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Chaos Solitons & Fractals

Hamiltonian chaos in the interaction of moving two-level atoms with cavity vacuum

Sergey V. Prants, Leonid E. Kon’kov

Published May 1, 2000• 2 citations
Publisher
DOI
10.1016/s0960-0779(98)00239-2

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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.

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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.

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Intermittency and dynamical chaos in reversible spontaneous emission

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Quantum Chaos of Atoms in a Resonant Cavity

Lecture notes in physics. M, Monographs•Jan 1, 1994•0 citations
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.
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.
It is shown that a type of reversible spontaneous emission, the chaotic vacuum Rabi oscillations, may occur in the interaction of two-level atoms strongly coupled with a single cavity mode under a modulation of the atom-field coupling. Such a modulation arises naturally if the atoms move through a cavity in maserlike experiments. The existence of homoclinic chaos in reversible spontaneous emission is proven analytically. Evidence of intermittency associated with the spatial modulation of the vacuum Rabi frequency is shown numerically for the values of parameters that are achievable in present-day experiments with Rydberg atoms moving through a high-Q microwave cavity in the strong-coupling regime.