Universal spectral form factor for chaotic dynamics

We consider the semiclassical limit of the spectral form factor $K(\tau)$ of fully chaotic dynamics. Starting from the Gutzwiller type double sum over classical periodic orbits we set out to recover the universal behavior predicted by random-matrix theory, both for dynamics with and without time reversal invariance. For times smaller than half the Heisenberg time $T_H\propto \hbar^{-f+1}$, we extend the previously known $\tau$-expansion to include the cubic term. Beyond confirming random-matrix behavior of individual spectra, the virtue of that extension is that the ``diagrammatic rules'' come in sight which determine the families of orbit pairs responsible for all orders of the $\tau$-expansion.Comment: 4 pages, 1 figur

Heat kernel of integrable billiards in a magnetic field

We present analytical methods to calculate the magnetic response of non-interacting electrons constrained to a domain with boundaries and submitted to a uniform magnetic field. Two different methods of calculation are considered - one involving the large energy asymptotic expansion of the resolvent (Stewartson-Waechter method) is applicable to the case of separable systems, and another based on the small time asymptotic behaviour of the heat kernel (Balian-Bloch method). Both methods are in agreement with each other but differ from the result obtained previously by Robnik. Finally, the Balian-Bloch multiple scattering expansion is studied and the extension of our results to other geometries is discussed.Comment: 13 pages, Revte

Semi-classical spectrum of integrable systems in a magnetic field

The quantum dynamics of an electron in a uniform magnetic field is studied for geometries corresponding to integrable cases. We obtain the uniform asymptotic approximation of the WKB energies and wavefunctions for the semi-infinite plane and the disc. These analytical solutions are shown to be in excellent agreement with the numerical results obtained from the Schrodinger equations even for the lowest energy states. The classically exact notions of bulk and edge states are followed to their semi-classical limit, when the uniform approximation provides the connection between bulk and edge.Comment: 17 pages, Revtex, 6 figure

Characterising two-sided quantum correlations beyond entanglement via metric-adjusted f-correlations

We introduce an infinite family of quantifiers of quantum correlations beyond entanglement which vanish on both classical-quantum and quantum-classical states and are in one-to-one correspondence with the metric-adjusted skew informations. The `quantum $f-$correlations' are defined as the maximum metric-adjusted $f-$correlations between pairs of local observables with the same fixed equispaced spectrum. We show that these quantifiers are entanglement monotones when restricted to pure states of qubit-qudit systems. We also evaluate the quantum $f-$correlations in closed form for two-qubit systems and discuss their behaviour under local commutativity preserving channels. We finally provide a physical interpretation for the quantifier corresponding to the average of the Wigner-Yanase-Dyson skew informations.Comment: 20 pages, 1 figure. Published versio

Field Theory Approach to Quantum Interference in Chaotic Systems

We consider the spectral correlations of clean globally hyperbolic (chaotic) quantum systems. Field theoretical methods are applied to compute quantum corrections to the leading (`diagonal') contribution to the spectral form factor. Far-reaching structural parallels, as well as a number of differences, to recent semiclassical approaches to the problem are discussed.Comment: 18 pages, 4 figures, revised version, accepted for publication in J. Phys A (Math. Gen.

Effect of pitchfork bifurcations on the spectral statistics of Hamiltonian systems

We present a quantitative semiclassical treatment of the effects of bifurcations on the spectral rigidity and the spectral form factor of a Hamiltonian quantum system defined by two coupled quartic oscillators, which on the classical level exhibits mixed phase space dynamics. We show that the signature of a pitchfork bifurcation is two-fold: Beside the known effect of an enhanced periodic orbit contribution due to its peculiar $\hbar$-dependence at the bifurcation, we demonstrate that the orbit pair born {\em at} the bifurcation gives rise to distinct deviations from universality slightly {\em above} the bifurcation. This requires a semiclassical treatment beyond the so-called diagonal approximation. Our semiclassical predictions for both the coarse-grained density of states and the spectral rigidity, are in excellent agreement with corresponding quantum-mechanical results.Comment: LaTex, 25 pp., 14 Figures (26 *.eps files); final version 3, to be published in Journal of Physics

Quantum correlations and distinguishability of quantum states

A survey of various concepts in quantum information is given, with a main emphasis on the distinguishability of quantum states and quantum correlations. Covered topics include generalized and least square measurements, state discrimination, quantum relative entropies, the Bures distance on the set of quantum states, the quantum Fisher information, the quantum Chernoff bound, bipartite entanglement, the quantum discord, and geometrical measures of quantum correlations. The article is intended both for physicists interested not only by collections of results but also by the mathematical methods justifying them, and for mathematicians looking for an up-to-date introductory course on these subjects, which are mainly developed in the physics literature.Comment: Review article, 103 pages, to appear in J. Math. Phys. 55 (special issue: non-equilibrium statistical mechanics, 2014

Shot noise from action correlations

We consider universal shot noise in ballistic chaotic cavities from a semiclassical point of view and show that it is due to action correlations within certain groups of classical trajectories. Using quantum graphs as a model system we sum these trajectories analytically and find agreement with random-matrix theory. Unlike all action correlations which have been considered before, the correlations relevant for shot noise involve four trajectories and do not depend on the presence of any symmetry.Comment: 4 pages, 2 figures (a mistake in version 1 has been corrected

Derivation of some translation-invariant Lindblad equations for a quantum Brownian particle

We study the dynamics of a Brownian quantum particle hopping on an infinite lattice with a spin degree of freedom. This particle is coupled to free boson gases via a translation-invariant Hamiltonian which is linear in the creation and annihilation operators of the bosons. We derive the time evolution of the reduced density matrix of the particle in the van Hove limit in which we also rescale the hopping rate. This corresponds to a situation in which both the system-bath interactions and the hopping between neighboring sites are small and they are effective on the same time scale. The reduced evolution is given by a translation-invariant Lindblad master equation which is derived explicitly.Comment: 28 pages, 4 figures, minor revisio

Describing semigroups with defining relations of the form xy=yz xy and yx=zy and connections with knot theory

We introduce a knot semigroup as a cancellative semigroup whose defining relations are produced from crossings on a knot diagram in a way similar to the Wirtinger presentation of the knot group; to be more precise, a knot semigroup as we define it is closely related to such tools of knot theory as the twofold branched cyclic cover space of a knot and the involutory quandle of a knot. We describe knot semigroups of several standard classes of knot diagrams, including torus knots and torus links T(2, n) and twist knots. The description includes a solution of the word problem. To produce this description, we introduce alternating sum semigroups as certain naturally defined factor semigroups of free semigroups over cyclic groups. We formulate several conjectures for future research