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arXiv:quant-ph/9903063 · Physical Review A

Dynamical stability and quantum chaos of ions in a linear trap

G. P. Berman, Daniel F. V. James, Richard Hughes, Mark S. Gulley, Michael H. Holzscheiter, Gustavo V. López

Published Jan 7, 2000• 16 citations
arXivHTMLPDF

Abstract

The realization of a paradigm chaotic system, namely, the harmonically driven oscillator, in the quantum domain using cold trapped ions driven by lasers is theoretically investigated. The simplest characteristics of regular and chaotic dynamics are calculated. The possibilities of experimental realization are discussed.
DOI
10.1103/physreva.61.023403

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We show how a nonlinear chaotic system, the parametrically kicked nonlinear oscillator, may be realized in the dynamics of a trapped, laser-cooled ion, interacting with a sequence of standing-wave pulses. Unlike the original optical scheme [G. J. Milburn and C. A. Holmes, Phys. Rev. A 44, 4704 (1991)], the trapped ion enables strongly quantum dynamics with minimal dissipation. This should permit an experimental test of one of the quantum signatures of chaos: irregular collapse and revival dynamics of the average vibrational energy.
We describe the transition to quantum chaos of an ion trapped in a linear ion trap and interacting with two laser fields. Under the conditions of adiabatic illumination of the upper level of the ion, and when the frequencies of the two laser beams are slightly different, the system is reduced to a quantum linear oscillator interacting with a monochromatic wave. The property of localization over the quantum resonance cells is proposed to exploit in order to facilitate the process of measurement of the probability distribution of an ion on the vibrational levels. In the regime of strong chaos the time-averaged values of the energy and dispersion of energy are computed and compared with the corresponding classical quantities for different values of the perturbation amplitude. In the exact resonance case, the classical analog of the system possesses an infinite inhomogeneous stochastic web. We analyze the quantum dynamics inside the inhomogeneous web. It is shown that the quantum system mimics on average the dynamics of the corresponding classical system. Formation of the quantum resonance cells is illustrated in the case of a finite detuning from the exact resonance, and under increasing of the wave amplitude. The parameters of the model and the initial conditions are close to the real physical situation which can be realized in the system of cold trapped ion perturbed by two lasers fields with close frequencies. (c) 2000 American Institute of Physics.
Summary form only given. We consider the semiclassical orbits of cold atoms in an optical lattice with a magnetically-induced harmonic trap. When the laser beams are aligned with a principal axis of the trap, the atoms follow regular classical paths. But when the laser beams are tilted at an angle to the axis, the atoms undergo a controllable transition from regular motion to strong classical chaos. Our work introduces atom traps in a tilted laser field as a new type of quantum chaotic system that is accessible to experiment.
We discuss recent experimental and theoretical investigations of quantum dynamics in periodically kicked chaotic systems. These investigations are centred on a modified version of the classic δ-kicked rotor, as realized using laser-cooled caesium atoms. Dramatic phenomena, principally quantum accelerator modes, have been observed and these have inspired novel theoretical approaches to quantum chaos. The latter work has, in turn, prompted further experiments and the combined experimental and theoretical effort continues to lead to exciting advances in this field.
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