Signature Of Chaos Research Articles

Quantum Signatures in Quench from Chaos to Superradiance.

The driven-dissipative Dicke model features normal, superradiant, and lasing steady states that may be regular or chaotic. We report quantum signatures of chaos in a quench protocol from the lasing states. Within the framework of a classical mean-field perspective, once quenched, the system relaxes either to the normal or to the superradiant state. Quench from chaos, unlike quench from a regular lasing state, exhibits erratic dependence on control parameters. In the quantum domain, this sensitivity implies an effect that is similar to universal conductance fluctuations.

Out-of-time-order correlator in the quantum Rabi model

We investigate signatures of chaos and equilibration in the quantum Rabi model, which exhibits a quantum phase transition when the ratio of the atomic level-splitting to bosonic frequency grows to infinity. We show that out-of-time-order correlator derived from the Loschmidt echo signal quickly saturates in the normal phase and reveals exponential growth in the superradiant phase which is associated with the onset of quantum chaos. Furthermore, we show that the effective time-averaged dimension of the quantum Rabi system can be large compared to the spin system size which leads to suppression of the temporal fluctuations and equilibration of the spin system.

Probing Many-Body Quantum Chaos with Quantum Simulators

A class of observables known as partial spectral form factors can efficiently measure signatures of chaos in existing quantum simulators, opening the door to future insights into chaos and thermalization.

Control of optical chaos spectrum in semiconductor laser for secure RoF communication

A critical requirement in optical chaos based secure radio over fiber (RoF) system design is the ability to control center frequency, spectral bandwidth, power level and signature of chaos to submerge message with sufficient horizontal and vertical margins both in time and frequency domains. Once frequency domain masking is completely achieved, time domain masking is met automatically, the former being more stringent. In a direct modulated semiconductor laser, the three control parameters are bias current (Ibias), modulation current (Imod) and modulation frequency (ωa). It is found that Imod increases bandwidth and amplitude dynamic range of chaotic pulses. Ibias increases the cavity power and hence average peak amplitude of laser chaotic pulses. The modulation frequency increases the speed of overall cavity dynamics and hence is used to increase the bandwidth of chaos but a corresponding increase in bias and modulation currents is required to support high repetition pulses. The results show relationship between three control parameters (bias current, modulation current and modulation frequency) in a direct modulated semiconductor laser and optical chaos bandwidth using regression.

Chaos in coupled Kerr-nonlinear parametric oscillators

A Kerr-nonlinear parametric oscillator (KPO) can generate a quantum superposition of two oscillating states, known as a Schr\"{o}dinger cat state, via quantum adiabatic evolution, and can be used as a qubit for gate-based quantum computing and quantum annealing. In this work, we investigate complex dynamics, i.e., chaos, in two coupled nondissipative KPOs at a few-photon level. After showing that a classical model for this system is nonintegrable and consequently exhibits chaotic behavior, we provide quantum counterparts for the classical results, which are quantum versions of the Poincar\'{e} surface of section and its lower-dimensional version defined with time integrals of the Wigner and Husimi functions, and also the initial and long-term behavior of out-of-time-ordered correlators. We conclude that some of them can be regarded as quantum signatures of chaos, together with energy-level spacing statistics (conventional signature). Thus, the system of coupled KPOs is expected to offer not only an alternative approach to quantum computing, but also a promising platform for the study on quantum chaos.

Imaging Quantum Interference in Stadium-Shaped Monolayer and Bilayer Graphene Quantum Dots.

Experimental realizations of graphene-based stadium-shaped quantum dots (QDs) have been few and have been incompatible with scanned probe microscopy. Yet, the direct visualization of electronic states within these QDs is crucial for determining the existence of quantum chaos in these systems. We report the fabrication and characterization of electrostatically defined stadium-shaped QDs in heterostructure devices composed of monolayer graphene (MLG) and bilayer graphene (BLG). To realize a stadium-shaped QD, we utilized the tip of a scanning tunneling microscope to charge defects in a supporting hexagonal boron nitride flake. The stadium states visualized are consistent with tight-binding-based simulations but lack clear quantum chaos signatures. The absence of quantum chaos features in MLG-based stadium QDs is attributed to the leaky nature of the confinement potential due to Klein tunneling. In contrast, for BLG-based stadium QDs (which have stronger confinement) quantum chaos is precluded by the smooth confinement potential which reduces interference and mixing between states.

Quantum coherence as a signature of chaos

We establish a rigorous connection between quantum coherence and quantum chaos by employing coherence measures originating from the resource theory framework as a diagnostic tool for quantum chaos. We quantify this connection at two different levels: quantum states and quantum channels. At the level of states, we show how several well-studied quantifiers of chaos are, in fact, quantum coherence measures in disguise (or closely related to them). We further this connection for all quantum coherence measures by using tools from majorization theory. Then we numerically study the coherence of chaotic-versus-integrable eigenstates and find excellent agreement with random matrix theory in the bulk of the spectrum. At the level of channels, we show that the coherence-generating power (CGP)—a measure of how much coherence a dynamical process generates on average—emerges as a subpart of the out-of-time-ordered correlator (OTOC), a measure of information scrambling in many-body systems. Via numerical simulations of the (nonintegrable) transverse-field Ising model, we show that the OTOC and CGP capture quantum recurrences in quantitatively the same way. Moreover, using random matrix theory, we analytically characterize the OTOC-CGP connection for the Haar and Gaussian ensembles. In closing, we remark on how our coherence-based signatures of chaos relate to other diagnostics, namely, the Loschmidt echo, OTOC, and the Spectral Form Factor.Received 3 November 2020Accepted 17 May 2021DOI:https://doi.org/10.1103/PhysRevResearch.3.023214Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Open access publication funded by the Max Planck Society.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasQuantum chaosQuantum coherence & coherence measuresQuantum entanglementQuantum information processingPhysical SystemsLattice models in statistical physicsTechniquesRandom matrix theoryStatistical PhysicsQuantum InformationCondensed Matter, Materials & Applied Physics

Observation of Wigner-Dyson level statistics in a classically integrable system

Resonances in particle transmission through a 1D finite lattice are studied in the presence of a finite number of impurities. Although this is a one-dimensional system that is classically integrable and has no chaos, studying the statistical properties of the spectrum such as the level spacing distribution and the spectral rigidity shows the same statistics as the one obtained for chaotic systems. Using a dimensionless parameter that reflects the degree of state localization, we demonstrate how the transition from Poisson-level statistics to the Wigner-Dyson is affected by state localization. The resonance positions are calculated using both the Wigner-Smith time delay and a Siegert state method, which are in good agreement. Our results show the dependence of the level statistics on the localization length as it evolves from a Poisson distribution to Wigner-Dyson.

Quantum signatures of chaos in a cavity-QED-based stimulated Raman adiabatic passage

Nonlinear stimulated Raman adiabatic passage (STIRAP) is a fascinating physical process that dynamically explores chaotic and nonchaotic phases. In a recent paper [A. Dey and M. Kulkarni, Phys. Rev. Res. 2, 042004(R) (2020)], such a phenomenon is realized in a cavity-QED platform. There, the emergence of chaos and its impact on STIRAP efficiency are mainly demonstrated in the semiclassical limit. In the present paper we treat the problem in a fully quantum many-body framework. With the aim of extracting quantum signatures of a classically chaotic system, it is shown that an out-of-time-ordered correlator (OTOC) measure precisely captures chaotic and nonchaotic features of the system. The prediction by OTOC is in precise matching with classical chaos quantified by Lyapunov exponent analysis. Furthermore, it is shown that the quantum route corresponding to the semiclassical followed state encounters a dip in single-particle purity within the chaotic phase, depicting a consequence of chaos. A dynamics through the chaotic phase is associated with spreading of the many-body quantum state and an irreversible increase in the number of participating adiabatic eigenstates.

Signatures of Chaos in Nonintegrable Models of Quantum Field Theories.

We study signatures of quantum chaos in (1+1)D quantum field theory (QFT) models. Our analysis is based on the method of Hamiltonian truncation, a numerical approach for the construction of low-energy spectra and eigenstates of QFTs that can be considered as perturbations of exactly solvable models. We focus on the double sine-Gordon, also studying the massive sine-Gordon and ϕ^{4} model, all of which are nonintegrable and can be studied by this method with sufficiently high precision from small to intermediate perturbation strength. We analyze the statistics of level spacings and of eigenvector components, which are expected to follow random matrix theory predictions. While level spacing statistics are close to the Gaussian orthogonal ensemble (GOE) as expected, on the contrary, the eigenvector components follow a distribution markedly different from the expected Gaussian. Unlike in the typical quantum chaos scenario, the transition of level spacing statistics to chaotic behavior takes place already in the perturbative regime. Moreover, the distribution of eigenvector components does not appear to change or approach Gaussian behavior, even for relatively large perturbations. Our results suggest that these features are independent of the choice of model and basis.

Pole skipping and chaos in anisotropic plasma: a holographic study

Recently, a direct signature of chaos in many body system has been realized from the energy density retarded Green’s function using the phenomenon of ‘pole skipping’. Moreover, special locations in the complex frequency and momentum plane are found, known as the pole skipping points such that the retarded Green’s function can not be defined uniquely there. In this paper, we compute the correction/shift to the pole skipping points due to a spatial anisotropy in a holographic system by performing near horizon analysis of EOMs involving different bulk field perturbations, namely the scalar, the axion and the metric field. For vector and scalar modes of metric perturbations we construct the gauge invariant variable in order to obtain the master equation. Two separate cases for every bulk field EOMs is considered with the fluctuation propagating parallel and perpendicular to the direction of anisotropy. We compute the dispersion relation for momentum diffusion along the transverse direction in the shear channel and show that it passes through the first three successive pole skipping points. The pole skipping phenomenon in the sound channel is found to occur in the upper half plane such that the parameters Lyapunov exponent λL and the butterfly velocity vB are explicitly obtained thus establishing the connection with many body chaos.

Searching for signatures of chaos in γ-ray light curves of selected Fermi-LAT blazars

ABSTRACT Blazar variability appears to be stochastic in nature. However, a possibility of low-dimensional chaos was considered in the past, but with no unambiguous detection so far. If present, it would constrain the emission mechanism by suggesting an underlying dynamical system. We rigorously searched for signatures of chaos in Fermi-Large Area Telescope light curves of 11 blazars. The data were comprehensively investigated using the methods of nonlinear time-series analysis: phase-space reconstruction, fractal dimension, and maximal Lyapunov exponent (mLE). We tested several possible parameters affecting the outcomes, in particular the mLE, in order to verify the spuriousness of the outcomes. We found no signs of chaos in any of the analysed blazars. Blazar variability is either truly stochastic in nature or governed by high-dimensional chaos that can often resemble randomness.

Quantum chaos dynamics on the surface of topological insulator attached to spiral multiferroic oxide in a magnetic field

Quantum dynamics of Dirac fermions on the surface of a 3D topological insulator attached to a spiral multiferroic oxide in an applied magnetic field is investigated. We find that the particle dynamics is significantly affected by interaction of its spin with the proximity-induced exchange field provided by a spiral multiferroic oxide, which changes considerably by changing the spin orientations, the strength of exchange field, and spiral wave vector. The strong dependence of dynamics on the initial conditions together with the unpredictable irregular behavior of cyclotron motion suggests that the dynamics is chaotic. The chaos signatures are diagnosed by evaluating the relevant out-of-time-ordered correlator and are analyzed using Poincaré maps. The appearance of chaotic behavior in the dynamics of Dirac fermions is attributed to the interplay of zitterbewegung, cyclotron oscillations, and proximity-induced exchange field oscillations, leading to nonlinear complicated oscillating modes that drive the system dynamics to chaotic regime.

Sir Isaac Newton Stranger in a Strange Land.

The theme of this essay is that the time of dominance of Newton’s world view in science is drawing to a close. The harbinger of its demise was the work of Poincaré on the three-body problem and its culmination into what is now called chaos theory. The signature of chaos is the sensitive dependence on initial conditions resulting in the unpredictability of single particle trajectories. Classical determinism has become increasingly rare with the advent of chaos, being replaced by erratic stochastic processes. However, even the probability calculus could not withstand the non-Newtonian assault from the social and life sciences. The ordinary partial differential equations that traditionally determined the evolution of probability density functions (PDFs) in phase space are replaced with their fractional counterparts. Allometry relation is proven to result from a system’s complexity using exact solutions for the PDF of the Fractional Kinetic Theory (FKT). Complexity theory is shown to be incompatible with Newton’s unquestioning reliance on an absolute space and time upon which he built his discrete calculus.

Signature of chaos and multistability in a Thomas-Fermi plasma

We propose a dynamical system governing from the nonlinear extension of the electrostatic ion-acoustic waves (IAWs) under the influence of external magnetic field in Thomas-Fermi (TM) consisting of hot electrons, cold electrons and mobile ions. The coupled lower dimensional model with a planar Hamiltonian exhibits periodic oscillations and conservative properties. The perturbed system with a trigonometric forcing can produce more rich phenomena in terms of quasiperiodic and chaotic patterns. The cosine excitation can also be observed as a source of multistability described by various coexisting attractors. The multistability in an external forcing TM plasma has never been reported before.

Dynamical Detection of Level Repulsion in the One-Particle Aubry-André Model

The analysis of level statistics provides a primary method to detect signatures of chaos in the quantum domain. However, for experiments with ion traps and cold atoms, the energy levels are not as easily accessible as the dynamics. In this work, we discuss how properties of the spectrum that are usually associated with chaos can be directly detected from the evolution of the number operator in the one-dimensional, noninteracting Aubry-André model. Both the quantity and the model are studied in experiments with cold atoms. We consider a single-particle and system sizes experimentally reachable. By varying the disorder strength within values below the critical point of the model, level statistics similar to those found in random matrix theory are obtained. Dynamically, these properties of the spectrum are manifested in the form of a dip below the equilibration point of the number operator. This feature emerges at times that are experimentally accessible. This work is a contribution to a special issue dedicated to Shmuel Fishman.

Manifestations of chaos in relativistic quantum systems - A study based on out-of-time-order correlator

Previous work in the field of relativistic quantum chaos has revealed initial evidence that the manifestations of classical chaos in relativistic quantum systems tend to be weakened as compared with those in nonrelativistic quantum systems. To place this finding on a firmer ground, we investigate the relativistic quantum fingerprints of classical chaos using the out-of-time-order correlator (OTOC). OTOC has recently been applied to a number of fields in physics and it holds special promises for the field of quantum chaos, but existing work focused exclusively on nonrelativistic quantum systems. Calculating and analyzing OTOC for relativistic quantum billiard systems described by the massless Dirac equation, we find that the signatures of classical chaos are indeed characteristically less pronounced in relativistic than in nonrelativistic quantum systems. The finding is substantiated by studying four different aspects of OTOC. Firstly, in the energy eigenspace, in the short time regime there is a complete lack of any signature of chaos associated with the evolution of OTOC in the Dirac billiard due to the relativistic quantum phenomenon of Zitterbewegung, in contrast to nonrelativistic quantum billiard systems where chaos leaves behind a quite distinct signature. Secondly, weakening of the relativistic quantum manifestations of classical chaos occurs in the long time regime as well. Thirdly, evolution of the wave packet based OTOC also reveals that the fingerprints of chaos are much less pronounced in the Dirac billiard systems as compared with the Schrödinger billiards. Fourthly, the level spacing statistics of the OTOC operators in systems with classically integrable and chaotic dynamics, which are characteristically distinct in the Schrödinger billiards, bear a strong similarity in the Dirac billiards, indicating again the dwindling effects of chaos. The OTOC based results suggest that the impacts of classical chaos are generally weaker in relativistic than in nonrelativistic quantum systems, a fundamental issue that warrants further investigation.

Fractal measures and nonlinear dynamics of overcontact binaries

Overcontact binary stars are systems of two stars where the component stars are in contact with each other. This implies that they share a common envelope of gas. In this work we seek signatures of nonlinearity and chaos in these stars by using time series analysis techniques. We use three main techniques, namely the correlation dimension, f(α) spectrum and the bicoherence. The former two are calculated from the reconstructed dynamics, while the latter is calculated from the Fourier transforms of the time series of intensity variations(light curves) of these stars. Our dataset consists of data from 463 overcontact binary stars in the Kepler field of view [1]. Our analysis indicates nonlinearity and signatures of chaos in almost all the light curves. We also explore whether the underlying nonlinear properties of the stars are related to their physical properties like fill-out-factor which is a measure of the extend of contact between the components of an overcontact binary system. We observe that significant correlations exist between the fill out factor and the nonlinear quantifiers. This correlation is more pronounced in specific subcategories constructed based on the mass ratios and effective temperatures of the binaries. The correlations observed can be indicative of variations in the nonlinear properties of the star as it ages. We believe that this study relating nonlinear and astrophysical properties of binary stars is the first of its kind and is an important starting point for such studies in other astrophysical objects displaying nonlinear dynamical behavior.

Effect of system energy on quantum signatures of chaos in the two-photon Dicke model.

We have studied entanglement entropy and Husimi Q distribution as a tool to explore chaos in the quantum two-photon Dicke model. With the increase of the energy of a system, the linear entanglement entropy of a coherent state prepared in the classical chaotic and regular regions becomes more distinguishable, and the corresponding relationship between the distribution of time-averaged entanglement entropy and the classical Poincaré section has clearly been improved. Moreover, Husimi Q distribution for the initial states corresponding to the points in the chaotic region in the higher-energy system disperses more quickly than that in the lower-energy system. Our results imply that higher system energy has contributed to distinguishing between the chaotic and regular behavior in the quantum two-photon Dicke model.

Quantum signatures of chaos, thermalization, and tunneling in the exactly solvable few-body kicked top.

Exactly solvable models that exhibit quantum signatures of classical chaos are both rare as well as important-more so in view of the fact that the mechanisms for ergodic behavior and thermalization in isolated quantum systems and its connections to nonintegrability are under active investigation. In this work, we study quantum systems of few qubits collectively modeled as a kicked top, a textbook example of quantum chaos. In particular, we show that the three- and four-qubit cases are exactly solvable and yet, interestingly, can display signatures of ergodicity and thermalization. Deriving analytical expressions for entanglement entropy and concurrence, we see agreement in certain parameter regimes between long-time average values and ensemble averages of random states with permutation symmetry. Comparing with results using the data of a recent transmons-based experiment realizing the three-qubit case, we find agreement for short times, including a peculiar steplike behavior in correlations of some states. In the case of four qubits we point to a precursor of dynamical tunneling between what in the classical limit would be two stable islands. Numerical results for larger number of qubits show the emergence of the classical limit including signatures of a bifurcation.