Canonical Typicality under General Quantum Channels.

With the control of increasingly complex quantum systems, the relevant degrees of freedom we are interested in may not be those traditionally addressed by statistical quantum mechanics. Here, we employ quantum channels to define generalized subsystems, capturing the pertinent degrees of freedom, and obtain their associated canonical state. We show that the generalized subsystem description from almost any microscopic pure state of the whole system will behave similarly to its corresponding canonical state.

Importance sampling with imperfect cloning for the computation of generalized Lyapunov exponents.

We revisit the numerical calculation of generalized Lyapunov exponents, L(q), in deterministic dynamical systems. The standard method consists of adding noise to the dynamics in order to use importance sampling algorithms. Then L(q) is obtained by taking the limit noise-amplitude → 0 after the calculation.

From synchronous to one-time delayed dynamics in coupled maps.

We study the completely synchronized states (CSSs) of a system of coupled logistic maps as a function of three parameters: interaction strength (ɛ), range of the interaction (α), that can vary from first neighbors to global coupling, and a parameter (β) that allows one to scan continuously from nondelayed to one-time delayed dynamics. In the α-ɛ plane we identify periodic orbits, limit cycles, and chaotic trajectories, and describe how these structures change with delay. These features can be explained by studying the bifurcation diagrams of a two-dimensional nondelayed map.

Generalized Lyapunov exponents of the random harmonic oscillator: cumulant expansion approach.

The cumulant expansion is used to estimate generalized Lyapunov exponents of the random-frequency harmonic oscillator. Three stochastic processes are considered: Gaussian white noise, Ornstein-Uhlenbeck, and Poisson shot noise. In some cases, nontrivial numerical difficulties arise.

Transfer matrices and circuit representation for the semiclassical traces of the baker map.

Because of a formal equivalence with the partition function of an Ising chain, the semiclassical traces of the quantum baker map can be calculated using the transfer-matrix method. We analyze the transfer matrices associated with the baker map and the symmetry-reflected baker map (the latter happens to be unitary but the former is not). In both cases simple quantum-circuit representations are obtained, which exhibit the typical structure of qubit quantum bakers.

Semiclassical description of wave packet revival.

We test the ability of semiclassical theory to describe quantitatively the revival of quantum wave packets-a long time phenomena-in the one dimensional quartic oscillator (a Kerr type Hamiltonian). Two semiclassical theories are considered: time-dependent WKB and Van Vleck propagation. We show that both approaches describe with impressive accuracy the autocorrelation function and wave function up to times longer than the revival time.

Semiclassical propagation of Gaussian wave packets.

We analyze the semiclassical evolution of Gaussian wave packets in chaotic systems. We show that after some short time a Gaussian wave packet becomes a primitive WKB state. From then on, the state can be propagated using the standard time-dependent WKB scheme.

Classical-quantum correspondence for the scattering dwell time.

Using results from the theory of dynamical systems, we derive a general expression for the classical average scattering dwell time tau . Remarkably, tau depends only on a ratio of phase space volumes. We further show that, for a wide class of systems, the average classical dwell time is not in correspondence with the energy average of the quantum Wigner time delay.

Random matrix ensembles from nonextensive entropy.

The classical Gaussian ensembles of random matrices can be constructed by maximizing Boltzmann-Gibbs-Shannon's entropy, S(BGS) = -integral dH[P(H)]ln[P(H)], with suitable constraints. Here, we construct and analyze random-matrix ensembles arising from the generalized entropy S(q) = [1- integral dH [P(H)](q)] /(q-1) (thus, S1 = S(BGS) ). The resulting ensembles are characterized by a parameter q measuring the degree of nonextensivity of the entropic form.

Lyapunov exponent of many-particle systems: testing the stochastic approach.

The stochastic approach to the determination of the largest Lyapunov exponent of a many-particle system is tested in the so-called mean-field XY Hamiltonians. In weakly chaotic regimes, the stochastic approach relates the Lyapunov exponent to a few statistical properties of the Hessian matrix of the interaction, which can be calculated as suitable thermal averages. We have verified that there is a satisfactory quantitative agreement between theory and simulations in the disordered phases of the XY models, either with attractive or repulsive interactions.

Theoretical estimates for the largest Lyapunov exponent of many-particle systems.

The largest Lyapunov exponent of an ergodic Hamiltonian system is the rate of exponential growth of the norm of a typical vector in the tangent space. For an N-particle Hamiltonian system with a smooth Hamiltonian of the type p(2)+V(q), the evolution of tangent vectors is governed by the Hessian matrix V of the potential. Ergodicity implies that the Lyapunov exponent is independent of initial conditions on the energy shell, which can then be chosen randomly according to the microcanonical distribution.

Scaling laws for the largest Lyapunov exponent in long-range systems: A random matrix approach.

We investigate the laws that rule the behavior of the largest Lyapunov exponent (LLE) in many particle systems with long-range interactions. We consider as a representative system the so-called Hamiltonian alpha-XY model where the adjustable parameter alpha controls the range of the interactions of N ferromagnetic spins in a lattice of dimension d. In previous work the dependence of the LLE with the system size N, for sufficiently high energies, was established through numerical simulations.