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

Lyapunov exponents and order-to-chaos transitions in the interaction of two-level atoms with their own radiation field

Leonid E. Kon’kov, Sergey V. Prants

Published May 6, 1996• 0 citations
Publisher

Abstract

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.

DOI
10.1117/12.239732

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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.
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 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.
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.
A system of N two-level atoms in a resonator is considered interacting with a resonant eigenmode field and with an external coherent field, with a frequency slightly different from the frequency of the atomic transition. A model Hamiltonian is constructed for describing the slow quantum dynamics of the system, and a set of closed c-number equations for time-dependent quantum expectation values is derived in the boson and spin coherent states. In the region of parameters for developed chaos in the semiclassical limit (when the radiation field is considered classically) we show that the semiclassical approximation is violated by quantum effects at the time scale ${\mathrm{\ensuremath{\tau}}}_{\mathrm{\ensuremath{\Elzxh}}}$\ensuremath{\sim}lnN. We also show that quantum correlations grow exponentially during the time interval 0${\mathrm{\ensuremath{\tau}}}_{\mathrm{\ensuremath{\Elzxh}}}$. An additional averaging over the initial density matrix is considered. The connection of our model with the Dicke model [Phys. Rev. 93, 99 (1956)] is demonstrated, and the possiblity of measuring the quantum chaos effects considered in this paper is discussed.
Abstract The interaction of a quantized electromagnetic field in a cavity with a set of two-level atoms inside it can be described with algebraic Hamiltonians of increasing complexity, from the Rabi to the Dicke models. Their algebraic character allows, through the use of coherent states, a semiclassical description in phase space, where the non-integrable Dicke model has regions associated with regular and chaotic motion. The appearance of classical chaos can be quantified calculating the largest Lyapunov exponent over the whole available phase space for a given energy. In the quantum regime, employing efficient diagonalization techniques, we are able to perform a detailed quantitative study of the regular and chaotic regions, where the quantum participation ratio ( P R ) of coherent states on the eigenenergy basis plays a role equivalent to the Lyapunov exponent. It is noted that, in the thermodynamic limit, dividing the participation ratio by the number of atoms leads to a positive value in chaotic regions, while it tends to zero in the regular ones.