Alejandro M. F. Rivas

The short periodic orbit approach for the quantum cat maps

The short periodic orbit approach is adapted for the quantum cat maps. The main objective is to explain, in a simple abstract model, the most relevant characteristics of this method which was originally developed for Hamiltonian fluxes. In particular, we describe a semiclassical Hamiltonian formulation to evaluate eigenphases and eigenstates of quantum cat maps. The main advantage of this formulation is that each eigenstate is described in terms of a small number, N/ln N, of short periodic orbits, with N the dimension of the Hilbert space. Moreover, matrix elements can be obtained semiclassically with high accuracy in terms of a very small number, of the order of ln2N, of homoclinic and heteroclinic orbits. From the computational point of view, this approach reduces the size of matrices used to the order N/ln N.

Correspondence behavior of classical and quantum dissipative directed transport via thermal noise.

We systematically study several classical-quantum correspondence properties of the dissipative modified kicked rotator, a paradigmatic ratchet model. We explore the behavior of the asymptotic currents for finite ℏ_{eff} values in a wide range of the parameter space. We find that the correspondence between the classical currents with thermal noise providing fluctuations of size ℏ_{eff} and the quantum ones without it is very good in general with the exception of specific regions. We systematically consider the spectra of the corresponding classical Perron-Frobenius operators and quantum superoperators. By means of an average distance between the classical and quantum sets of eigenvalues we find that the correspondence is unexpectedly quite uniform. This apparent contradiction is solved with the help of the Weyl-Wigner distributions of the equilibrium eigenvectors, which reveal the key role of quantum effects by showing surviving coherences in the asymptotic states.

Semiclassical scar functions in phase space

We develop a semiclassical approximation for the scar function in the Weyl?Wigner representation in the neighborhood of a classically unstable periodic orbit of chaotic two-dimensional systems. The prediction of hyperbolic fringes, asymptotic to the stable and unstable manifolds, is verified computationally for a (linear) cat map, after the theory is adapted to a discrete phase space appropriate to a quantized torus. Characteristic fringe patterns can be distinguished even for quasi-energies where the fixed point is not Bohr-quantized. Also the patterns are highly localized in the neighborhood of the periodic orbit and along its stable and unstable manifolds without any long distance patterns that appear for the case of the spectral Wigner function.

Quantum and classical complexity in coupled maps

We study a generic and paradigmatic two-degrees-of-freedom system consisting of two coupled perturbed cat maps with different types of dynamics. The Wigner separability entropy (WSE)-equivalent to the operator space entanglement entropy-and the classical separability entropy (CSE) are used as measures of complexity. For the case where both degrees of freedom are hyperbolic, the maps are classically ergodic and the WSE and the CSE behave similarly, growing to higher values than in the doubly elliptic case. However, when one map is elliptic and the other hyperbolic, the WSE reaches the same asymptotic value than that of the doubly hyperbolic case but at a much slower rate. The CSE only follows the WSE for a few map steps, revealing that classical dynamical features are not enough to explain complexity growth.

Signatures of classical structures in the leading eigenstates of quantum dissipative systems

By analyzing a paradigmatic example of the theory of dissipative systems-the classical and quantum dissipative standard map-we are able to explain the main features of the decay to the quantum equilibrium state. The classical isoperiodic stable structures typically present in the parameter space of these kinds of systems play a fundamental role. In fact, we have found that the period of stable structures that are near in this space determines the phase of the leading eigenstates of the corresponding quantum superoperator. Moreover, the eigenvectors show a strong localization on the corresponding periodic orbits (limit cycles). We show that this sort of scarring phenomenon (an established property of Hamiltonian and projectively open systems) is present in the dissipative case and it is of extreme simplicity.

Classical to quantum correspondence in dissipative directed transport.

We compare the quantum and classical properties of the (quantum) isoperiodic stable structures [(Q)ISSs], which organize the parameter space of a paradigmatic dissipative ratchet model, i.e., the dissipative modified kicked rotator. We study the spectral behavior of the corresponding classical Perron-Frobenius operators with thermal noise and the quantum superoperators without it for small ℏ(eff) values. We find a remarkable similarity between the classical and quantum spectra. This finding significantly extends previous results-obtained for the mean currents and asymptotic distributions only-and, on the other hand, unveils a classical to quantum correspondence mechanism where the classical noise is qualitatively different from the quantum one. This is crucial not only for simple attractors but also for chaotic ones, where just analyzing the asymptotic distribution is revealed as insufficient. Moreover, we provide with a detailed characterization of relevant eigenvectors by means of the corresponding Weyl-Wigner distributions, in order to better identify similarities and differences. Finally, this model being generic, it allows us to conjecture that this classical to quantum correspondence mechanism is a universal feature of dissipative systems.

The Weyl law for contractive maps

We find an empirical Weyl law followed by the eigenvalues of contractive maps. An important property is that it is mainly insensitive to the dimension of the corresponding invariant classical set, the strange attractor. The usual explanation for the fractal Weyl law emergence in scattering systems (i.e., having a projective opening) is based on the classical phase space distributions evolved up to the quantum to classical correspondence (Ehrenfest) time. In the contractive case this reasoning fails to describe it. Instead, we conjecture that the support for this behavior is essentially given by the strong non-orthogonality of the eigenvectors of the contractive superoperator. We test the validity of the Weyl law and this conjecture on two paradigmatic systems, the dissipative baker and kicked top maps.

Semiclassical matrix elements for a chaotic propagator in the scar function basis

A semiclassical approximation for the matrix elements of a quantum chaotic propagator in the scar function basis has been derived. The obtained expression is solely expressed in terms of canonical invariant objects. For our purpose, we have used the recently developed, semiclassical matrix elements of the propagator in coherent states, together with the linearization of the flux in the neighborhood of a classically unstable periodic orbit of chaotic two-dimensional systems. The expression derived here is successfully verified to be exact for a (linear) cat map, after the theory is adapted to a discrete phase space appropriate to a quantized torus.