Department of Physics
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Item Stochasticity in Complex Networks: A random matrix analysis(2006-08) Bandyopadhyay, Jayendra N.Following random matrix theory, we study nearest neighbor spacing distribution (NNSD) of the eigenvalues of the adjacency matrix of various model networks, namely scale-free, small-world and random networks. Our analysis shows that, though spectral densities of these model networks are different, their eigenvalue fluctuations are same and follow Gaussian orthogonal ensemble (GOE) statistics. Secondly we show the analogy between the onset of small-world behavior (quantified by small diameter and large clustering coefficients) and the transition from Poisson to GOE statistics (quantified by Brody parameter). We also present our analysis for a protein-protein interaction network in budding yeast.Item Randomness of random networks: A random matrix analysis(Sci Egngine, 2009-08) Bandyopadhyay, Jayendra N.We analyze complex networks under the random matrix theory framework. Particularly, we show that statistics, which gives information about the long-range correlations among eigenvalues, provides a measure of randomness in networks. As networks deviate from the regular structure, follows the random matrix prediction of logarithmic behavior (i.e., ) for longer scale.Item Entanglement measures in quantum and classical chaos(ARXIV, 2005-09) Bandyopadhyay, Jayendra N.Entanglement is a Hilbert-space based measure of nonseparability of states that leads to unique quantum possibilities such as teleportation. It has been at the center of intense activity in the area of quantum information theory and computation. In this paper we discuss the implications of quantum chaos on entanglement, showing how chaos can lead to large entanglement that is universal and describable using random matrix theory. We also indicate how this measure can be used in the Hilbert space formulation of classical mechanics. This leads us to consider purely Hilbert-space based measures of classical chaos, rather than the usual phase-space based ones such as the Lyapunov exponents, and can possibly lead to understanding of partial differential equations and nonintegrable classical field theories.Item Entanglement and level crossings in frustrated ferromagnetic rings(APS, 2009-04) Bandyopadhyay, Jayendra N.We study the entanglement content of a class of mesoscopic tunable magnetic systems. The systems are closed finite spin-1/2 chains with ferromagnetic nearest-neighbor interactions frustrated by antiferromagnetic next-nearest-neighbor interactions. The finite chains display a series of level crossings reflecting the incommensurate physics of the corresponding infinite-size chain. We present dramatic entanglement signatures characterizing these unusual level crossings. We focus on multispin and global measures of entanglement rather than only one-spin or two-spin entanglements. We compare and contrast the information obtained from these measures to that obtained from traditional condensed-matter quantities such as correlation functions.Item Entanglement production in quantized chaotic systems(Springer, 2005-04) Bandyopadhyay, Jayendra N.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, chaos in the system produces more entanglement. However, coupling strength between two subsystems is also a very important parameter for 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.Item Algebraic approach in the study of time-dependent nonlinear integrable systems: Case of the singular oscillator(APS, 2001-03) Bandyopadhyay, Jayendra N.The classical and the quantal problem of a particle interacting in one dimension with an external time-dependent quadratic potential and a constant inverse square potential is studied from the Lie-algebraic point of view. The integrability of this system is established by evaluating the exact invariant closely related to the Lewis and Riesenfeld invariant for the time-dependent harmonic oscillator. We study extensively the special and interesting case of a kicked-quadratic potential from which we derive a new integrable, nonlinear, area preserving, two-dimensional map that may, for instance, be used in numerical algorithms that integrate the Calogero-Sutherland-Moser Hamiltonian. The dynamics, both classical and quantal, is studied via the time-evolution operator that we evaluate using a recent method of integrating the quantum Liouville-Bloch equations [A. R. P. Rau, Phys. Rev. Lett. 81, 4785 (1990)]. The results show the exact one-to-one correspondence between the classical and the quantal dynamics. Our analysis also sheds light on the connection between properties of the su(1,1) algebra and that of simple dynamical systems.Item Entangling power of quantum chaotic evolutions via operator entanglement(ARXIV, 2005-04) Bandyopadhyay, Jayendra N.We study operator entanglement of the quantum chaotic evolutions. This study shows that properties of the operator entanglement production are qualitatively similar to the properties reported in literature about the pure state entanglement production. This similarity establishes that the operator entanglement quantifies {\it intrinsic} entangling power of an operator. The term `intrinsic' suggests that this measure is independent of any specific choice of initial states.Item European Physical Society logo. Italian Physical Society logo. EDP Sciences logo. The Institute of Physics logo. Quantum chaotic system as a model of decohering environment(IOP, 2009-03) Bandyopadhyay, Jayendra N.As a model of decohering environment, we show that a quantum chaotic system behaves equivalently as a many-body system. An approximate formula for the time evolution of the reduced density matrix of a system interacting with a quantum chaotic environment is derived. This theoretical formulation is substantiated by the numerical study of the decoherence of two qubits interacting with a quantum chaotic environment modeled by a chaotic kicked top. Like the many-body model of environment, the quantum chaotic system is an efficient decoherer, and it can generate entanglement between the two qubits which have no direct interaction.Item Random matrix analysis of network Laplacians(Elsevier, 2008-01) Bandyopadhyay, Jayendra N.We analyse the eigenvalue fluctuations of the Laplacian of various networks under the random matrix theory framework. Analyses of random networks, scale-free networks and small-world networks show that the nearest neighbor spacing distribution of the Laplacian of these networks follow Gaussian orthogonal ensemble statistics of the random matrix theory. Furthermore, we study the nearest neighbor spacing distribution as a function of the random connections and find that the transition to the Gaussian orthogonal ensemble statistics occurs at the small-world transition.Item Quantum spectrum as a time series: Fluctuation measures(APS, 2006-01) Bandyopadhyay, Jayendra N.The fluctuations in the quantum spectrum could be treated like a time series. In this framework, we explore the statistical self-similarity in the quantum spectrum using the detrended fluctuation analysis (DFA) and random matrix theory (RMT). We calculate the Hausdorff measure for the spectra of atoms and Gaussian ensembles and study their self-affine properties. We show that DFA is equivalent to the Δ3 statistics of RMT, unifying two different approaches. We exploit this connection to obtain theoretical estimates for the Hausdorff measure.