Light propagation on a two-dimensional curved surface embedded in a three-dimensional space has attracted increasing attention as an analog model of four-dimensional curved spacetime in laboratory. Despite recent developments in modern cosmology on the dynamics and evolution of the universe, investigation of nonlinear dynamics of light in non-Euclidean geometry is still scarce and remains challenging. Here, we study classical and wave chaotic dynamics on a family of surfaces of revolution by considering its equivalent conformally transformed flat billiard, with nonuniform distribution of refractive index. This equivalence is established by showing how these two systems have the same equations and the same dynamics. By exploring the Poincaré surface of section, the Lyapunov exponent and the statistics of eigenmodes and eigenfrequency spectrum in the transformed inhomogeneous table billiard, we find that the degree of chaos is fully controlled by a single geometric parameter of the curved surface. A simple interpretation of our findings in transformed billiards, the "fictitious force", allows to extend our prediction to other class of curved surfaces. This powerful analogy between two a prior unrelated systems not only brings forward a novel approach to control the degree of chaos, but also provides potentialities for further studies and applications in various fields, such as billiards design, optical fibers, or laser microcavities.

We study stability and chaotic-transport features of paradigmatic nonequilibrium many-body systems, i.e., periodically kicked and interacting particles, for arbitrary number of particles, nonintegrability strength unbounded from above, and different interaction cases. We rigorously show that under the latter general conditions and in strong nonintegrability regimes there exist fully stable orbits, accelerator-mode (AM) fixed points, performing ballistic motion in momentum. These orbits exist despite of the completely and strongly chaotic phase space with generally fast Arnol'd diffusion. It is numerically shown that an "isolated chaotic zone" (ICZ), separated from the rest of the phase space, remains localized around an AM fixed point for long times even when this point is partially stable in only a few phase-space directions. This localization should reflect an Arnol'd diffusion in an ICZ much slower than that in the rest of phase space. The time evolution of the mean kinetic energy of an initial ensemble containing an ICZ exhibits superdiffusion instead of normal chaotic diffusion.

We study the quantum topological properties of Floquet (time-periodic) systems exhibiting Hall effects due to perpendicular magnetic and electric fields. The systems are charged particles periodically kicked by a one-dimensional cosine potential in the presence of such fields, where the magnetic field is perpendicular also to the kicking direction. We consider parameter values including small kicking strength, which enables the quantum evolution to be described by effective Floquet Hamiltonians. We also assume the semiclassical regime of a small scaled Planck constant. In the case of an electric field parallel to the kicking direction, one observes, as the electric-field strength is varied, a series of topological phase transitions, causing changes in the Chern numbers of quasienergy (QE) bands. These transitions are due to band degeneracies which, in an initial wide range of electric-field strengths, are shown to occur when the energy of the classical separatrix is approximately equal to the average QE of the degenerating bands. For larger electric-field strengths, the degeneracies reflect changes in the classical phase-space structure. In the case of an electric field not parallel to the kicking direction and satisfying some resonance conditions, we show that the topological properties of the QE bands are characterized by universal (electric-field independent) Chern numbers. These same numbers also characterize the QE band spectrum in the first case, at the end of the basic electric-field interval where the topological phase transitions take place. The two cases above are known to exhibit significantly different dynamical rates, both classical and quantum. This is well reflected by the different topological properties in the two cases.

We introduce well-defined characterizations of prethermal states in realistic periodically driven many-body systems with unbounded chaotic diffusion of the kinetic energy. These systems, interacting arrays of periodically kicked rotors, are paradigmatic models of many-body chaos theory. We show that the prethermal states in these systems are well described by a generalized Gibbs ensemble based essentially on the average Hamiltonian. The latter is the quasiconserved quantity in the prethermal state and the ensemble is characterized by the temperature of the state. An explicit exact expression for this temperature is derived. Using also arguments based on chaos theory, we demonstrate that the lifetime of the prethermal state is exponentially long in the inverse of the temperature, in units of the driving frequency squared. Our analytical results, in particular those for the temperature and the lifetime of the prethermal state, agree well with numerical observations.

Classical "kicked Hall systems" (KHSs), i.e., periodically kicked charges in the presence of uniform magnetic and electric fields that are perpendicular to each other and to the kicking direction, have been introduced and studied recently. It was shown that KHSs exhibit, under generic conditions, the phenomenon of "superweak chaos" (SWC), i.e., for small kick strength κ a KHS behaves as if this strength were effectively κ^{2} rather than κ. Here we investigate quantum-dynamical and spectral manifestations of this generic SWC. We first derive general expressions for quantum effective Hamiltonians for the KHSs. We then show that the phenomenon of quantum antiresonance (QAR), i.e., "frozen" quantum dynamics with flat quasienergy (QE) bands, takes place for integer values of a scaled Planck constant ℏ_{s} and under the same generic conditions for SWC. This appears to be the most generic occurrence of QAR in quantum systems. The vicinity of QAR is shown to correspond semiclassically to SWC. A global spectral manifestation of SWC is the fact that a scaled QE spectrum as function of ℏ_{s}, at fixed small value of κ/ℏ_{s}, features an approximately "doubled" structure. In the case of standard (cosine) potentials, this structure is that of a universal (parameters-independent) double Hofstadter butterfly. Also, for standard potentials and for small ℏ_{s} (semiclassical regime), the evolution of the kinetic-energy expectation value exhibits a relatively slow quantum-diffusive behavior having universal features. These approximate spectral and quantum-dynamical universalities agree with predictions from the effective Hamiltonian.

Topological properties of Harper and generalized Fibonacci chains are studied in crystalline cases, i.e., for rational values of the modulation frequency. The Harper and Fibonacci crystals at fixed frequency are connected by an interpolating one-parameter Hamiltonian. As the parameter is varied, one observes topological phase transitions, i.e., changes in the Chern integers of two bands due to the degeneracy of these bands at some parameter value. For small frequency, corresponding to a semiclassical regime, the degeneracies are shown to occur when the average energy of the two bands is approximately equal to the energy of the classical separatrix. Spectral and topological features of the Fibonacci crystal for small frequency leave a clear imprint on the corresponding Hofstadter butterfly for arbitrary frequency.

Energy or quasienergy (QE) band spectra depending on two parameters may have a nontrivial topological characterization by Chern integers. Band spectra of one-dimensional (1D) systems that are spanned by just one parameter, a Bloch phase, are topologically trivial. Recently, an ensemble of 1D Floquet (time-periodic) systems, double-kicked rotors (DKRs) that are classically nonintegrable and depend on an external parameter, has been studied. It was shown that a QE band spanned by both the Bloch phase and the external parameter is characterized by a Chern integer. The latter determines the change in the mean angular momentum of a state in the band when the external parameter is adiabatically varied by a natural period. We show here, under conditions much more general than in previous works, that the ensemble of DKRs for all values of the external parameter corresponds to a 1D double-kicked particle (DKP) having translational invariance in the position-momentum phase plane. This DKP can be characterized by a second Chern integer, which is shown to be connected with the integer above for the DKR ensemble. This connection is expressed by a Diophantine equation (DE), which we derive. The DE, involving the band degeneracies of the DKR ensemble and of the DKP system, determines the allowed values of the DKR-ensemble integer. In particular, this integer is generically nonzero, showing the general topological nontriviality of the DKR ensemble.

We introduce and study the "kicked Hall system" (KHS), i.e., charged particles periodically kicked in the presence of uniform magnetic (B) and electric (E) fields that are perpendicular to each other and to the kicking direction. We show that for resonant values of B and E and in the weak-chaos regime of sufficiently small nonintegrability parameter κ (the kicking strength), there exists a generic family of periodic kicking potentials for which the Hall effect from B and E significantly suppresses the weak chaos, replacing it by "superweak" chaos (SWC). This means that the system behaves as if the kicking strength were κ^{2} rather than κ. For E=0, SWC is known to be a classical fingerprint of quantum antiresonance, but it occurs under much less generic conditions, in particular only for very special kicking potentials. Manifestations of SWC are a decrease in the instability of periodic orbits and a narrowing of the chaotic layers, relative to the ordinary weak-chaos case. Also, for global SWC, taking place on an infinite "stochastic web" in phase space, the chaotic diffusion on the web is much slower than the weak-chaos one. Thus, the Hall effect can be relatively stabilizing for small κ. In some special cases, the effect is shown to cause ballistic motion for almost all parameter values. The generic global SWC on stochastic webs in the KHS appears to be the two-dimensional closest analog to the Arnol'd web in higher dimensional systems.

Double kicked rotors (DKRs) appear to be the simplest nonintegrable Hamiltonian systems featuring classical translational symmetry in phase space (i.e., in angular momentum) for an infinite set of values (the rational ones) of a parameter η. The experimental realization of quantum DKRs by atom-optics methods motivates the study of the double kicked particle (DKP). The latter reduces, at any fixed value of the conserved quasimomentum βℏ, to a generalized DKR, the "β-DKR." We determine general quantum properties of β-DKRs and DKPs for arbitrary rational η. The quasienergy problem of β-DKRs is shown to be equivalent to the energy eigenvalue problem of a finite strip of coupled lattice chains. Exact connections are then obtained between quasienergy spectra of β-DKRs for all β in a generically infinite set. The general conditions of quantum resonance for β-DKRs are shown to be the simultaneous rationality of η,β, and a scaled Planck constant ℏ(S). For rational ℏ(S) and generic values of β, the quasienergy spectrum is found to have a staggered-ladder structure. Other spectral structures, resembling Hofstadter butterflies, are also found. Finally, we show the existence of particular DKP wave-packets whose quantum dynamics is free, i.e., the evolution frequencies of expectation values in these wave-packets are independent of the nonintegrability. All the results for rational ℏ(S) exhibit unique number-theoretical features involving η,ℏ(S), and β.

This paper is a brief review of classical and quantum transport phenomena, as well as related spectral properties, exhibited by one-dimensional periodically kicked systems. Two representative and fundamentally different classes of systems will be considered, those satisfying the classical Kolmogorov-Arnol'd-Moser scenario and those which not. The experimental realization of some of these systems using atom-optics methods will be mentioned.

Topological properties of energy spectra of general one-dimensional quasiperiodic systems, describing also Bloch electrons in magnetic fields, are studied for an infinity of irrational modulation frequencies corresponding to irrational numbers of flux quanta per unit cell. These frequencies include well-known ones considered in works on Fibonacci quasicrystals. It is shown that the spectrum for any such frequency exhibits a self-similar hierarchy of clusters characterized by universal (system-independent) values of Chern topological integers which are exactly determined. The cluster hierarchy provides a simple and systematic organization of all the spectral gaps, labeled by universal topological numbers which are exactly determinable, thus avoiding their numerical evaluation using rational approximants of the irrational frequency. These numbers give both the quantum Hall conductance of the system and the winding number of the edge-state energy traversing a gap as a Bloch quasimomentum is varied.

A non-Poisson regular quasienergy spectrum is found for the generalized kicked particle under quantum-resonance conditions at generic quasimomentum, a quantity most relevant in atom-optics experimental realizations of kicked-rotor systems. The regular spectrum has the structure of a staggered ladder, i.e., it is the superposition of a finite number of ladder subspectra all having the same spacing, which is independent of the nonintegrability of the system. This spectral structure is shown to have distinct quantum-dynamical manifestations: a suppression of quantum resonances and a dynamical localization characterized by unique features such as traveling-wave components in the time evolution. These phenomena are found to be robust under small variations of the quasimomentum and should therefore be experimentally observable using Bose-Einstein condensates with sufficiently small quasimomentum width.

Generic and significant regularities are shown to occur in the quasienergy spectra of the generalized quantum kicked particle for arbitrary quasimomentum, a quantity most relevant in atom-optics experimental realizations of this paradigmatic nonintegrable system. The spectral regularities are basically different from the usual ones associated with ordered regions in a mixed phase space, since they are \emph{completely independent} of the chaos strength. Their origin are free-motion features unaffected by the nonintegrability. The generic regularities are \emph{dense} subspectra, each being either a regular level sequence with approximately a Poisson level-spacing distribution or a ladder of equally-spaced levels. The quantum dynamics associated with each regular subspectrum is essentially \emph{free} for \emph{arbitrarily strong} chaos.

Classical Hamiltonian systems with a mixed phase space and some asymmetry may exhibit chaotic ratchet effects. The most significant such effect is a directed momentum current or acceleration. In known model systems, this effect may arise only for sufficiently strong chaos. In this paper, a Hamiltonian ratchet accelerator is introduced, featuring a momentum current for arbitrarily weak chaos. The system is a realistic, generalized kicked rotor and is exactly solvable to some extent, leading to analytical expressions for the momentum current. While this current arises also for relatively strong chaos, the maximal current is shown to occur, at least in one case, precisely in a limit of arbitrarily weak chaos.

This paper is a brief review of a new approach to the quantum-chaotic ratchet effect, introduced recently to address for the first time the sensitivity of the effect of the initial state in a global fashion. This is done by studying statistical properties of the ratchet current over well-defined sets of initial states. First results concern the semiclassical full-chaos regime, where the current is strongly sensitive to the initial state. Natural initial states in this regime are those that are phase-space uniform with the maximal possible resolution of the one Planck cell. General arguments, for a class of paradigmatic model systems and for special quantum-resonance values of a scaled Planck constant ħ, predict that the distribution of the momentum current over all such states is a zero-mean Gaussian with variance ~ Dħ2∕(2π2), where D is the chaotic-diffusion coefficient. This prediction is well supported by extensive numerical evidence. The average strength of the effect, measured by the variance above, is significantly larger than that for the usual momentum states and other states. Open problems, concerning extensions of these first results in different directions, are discussed.

The quantum ratchet effect in fully chaotic systems is approached by studying statistical properties of the ratchet current over well-defined sets of initial states. Natural initial states in a semiclassical regime are those that are phase-space uniform with the maximal possible resolution of one Planck cell. General arguments in this regime, for quantum-resonance values of a scaled Planck constant variant Planck's over 2pi , predict that the distribution of the current over all such states is a zero-mean Gaussian with variance approximately Dvariant Planck's over 2pi2/(2pi2) , where D is the chaotic-diffusion coefficient. This prediction is well supported by extensive numerical evidence. The average strength of the effect, measured by the variance above, is significantly larger than that for the usual momentum states and other states. Such strong effects should be experimentally observable.

A generic kind of quantum chaotic ratchet is introduced, based on initial states that are \emph{uniform} in phase space with the \emph{maximal possible} resolution of one Planck cell. Unlike a classical phase-space uniform density, such a state usually carries a \emph{nonzero} ratchet current, even in \emph{symmetric} systems. This quantum ratchet effect basically emerges from the generic asymmetry of the state quasicoordinates in the Planck cell. It is shown, on the basis of exact results, general arguments, and extensive numerical evidence, that in a semiclassical full-chaos regime the variance of the current over all the states is nearly proportional to the chaotic-diffusion rate and to the square of the scaled Planck constant. Experimental realizations are suggested.

Quantum resonance (QR) is defined in the free-falling frame of the quantum kicked particle subjected to gravity. The general QR conditions are derived. They imply the rationality of the gravity parameter eta , the kicking-period parameter tau/(2pi) , and the quasimomentum beta . Exact results are obtained concerning wave-packet evolution for arbitrary periodic kicking potentials in the case of integer tau/(2pi) (the main QRs). It is shown that a quantum ratchet generally arises in this case for resonant beta . The noninertial nature of the free-falling frame affects the ratchet by effectively changing the kicking potential to one depending on (beta,eta) . For a simple class of initial wave packets, it is explicitly shown that the ratchet characteristics are determined to a large extent by symmetry properties and by number-theoretical features of eta .

The experimental study of cold atoms kicked by a pulsed optical standing wave and falling under gravity has led to the important discovery of the accelerator modes (QAMs). It is shown that QAMs in the absence of gravity can feature a rotational quasiregularity which cannot be found in the usual QAMs. Thus, QAMs with non-accelerating quasiregular segments may be used to control the quantum motion of atoms in a way which is not possible in the presence of gravity. Copyright © 2007 Praise Worthy Prize S.r.l. - All rights reserved.