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arXiv:nlin/0205006 · Physical Review Letters

Stable Quantum Resonances in Atom Optics

Shmuel Fishman, Italo Guarneri, Laura Rebuzzini

Published Aug 5, 2002• 91 citations
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Abstract

A theory for stabilization of quantum resonances by a mechanism similar to one leading to classical resonances in nonlinear systems is presented. It explains recent surprising experimental results, obtained for cold cesium atoms when driven in the presence of gravity, and leads to further predictions. The theory makes use of invariance properties of the system allowing for separation into independent kicked rotor problems. The analysis relies on a fictitious classical limit where the small parameter is not Planck's constant, but rather the detuning from the frequency that is resonant in the absence of gravity.

DOI
10.1103/physrevlett.89.084101

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A theory for stabilization of quantum resonances by a mechanism similar to one leading to classical resonances in nonlinear systems is presented. It explains recent surprising experimental results, obtained for cold cesium atoms when driven in the presence of gravity, and leads to further predictions. The theory makes use of invariance properties of the system allowing for separation into independent kicked rotor problems. The analysis relies on a fictitious classical limit where the small parameter is not Planck's constant, but rather the detuning from the frequency that is resonant in the absence of gravity.
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We present experimental results for the dynamics of cold atoms in a far detuned amplitude-modulated optical standing wave. Phase-space resonances constitute distinct peaks in the atomic momentum distribution containing up to 65% of all atoms resulting from a mixed quantum chaotic phase space. We characterize the atomic behavior in classical and quantum regimes and we present the applicable quantum and classical theory, which we have developed and refined. We show experimental proof that the size and the position of the resonances in phase space can be controlled by varying several parameters, such as the modulation frequency, the scaled well depth, the modulation amplitude, and the scaled Planck's constant of the system. We have found a surprising stability against amplitude noise. We present methods to accurately control the momentum of an ensemble of atoms using these phase-space resonances which could be used for efficient phase-space state preparation.
This thesis is devoted to the study of some theoretical aspects of the problem of Quantum Accelerator Modes (QAMs), which has received considerable interest in recent years both by experimentalists and theorists in the fields of Quantum Chaos, Quantum Optics and cold-ultracold atomic physics.\nQAMs, recently observed in cold atom optics, are formed by exposing cold alkali atoms to periodic kicks in the direction of the gravitational field. The kicks are generated by a pulsed standing wave of light. A QAM is characterized by a momentum transfer, which increases linearly with the number of pulses, to a substantial fraction of the atoms. QAMs arise for values of the pulse period close to a integer multiple of half of a characteristic time TB (the Talbot time), typical of the kind of atoms used.\nA detailed theoretical explanation of this quantum effect is presented. The system can be modelled by a variant of the well-known quantum Kicked Rotator, in which the effects of a static force, produced by the earth gravitational field, are taken into account. A pseudo-classical theory has been formulated, in which the role of Planck constant is played by the detuning of the kicking period from the resonant value: in the asymptotic limit for a vanishing detuning, QAMs are described by the stable periodic orbit of a “formally” classical map on the 2-torus. An analysis of parametric dependence of the map is performed perturbatively; the parameter regions where the motion is stable and periodic exhibit Arnol’d tongues-like structures. The ordering of Arnol’d tongues and hence the classification of experimentally observed accelerator modes is provided by a number-theoretic construction known as the Farey Tree.\nQuantum decay of the modes from classical stable islands immersed in a chaotic sea is analysed and its relations to the famous Wannier-Stark problem are investigated. Theoretical estimates for the lifetimes of the metastable states as a function of the effective Planck constant are obtained.\nQAM dynamics in a Bose-Einstein condensate is then considered. Modifications on transport phenomena, imposed by nonlinearities of Gross-Pitaevskii type, which describe the atomic interactions in a mean-field approach, are examined by computer-assisted analysis.\nFinally, the problem of QAMs in the vicinity of a generic quantum resonance, namely for kicking period sufficiently close to any rational multiple of the Talbot time, is studied.\nThe theoretical framework is nontrivially generalized in terms of spinor dynamics: an ansatz of the Born-Oppenheimer type allows to decouple the dynamics of spinor and orbital degrees of freedom and a description of the orbital motion by means of “formally” classical equations is achieved. New rich families of experimental observable QAMs are predicted.
Controlling the translational motion of cold atoms using optical lattice potentials is of both theoretical and experimental interest. By designing two on-resonance time sequences of kicking optical lattice potentials, a novel connection between two paradigms of nonlinear mapping systems, i.e. the kicked rotor model and the kicked Harper model, is established. In particular, it is shown that Hofstadter's butterfly quasi-energy spectrum in periodically driven quantum systems may soon be realized experimentally, with the effective Planck constant tunable by varying the time delay between two sequences of control fields. Extensions of this study are also discussed. The results are intended to open up a new generation of cold-atom experiments of quantum nonlinear dynamics.
Quantum accelerator modes were discovered in experiments with kicked cold atoms in the presence of gravity. They were shown to be tightly related to resonances of the quantum kicked rotor. In this paper a spinor formalism is developed for the analysis of modes associated with resonances of arbitrary order $q\ensuremath{\ge}1$. Decoupling of spin variables from the orbital ones is achieved by means of an ansatz of the Born-Oppenheimer type that generates qqq independent band dynamics. Each of these is described, in classical terms, by a map, and the stable periodic orbits of this map give rise to quantum accelerator modes, which are potentially observable in experiments. The arithmetic organization of such periodic orbits is briefly discussed.
Kicked atoms under a constant Stark or gravity field are investigated for experimental setups with cold and ultracold atoms. The parametric stability of the quantum dynamics is studied using the fidelity. In the case of a quantum resonance, it is shown that the behavior of the fidelity depends on arithmetic properties of the gravity parameter. Close to a quantum resonance, the long-time asymptotics of the fidelity is studied by means of a pseudoclassical approximation introduced by Fishman et al. [J. Stat. Phys. 110, 911 (2003)]. The long-time decay of fidelity arises from the tunneling out of pseudoclassical stable islands, and a simple ansatz is proposed which satisfactorily reproduces the main features observed in numerical simulations.