arXiv subjects

Arul Lakshminarayan

Publications and source records attributed to Arul Lakshminarayan.

At least 91 records · Page 5Linked to original sources

arXiv2005

Entanglement production in Quantized Chaotic Systems

Quantum chaos is a subject whose major goal is to identify and to investigate different quantum signatures of classical chaos. Here we study entanglement production in coupled chaotic systems as a possible quantum indicator of classical chaos. We use coupled kicked tops as a model for our extensive numerical studies. We find that, in general, presence of chaos in the system produces more entanglement. However, coupling strength between two subsystems is also very important parameter for the entanglement production. Here we show how chaos can lead to large entanglement which is universal and describable by random matrix theory (RMT). We also explain entanglement production in coupled strongly chaotic systems by deriving a formula based on RMT. This formula is valid for arbitrary coupling strengths, as well as for sufficiently long time. Here we investigate also the effect of chaos on the entanglement production for the mixed initial state. We find that many properties of the mixed state entanglement production are qualitatively similar to the pure state entanglement production. We however still lack an analytical understanding of the mixed state entanglement production in chaotic systems.

quant-ph↗

arXiv2004

Transport of Entanglement Through a Heisenberg-XY Spin Chain

The entanglement dynamics of spin chains is investigated using Heisenberg-XY spin Hamiltonian dynamics. The various measures of two-qubit entanglement are calculated analytically in the time-evolved state starting from initial states with no entanglement and exactly one pair of maximally-entangled qubits. The localizable entanglement between a pair of qubits at the end of chain captures the essential features of entanglement transport across the chain, and it displays the difference between an initial state with no entanglement and an initial state with one pair of maximally-entangled qubits.

quant-ph↗

arXiv2004

Cyclic Identities Involving Ratios of Jacobi Theta Functions

Identities involving cyclic sums of terms composed from Jacobi elliptic functions evaluated at $p$ equally shifted points were recently found. The purpose of this paper is to re-express these cyclic identities in terms of ratios of Jacobi theta functions, since many physicists prefer using Jacobi theta functions rather than Jacobi elliptic functions.

math-ph↗

arXiv2003

Entanglement production in coupled chaotic systems : Case of the Kicked Tops

Entanglement production in coupled chaotic systems is studied with the help of kicked tops. Deriving the correct classical map, we have used the reduced Husimi function, the Husimi function of the reduced density matrix, to visualize the possible behaviors of a wavepacket. We have studied a phase space based measure of the complexity of a state and used random matrix theory (RMT) to model the strongly chaotic cases. Extensive numerical studies have been done for the entanglement production in coupled kicked tops corresponding to different underlying classical dynamics and different coupling strengths. An approximate formula, based on RMT, is derived for the entanglement production in coupled strongly chaotic systems. This formula, applicable for arbitrary coupling strengths and also valid for long time, complements and extends significantly recent perturbation theories for strongly chaotic weakly coupled systems.

quant-ph↗

arXiv2003

Recurrence of fidelity in near integrable systems

Within the framework of simple perturbation theory, recurrence time of quantum fidelity is related to the period of the classical motion. This indicates the possibility of recurrence in near integrable systems. We have studied such possibility in detail with the kicked rotor as an example. In accordance with the correspondence principle, recurrence is observed when the underlying classical dynamics is well approximated by the harmonic oscillator. Quantum revivals of fidelity is noted in the interior of resonances, while classical-quantum correspondence of fidelity is seen to be very short for states initially in the rotational KAM region.

nlin.CD↗

arXiv2003

Local Identities Involving Jacobi Elliptic Functions

We derive a number of local identities of arbitrary rank involving Jacobi elliptic functions and use them to obtain several new results. First, we present an alternative, simpler derivation of the cyclic identities discovered by us recently, along with an extension to several new cyclic identities of arbitrary rank. Second, we obtain a generalization to cyclic identities in which successive terms have a multiplicative phase factor exp(2iπ/s), where s is any integer. Third, we systematize the local identities by deriving four local ``master identities'' analogous to the master identities for the cyclic sums discussed by us previously. Fourth, we point out that many of the local identities can be thought of as exact discretizations of standard nonlinear differential equations satisfied by the Jacobian elliptic functions. Finally, we obtain explicit answers for a number of definite integrals and simpler forms for several indefinite integrals involving Jacobi elliptic functions.

math-ph↗

arXiv2003

What is random about a quantum random walk?

We use simple deterministic dynamical systems as coins in studying quantum walks. These dynamical systems can be chosen to display, in the classical limit, a range of behaviors from the integrable to chaotic, or deterministically random. As an example of an integrable coin we study the Fourier walk that generalizes the Hadamard walk and show that the walker slows down with coin dimensionality, which controls the effective Planck constant. Introducing multi-Harper maps as deterministic models of random walks we study the effect of coin chaos on the quantum walk. We also demonstrate that breaking time-reversal symmetry in the coin dynamics effectively slows down the walk.

quant-ph↗

arXiv2003

Entanglement sharing in one-particle states

Entanglement sharing among sites of one-particle states is considered using the measure of concurrence. These are the simplest in an hierarchy of number-specific states of many qubits and corresponds to ``one-magnon'' states of spins. We study the effects of onsite potentials that are both integrable and nonintegrable. In the integrable case we point to a metal-insulator transition that reflects on the way entanglement is shared. In the nonintegrable case the average entanglement content increases and saturates along with a transition to classical chaos. Such quantum chaotic states are shown to have universal concurrence distributions that are modified Bessel functions derivable within random matrix theory. Time-reversal breaking and time evolving states are shown to possess significantly higher entanglement sharing capacity that eigenstates of time-reversal symmetric systems. We use the ordinary Harper and kicked Harper Hamiltonians as model systems.

quant-ph↗

arXiv2003

Cyclic Identities Involving Jacobi Elliptic Functions. II

Identities involving cyclic sums of terms composed from Jacobi elliptic functions evaluated at $p$ equally shifted points on the real axis were recently found. These identities played a crucial role in discovering linear superposition solutions of a large number of important nonlinear equations. We derive four master identities, from which the identities discussed earlier are derivable as special cases. Master identities are also obtained which lead to cyclic identities with alternating signs. We discuss an extension of our results to pure imaginary and complex shifts as well as to the ratio of Jacobi theta functions.

math-ph↗

arXiv2002

Testing statistical bounds on entanglement using quantum chaos

Previous results indicate that while chaos can lead to substantial entropy production, thereby maximizing dynamical entanglement, this still falls short of maximality. Random Matrix Theory (RMT) modeling of composite quantum systems, investigated recently, entails an universal distribution of the eigenvalues of the reduced density matrices. We demonstrate that these distributions are realized in quantized chaotic systems by using a model of two coupled and kicked tops. We derive an explicit statistical universal bound on entanglement, that is also valid for the case of unequal dimensionality of the Hilbert spaces involved, and show that this describes well the bounds observed using composite quantized chaotic systems such as coupled tops.

quant-ph↗

arXiv2001

The classical counterpart of entanglement

We define and explore the classical counterpart of entanglement in complete analogy with quantum mechanics. Using a basis independent measure of entropy in the classical Hilbert space of densities that are propagated by the Frobenius-Perron operator, we demonstrate that at short times the quantum and classical entropies share identical power laws and qualitative behaviors.

quant-ph↗

arXiv2000

Entangling power of quantized chaotic systems

We study the quantum entanglement caused by unitary operators that have classical limits that can range from the near integrable to the completely chaotic. Entanglement in the eigenstates and time-evolving arbitrary states is studied through the von Neumann entropy of the reduced density matrices. We demonstrate that classical chaos can lead to substantially enhanced entanglement. Conversely, entanglement provides a novel and useful characterization of quantum states in higher dimensional chaotic or complex systems. Information about eigenfunction localization is stored in a graded manner in the Schmidt vectors, and the principal Schmidt vectors can be scarred by the projections of classical periodic orbits onto subspaces. The eigenvalues of the reduced density matrices are sensitive to the degree of wavefunction localization, and are roughly exponentially arranged. We also point out the analogy with decoherence, as reduced density matrices corresponding to subsystems of fully chaotic systems are diagonally dominant.

nlin.CD↗

arXiv2000

Barnett-Pegg formalism of angle operators, revivals, and flux lines

We use the Barnett-Pegg formalism of angle operators to study a rotating particle with and without a flux line. Requiring a finite dimensional version of the Wigner function to be well defined we find a natural time quantization that leads to classical maps from which the arithmetical basis of quantum revivals is seen. The flux line, that fundamentally alters the quantum statistics, forces this time quantum to be increased by a factor of a winding number and determines the homotopy class of the path. The value of the flux is restricted to the rational numbers, a feature that persists in the infinite dimensional limit.

quant-ph↗

arXiv2000

Exploring phase space localization of chaotic eigenstates via parametric variation

In a previous Letter [Phys. Rev. Lett. 77, 4158 (1996)], a new correlation measure was introduced that sensitively probes phase space localization properties of eigenstates. It is based on a system's response to varying an external parameter. The measure correlates level velocities with overlap intensities between the eigenstates and some localized state of interest. Random matrix theory predicts the absence of such correlations in chaotic systems whereas in the stadium billiard, a paradigm of chaos, strong correlations were observed. Here, we develop further the theoretical basis of that work, extend the stadium results to the full phase space, study the hbar-dependence, and demonstrate the agreement between this measure and a semiclassical theory based on homoclinic orbits.

nlin.CD↗

arXiv2000

Phase space localization of chaotic eigenstates: Violating ergodicity

The correlation between level velocities and eigenfunction intensities provides a new way of exploring phase space localization in quantized non-integrable systems. It can also serve as a measure of deviations from ergodicity due to quantum effects for typical observables. This paper relies on two well known paradigms of quantum chaos, the bakers map and the standard map, to study correlations in simple, yet chaotic, dynamical systems. The behaviors are dominated by the presence of several classical structures. These primarily include short periodic orbits and their homoclinic excursions. The dependences of the correlations deriving from perturbations allow for eigenfunction features violating ergodicity to be selectively highlighted. A semiclassical theory based on periodic orbit sums leads to certain classical correlations that are super-exponentially cut off beyond a logarithmic time scale. The theory is seen to be quite successful in reproducing many of the quantum localization features.

nlin.CD↗

arXiv1999

Classical Diffusion and Quantum Level Velocities: Systematic Deviations from Random Matrix Theory

We study the response of the quasi-energy levels in the context of quantized chaotic systems through the level velocity variance and relate them to classical diffusion coefficients using detailed semiclassical analysis. The systematic deviations from random matrix theory, assuming independence of eigenvectors from eigenvalues, is shown to be connected to classical higher order time correlations of the chaotic system. We study the standard map as a specific example, and thus the well known oscillatory behavior of the diffusion coefficient with respect to the parameter is reflected exactly in the oscillations of the variance of the level velocities. We study the case of mixed phase-space dynamics as well and note a transition in the scaling properties of the variance that occurs along with the classical transition to chaos.

chao-dyn↗

arXiv1998

Relaxation Fluctuations in Quantum Chaos

Relaxation in the time correlation between operators is studied. Quantized chaotic systems are shown to have distinct relaxation fluctuations that are universal and can be usefully modelled by Random Matrix Theory. Various quantized maps are used to demonsrate these results.

chao-dyn↗

arXiv1997

Relaxation fluctuations about an equilibrium in quantum chaos

Classically chaotic systems relax to coarse grained states of equilibrium. Here we numerically study the quantization of such bounded relaxing systems, in particular the quasi-periodic fluctuations associated with the correlation between two density operators. We find that when the operators, or their Wigner-Weyl transforms, have obvious classical limits that can be interpreted as piecewise continuous functions on phase space, the variance of the fluctuations can distinguish classically chaotic and regular motions, thus providing a novel diagnostic devise of quantum chaos. We uncover several features of the relaxation fluctuations that are shared by disparate systems thus establishing restricted universality. If we consider the nonlinearity driving the chaos as pseudo-time, we find that the onset of classical chaos is indicated quantally as the relaxation of the relaxation fluctuations to a Gaussian distribution.

chao-dyn↗