behavior and squeezing degree time dependence were obtained in computer simulations for coupling constant values or order of the atomic frequency.
The influence of chaotic behavior of coherent state (CS) parameters on entanglement dynamics in system of two level atoms and photons in cavity is investigated. The initial state of the system is chosen as factorized production of photon and atomic coherent states. To find a solution of the Schrödinger equation, we presented the time-dependent wave function in the form of a wave packet in CS representation. The phase portraits and Poincare sections for classical dynamics of CS were computed, and time dependence of the mean number of photons level populations and atomic entropy were calculated in quantum case. The analysis showed that the chaotic behavior in quantum-optical Dicke model increases the degree of entanglement in the atomic subsystem. At the same time greater entanglement is generated when selecting the initial atomic CS parameters near the center of the corresponding Poincare sections. It is considered dynamics of two identical two-level atoms in non-ideal cavity (nonideal Tavis-Cummings model). Accounting for decay showed that the generation of atomic entanglement is accompanied by the return of the initial separable state (”death” of entanglement), but this behavior with the increase of the initial parameter of the field CS is only at sufficiently small times. Influence of detuning and damping constants are considered.
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.
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.
We study the nonlinear dynamics of the interaction of two-level atoms and a selected mode of a high- Q cavity with frequency modulation analytically and numerically. In the absence of modulation, the corresponding semiclassical Heisenberg equations for the expectation values of the collective atomic observables and the field-mode amplitudes allow, in the rotating wave approximation and in the strong-coupling limit, an exact solution with arbitrary detuning. Using this solution, we detect the coherent effect of trapping of the population of atomic levels and of trapping of the number of photons in the cavity. The explanation for this effect lies in the destructive interference of the atomic dipoles and the field mode. The integrable version of the system of equations exhibits a separatrix near which a stochastic layer is formed when modulation is introduced. The width of the layer is found to gradually increase with degree of modulation, and finally it fills the entire energy-permissible volume of the phase space. We show that the rotating wave approximation does not hinder the formation of Hamiltonian chaos in cavity semiclassical electrodynamics. The calculation of the maximum Lyapunov indices of nonlinear (in this approximation) equations of motion as functions of the modulation frequency δ and the frequency of natural Rabi oscillations of the atom-field system, Ω, suggests that Hamiltonian chaos appears first in the area of the fundamental parametric resonance, δ /2Ω≃1. Parametric instability increases with increasing modulation and decreasing detuning from the atom-field resonance, generating at exact resonance new areas of chaos corresponding to multiple parametric resonances. The results of numerical experiments and estimates of the characteristic parameters show that Rydberg atoms placed in a high- Q microwave cavity are possible objects for observing parametric instability and dynamical chaos.
The properties of a system consisting of two-level atoms interacting with a mode of the electromagnetic field in a cavity are considered for the case when the cavity detuning or the coefficient of the atom-field interaction is modulated. The consideration is performed with account taken of the Hamiltonian terms that are neglected in the rotating-wave approximation. It is shown that in the semiclassical equations for such a model, the effect of extension (compared to the autonomous system) of the range of variation of the quantity characterizing the number of photons can manifest itself; in this case, the energy oscillations have a chaotic character. The dependence of this phenomenon on parameters characterizing the model is studied. It is numerically demonstrated that with account taken for the relaxation, the system studied can have attractors different from the equilibrium positions, i.e., the number of photons in the mode does not tend to a constant value. The limits of validity of the rotating-wave approximation in the parametrically perturbed Dicke model are discussed.