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 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.
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.
We study the coupled translational, electronic, and field dynamics of the combined system "a two-level atom + a single-mode quantized field + a standing-wave ideal cavity". We derive Hamilton -- Schr\\"odinger equations for probability amplitudes and averaged position and momentum of a point-like atom interacting with the quantized field in a standing-wave cavity. They constitute, in general, an infinite-dimensional set of equations with an infinite number of integrals of motion which may be reduced to a dynamical system with four degrees of freedom if the quantized field is supposed to be initially prepared in a Fock state. This system is found to produce semiquantum chaos with positive values of the maximal Lyapunov exponent. At large values of detuning
∣delta∣gg1, the Rabi atomic oscillations are usually shallow, and the dynamics is found to be almost regular. The Doppler -- Rabi resonance, deep Rabi oscillations that may occur at any large value of
∣delta∣ to be equal to
∣alphap0∣, is found numerically and described analytically (with
alpha to be the normalized recoil frequency and
p0 the initial atomic momentum). Two gedanken experiments are proposed to detect manifestations of semiquantum chaos in real experiments. In the chaotic regime values of the population inversion
zout, measured with atoms after transversing a cavity, are so sensitive to small changes in the initial inversion
zin that the probability of detecting any value of
zout in the admissible interval becomes almost unity in a short time. Chaotic wandering of a two-level atom in a quantized Fock field is shown to be fractal. Fractal-like structures, typical for chaotic scattering, are numerically found in the dependence of the time of exit of atoms from the cavity on their initial momenta.
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.