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Physical Review A

Chaotic dynamics of a single two-level atom in the field of a plane standing electromagnetic wave

Vladimir Gubernov

Published Jul 24, 2002• 1 citation
Publisher

Abstract

In this work we undertake an analytical and numerical investigation of the motion of a neutral two-level atom in a plane standing electromagnetic wave using a semiclassical approximation. In our study we neglect the energy loss from radiation. We show that, for experimentally achievable parameter values, the dynamics of the atom can be chaotic.

DOI
10.1103/physreva.66.013408

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In this work we undertake an analytical and numerical investigation of the motion of a neutral two-level atom in a plane standing electromagnetic wave using a semiclassical approximation. In our study we neglect the energy loss from radiation. We show that, for experimentally achievable parameter values, the dynamics of the atom can be chaotic.
The properties of a system consisting of two-level atoms interacting with a mode of the electromagnetic field in a cavity are considered for the case when the cavity detuning or the coefficient of the atom-field interaction is modulated. The consideration is performed with account taken of the Hamiltonian terms that are neglected in the rotating-wave approximation. It is shown that in the semiclassical equations for such a model, the effect of extension (compared to the autonomous system) of the range of variation of the quantity characterizing the number of photons can manifest itself; in this case, the energy oscillations have a chaotic character. The dependence of this phenomenon on parameters characterizing the model is studied. It is numerically demonstrated that with account taken for the relaxation, the system studied can have attractors different from the equilibrium positions, i.e., the number of photons in the mode does not tend to a constant value. The limits of validity of the rotating-wave approximation in the parametrically perturbed Dicke model are discussed.
The motion of a two-level atom with an internal degree of freedom that interacts with the field of a single-mode standing light wave in a high- Q cavity is treated as a scattering problem. At detunings between atomic and field resonances δ, atomic recoil frequencies α, initial atomic momenta ρ 0 , and numbers of excitations N at which the semiclassical equations have chaotic solutions (in the sense of their exponential sensitivity to small changes in the initial conditions), atomic scattering is fractal with strongly pronounced self-similarity of the dependence of time T at which the atom leaves the cavity at its initial momentum; the fractal dimension of this curve is 1.84. In the chaotic regime, there are two infinite sets of atomic initial momenta for which T =∞ (in the idealized case of the absence of any loss). They correspond to separatrix-like trajectories along which the atom asymptotically approaches certain configuration space points and to the trajectories of infinitely long chaotic wandering of atoms in the cavity. These trajectories make up countable and uncountable fractals, respectively. Correlations between Rabi atomic oscillations and atomic motion can lead to Doppler-Rabi resonances, that is, deep oscillations of the internal energy of the atom, for large detunings δ that satisfy the condition |αρ 0 |≈|δ|.
We study dynamics of the atom-photon interaction in cavity quantum electrodynamics, considering a cold two-level atom in a single-mode high-finesse standing-wave cavity as a nonlinear Hamiltonian system with three coupled degrees of freedom: translational, internal atomic, and the field. The system proves to have different types of motion including Lévy flights and chaotic walkings of an atom in a cavity. The corresponding equations of motion for expectation values of the atom and field variables have two characteristic time scales: fast Rabi oscillations of the internal atomic and field quantities and slow translational oscillations of the center of the atom mass. It is shown that the translational motion, related to the atom recoils, is governed by an equation of a parametric nonlinear pendulum with a frequency modulated by the Rabi oscillations. This type of dynamics is chaotic with some width of the stochastic layer that is estimated analytically. The width is fairly small for realistic values of the control parameters, the normalized detuning delta and atomic recoil frequency alpha. We consider the Poincaré sections of the dynamics, compute the Lyapunov exponents, and find a range of the detuning, |delta| less, similar 3, where chaos is prominent. It is demonstrated how the atom-photon dynamics with a given value of alpha depends on the values of delta and initial conditions. Two types of Lévy flights, one corresponding to the ballistic motion of the atom and the other corresponding to small oscillations in a potential well, are found. These flights influence statistical properties of the atom-photon interaction such as distribution of Poincaré recurrences and moments of the atom position x. The simulation shows different regimes of motion, from slightly abnormal diffusion with <x(2)> approximately tau(1.13) at delta=1.2 to a superdiffusion with <x(2)> approximately tau(2.2) at delta=1.92 that corresponds to a superballistic motion of the atom with an acceleration. The obtained results can be used to find new ways to manipulate atoms, to cool and trap them by adjusting the detuning delta.
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 group-theoretical approach is applied to study the behavior of lossless two-level atoms in a standing-wave laser field. Due to the recoil effect, the internal and external atomic degrees of freedom become coupled. The internal dynamics is described quantum mechanically in terms of the SU (2) group parameters. The evolution operator is found in an explicit way after solving a single ODE for one of the group parameters. The translational motion in a standing wave is governed by the classical Hamilton equations which are coupled to the SU (2) group equations. It is shown that the full set of equations may be chaotic in some ranges of the control parameters and initial conditions. It means physically that there are regimes of motion with chaotic center-of-mass motion and irregular internal dynamics. It is established that the chaotic regime is specified by the character of oscillations of the group parameter characterizing the mean interaction energy between the atom and the laser field. It is shown that the effect of chaotic walking can be observed in a real experiment with cold atoms crossing a standing-wave laser field.