Numerical verification of Percival's conjecture in a quantum billiard

In order to verify Percival's conjecture [J. Phys. B 6,L229 (1973)] we study a planar billiard in its classical and quantum versions. We provide an evaluation of the nearest-neighbor level-spacing distribution for the Cassini oval billiard, taking into account relations with classical results. The statistical behavior of integrable and ergodic systems has been extensively confirmed numerically, but that is not the case for the transition between these two extremes. Our system's classical dynamics undergoes a transition from integrability to chaos by varying a shape parameter. This feature allows us to investigate the spectral fluctuations, comparing numerical results with semiclassical predictions founded on Percival's conjecture. We obtain good $global$ agreement with those predictions, in clear contrast with similar comparisons for other systems found in the literature. The structure of some eigenfunctions, displayed in the quantum Poincar\'e section, provides a clear explanation of the conjecture.Comment: 8 pages, 9 figures, to appear in Physical Review E, vol. 57, issue 5 (01 May 1998

Scarring in open quantum systems

We study scarring phenomena in open quantum systems. We show numerical evidence that individual resonance eigenstates of an open quantum system present localization around unstable short periodic orbits in a similar way as their closed counterparts. The structure of eigenfunctions around these classical objects is not destroyed by the opening. This is exposed in a paradigmatic system of quantum chaos, the cat map.Comment: 4 pages, 4 figure

Short periodic orbits theory for partially open quantum maps

We extend the semiclassical theory of short periodic orbits [Phys. Rev. E {\bf 80}, 035202(R) (2009)] to partially open quantum maps. They correspond to classical maps where the trajectories are partially bounced back due to a finite reflectivity $R$. These maps are representative of a class that has many experimental applications. The open scar functions are conveniently redefined, providing a suitable tool for the investigation of these kind of systems. Our theory is applied to the paradigmatic partially open tribaker map. We find that the set of periodic orbits that belong to the classical repeller of the open map ($R=0$) are able to support the set of long-lived resonances of the partially open quantum map in a perturbative regime. By including the most relevant trajectories outside of this set, the validity of the approximation is extended to a broad range of $R$ values. Finally, we identify the details of the transition from qualitatively open to qualitatively closed behaviour, providing an explanation in terms of short periodic orbits.Comment: 6 pages, 4 figure

Localization of resonance eigenfunctions on quantum repellers

We introduce a new phase space representation for open quantum systems. This is a very powerful tool to help advance in the study of the morphology of their eigenstates. We apply it to two different versions of a paradigmatic model, the baker map. This allows to show that the long-lived resonances are strongly scarred along the shortest periodic orbits that belong to the classical repeller. Moreover, the shape of the short-lived eigenstates is also analyzed. Finally, we apply an antiunitary symmetry measure to the resonances that permits to quantify their localization on the repeller.Comment: 4 pages, 4 figure

OTOC, complexity and entropy in bi-partite systems

There is a remarkable interest in the study of Out-of-time ordered correlators (OTOCs) that goes from many body theory and high energy physics to quantum chaos. In this latter case there is a special focus on the comparison with the traditional measures of quantum complexity such as the spectral statistics, for example. The exponential growth has been verified for many paradigmatic maps and systems. But less is known for multi-partite cases. On the other hand the recently introduced Wigner separability entropy (WSE) and its classical counterpart (CSE) provide with a complexity measure that treats equally quantum and classical distributions in phase space. We have compared the behavior of these measures in a system consisting of two coupled and perturbed cat maps with different dynamics: double hyperbolic (HH), double elliptic (EE) and mixed (HE). In all cases, we have found that the OTOCs and the WSE have essentially the same behavior, providing with a complete characterization in generic bi-partite systems and at the same time revealing them as very good measures of quantum complexity for phase space distributions. Moreover, we establish a relation between both quantities by means of a recently proven theorem linking the second Renyi entropy and OTOCs.Comment: 6 pages, 5 figure

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.Fil: Bergamasco, Pablo D.. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Física; ArgentinaFil: Carlo, Gabriel Gustavo. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Física; ArgentinaFil: Rivas, Alejandro Mariano Fidel. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Física; Argentin

Entanglement Across a Transition to Quantum Chaos

We study the relation between entanglement and quantum chaos in one- and two-dimensional spin-1/2 lattice models, which exhibit mixing of the noninteracting eigenfunctions and transition from integrability to quantum chaos. Contrary to what occurs in a quantum phase transition, the onset of quantum chaos is not a property of the ground state but take place for any typical many-spin quantum state. We study bipartite and pairwise entanglement measures, namely the reduced Von Neumann entropy and the concurrence, and discuss quantum entanglement sharing. Our results suggest that the behavior of the entanglement is related to the mixing of the eigenfunctions rather than to the transition to chaos.Comment: 14 pages, 14 figure