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Physical Review A

Generalized coherent-state analysis of semiclassical quantum chaos for an angular momentumJin a resonant cavity

Ronald F. Fox

Published Nov 1, 1991• 7 citations
Publisher

Abstract

An angular momentum J in a resonant-cavity electromagnetic field exhibits chaos in the time dependence of the expectation value of ${\mathit{J}}_{\mathit{z}}$. This fact is transparently demonstrated using the SU(2) generalized coherent-state picture. Using Hale's averaging theorem, the mechanism of chaos is elucidated and found to be equivalent to a periodically modulated, near-separatrix, pendulum motion. The rotating-wave approximation is also elucidated, and its nonchaos is explained. The modulations that cause chaos result from virtual transitions, and the variance of ${\mathit{J}}_{\mathit{z}}$ initially grows exponentially.

DOI
10.1103/physreva.44.6193

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An angular momentum J in a resonant-cavity electromagnetic field exhibits chaos in the time dependence of the expectation value of Jz{\mathit{J}}_{\mathit{z}}Jz​. This fact is transparently demonstrated using the SU(2) generalized coherent-state picture. Using Hale's averaging theorem, the mechanism of chaos is elucidated and found to be equivalent to a periodically modulated, near-separatrix, pendulum motion. The rotating-wave approximation is also elucidated, and its nonchaos is explained. The modulations that cause chaos result from virtual transitions, and the variance of Jz{\mathit{J}}_{\mathit{z}}Jz​ initially grows exponentially.
Quantum chaos has been demonstrated in several studies of the interaction between N two-level quantum systems and a single-mode radiation field interacting in a resonant cavity. It has been shown that the mechanism for this chaos is dynamically identical to a periodically perturbed, near-separatrix motion of a pendulum. Identification of this mechanism followed from examination of the rotating-wave approximation and corrections to this approximation. Our earlier study established the nature of the corrections on the basis of numerical simulation experiments and heuristic arguments. In this paper we invoke an averaging procedure which enables us to systematically establish the form of these corrections. We first apply this procedure to the Rabi model in order to illustrate its utility and exhibit its structure, and then we apply it to the more complicated problem of quantum chaos.
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.
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.
A system of atoms interacting with a radiation field in a resonant cavity is studied under conditions when the dynamics in the classical limit is stochastic. This situation is called quantum chaos. Equations of motion are obtained for the quantum-mechanical expectation values which take into account the quantum correlation functions. It is shown that in a situation corresponding to quantum chaos, the quantum corrections grow exponentially, making the evolution of the system essentially quantal after a certain time tau( variant Planck's over 2pi ) has elapsed. Analytical and numerical analysis show that in this regime the time tau( variant Planck's over 2pi ) obeys the logarithmic law tau( variant Planck's over 2pi ) approximately ln N (N is the number of atoms), and not the law tau( variant Planck's over 2pi ) approximately N(alpha) (alpha is a certain constant of order unity), as would be the case in the absence of chaos.
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.
A mechanism of emergence of Hamiltonian chaos is considered for the model describing the interaction between two-level atoms and their own radiation field in an ideal single-mode cavity. The analysis of the semiclassical Maxwell-Bloch equations shows that the Hamiltonian terms that are neglected in the rotating-wave approximation (RWA) give rise to the formation of a stochastic layer near the RWA-system separatrix. The Mel’nikov method is used to prove that the splitting of the separatrix takes place for arbitrarily small vacuum Rabi frequencies Ω N . The computation of Poincare sections shows that the stochastic layer, which is exponentially narrow for small Ω N , expands with increasing Ω N , and at Ω N ≃ 1, the system exhibits global chaos that manifests itself in irregular oscillation of the atomic population inversion and the broadening of the power spectrum. Promising candidates for observing manifestations of dynamic chaos in this basic quantum-optical system are Rydberg atoms placed in a high-Q superconducting microwave cavity.