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

Calculation of the maximal Lyapunov coefficient for the coherent dynamics of three-level atoms in an ideal cavity

Sergey N. Agapov, Alexander V. Gorokhov

Published Apr 21, 2016• 0 citations
Publisher

Abstract

A coherent dynamics of a three-level atoms in the ideal cavity were investigated. This model evidently generalizes the well-known Tavis-Cummings model of two-level atoms. We have derived a system of equations for the parameters of coherent states associated with the group G = W2˄SU(3). It is shown that this model gives a nontrivial example of the chaotic behavior of the parameters of coherent states, taking into account non-resonant terms in the Hamiltonian. Time dependencies of mean number of the cavity photons and level populations of atoms are computed and maximal Lyapunov coefficient is calculated.

DOI
10.1117/12.2229838

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A coherent dynamics of a three-level atoms in the ideal cavity were investigated. This model evidently generalizes the well-known Tavis-Cummings model of two-level atoms. We have derived a system of equations for the parameters of coherent states associated with the group G = W2˄SU(3). It is shown that this model gives a nontrivial example of the chaotic behavior of the parameters of coherent states, taking into account non-resonant terms in the Hamiltonian. Time dependencies of mean number of the cavity photons and level populations of atoms are computed and maximal Lyapunov coefficient is calculated.
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.
A coherent dynamics of two-and three-level atoms in the ideal cavity were investigated. In particular, the investigation of the dynamics of three-level atoms (qutrits) interacting with photons in the cavity having two resonant frequencies close to the frequencies of the two allowed atomic transitions were performed. This model generalizes the well-known Tavis-Cummings model. Time dependencies of mean number of the cavity photons and level populations of atoms are computed and entanglement of atoms and photon fields states are discussed.
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.
The dynamics of two identical two-level atoms in non-ideal cavity it is considered. Using approach developed before by us for case of one two-level atom, the analytical expression for density matrix of the system is presented. Dynamics of level populations, mean number of photons and correlation function for photons are calculated. Using Peres - Horodecki criteria exact expression for entanglement dynamics of two identical two-level atoms in cavity is given. Influence of detuning and damping constants are investigated.
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.