Classical versus Quantum Time Evolution of Densities at Limited Phase-Space Resolution

We study the interrelations between the classical (Frobenius-Perron) and the quantum (Husimi) propagator for phase-space (quasi-)probability densities in a Hamiltonian system displaying a mix of regular and chaotic behavior. We focus on common resonances of these operators which we determine by blurring phase-space resolution. We demonstrate that classical and quantum time evolution look alike if observed with a resolution much coarser than a Planck cell and explain how this similarity arises for the propagators as well as their spectra. The indistinguishability of blurred quantum and classical evolution implies that classical resonances can conveniently be determined from quantum mechanics and in turn become effective for decay rates of quantum correlations.Comment: 10 pages, 3 figure

Fidelity Decay as an Efficient Indicator of Quantum Chaos

Recent work has connected the type of fidelity decay in perturbed quantum models to the presence of chaos in the associated classical models. We demonstrate that a system's rate of fidelity decay under repeated perturbations may be measured efficiently on a quantum information processor, and analyze the conditions under which this indicator is a reliable probe of quantum chaos and related statistical properties of the unperturbed system. The type and rate of the decay are not dependent on the eigenvalue statistics of the unperturbed system, but depend on the system's eigenvector statistics in the eigenbasis of the perturbation operator. For random eigenvector statistics the decay is exponential with a rate fixed precisely by the variance of the perturbation's energy spectrum. Hence, even classically regular models can exhibit an exponential fidelity decay under generic quantum perturbations. These results clarify which perturbations can distinguish classically regular and chaotic quantum systems.Comment: 4 pages, 3 figures, LaTeX; published version (revised introduction and discussion

Entanglement in the classical limit: quantum correlations from classical probabilities

We investigate entanglement for a composite closed system endowed with a scaling property allowing to keep the dynamics invariant while the effective Planck constant hbar_eff of the system is varied. Entanglement increases as hbar_eff goes to 0. Moreover for sufficiently low hbar_eff the evolution of the quantum correlations, encapsulated for example in the quantum discord, can be obtained from the mutual information of the corresponding \emph{classical} system. We show this behavior is due to the local suppression of path interferences in the interaction that generates the entanglement. This behavior should be generic for quantum systems in the classical limit.Comment: 10 pages 3 figure

Periodic-Orbit Theory of Level Correlations

We present a semiclassical explanation of the so-called Bohigas-Giannoni-Schmit conjecture which asserts universality of spectral fluctuations in chaotic dynamics. We work with a generating function whose semiclassical limit is determined by quadruplets of sets of periodic orbits. The asymptotic expansions of both the non-oscillatory and the oscillatory part of the universal spectral correlator are obtained. Borel summation of the series reproduces the exact correlator of random-matrix theory.Comment: 4 pages, 1 figure (+ web-only appendix with 2 pages, 1 figure

Matrix Element Distribution as a Signature of Entanglement Generation

We explore connections between an operator's matrix element distribution and its entanglement generation. Operators with matrix element distributions similar to those of random matrices generate states of high multi-partite entanglement. This occurs even when other statistical properties of the operators do not conincide with random matrices. Similarly, operators with some statistical properties of random matrices may not exhibit random matrix element distributions and will not produce states with high levels of multi-partite entanglement. Finally, we show that operators with similar matrix element distributions generate similar amounts of entanglement.Comment: 7 pages, 6 figures, to be published PRA, partially supersedes quant-ph/0405053, expands quant-ph/050211

Universal features of spin transport and breaking of unitary symmetries

When time-reversal symmetry is broken, quantum coherent systems with and without spin rotational symmetry exhibit the same universal behavior in their electric transport properties. We show that spin transport discriminates between these two cases. In systems with large charge conductance, spin transport is essentially insensitive to the breaking of time-reversal symmetry, while in the opposite limit of a single exit transport channel, spin currents vanish identically in the presence of time-reversal symmetry but can be turned on by breaking it with an orbital magnetic field

Finite-difference distributions for the Ginibre ensemble

The Ginibre ensemble of complex random matrices is studied. The complex valued random variable of second difference of complex energy levels is defined. For the N=3 dimensional ensemble are calculated distributions of second difference, of real and imaginary parts of second difference, as well as of its radius and of its argument (angle). For the generic N-dimensional Ginibre ensemble an exact analytical formula for second difference's distribution is derived. The comparison with real valued random variable of second difference of adjacent real valued energy levels for Gaussian orthogonal, unitary, and symplectic, ensemble of random matrices as well as for Poisson ensemble is provided.Comment: 8 pages, a number of small changes in the tex

Semiclassical spectral correlator in quasi one-dimensional systems

We investigate the spectral statistics of chaotic quasi one dimensional systems such as long wires. To do so we represent the spectral correlation function $R(\epsilon)$ through derivatives of a generating function and semiclassically approximate the latter in terms of periodic orbits. In contrast to previous work we obtain both non-oscillatory and oscillatory contributions to the correlation function. Both types of contributions are evaluated to leading order in $1/\epsilon$ for systems with and without time-reversal invariance. Our results agree with expressions from the theory of disordered systems.Comment: 10 pages, no figure

Fluctuations and Ergodicity of the Form Factor of Quantum Propagators and Random Unitary Matrices

We consider the spectral form factor of random unitary matrices as well as of Floquet matrices of kicked tops. For a typical matrix the time dependence of the form factor looks erratic; only after a local time average over a suitably large time window does a systematic time dependence become manifest. For matrices drawn from the circular unitary ensemble we prove ergodicity: In the limits of large matrix dimension and large time window the local time average has vanishingly small ensemble fluctuations and may be identified with the ensemble average. By numerically diagonalizing Floquet matrices of kicked tops with a globally chaotic classical limit we find the same ergodicity. As a byproduct we find that the traces of random matrices from the circular ensembles behave very much like independent Gaussian random numbers. Again, Floquet matrices of chaotic tops share that universal behavior. It becomes clear that the form factor of chaotic dynamical systems can be fully faithful to random-matrix theory, not only in its locally time-averaged systematic time dependence but also in its fluctuations.Comment: 12 pages, RevTEX, 4 figures in eps forma

Universal spectral statistics in quantum graphs

We prove that the spectrum of an individual chaotic quantum graph shows universal spectral correlations, as predicted by random--matrix theory. The stability of these correlations with regard to non--universal corrections is analyzed in terms of the linear operator governing the classical dynamics on the graph.Comment: 4 pages, reftex, 1 figure, revised version to be published in PR