Regular Islands Research Articles

Fractal structures in a generalized square map with exponential terms

Fractal structures in a generalized squared map with exponential terms are expanded in this paper. We describe how complex behaviors can arise as the parameters change. The appearances of different kinds of fractal structures, in both the attractive and the divergent regions, and most interestingly, on small regular islands embedded in the chaotic region, are manifested to have a variety of extraordinary geometries in the parameter plane.

Effect of Coulomb blockade, gold resistance, and thermal expansion on the electrical resistance of ultrathin gold films

Controlling the electrical resistance of granular thin films is of great importance for many applications, yet a full understanding of electron transport in such films remains a major challenge. We have studied experimentally and by model calculations the temperature dependence of the electrical resistance of ultrathin gold films at temperatures between 2 K and 300 K. Using sputter deposition, the film morphology was varied from a discontinuous film of weakly coupled meandering islands to a continuous film of strongly coupled coalesced islands. In the weak-coupling regime, we compare the regular island array model, the cotunneling model, and the conduction percolation model with our experimental data. We show that the tunnel barriers and the Coulomb blockade energies are important at low temperatures and that the thermal expansion of the substrate and the island resistance affect the resistance at high temperatures. At low temperatures our experimental data show evidence for a transition from electron cotunneling to sequential tunneling but the data can also be interpreted in terms of conduction percolation. The resistivity and temperature coefficient of resistance of the meandering gold islands are found to resemble those of gold nanowires. We derive a simple expression for the temperature at which the resistance changes from non-metal-like behavior into metal-like behavior. In the case of strong island coupling, the total resistance is solely determined by the Ohmic island resistance.

Light emission of a scarlike mode with assistance of quasiperiodicity

In an elliptic In${}_{x}$Ga${}_{1\ensuremath{-}x}$AsP microcavity laser, various scarlike modes are experimentally observed. Below the lasing threshold, a bouncing-ball, a triangle, a double, and a triple bow-tie mode spontaneously emit. Above the threshold, a bow-tie scarlike mode lases alone. Our numerical analysis reveals that the bow-tie scarlike mode is not caused by regular islands in phase space due to the Goos-H\anchen shift, but by unstable periodic orbits and that the light emission is assisted by a quasiperiodic orbit.

CLASSICAL DYNAMICS OF ESCAPE IN A PERIODICALLY DRIVEN UNBOUND HAMILTONIAN SYSTEM: INITIAL PHASE EFFECTS

We consider the problem of a classical Morse oscillator driven by an external periodic field of controllable characteristics and study systematically the effects of the initial phase of the external field on the probability of escape upon molecular dissociation at the end of the interaction time. First, it is shown that such effects indeed exist and may become important depending on the field amplitude and frequency. Then, an explanation is given in terms of the energy exchanges between the basic periodic orbits and the external field and of the associated modifications of the shape of regular islands in phase space. Finally, we exploit this explanation in order to make predictions regarding the behavior of the escape probability as a function of the initial phase and frequency of the external field.

Direct regular-to-chaotic tunneling rates using the fictitious-integrable-system approach

We review the fictitious integrable system approach which predicts dynamical tunneling rates from regular states to the chaotic region in systems with a mixed phase space. It is based on the introduction of a fictitious integrable system that resembles the regular dynamics within the regular island. We focus on the direct regular-to-chaotic tunneling process which dominates if nonlinear resonances within the regular island are not relevant. For quantum maps, billiard systems, and optical microcavities, we find excellent agreement with numerical rates for all regular states.

Control of mixing in a Stokes’ fluid flow

Mixing problems are common in science and engineering, the aim being to combine fluids as quickly and efficiently as possible. We consider the design of a simple controller to promote mixing in a Stokes’ fluid flow. In this paper, a controlled stirring motion is represented by a velocity field consisting of the superposition of a steady base flow and a second field modulated by a saturating, time-dependent control variable. The problem can be formulated as an optimal control one, but the presence of a nonlinearity in the state dynamics and an input constraint make the construction of a feedback law difficult. The size of the problem means that receding horizon schemes, revolving around real-time optimization of even a simplified model, are currently not feasible for fast applications. To address this problem, we exploit theory for the control of bilinear systems to propose a simple, well-performing, explicit feedback law. There are several interesting design issues associated with applying this approach to a fluid mixing application. We demonstrate these by designing a controller for a two-dimensional fluid mixing problem with simple cellular flows and discuss the relevant implementation decisions. The closed-loop forcing fields have many features that are as expected: regions of the flow with large spatial velocity gradients target regular islands of concentration and the input favors velocity fields with contours aligned perpendicular to the scalar field.

Statistical mechanics of Floquet systems with regular and chaotic states

We investigate the asymptotic state of time-periodic quantum systems with regular and chaotic Floquet states weakly coupled to a heat bath. The asymptotic occupation probabilities of these two types of states follow fundamentally different distributions. Among regular states the probability decreases from the state in the center of a regular island to the outermost state by orders of magnitude, while chaotic states have almost equal probabilities. We derive an analytical expression for the occupations of regular states of kicked systems, which depends on the winding numbers of the regular tori and the parameters temperature and driving frequency. For a constant winding number within a regular island it simplifies to Boltzmann-like weights exp(-β{eff}E{m}^{reg}) , similar to time-independent systems. For this we introduce the regular energies E{m}^{reg} of the quantizing tori and an effective winding-number-dependent temperature 1/β{eff} , different from the actual bath temperature. Furthermore, the occupations of other typical Floquet states in a mixed phase space are studied, i.e., regular states on nonlinear resonances, beach states, and hierarchical states, giving rise to distinct features in the occupation distribution. Avoided crossings involving a regular state lead to drastic consequences for the entire set of occupations. We introduce a simplified rate model whose analytical solutions describe these occupation changes quite accurately.

Supramolecular Organization and Chiral Resolution of p‐Terphenyl‐m‐Dicarbonitrile on the Ag(111) Surface

The supramolecular organization and layer formation of the non-linear, prochiral molecule [1, 1';4',1'']-terphenyl-3,3"-dicarbonitrile adsorbed on the Ag(111) surface is investigated by scanning tunneling microscopy (STM) and near-edge X-ray absorption fine-structure spectroscopy (NEXAFS). Upon two-dimensional confinement the molecules are deconvoluted in three stereoisomers, that is, two mirror-symmetric trans- and one cis-species. STM measurements reveal large and regular islands following room temperature deposition, whereby NEXAFS confirms a flat adsorption geometry with the electronic pi-system parallel to the surface plane. The ordering within the expressed supramolecular arrays reflects a substrate templating effect, steric constraints and the operation of weak lateral interactions mainly originating from the carbonitrile endgroups. High-resolution data at room temperature reveal enantiormorphic characteristics of the molecular packing schemes in different domains of the arrays, indicative of chiral resolution during the 2D molecular self-assembly process. At submonolayer coverage supramolecular islands coexist with a disordered fluid phase of highly mobile molecules. Following thermal quenching (down to 6 K) we find extended supramolecular ribbons stabilised again by attractive and directional noncovalent interactions, the formation of which reflects a chiral resolution of trans-species.

Evolutionary phase space in driven elliptical billiards

We perform the first long-time exploration of the classical dynamics of a driven billiard with a four-dimensional phase space. With increasing velocity of the ensemble, we observe an evolution from a large chaotic sea with stickiness due to regular islands to thin chaotic channels with diffusive motion leading to Fermi acceleration. As a surprising consequence, we encounter a crossover, which is not parameter induced but rather occurs dynamically, from amplitude-dependent tunable subdiffusion to universal normal diffusion in momentum space.

Universality of Algebraic Laws in Hamiltonian Systems

Hamiltonian mixed systems with unbounded phase space are typically characterized by two asymptotic algebraic laws: decay of recurrence time statistics (gamma) and superdiffusion (beta). We conjecture the universal exponents gamma=beta=3/2 for trapping of trajectories to regular islands based on our analytical results for a wide class of area-preserving maps. For Hamiltonian mixed systems with a bounded phase space the interval 3/2< or =gamma_{b}< or =3 is obtained, given that trapping takes place. A number of simulations and experiments by other authors give additional support to our claims.

Entanglement and the generation of random states in the quantum chaotic dynamics of kicked coupled tops

We study the dynamical generation of entanglement as a signature of chaos in a system of periodically kicked coupled tops, where chaos and entanglement arise from the same physical mechanism. The long-time-averaged entanglement as a function of the position of an initially localized wave packet very closely correlates with the classical phase space surface of section--it is nearly uniform in the chaotic sea, and reproduces the detailed structure of the regular islands. The uniform value in the chaotic sea is explained by the random state conjecture. As classically chaotic dynamics take localized distributions in phase space to random distributions, quantized versions take localized coherent states to pseudorandom states in Hilbert space. Such random states are highly entangled, with an average value near that of the maximally entangled state. For a map with global chaos, we derive that value based on analytic results for the entropy of random states. For a mixed phase space, we use the Percival conjecture to identify a "chaotic subspace" of the Hilbert space. The typical entanglement, averaged over the unitarily invariant Haar measure in this subspace, agrees with the long-time-averaged entanglement for initial states in the chaotic sea. In all cases the dynamically generated entanglement is that of a random complex vector, even though the system is time-reversal invariant, and the Floquet operator is a member of the circular orthogonal ensemble.

Soft wall effects on interacting particles in billiards

The effect of physically realizable wall potentials (soft walls) on the dynamics of two interacting particles in a one-dimensional (1D) billiard is examined numerically. The 1D walls are modeled by the error function and the transition from hard to soft walls can be analyzed continuously by varying the softness parameter sigma . For sigma-->0 the 1D hard wall limit is obtained and the corresponding wall force on the particles is the delta function. In this limit the interacting particle dynamics agrees with previous results obtained for the 1D hard walls. We show that the two interacting particles in the 1D soft walls model is equivalent to one particle inside a soft right triangular billiard. Very small values of sigma substantiously change the dynamics inside the billiard and the mean finite-time Lyapunov exponent decreases significantly as the consequence of regular islands which appear due to the low-energy double collisions (simultaneous particle-particle-1D wall collisions). The rise of regular islands and sticky trajectories induced by the 1D wall softness is quantified by the number of occurrences of the most probable finite-time Lyapunov exponent. On the other hand, chaotic motion in the system appears due to the high-energy double collisions. In general we observe that the mean finite-time Lyapunov exponent decreases when sigma increases, but the number of occurrences of the most probable finite-time Lyapunov exponent increases, meaning that the phase-space dynamics tends to be more ergodiclike. Our results suggest that the transport efficiency of interacting particles and heat conduction in periodic structures modeled by billiards will strongly be affected by the smoothness of physically realizable walls.

Formation and temperature evolution of Au nanoparticles supported on the h-BN nanomesh

Sub-monolayers of gold have been grown at room temperature on the hexagonal boron nitride (h-BN) nanomesh formed on Rh(1 1 1), and studied systematically with core level and angle-resolved valence band photoelectron spectroscopy. The results are compared with those for the Au/Rh(1 1 1) and Au/h-BN/Pt(1 1 1) interfaces. It has been found that on clean Rh(1 1 1) substrate gold starts to grow two-dimensionally (2D), while in the presence of a h-BN interlayer it forms islands from the very beginning. In the case of flat h-BN monolayer (on Pt) these islands are essentially three-dimensional (3D) and irregular in size. In contrast, on the h-BN nanomesh (on Rh) gold grows initially as regular islands (predominantly 2D), probably filling the pores of the nanomesh. In addition, the evolution of the Au islands in h-BN/Rh(1 1 1) upon annealing has been investigated with core level photoemission and X-ray absorption. The annealing at temperatures between RT and 300 °C does not affect the 2D character of Au islands, while further increase in temperature results in formation of 3D islands. At higher temperatures (above 500 °C), the gold atoms diffuse through h-BN into the Rh substrate and/or desorb from the surface. However, they do not influence the chemical bond between rhodium and h-BN at any stage of annealing: the nanomesh remains intact.

Kicked Bose–Hubbard systems and kicked tops—destruction and stimulation of tunneling

In a two-mode approximation, Bose–Einstein condensates (BEC) in a double-well potential can be described by a many particle Hamiltonian of Bose–Hubbard type. We focus on such a BEC whose interatomic interaction strength is modulated periodically by δ-kicks which represents a realization of a kicked top. In the (classical) mean-field approximation it provides a rich mixed phase space dynamics with regular and chaotic regions. By increasing the kick-strength a bifurcation leads to the appearance of self-trapping states localized on regular islands. This self-trapping is also found for the many particle system, however, in general suppressed by coherent many particle tunneling oscillations. The tunneling time can be calculated from the quasi-energy splitting of the corresponding Floquet states. By varying the kick-strength these quasi-energy levels undergo both avoided and even actual crossings. Therefore, stimulation or complete destruction of tunneling can be observed for this many particle system.

Electron emission from carbon film on island-like tin oxide layer

Electron emission from carbon film on top of island-like tin oxide layer is proposed and demonstrated in this Letter. The tin oxide layer, consisting of regular islands in size of tens of nanometers, induces local heating to the carbon film and helps in forming the electron emission area. Electron emission with good stability and uniformity is reproducibly obtained with emission efficiency up to 1% under an anode voltage of 3 KV applied at 2 mm away.

Universality in the flooding of regular islands by chaotic states

We investigate the structure of eigenstates in systems with a mixed phase space in terms of their projection onto individual regular tori. Depending on dynamical tunneling rates and the Heisenberg time, regular states disappear and chaotic states flood the regular tori. For a quantitative understanding we introduce a random matrix model. The resulting statistical properties of eigenstates as a function of an effective coupling strength are in very good agreement with numerical results for a kicked system. We discuss the implications of these results for the applicability of the semiclassical eigenfunction hypothesis.

Hypothesis of strong chaos and anomalous diffusion in coupled symplectic maps

We investigate the high-dimensional Hamiltonian chaotic dynamics in N coupled area-preserving maps. We show the existence of an enhanced trapping regime caused by trajectories performing a random walk inside the area corresponding to regular islands of the uncoupled maps. As a consequence, we observe long intermediate regimes of power law decay of the recurrence time statistics (with exponent γ = 0.5) and of ballistic motion. The asymptotic decay of correlations and anomalous diffusion depend on the stickiness of the N-dimensional invariant tori. Detailed numerical simulations show weaker stickiness for increasing N suggesting that such paradigmatic class of Hamiltonian systems asymptotically fulfill the demands of the usual hypotheses of strong chaos.

Influence of classical resonances on chaotic tunneling

Dynamical tunneling between symmetry-related stable modes is studied in the periodically driven pendulum. We present strong evidence that the tunneling process is governed by nonlinear resonances that manifest within the regular phase-space islands on which the stable modes are localized. By means of a quantitative numerical study of the corresponding Floquet problem, we identify the trace of such resonances not only in the level splittings between near-degenerate quantum states, where they lead to prominent plateau structures, but also in overlap matrix elements of the Floquet eigenstates, which reveal characteristic sequences of avoided crossings in the Floquet spectrum. The semiclassical theory of resonance-assisted tunneling yields good overall agreement with the quantum-tunneling rates, and indicates that partial barriers within the chaos might play a prominent role.

Nonlinear coherent dynamics of an atom in an optical lattice

We consider a simple model of the lossless interaction between a two-level single atom and a standing-wave single-mode laser field which creates a one-dimensional optical lattice. The internal dynamics of the atom is governed by the laser field, which is treated as classical with a large number of photons. The center-of-mass classical atomic motion is governed by the optical potential and the internal atomic degrees of freedom. The resulting Hamilton-Schro dinger equations of motion are a five-dimensional nonlinear dynamical system with two integrals of motion, and the total atomic energy and the Bloch vector length are conserved during the interaction. In our previous papers, the motion of the atom has been shown to be regular or chaotic (in the sense of exponential sensitivity to small variations of the initial conditions and/or the system’s control parameters) depending on the values of the control parameters, atom-field detuning, and recoil frequency. At the exact atom-field resonance, the exact solutions for both the external and internal atomic degrees of freedom can be derived. The center-of-mass motion does not depend in this case on the internal variables, whereas the Rabi oscillations of the atomic inversion is a frequency-modulated signal with the frequency defined by the atomic position in the optical lattice. We study analytically the correlations between the Rabi oscillations and the center-of-mass motion in two limiting cases of a regular motion out of the resonance: (1) far-detuned atoms and (2) rapidly moving atoms. This paper is concentrated on chaotic atomic motion that may be quantified strictly by positive values of the maximal Lyapunov exponent. It is shown that an atom, depending on the value of its total energy, can either oscillate chaotically in a well of the optical potential, or fly ballistically with weak chaotic oscillations of its momentum, or wander in the optical lattice, changing the direction of motion in a chaotic way. In the regime of chaotic wandering, the atomic motion is shown to have fractal properties. We find a useful tool to visualize complicated atomic motion-Poincare mapping of atomic trajectories in an effective three-dimensional phase space onto planes of atomic internal variables and momentum. The Poincare mappings are constructed using the translational invariance of the standing laser wave. We find common features with typical nonhyperbolic Hamiltonian systems-chains of resonant islands of different sizes imbedded in a stochastic sea, stochastic layers, bifurcations, and so on. The phenomenon of the atomic trajectories sticking to boundaries of regular islands, which should have a great influence on atomic transport in optical lattices, is found and demonstrated numerically.

The classical-quantum correspondence of a kicked particle in an infinite potential well

Abstract The classical-quantum correspondence of a periodically kicked particle in a 1-D infinite potential well is investigated. Stroboscopic state space portraits are presented for various kick strengths and a classical diffusion study reveals anomalous behaviour and the presence of both regular islands of stability and accelerator modes. Quantum diffusion is subsequently studied and the quantum diffusion coefficient is found to mimic the classical diffusion coefficient by rescaling k . Wigner and Husimi distribution functions are derived and comparisons are made between the classical stroboscopic state space portraits and these quantum quasi-probability distribution functions.