E. Vergini

Signatures of homoclinic motion in quantum chaos.

Homoclinic motion plays a key role in the organization of classical chaos in Hamiltonian systems. In this Letter, we show that it also imprints a clear signature in the corresponding quantum spectra. By numerically studying the fluctuations of the widths of wave functions localized along periodic orbits we reveal the existence of an oscillatory behavior that is explained solely in terms of the primary homoclinic motion. Furthermore, our results indicate that it survives the semiclassical limit.

Sensitivity to perturbations in a quantum chaotic billiard.

The Loschmidt echo (LE) measures the ability of a system to return to the initial state after a forward quantum evolution followed by a backward perturbed one. It has been conjectured that the echo of a classically chaotic system decays exponentially, with a decay rate given by the minimum between the width Gamma of the local density of states and the Lyapunov exponent. As the perturbation strength is increased one obtains a crossover between both regimes. These predictions are based on situations where the Fermi golden rule (FGR) is valid. By considering a paradigmatic fully chaotic system, the Bunimovich stadium billiard, with a perturbation in a regime for which the FGR manifestly does not work, we find a crossover from Gamma to Lyapunov decay. We find that, challenging the analytic interpretation, these conjectures are valid even beyond the expected range.

Semiclassical theory of short periodic orbits in quantum chaos

We have developed a semiclassical theory of short periodic orbits to obtain all the quantum information of a bounded chaotic Hamiltonian system. If T1 is the period of the shortest periodic orbit, T2 the period of the next one and so on, the number Npo of periodic orbits required in the calculation is such that T1 + + TNpo TH, with TH the Heisenberg time. As a result Npo hTH / ln (hTH), where h is the topological entropy. For methods related to the trace formula Npo exp (hTH) / (hTH).

Universal response of quantum systems with chaotic dynamics.

The prediction of the response of a closed system to external perturbations is one of the central problems in quantum mechanics, and in this respect, the local density of states (LDOS) provides an in-depth description of such a response. The LDOS is the distribution of the overlaps squared connecting the set of eigenfunctions with the perturbed one. Here, we show that in the case of closed systems with classically chaotic dynamics, the LDOS is a Breit-Wigner distribution under very general perturbations of arbitrary high intensity. Consequently, we derive a semiclassical expression for the width of the LDOS which is shown to be very accurate for paradigmatic systems of quantum chaos. This Letter demonstrates the universal response of quantum systems with classically chaotic dynamics.

Localization properties of groups of eigenstates in chaotic systems

In this paper we study in detail the localized wave functions defined in Phys. Rev. Lett. 76, 1613 (1994), in connection with the scarring effect of unstable periodic orbits in highly chaotic Hamiltonian system. These functions appear highly localized not only along periodic orbits but also on the associated manifolds. Moreover, they show in phase space the hyperbolic structure in the vicinity of the orbit, something that translates in configuration space into the structure induced by the corresponding self-focal points. On the other hand, the- quantum dynamics of these functions are also studied. Our results indicate that the probability density first evolves along the unstable manifold emanating from the periodic orbit, and localizes temporarily afterwards on only a few, short related periodic orbits. We believe that this type of study can provide some keys to disentangle the complexity associated with the quantum mechanics of these kind of systems, which permits the construction of a simple explanation in terms of the dynamics of a few classical structures.

Semiclassical construction of resonances with hyperbolic structure: the scar function

The formalism of resonances in quantum chaos is improved by using conveniently defined creation-annihilation operators. With these operators at hand, we are able to construct transverse excited resonances at a given Bohr-quantized energy. Then, by requiring minimum energy dispersion we obtain solutions in terms of even or odd transverse excitations. These wavefunctions, which are constructed in the vicinity of a periodic orbit with maximum energy localization, provide a precise definition of a scar function. These scar functions acquire, in the semiclassical limit, the hyperbolic structure characteristic of unstable periodic orbits.

Semiclassical quantization with short periodic orbits

We apply a recently developed semiclassical theory of short periodic orbits to the stadium billiard. We give explicit expressions for the resonances of periodic orbits and for the application of the semiclassical Hamiltonian operator to them. Then, by using the three shortest periodic orbits and two more living in the bouncing-ball region, we obtain the first 25 odd-odd eigenfunctions with surprising accuracy.

Quantum localization through interference on homoclinic and heteroclinic circuits

Localization effects due to scarring constitute one of the clearest indications of the relevance of interference in the transport of quantum probability density along quantized closed circuits in phase space. The corresponding path can be obvious, such as the scarring periodic orbit (PO) itself which produces time recurrences at multiples of the period. However, there are others more elaborate which only close asymptotically, for example, those associated with homoclinic and heteroclinic orbits. In this paper, we demonstrate that these circuits are also able to produce recurrences but at (semiclassically) longer times, of the order of the Ehrenfest time. The most striking manifestation of this phenomenon is the accumulation of quantum probability density along the corresponding circuits. The discussion is illustrated with an example corresponding to a typical PO of the quartic two-dimensional oscillator.

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.

Scar functions in the Bunimovich stadium billiard

In the context of the semiclassical theory of short periodic orbits, scar functions play a crucial role. These wavefunctions live in the neighbourhood of the trajectories, resembling the hyperbolic structure of the phase space in their immediate vicinity. This property makes them extremely suitable for investigating chaotic eigenfunctions. On the other hand, for all practical purposes reductions to Poincare sections become essential. Here we give a detailed explanation of resonance and scar function construction in the Bunimovich stadium billiard and the corresponding reduction to the boundary. Moreover, we develop a method that takes into account the departure of the unstable and stable manifolds from the linear regime. This new feature extends the validity of the expressions.