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Optics and Spectroscopy

A mechanism of emergence of Hamiltonian chaos in a basic quantum-optical model

Igor Leonidovich Kirilyuk, Sergey V. Prants

Published Dec 1, 2000• 2 citations
Publisher

Abstract

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.

DOI
10.1134/1.1335042

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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.
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.
We study analytically and numerically vacuum Rabi oscillations of N identical two-level atoms moving through a single-mode lossless cavity. Equations of motion which take into account the atomic quantum correlations are obtained for the quantum mechanical expectation values in the strong-coupling, rotating-wave, pointlike, and Raman-Nath approximations. It is shown that moving atoms may demonstrate an unusual type of spontaneous emission, the chaotic vacuum Rabi oscillations. This manifestation of quantum dynamical chaos in the matter-vacuum interaction is caused by a spatial inhomogeneity of the cavity mode that modulates the vacuum Rabi frequency of moving atoms. For small values of the depth of this modulation we use the Melnikov method and show analytically the presence of homoclinic chaos in this interaction. Transition to global chaos and global phase space stochasticity under conditions of the strong modulation are studied numerically by computing the maximal Lyapunov exponent lambda of the atom-field dynamical system as a function of the number and velocity of atoms and detuning. We find a curious structure of the lambda(N) dependence reflecting an intermittent route to global chaos. The strength of chaos depends strongly on the initial state preparation of atoms just before injecting into a cavity. It is shown that initially fully inverted atoms, which are in a superfluorescent Dicke state, demonstrate much stronger chaos (under other equal conditions) than the atoms prepared initially in a superradiant state with macroscopic polarization. A maser operating with two-level Rydberg atoms to be injected into a high-Q superconducting microwave cavity seems to be a realistic device for observing some manifestations of the chaotic vacuum Rabi oscillations.
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.
In the strong-coupling limit, the nonlinear dynamics of interaction of two-level atoms with their own emission field in a cavity with a periodically modulated length was studied. An analytical analysis was made that shows the appearance of regions in the pseudophase space of Maxwell-Bloch equations that are responsible for the origin of dynamic chaos for the smallest possible modulation depth, and the homoclinic mechanism of their formation was revealed. In the model of a Rydberg maser with a modulated cavity, a special type of chaos was numerically discovered. This chaos, which was called structural, manifests itself under certain conditions in parametric oscillations of atomic population. On the one hand, these Rabi oscillations demonstrate sensitivity to initial conditions with a positive maximum Lyapunov exponent, and, on the other hand, they represent a signal with a long-term structure that is caused by modulation.
We study nonlinear dynamics in the fundamental cavity QED system, consisting of a point-like collection of two-level atoms moving through a single-mode cavity, with a spatially inhomogenous field. In the semiclassical, strong-coupling and rotating-wave approximations, the model is shown to be integrable with atoms moving at exact resonance through a cavity with an arbitrary spatial profile of the mode along the propagation axis. The general exact semiclassical solution is derived in an explicit form, in terms of Jacobian elliptic functions. Numerical simulation confirms that perturbations, which are produced by a modulation of the coupling between moving atoms and a cavity mode, provide, our of resonance, a mechanism responsible for Hamiltonian chaos. Taking into account the interatomic quantum correlations, we go beyond the semiclassical model and derive a dynamical system, which is able to describe vacuum Rabi oscillation with moving atoms. At resonance, this system is again shown to be integrable with a general semiquantum solution found. Out of resonance, it can demonstrate chaotic vacuum Rabi oscillations, which may be considered as a new kind of reversible spontaneous emission.
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.