Signature Of Chaos Research Articles

Analysis of delay time signature in broadband chaos generated by an optoelectronic oscillator

Chaotic laser characteristics of an optoelectronic oscillator are investigated theoretically and numerically, and the influences of the offset phase and the feedback strength on the time-delay signature of chaos which is generated by an optoelectronic oscillator are studied, based on the method of autocorrelation function. Numerical analyses show that the increase of the feedback strength can further suppress or even conceal the time-delay signature. The closer to the extreme point of the transmission curve the operating point corresponding to the DC offset phase, the weaker the time-delay signature is. The time-delay signature can be suppressed obviously as the offset phase is set to be 0. We also find that the sign of the correlation coefficient corresponding to the time-delay signature is changed when the offset phase and the phase shift caused by the delay feedback produce a phase-shift value of π/2.

Constrained quantum mechanics: chaos in non-planar billiards

We illustrate some of the techniques to identify chaos signatures at the quantum level using as guiding examples some systems where a particle is constrained to move on a radial symmetric, but non-planar, surface. In particular, two systems are studied: the case of a cone with an arbitrary contour or dunce hat billiard and the rectangular billiard with an inner Gaussian surface.

Recurrence analysis and phase space reconstruction of irregular vibration in friction brakes: Signatures of chaos in steady sliding

Irregular friction brake vibration data have been collected with sampling rates of up to 200kHz. The measured time series have been subjected to recurrence analysis and phase space reconstruction. The recurrence analysis indicates that irregular vibration states of friction brakes are strongly dominated by intermittency phenomena. Phase space reconstruction suggests that this intermittency is dominated by low-dimensional irregular deterministic dynamics rather than by high-dimensional stochastic processes.

Evolution of time delay signature of chaos generated in a mutually delay-coupled semiconductor lasers system

In this paper, evolution of time delay (TD) signature of chaos generated in a mutual delay-coupled semiconductor lasers (MDC-SLs) system is investigated experimentally and theoretically. Two statistical methods, including self-correlation function (SF) and permutation entropy (PE), are used to estimate the TD signature of chaotic time series. Through extracting the characteristic peak from the SF curve, a series of TD signature evolution maps are firstly obtained in the parameter space of coupled strength and frequency detuning. Meantime, the influences of injection current on the evolution maps of TD signature have been discussed, and the optimum scope of TD signature suppression is also specified. An overall qualitative agreement between our theoretical and experimental results is obtained.

The Quantum Signature of Chaos through the Dynamics of Entanglement in Classically Regular and Chaotic Systems

We demonstrate the dependence of the entanglement dynamics on the global classical regime of regular, mixed and chaotic dynamics with initial squeezed coherent states. By squeezing the initial states, we are able to enhance or suppress the entanglement entropy of the dynamically generated quantum states relative to those produced via initial coherent states. This suggests the possibility of exploiting a judicious selection of initial squeezed states to create quantum states with enhanced entanglement entropy dynamically for the purpose of robust quantum information processing.

Thermalization in the two-body random ensemble

Using the ergodicity principle for the expectation values of several types of observables, weinvestigate the thermalization process in isolated fermionic systems. These are described bythe two-body random ensemble, which is a paradigmatic model to study quantum chaosand especially the dynamical transition from integrability to chaos. By means of exactdiagonalizations we analyze the relevance of the eigenstate thermalization hypothesis aswell as the influence of other factors, such as the energy and structure of the initial state,or the dimension of the Hilbert space. We also obtain analytical expressions linkingthe degree of thermalization for a given observable with the so-called number ofprincipal components for transition strengths originating at a given energy, with thedimensions of the whole Hilbert space and microcanonical energy shell, and with thecorrelations generated by the observable. As the strength of the residual interaction isincreased, an order-to-chaos transition takes place, and we show that the onset ofWigner spectral fluctuations, which is the standard signature of chaos, is notsufficient to guarantee thermalization in finite systems. When all the signatures ofchaos are fulfilled, including the quasicomplete delocalization of eigenfunctions,the eigenstate thermalization hypothesis is the mechanism responsible for thethermalization of certain types of observables, such as (linear combinations of)occupancies and strength function operators. Our results also suggest that fullychaotic systems will thermalize relative to most observables in the thermodynamiclimit.

Chaos and the quantum: how nonlinear effects can explain certain quantum paradoxes

In recent years we have suggested that many of the so-called paradoxes resulting from the Copenhagen interpretation of quantum mechanics could well have more logical parallels based in nonlinear dynamics and chaos theory. Perhaps quantum mechanics might not be strictly linear as has been commonly postulated, and indeed, during the past year experimentalists have discovered signatures of chaos in a definitely quantum system. As an illustration of what can go wrong when quantum effects are forced into a linear interpretation, I examine Bell-type inequalities. In conventional derivations of such inequalities, classical systems are found to impose upper limits on the statistical correlations between, say, the properties of a pair of separated but entangled particles, whereas quantum systems allow greater correlations. Numerous experiments have upheld the quantum predictions (greater statistical correlations than allowed classically), which has led to inferences such as the instantaneous transmission of information between effectively infinitely separated particles — Einstein's "spooky action-at-a-distance," incompatible with relativity. I argue that there is nothing wrong with the quantum mechanical side of such derivations (the usual point of attack by those attempting to debunk Bell-type arguments), but implicit in the derivations on the classical side is the assumption of independent, uncorrelated particles. As a result, one is comparing uncorrelated probabilities versus conditional probabilities rather than comparing classical versus quantum mechanics, making moot the experimental inferences. Further, nonlinear classical systems are known to exhibit correlations that can easily be as great as and overlap with quantum correlations — so-called nonextensive thermodynamics with its nonadditive entropy has verified this with numerous examples. Perhaps quantum mechanics does contain fundamental nonlinear elements. Nonlinear dynamics and chaos theory could well provide a bridge between the determinism so dear to Einstein and the statisical interpretation of the Copenhagen school. Einstein and Bohr both could have been right in their debates.

Improved unfolding by detrending of statistical fluctuations in quantum spectra

A fundamental relation exists between the statistical properties of the fluctuations of the energy-level spectrum of a Hamiltonian and the chaotic properties of the physical system it describes. This relationship has been addressed previously as a signature of chaos in quantum dynamical systems. In order to properly analyze these fluctuations, however, it is necessary to separate them from the general tendency, namely, its secular part. Unfortunately this process, called unfolding, is not trivial and can lead to erroneous conclusions about the chaoticity of a system. In this paper we propose a technique to improve the unfolding procedure for the purpose of minimizing the dependence on the particular procedure. This technique is based on detrending the fluctuations of the unfolded spectra through the empirical mode decomposition method.

Quantifying chaos: A tale of two maps

In many applications, there is a desire to determine if the dynamics of interest are chaotic or not. Since positive Lyapunov exponents are a signature for chaos, they are often used to determine this. Reliable estimates of Lyapunov exponents should demonstrate evidence of convergence; but literature abounds in which this evidence lacks. This Letter presents two maps through which it highlights the importance of providing evidence of convergence of Lyapunov exponent estimates. The results suggest cautious conclusions when confronted with real data. Moreover, the maps are interesting in their own right.

双光反馈半导体激光混沌系统的外腔反馈强度对延时特征的影响

Through constructing a double optical feedback semiconductor laser chaotic system,the time delay signatures of chaos are investigated experimentally under different working conditions.The results show that,time delay signatures of chaotic output can be suppressed efficiently through adjusting the feedback strength of one cavity carefully and fixing the feedback strength of another cavity.Meantime,there is a big time shift which is the deviation between extracted delay time and real system delay time.Therefore,double optical feedback semiconductor laser chaotic system can efficiently hide the delay time signature of chaotic laser output,and also improve the performance in related application fields.

Chaotic dynamics in collective models of nuclei

We present results of an extensive analysis of classical and quantum signatures of chaos in the geometric collective model (GCM) and the interacting boson model (IBM) of nuclei. Apart from comparing the regular fraction of the classical phase space and the Brody parameter for the nearest neighbor spacing distribution in the quantum case, we also adopt (i) the Peres lattices allowing one to distinguish ordered and disordered parts of spectra and to reveal main ordering principles of quantum states, (ii) the geometrical method to determine the position where the transition from order to chaos occurs, and (iii) we look for the 1/fα power law in the power spectrum of energy level fluctuations. The Peres method demonstrates the adiabatic separation of collective rotations in the IBM.

Dynamical entanglement as a signature of chaos in the semiclassical limit

The relationship between classically chaotic dynamics and the entanglement properties of the corresponding quantum system is examined in the semiclassical limit. Numerical results are computed for a modified kicked top, keeping the classical dynamics constant while investigating the entanglement for several versions of the corresponding quantum system characterized by a different value of the effective Planck constant ℏ eff. Our findings indicate that as ℏ eff → 0, the apparent signatures of classical chaos in the entanglement properties, such as characteristic oscillations in the time-dependence of the linear entropy, can also be obtained in the regular regime. These results suggest that entanglement is not a universal marker of chaotic dynamics of the corresponding classical system.

Signature of chaos in the semi quantum behavior of a classically regular triple well heterostructure

We analyze the phenomenon of semiquantum chaos in the classically regular triple well model from classical to quantum. His dynamics is very rich because it provides areas of regular be-havior, chaotic ones and multiple quantum tun-neling depending on the energy of the system as the Planck’s constant varies from 0 to 1. The Time Dependent Variational Principle TDVP using generalized Gaussian trial wave function, which, in many-body theory leads to the Hartree Fock Approximation TDHF, is added to the tech-niques of Gaussian effective potentials and both are used to study the system. The extended classical system with fluctuation variables non- linearly coupled to the average variables exhibit energy dependent transitions between regular behavior and semi quantum chaos monitored by bifurcation diagram together with some numerical indicators.

Quantum signatures of chaos in a kicked top

Chaotic behaviour is ubiquitous and plays an important part in most fields of science. In classical physics, chaos is characterized by hypersensitivity of the time evolution of a system to initial conditions. Quantum mechanics does not permit a similar definition owing in part to the uncertainty principle, and in part to the Schrödinger equation, which preserves the overlap between quantum states. This fundamental disconnect poses a challenge to quantum-classical correspondence, and has motivated a long-standing search for quantum signatures of classical chaos. Here we present the experimental realization of a common paradigm for quantum chaos-the quantum kicked top- and the observation directly in quantum phase space of dynamics that have a chaotic classical counterpart. Our system is based on the combined electronic and nuclear spin of a single atom and is therefore deep in the quantum regime; nevertheless, we find good correspondence between the quantum dynamics and classical phase space structures. Because chaos is inherently a dynamical phenomenon, special significance attaches to dynamical signatures such as sensitivity to perturbation or the generation of entropy and entanglement, for which only indirect evidence has been available. We observe clear differences in the sensitivity to perturbation in chaotic versus regular, non-chaotic regimes, and present experimental evidence for dynamical entanglement as a signature of chaos.

Wave chaos techniques to analyze a modeled reverberation chamber

A Reverberation Chamber (RC) is analyzed with techniques issued from the wave chaos domain. The first 200 modes are determined numerically: their cumulated number is separated into a smooth part, predicted by the Weyl formula, and an oscillating part, interpreted in term of periodic orbits. The technique of the nearest-neighbor spacing distribution is also presented, showing the signature of chaos. Eigenfield distributions are examined for two RC geometries and compared to the Gaussian ideal case. Finally, the notion of avoided crossing is illustrated for an almost chaotic RC, leading to a statement for frequency sweeps induced by the stirrer displacement. To cite this article: G. Orjubin et al., C R. Physique 10 (2009). (C) 2009 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.

Suppression of photon-echo as a signature of chaos.

The paper discusses the effect of quantum chaos on photon-echo signals of two-electronic-state molecular systems. The temporal profile of photon-echo signals is shown to reveal key information about nuclear dynamics on the excited electronic state surface. Specifically, the suppression of echo signals at a particular value of the delay time tau1 between the first and second excitation pulses is demonstrated as a signature of quantum level statistics that corresponds to the classically chaotic nuclear motion in the excited electronic state surface.

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.

Chaos, entanglement, and decoherence in the quantum kicked top

We analyze the interplay of chaos, entanglement, and decoherence in a system of qubits whose collective behavior is that of a quantum kicked top. The dynamical entanglement between a single qubit and the rest can be calculated from the mean of the collective spin operators. This allows the possibility of efficiently measuring entanglement dynamics in an experimental setting. We consider a deeply quantum regime and show that signatures of chaos are present in the dynamical entanglement for parameters accessible in an experiment that we propose using cold atoms. The evolution of the entanglement depends on the support of the initial state on regular versus chaotic Floquet eigenstates, whose phase-space distributions are concentrated on the corresponding regular or chaotic eigenstructures. We include the effect of decoherence via a realistic model and show that the signatures of chaos in the entanglement dynamics persist in the presence of decoherence. In addition, the classical chaos affects the decoherence rate itself.

PRECESSION AND CHAOS IN THE CLASSICAL TWO-BODY PROBLEM IN A SPHERICAL UNIVERSE

We generalize the classical two-body problem from flat space to spherical space and realize much of the complexity of the classical three-body problem with only two bodies. We show analytically, by perturbation theory, that small, nearly circular orbits of identical particles in a spherical universe precess at rates proportional to the square root of their initial separations and inversely proportional to the square of the universe's radius. We show computationally, by graphically displaying the outcomes of large open sets of initial conditions, that large orbits can exhibit extreme sensitivity to initial conditions, the signature of chaos. Although the spherical curvature causes nearby geodesics to converge, the compact space enables infinitely many close encounters, which is the mechanism of the chaos.

Quantum Signatures of Chaos in Adiabatic Interaction Between a Trapped Ion and a Laser Standing Wave

We study quantum motion around a classical heteroclinic point of a single trapped ion interacting with a strong laser standing wave. We construct a set of exact coherent states of the quantum system and from the exact solutions reveal that quantum signatures of chaos can be induced by the adiabatic interaction between the trapped ion and the laser standing wave, where the quantum expectation values of position and momentum correspond to the classically chaotic orbit. The chaotic region on the phase space is illustrated. The energy crossing and quantum resonance in time evolution and the exponentially increased Heisenberg uncertainty are found. The results suggest a theoretical scheme for controlling the unstable regular and chaotic motions.