Entanglement and chaos in the kicked top

Out-of-time ordered correlators in kicked coupled tops: Information scrambling in mixed phase space and the role of conserved quantities

We study operator growth in a bipartite kicked coupled tops (KCTs) system using out-of-time ordered correlators (OTOCs), which quantify "information scrambling" due to chaotic dynamics and serve as a quantum analog of classical Lyapunov exponents. In the KCT system, chaos arises from the hyper-fine coupling between the spins. Due to a conservation law, the system's dynamics decompose into distinct invariant subspaces. Focusing initially on the largest subspace, we numerically verify that the OTOC growth rate aligns well with the classical Lyapunov exponent for fully chaotic dynamics. While previous studies have largely focused on scrambling in fully chaotic dynamics, works on mixed-phase space scrambling are sparse. We explore scrambling behavior in both mixed-phase space and globally chaotic dynamics. In the mixed-phase space, we use Percival's conjecture to partition the eigenstates of the Floquet map into "regular" and "chaotic." Using these states as the initial states, we examine how their mean phase space locations affect the growth and saturation of the OTOCs. Beyond the largest subspace, we study the OTOCs across the entire system, including all other smaller subspaces. For certain initial operators, we analytically derive the OTOC saturation using random matrix theory (RMT). When the initial operators are chosen randomly from the unitarily invariant random matrix ensembles, the averaged OTOC relates to the linear entanglement entropy of the Floquet operator, as found in earlier works. For the diagonal Gaussian initial operators, we provide a simple expression for the OTOC.

Classical route to ergodicity and scarring in collective quantum systems

Ergodicity, a fundamental concept in statistical mechanics, is not yet a fully understood phenomena for closed quantum systems, particularly its connection with the underlying chaos. In this review, we consider a few examples of collective quantum systems to unveil the intricate relationship of ergodicity as well as its deviation due to quantum scarring phenomena with their classical counterpart. A comprehensive overview of classical and quantum chaos is provided, along with the tools essential for their detection. Furthermore, we survey recent theoretical and experimental advancements in the domain of ergodicity and its violations. This review aims to illuminate the classical perspective of quantum scarring phenomena in interacting quantum systems.

Exposing hypersensitivity in quantum chaotic dynamics

We study hypersensitivity to initial-state perturbation in the unitary dynamics of a multiqubit system. We use the quantum state metric, introduced by Girolami and Anza [Phys. Rev. Lett. 126, 170502 (2021)0031-900710.1103/PhysRevLett.126.170502], which can be interpreted as a quantum Hamming distance. To provide a proof of principle, we take the multiqubit implementation of the quantum kicked top, a paradigmatic system known to exhibit quantum chaotic behavior. Our findings confirm that the observed hypersensitivity corresponds to commonly used signatures of quantum chaos. Furthermore, we demonstrate that the proposed metric can detect quantum chaos in the same regime and under analogous initial conditions as in the corresponding classical case.

Multifractality in Quasienergy Space of Coherent States as a Signature of Quantum Chaos

We present the multifractal analysis of coherent states in kicked top model by expanding them in the basis of Floquet operator eigenstates. We demonstrate the manifestation of phase space structures in the multifractal properties of coherent states. In the classical limit, the classical dynamical map can be constructed, allowing us to explore the corresponding phase space portraits and to calculate the Lyapunov exponent. By tuning the kicking strength, the system undergoes a transition from regularity to chaos. We show that the variation of multifractal dimensions of coherent states with kicking strength is able to capture the structural changes of the phase space. The onset of chaos is clearly identified by the phase-space-averaged multifractal dimensions, which are well described by random matrix theory in a strongly chaotic regime. We further investigate the probability distribution of expansion coefficients, and show that the deviation between the numerical results and the prediction of random matrix theory behaves as a reliable detector of quantum chaos.

Star-topology registers: NMR and quantum information perspectives

Quantum control of large spin registers is crucial for many applications ranging from spectroscopy to quantum information. A key factor that determines the efficiency of a register for implementing a given information processing task is its network topology. One particular type, called star-topology, involves a central qubit uniformly interacting with a set of ancillary qubits. A particular advantage of the star-topology quantum registers is in the efficient preparation of large entangled states, called NOON states, and their generalized variants. Thanks to the robust generation of such correlated states, spectral simplicity, ease of polarization transfer from ancillary qubits to the central qubit, as well as the availability of large spin-clusters, the star-topology registers have been utilized for several interesting applications over the last few years. Here we review some recent progress with the star-topology registers, particularly via nuclear magnetic resonance methods.

Nonlinear dynamics and quantum chaos of a family of kicked p-spin models

We introduce kicked p-spin models describing a family of transverse Ising-like models for an ensemble of spin-1/2 particles with all-to-all p-body interaction terms occurring periodically in time as delta-kicks. This is the natural generalization of the well-studied quantum kicked top (p=2) [Haake, Kuś, and Scharf, Z. Phys. B 65, 381 (1987)10.1007/BF01303727]. We fully characterize the classical nonlinear dynamics of these models, including the transition to global Hamiltonian chaos. The classical analysis allows us to build a classification for this family of models, distinguishing between p=2 and p>2, and between models with odd and even p's. Quantum chaos in these models is characterized in both kinematic and dynamic signatures. For the latter, we show numerically that the growth rate of the out-of-time-order correlator is dictated by the classical Lyapunov exponent. Finally, we argue that the classification of these models constructed in the classical system applies to the quantum system as well.

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.

NMR studies of quantum chaos in a two-qubit kicked top

Quantum chaotic kicked top model is implemented experimentally in a two-qubit system comprising of a pair of spin-1/2 nuclei using nuclear magnetic resonance techniques. The essential nonlinear interaction was realized using indirect spin-spin coupling, while the linear kicks were realized using radio-frequency pulses. After a variable number of kicks, quantum state tomography was employed to reconstruct the single-qubit reduced density matrices, using which we could extract von Neumann entropies and Husimi distributions. These measures enabled the study of correspondence with classical phase space as well as probing distinct features of quantum chaos, such as symmetries and temporal periodicity in the two-qubit kicked top.

Quantum chaos and entanglement in ergodic and nonergodic systems

We study entanglement entropy (EE) as a signature of quantum chaos in ergodic and nonergodic systems. In particular we look at the quantum kicked top and kicked rotor as multispin systems and investigate the single-spin EE which characterizes bipartite entanglement of this spin with the rest of the system. We study the correspondence of the Kolmogorov-Sinai entropy of the classical kicked systems with the EE of their quantum counterparts. We find that EE is a signature of global chaos in ergodic systems and local chaos in nonergodic systems. In particular, we show that EE can be maximized even when systems are highly nonergodic, when the corresponding classical system is locally chaotic. In contrast, we find evidence that the quantum analog of Kolmogorov-Arnol'd-Moser (KAM) tori are tori of low entanglement entropy. We conjecture that entanglement should play an important role in any quantum KAM theory.

Quantum-classical correspondence in the vicinity of periodic orbits

Quantum-classical correspondence in chaotic systems is a long-standing problem. We describe a method to quantify Bohr's correspondence principle and calculate the size of quantum numbers for which we can expect to observe quantum-classical correspondence near periodic orbits of Floquet systems. Our method shows how the stability of classical periodic orbits affects quantum dynamics. We demonstrate our method by analyzing quantum-classical correspondence in the quantum kicked top (QKT), which exhibits both regular and chaotic behavior. We use our correspondence conditions to identify signatures of classical bifurcations even in a deep quantum regime. Our method can be used to explain the breakdown of quantum-classical correspondence in chaotic systems.

Quantum and classical complexity in coupled maps

We study a generic and paradigmatic two-degrees-of-freedom system consisting of two coupled perturbed cat maps with different types of dynamics. The Wigner separability entropy (WSE)-equivalent to the operator space entanglement entropy-and the classical separability entropy (CSE) are used as measures of complexity. For the case where both degrees of freedom are hyperbolic, the maps are classically ergodic and the WSE and the CSE behave similarly, growing to higher values than in the doubly elliptic case. However, when one map is elliptic and the other hyperbolic, the WSE reaches the same asymptotic value than that of the doubly hyperbolic case but at a much slower rate. The CSE only follows the WSE for a few map steps, revealing that classical dynamical features are not enough to explain complexity growth.

Entanglement and its relationship to classical dynamics

We present an analysis of the entangling quantum kicked top focusing on the few qubit case and the initial condition dependence of the time-averaged entanglement S_{Q} for spin-coherent states. We show a very strong connection between the classical phase space and the initial condition dependence of S_{Q} even for the extreme case of two spin-1/2 qubits. This correlation is not related directly to chaos in the classical dynamics. We introduce a measure of the behavior of a classical trajectory which correlates far better with the entanglement and show that the maps of classical and quantum initial-condition dependence are both organized around the symmetry points of the Hamiltonian. We also show clear (quasi-)periodicity in entanglement as a function of number of kicks and of kick strength.

Signatures of bifurcation on quantum correlations: Case of the quantum kicked top

Quantum correlations reflect the quantumness of a system and are useful resources for quantum information and computational processes. Measures of quantum correlations do not have a classical analog and yet are influenced by classical dynamics. In this work, by modeling the quantum kicked top as a multiqubit system, the effect of classical bifurcations on measures of quantum correlations such as the quantum discord, geometric discord, and Meyer and Wallach Q measure is studied. The quantum correlation measures change rapidly in the vicinity of a classical bifurcation point. If the classical system is largely chaotic, time averages of the correlation measures are in good agreement with the values obtained by considering the appropriate random matrix ensembles. The quantum correlations scale with the total spin of the system, representing its semiclassical limit. In the vicinity of trivial fixed points of the kicked top, the scaling function decays as a power law. In the chaotic limit, for large total spin, quantum correlations saturate to a constant, which we obtain analytically, based on random matrix theory, for the Q measure. We also suggest that it can have experimental consequences.

Effect of geometry on the classical entanglement in a chaotic optical fiber

The effect of boundary deformation on the classical entanglement which appears in the classical electromagnetic field is considered. A chaotic billiard geometry is used to explore the influence of the mechanical modification of the optical fiber cross-sectional geometry on the production of classical entanglement within the electromagnetic fields. For the experimental realization of our idea, we propose an optical fiber with a cross section that belongs to the family of Robnik chaotic billiards. Our results show that a modification of the fiber geometry from a regular to a chaotic regime can enhance the transverse mode classical entanglement.

Comment on "entanglement and chaos in the kicked top"

We comment on the investigation of the connection between chaos and dynamically generated entanglement in Lombardi and Matzkin [Phys. Rev. E 83, 016207 (2011)PRESCM1539-375510.1103/PhysRevE.83.016207]. Whereas, in the referred paper, the authors give an explicit example of a state initially localized in the regular region and still has entanglement properties similar to the states localized in the chaotic region, a few clarifications related to previous works are in order. First, it is crucial to point out that such a behavior can occur for states initially localized near the border between the chaotic region and the regular island, which seems to be the case in the example provided in the referred paper. We comment on the time evolution of such states and the extent to which these can be regarded as having "regular dynamics." Second, the degree to which entanglement is correlated with the chaos in the system is better understood when we analyze the same initial state and increase the chaoticity in the system gradually. We also discuss in what capacity entanglement can be regarded as a signature of chaos in such studies.

Reply to "comment on 'entanglement and chaos in the kicked top' "

We reply to the preceding Comment that attempts to clarify the connection between chaos and entanglement exposed in our previous paper [Phys. Rev. E 83, 016207 (2011)PRESCM1539-375510.1103/PhysRevE.83.016207]. We present additional computations that show the argument exposed in the Comment to explain the entangling power of some regular states is not important in the present case. More fundamentally we argue that the example chosen in the Comment is not the most significant in order to understand why specific regular dynamics can entangle as efficiently as when the corresponding classical dynamics is chaotic.

Signatures of chaos in the dynamics of quantum discord

We identify signatures of chaos in the dynamics of discord in a multiqubit system collectively modelled as a quantum kicked top. The evolution of discord between any two qubits is quasiperiodic in regular regions, while in chaotic regions the quasiperiodicity is lost. As the initial wave function is varied from the regular regions to the chaotic sea, a contour plot of the time-averaged discord remarkably reproduces the structures of the classical stroboscopic map. We also find surprisingly opposite behavior of two-qubit discord versus entanglement of the two qubits as measured by the concurrence. Our results provide evidence of signatures of chaos in dynamically generated discord.

Classical dynamics of quantum entanglement

We analyze numerically the dynamical generation of quantum entanglement in a system of two interacting particles, started in a coherent separable state, for decreasing values of ℏ. As ℏ→0 the entanglement entropy, computed at any finite time, converges to a finite nonzero value. The limit law that rules the time dependence of entropy is well reproduced by purely classical computations. Its general features can be explained by simple classical arguments, which expose the different ways entanglement is generated in systems that are classically chaotic or regular.