Localization and fluctuations in quantum kicked rotors

We address the issue of fluctuations, about an exponential line shape, in a pair of one-dimensional kicked quantum systems exhibiting dynamical localization. An exact renormalization scheme establishes the fractal character of the fluctuations and provides a method to compute the localization length in terms of the fluctuations. In the case of a linear rotor, the fluctuations are independent of the kicking parameter k and exhibit self-similarity for certain values of the quasienergy. For given k, the asymptotic localization length is a good characteristic of the localized line shapes for all quasienergies. This is in stark contrast to the quadratic rotor, where the fluctuations depend upon the strength of the kicking and exhibit local "resonances." These resonances result in strong deviations of the localization length from the asymptotic value. The consequences are particularly pronounced when considering the time evolution of a packet made up of several quasienergy states.     

Spin dynamics in a one-dimensional ferromagnetic bose gas

We investigate the propagation of spin excitations in a one-dimensional ferromagnetic Bose gas. While the spectrum of longitudinal spin waves in this system is soundlike, the dispersion of transverse spin excitations is quadratic, making a direct application of the Luttinger liquid theory impossible. By using a combination of different analytic methods we derive the large time asymptotic behavior of the spin-spin dynamical correlation function for strong interparticle repulsion. The result has an unusual structure associated with a crossover from the regime of trapped spin wave to an open regime and does not have analogues in known low-energy universality classes of quantum 1D systems.     

A super-Ohmic energy absorption in driven quantum chaotic systems

We consider energy absorption by driven chaotic systems of the symplectic symmetry class. According to our analytical perturbative calculation, at the initial stage of evolution the energy growth with time can be faster than linear. This appears to be an analog of weak anti-localization in disordered systems with spin-orbit interaction. Our analytical result is also confirmed by numerical calculations for the symplectic quantum kicked rotor.     

Quantum-to-classical crossover of quasibound states in open quantum systems

In the semiclassical limit of open ballistic quantum systems, we demonstrate the emergence of instantaneous decay modes guided by classical escape faster than the Ehrenfest time. The decay time of the associated quasibound states is smaller than the classical time of flight. The remaining long-lived quasibound states obey random-matrix statistics, renormalized in compliance with the recently proposed fractal Weyl law for open systems [W.T. Lu, S. Sridhar, and M. Zworski, Phys. Rev. Lett. 91, 154101 (2003)]. We validate our theory numerically for a model system, the open kicked rotator.     

Hierarchical structures in the phase space and fractional kinetics: II. Immense delocalization in quantized systems

Anomalous transport due to Levy-type flights in quantum kicked systems is studied. These systems are kicked rotor and kicked Harper model. It is confirmed for a kicked rotor that there exist special "magic" values of a control parameter of chaos K=K(*)=6.908 745 em leader for which an essential increasing of a localization length is obtained. Functional dependence of the localization length on both parameter of chaos and quasiclassical parameter h is studied. We also observe immense delocalization of the order of 10(9) for a kicked Harper model when a control parameter K is taken to be K(*)=6.349 972. This "magic" value corresponds to special phase space topology in the classical limit, when a hierarchical self-similar set of sticky islands emerges. The origin of the effect is of the general nature and similar immense delocalization as well as increasing of localization length can be found in other systems. (c) 2000 American Institute of Physics.     

A variational theory of hyperbolic Lagrangian Coherent Structures

We develop a mathematical theory that clarifies the relationship between observable Lagrangian Coherent Structures (LCSs) and invariants of the Cauchy-Green strain tensor field. Motivated by physical observations of trajectory patterns, we define hyperbolic LCSs as material surfaces (i.e., codimension-one invariant manifolds in the extended phase space) that extremize an appropriate finite-time normal repulsion or attraction measure over all nearby material surfaces. We also define weak LCSs (WLCSs) as stationary solutions of the above variational problem. Solving these variational problems, we obtain computable sufficient and necessary criteria for WLCSs and LCSs that link them rigorously to the Cauchy-Green strain tensor field. We also prove a condition for the robustness of an LCS under perturbations such as numerical errors or data imperfection. On several examples, we show how these results resolve earlier inconsistencies in the theory of LCS. Finally, we introduce the notion of a Constrained LCS (CLCS) that extremizes normal repulsion or attraction under constraints. This construct allows for the extraction of a unique observed LCS from linear systems, and for the identification of the most influential weak unstable manifold of an unstable node.     

Symmetry breaking in quantum chaotic systems

We show, using semiclassical methods, that as a symmetry is broken, the transition between universality classes for the spectral correlations of quantum chaotic systems is governed by the same parametrization as in the theory of random matrices. The theory is quantitatively verified for the kicked rotor quantum map. We also provide an explicit substantiation of the random matrix hypothesis, namely that in the symmetry-adapted basis the symmetry-violating operator is random.     

Effect of temporal disorder on wave packet dynamics in one-dimensional kicked lattices

Inspired by the recent experimental progress in noisy kicked rotor systems,we investigate the effect of temporal disorder or quasi-periodicity in one-dimensional kicked lattices with pulsed on-site potential.We found that,unlike the spatial disorder or quasi-periodicity which usually leads to localization,the effect of the temporal one is more complex and depends on the spatial configuration.If the kicked on-site potential is periodic in real space,then the wave packet will stay diffusive in the presence of temporal disorder or quasi-periodicity.On the other hand,if the kicked on-site potential is spatially quasi-periodic,then the temporal disorder or quasi-periodicity may lead to a shift of the transition point of the dynamical localization and destroy the dynamical localization in a certain parameter range.The results we obtained can be readily tested by experiments and may help us better understand the dynamical localization.     

Real-time renormalization group and charge fluctuations in quantum dots

We develop a perturbative renormalization-group method in real time to describe nonequilibrium properties of discrete quantum systems coupled linearly to an environment. We include energy broadening and dissipation and develop a cutoff-independent formalism. We present quantitatively reliable results for the linear and nonlinear conductance in the mixed-valence and empty-orbital regime of the nonequilibrium Anderson impurity model with finite on-site Coulomb repulsion.     

Strange Attractors in Periodically-Kicked Limit Cycles and Hopf Bifurcations

We prove the emergence of chaotic behavior in the form of horseshoes and strange attractors with SRB measures when certain simple dynamical systems are kicked at periodic time intervals. The settings considered include limit cycles and stationary points undergoing Hopf bifurcations.     

Darboux First Integral Conditions and Integrability of the 3D Lotka-Volterra System

Abstract

We apply the Darboux theory of integrability to polynomial ODE’s of dimension 3. Using this theory and computer algebra, we study the existence of first integrals for the 3–dimensional Lotka–Volterra systems with polynomial invariant algebraic solutions linear and quadratic and determine numerous cases of integrability.     

Ballistic and localized transport for the atom optics kicked rotor in the limit of a vanishing kicking period

We present mean energy measurements for the atom optics kicked rotor as the kicking period tends to zero. A narrow resonance is observed marked by quadratic energy growth, in parallel with a complete freezing of the energy absorption away from the resonance peak. Both phenomena are explained by classical means, taking proper account of the atoms' initial momentum distribution.     

Theory of the Anderson transition in the quasiperiodic kicked rotor

We present the first microscopic theory of transport in quasiperiodically driven environments ("kicked rotors"), as realized in recent atom optic experiments. We find that the behavior of these systems depends sensitively on the value of a dimensionless Planck constant h: for irrational values of h/(4π) they fall into the universality class of disordered electronic systems and we describe the corresponding localization phenomena. In contrast, for rational values the rotor-Anderson insulator acquires an infinite (static) conductivity and turns into a "supermetal." We discuss the ensuing possibility of a metal-supermetal quantum phase transition.     

Intensity-controlled interactions between vectorial spatial solitary waves in quadratic nonlinear media

We numerically investigate the interaction of vectorial solitary waves propagating in a bulk crystal with a phase-matchable quadratic nonlinearity. We obtain coalescence, steering, crossing, or repulsion by controlling the launched intensities in two orthogonal polarizations without any coherent seed at the second harmonic.     

Directed motion for delta-kicked atoms with broken symmetries: comparison between theory and experiment

We report an experimental investigation of momentum diffusion in the delta-function kicked rotor where time symmetry is broken by a two-period kicking cycle and spatial symmetry by an alternating linear potential. We exploit this, and a technique involving a moving optical potential, to create an asymmetry in the momentum diffusion that is due to the classical chaotic diffusion. This represents a realization of a type of Hamiltonian quantum ratchet.     

Testing statistical bounds on entanglement using quantum chaos

Previous results indicate that while chaos can lead to substantial entropy production, thereby maximizing dynamical entanglement, this still falls short of maximality. Random matrix theory modeling of composite quantum systems, investigated recently, entails a universal distribution of the eigenvalues of the reduced density matrices. We demonstrate that these distributions are realized in quantized chaotic systems by using a model of two coupled and kicked tops. We derive an explicit statistical universal bound on entanglement, which is also valid for the case of unequal dimensionality of the Hilbert spaces involved, and show that this describes well the bounds observed using composite quantized chaotic systems such as coupled tops.     

Entanglement production in quantized chaotic systems

Quantum chaos is a subject whose major goal is to identify and to investigate different quantum signatures of classical chaos. Here we study entanglement production in coupled chaotic systems as a possible quantum indicator of classical chaos. We use coupled kicked tops as a model for our extensive numerical studies. We find that, in general, chaos in the system produces more entanglement. However, coupling strength between two subsystems is also a very important parameter for entanglement production. Here we show how chaos can lead to large entanglement which is universal and describable by random matrix theory (RMT). We also explain entanglement production in coupled strongly chaotic systems by deriving a formula based on RMT. This formula is valid for arbitrary coupling strengths, as well as for sufficiently long time. Here we investigate also the effect of chaos on the entanglement production for the mixed initial state. We find that many properties of the mixed-state entanglement production are qualitatively similar to the pure state entanglement production. We however still lack an analytical understanding of the mixed-state entanglement production in chaotic systems.     

A spectroscopical measure for the exploration of phase space

Many nonintegrable systems have eigenstates that typically require numerous basis states to represent them. We develop a criterion to judge the extent to which phase space is explored by the spectrum of such a Hamiltonian. Our criterion uses the eigenvalues rather than the eigenfunctions and is based on identifying a direct relation between the intensity of Shnirelman's peak and the localization length. We illustrate our procedure by applying it to the spectrum of two prototypical nonintegrable systems, the kicked rotor and the kicked top. (c) 1998 American Institute of Physics.     

Ehrenfest-time dependence of weak localization in open quantum dots

Semiclassical theory predicts that the weak localization correction to the conductance of a ballistic chaotic cavity is suppressed if the Ehrenfest time exceeds the dwell time in the cavity [I. L. Aleiner and A. I. Larkin, Phys. Rev. B 54, 14423 (1996)]. We report numerical simulations of weak localization in the open quantum kicked rotator that confirm this prediction. Our results disagree with the "effective random matrix theory" of transport through ballistic chaotic cavities.     

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.