Gross-Pitaevski map as a chaotic dynamical system.

The Gross-Pitaevski map is a discrete time, split-operator version of the Gross-Pitaevski dynamics in the circle, for which exponential instability has been recently reported. Here it is studied as a classical dynamical system in its own right. A systematic analysis of Lyapunov exponents exposes strongly chaotic behavior. Exponential growth of energy is then shown to be a direct consequence of rotational invariance and for stationary solutions the full spectrum of Lyapunov exponents is analytically computed. The present analysis includes the "resonant" case, when the free rotation period is commensurate to 2π, and the map has countably many constants of the motion. Except for lowest-order resonances, this case exhibits an integrable-chaotic transition.

Classical dynamical localization.

We consider classical models of the kicked rotor type, with piecewise linear kicking potentials designed so that momentum changes only by multiples of a given constant. Their dynamics display quasilocalization of momentum, or quadratic growth of energy, depending on the arithmetic nature of the constant. Such purely classical features mimic paradigmatic features of the quantum kicked rotor, notably dynamical localization in momentum, or quantum resonances. We present a heuristic explanation, based on a classical phase space generalization of a well-known argument, that maps the quantum kicked rotor on a tight-binding model with disorder. Such results suggest reconsideration of generally accepted views that dynamical localization and quantum resonances are a pure result of quantum coherence.

Exponential quantum spreading in a class of kicked rotor systems near high-order resonances.

Long-lasting exponential quantum spreading was recently found in a simple but very rich dynamical model, namely, an on-resonance double-kicked rotor model [J. Wang, I. Guarneri, G. Casati, and J. B. Gong, Phys. Rev. Lett. 107, 234104 (2011)]. The underlying mechanism, unrelated to the chaotic motion in the classical limit but resting on quasi-integrable motion in a pseudoclassical limit, is identified for one special case. By presenting a detailed study of the same model, this work offers a framework to explain long-lasting exponential quantum spreading under much more general conditions. In particular, we adopt the so-called "spinor" representation to treat the kicked-rotor dynamics under high-order resonance conditions and then exploit the Born-Oppenheimer approximation to understand the dynamical evolution. It is found that the existence of a flat band (or an effectively flat band) is one important feature behind why and how the exponential dynamics emerges. It is also found that a quantitative prediction of the exponential spreading rate based on an interesting and simple pseudoclassical map may be inaccurate. In addition to general interests regarding the question of how exponential behavior in quantum systems may persist for a long time scale, our results should motivate further studies toward a better understanding of high-order resonance behavior in δ-kicked quantum systems.

Fidelity for kicked atoms with gravity near a quantum resonance.

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.

Classical dynamics of quantum entanglement.

We analyze numerically the dynamical generation of quantum entanglement in a system of two interacting particles, started in a coherent separable state, for decreasing values of ℏ. As ℏ→0 the entanglement entropy, computed at any finite time, converges to a finite nonzero value. The limit law that rules the time dependence of entropy is well reproduced by purely classical computations. Its general features can be explained by simple classical arguments, which expose the different ways entanglement is generated in systems that are classically chaotic or regular.

Long-lasting exponential spreading in periodically driven quantum systems.

Using a dynamical model relevant to cold-atom experiments, we show that long-lasting exponential spreading of wave packets in momentum space is possible. Numerical results are explained via a pseudoclassical map, both qualitatively and quantitatively. Possible applications of our findings are also briefly discussed.

Pseudoclassical theory for fidelity of nearly resonant quantum rotors.

Using a semiclassical ansatz we analytically predict for the fidelity of delta-kicked rotors the occurrence of revivals and the disappearance of intermediate revival peaks arising from the breaking of a symmetry in the initial conditions. A numerical verification of the predicted effects is given and experimental ramifications are discussed.

Quantum accelerator modes near higher-order resonances.

Quantum accelerator modes have been experimentally observed, and theoretically explained, in the dynamics of kicked cold atoms in the presence of gravity, when the kicking period is close to a half-integer multiple of the Talbot time. We generalize the theory to the case when the kicking period is sufficiently close to any rational multiple of the Talbot time, and thus predict new rich families of experimentally observable quantum accelerator modes.

Decoherence as a probe of coherent quantum dynamics.

The effect of decoherence, induced by spontaneous emission, on the dynamics of cold atoms periodically kicked by an optical lattice is experimentally and theoretically studied. Ideally, the mean energy growth is essentially unaffected by weak decoherence, but the resonant momentum distributions are fundamentally altered. It is shown that experiments are inevitably sensitive to certain nontrivial features of these distributions, in a way that explains the puzzle of the observed enhancement of resonances by decoherence [Phys. Rev. Lett. 87, 074102 (2001)]. This clarifies both the nature of the coherent evolution, and the way in which decoherence disrupts it.

Classical scaling theory of quantum resonances.

The quantum resonances occurring with delta-kicked particles are studied with the help of a fictitious classical limit, establishing a direct correspondence between the nearly resonant quantum motion and the classical resonances of a related system. A scaling law which characterizes the structure of the resonant peaks is derived and numerically demonstrated.

Stable quantum resonances in atom optics.

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.

Fractal fluctuations in quantum integrable scattering.

We theoretically and numerically demonstrate that completely integrable scattering processes may exhibit fractal transmission fluctuations, due to typical spectral properties of integrable systems. Similar properties also occur with scattering processes in the presence of strong dynamical localization, thus explaining recent numerical observations of fractality in the latter class of systems.

Spectral properties and anomalous transport in a polygonal billiard.

We analyze a class of polygonal billiards, whose behavior is conjectured to exhibit a variety of interesting dynamical features. Correlation functions are numerically investigated, and in a subclass of billiard tables they give indications about a singular continuous spectral measure. By lifting billiard dynamics we are also able to study transport properties: the (normal or anomalous) diffusive behavior is theoretically connected to a scaling index of the spectral measure; the proposed identity is shown to agree with numerical simulations. (c) 2000 American Institute of Physics.