Multifractality and intermediate statistics in quantum maps

We study multifractal properties of wave functions for a one-parameter family of quantum maps displaying the whole range of spectral statistics intermediate between integrable and chaotic statistics. We perform extensive numerical computations and provide analytical arguments showing that the generalized fractal dimensions are directly related to the parameter of the underlying classical map, and thus to other properties such as spectral statistics. Our results could be relevant for Anderson and quantum Hall transitions, where wave functions also show multifractality.Comment: 4 pages, 4 figure

Edge effects in graphene nanostructures: II. Semiclassical theory of spectral fluctuations and quantum transport

We investigate the effect of different edge types on the statistical properties of both the energy spectrum of closed graphene billiards and the conductance of open graphene cavities in the semiclassical limit. To this end, we use the semiclassical Green's function for ballistic graphene flakes that we have derived in Reference 1. First we study the spectral two point correlation function, or more precisely its Fourier transform the spectral form factor, starting from the graphene version of Gutzwiller's trace formula for the oscillating part of the density of states. We calculate the two leading order contributions to the spectral form factor, paying particular attention to the influence of the edge characteristics of the system. Then we consider transport properties of open graphene cavities. We derive generic analytical expressions for the classical conductance, the weak localization correction, the size of the universal conductance fluctuations and the shot noise power of a ballistic graphene cavity. Again we focus on the effects of the edge structure. For both, the conductance and the spectral form factor, we find that edge induced pseudospin interference affects the results significantly. In particular intervalley coupling mediated through scattering from armchair edges is the key mechanism that governs the coherent quantum interference effects in ballistic graphene cavities

Classical bifurcations and entanglement in smooth Hamiltonian system

We study entanglement in two coupled quartic oscillators. It is shown that the entanglement, as measured by the von Neumann entropy, increases with the classical chaos parameter for generic chaotic eigenstates. We consider certain isolated periodic orbits whose bifurcation sequence affects a class of quantum eigenstates, called the channel localized states. For these states, the entanglement is a local minima in the vicinity of a pitchfork bifurcation but is a local maxima near a anti-pitchfork bifurcation. We place these results in the context of the close connections that may exist between entanglement measures and conventional measures of localization that have been much studied in quantum chaos and elsewhere. We also point to an interesting near-degeneracy that arises in the spectrum of reduced density matrices of certain states as an interplay of localization and symmetry.Comment: 7 pages, 6 figure

Localization and the effects of symmetries in the thermalization properties of one-dimensional quantum systems

We study how the proximity to an integrable point or to localization as one approaches the atomic limit, as well as the mixing of symmetries in the chaotic domain, may affect the onset of thermalization in finite one-dimensional systems. We consider systems of hard-core bosons at half-filling with nearest neighbor hopping and interaction, and next-nearest neighbor interaction. The latter breaks integrability and induces a ground-state superfluid to insulator transition. By full exact diagonalization, we study chaos indicators and few-body observables. We show that when different symmetry sectors are mixed, chaos indicators associated with the eigenvectors, contrary to those related to the eigenvalues, capture the onset of chaos. The results for the complexity of the eigenvectors and for the expectation values of few-body observables confirm the validity of the eigenstate thermalization hypothesis in the chaotic regime, and therefore the occurrence of thermalization. We also study the properties of the off-diagonal matrix elements of few-body observables in relation to the transition from integrability to chaos and from chaos to localization.Comment: 12 pages, 13 figures, as published (Fig.09 was corrected in this final version

Spectral fluctuations and 1/f noise in the order-chaos transition regime

Level fluctuations in quantum system have been used to characterize quantum chaos using random matrix models. Recently time series methods were used to relate level fluctuations to the classical dynamics in the regular and chaotic limit. In this we show that the spectrum of the system undergoing order to chaos transition displays a characteristic $f^{-\gamma}$ noise and $\gamma$ is correlated with the classical chaos in the system. We demonstrate this using a smooth potential and a time-dependent system modeled by Gaussian and circular ensembles respectively of random matrix theory. We show the effect of short periodic orbits on these fluctuation measures.Comment: 4 pages, 5 figures. Modified version. To appear in Phys. Rev. Let

Universality of Decoherence

We consider environment induced decoherence of quantum superpositions to mixtures in the limit in which that process is much faster than any competing one generated by the Hamiltonian $H_{\rm sys}$ of the isolated system. While the golden rule then does not apply we can discard $H_{\rm sys}$. By allowing for simultaneous couplings to different reservoirs, we reveal decoherence as a universal short-time phenomenon independent of the character of the system as well as the bath and of the basis the superimposed states are taken from. We discuss consequences for the classical behavior of the macroworld and quantum measurement: For the decoherence of superpositions of macroscopically distinct states the system Hamiltonian is always negligible.Comment: 4 revtex pages, no figure

Statistical properties of energy levels of chaotic systems: Wigner or non-Wigner

For systems whose classical dynamics is chaotic, it is generally believed that the local statistical properties of the quantum energy levels are well described by Random Matrix Theory. We present here two counterexamples - the hydrogen atom in a magnetic field and the quartic oscillator - which display nearest neighbor statistics strongly different from the usual Wigner distribution. We interpret the results with a simple model using a set of regular states coupled to a set of chaotic states modeled by a random matrix.Comment: 10 pages, Revtex 3.0 + 4 .ps figures tar-compressed using uufiles package, use csh to unpack (on Unix machine), to be published in Phys. Rev. Let

Quantum Chaotic Dynamics and Random Polynomials

We investigate the distribution of roots of polynomials of high degree with random coefficients which, among others, appear naturally in the context of "quantum chaotic dynamics". It is shown that under quite general conditions their roots tend to concentrate near the unit circle in the complex plane. In order to further increase this tendency, we study in detail the particular case of self-inversive random polynomials and show that for them a finite portion of all roots lies exactly on the unit circle. Correlation functions of these roots are also computed analytically, and compared to the correlations of eigenvalues of random matrices. The problem of ergodicity of chaotic wave-functions is also considered. For that purpose we introduce a family of random polynomials whose roots spread uniformly over phase space. While these results are consistent with random matrix theory predictions, they provide a new and different insight into the problem of quantum ergodicity. Special attention is devoted all over the paper to the role of symmetries in the distribution of roots of random polynomials.Comment: 33 pages, Latex, 6 Figures not included (a copy of them can be requested at [email protected]); to appear in Journal of Statistical Physic

Fano interference and cross-section fluctuations in molecular photodissociation

We derive an expression for the total photodissociation cross section of a molecule incorporating both indirect processes that proceed through excited resonances, and direct processes. We show that this cross section exhibits generalized Beutler-Fano line shapes in the limit of isolated resonances. Assuming that the closed system can be modeled by random matrix theory, we derive the statistical properties of the photodissociation cross section and find that they are significantly affected by the direct processes. We identify a unique signature of the direct processes in the cross-section distribution in the limit of isolated resonances.Comment: 4 pages, 4 figure

Multifractal eigenstates of quantum chaos and the Thue-Morse sequence

We analyze certain eigenstates of the quantum baker's map and demonstrate, using the Walsh-Hadamard transform, the emergence of the ubiquitous Thue-Morse sequence, a simple sequence that is at the border between quasi-periodicity and chaos, and hence is a good paradigm for quantum chaotic states. We show a family of states that are also simply related to Thue-Morse sequence, and are strongly scarred by short periodic orbits and their homoclinic excursions. We give approximate expressions for these states and provide evidence that these and other generic states are multifractal.Comment: Substantially modified from the original, worth a second download. To appear in Phys. Rev. E as a Rapid Communicatio