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 resonances and scar functions 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.Comment: 21 pages, 10 figure

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 cross-over 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 cross over from $\Gamma$ to Lyapunov decay. We find that, challenging the analytic interpretation, these conjetures are valid even beyond the expected range.Comment: Significantly revised version. To appear in Physical Review E Rapid Communication

Semiclassical quantization of highly excited scar states

The semiclassical quantization of Hamiltonian systems with classically chaotic dynamics is restricted to low excited states, close to the ground state, because the number of required periodic orbits grows exponentially with energy. Nevertheless, here we demonstrate that it is possible to find eigenenergies of highly excited states scarred by a short periodic orbit. Specifically, by using 18146 homoclinic orbits (HO)s of the shortest periodic orbit of the hyperbola billiard, we find eigenenergies of the strongest scars over a range which includes 630 even eigenfunctions. The analysis of data reveals that the used semiclassical formula presents two regimes. First, when all HOs with excursion time smaller than the Heisenberg time tH are included, the error is around 3.3% of the mean level spacing. Second, in the energy region defined by $\tilde{t}/ t_H > 0.13$ , where $\tilde{t}$ is the maximum excursion time included in the calculation, the error is around 15% of the mean level spacing

The semiclassical limit of scar intensities

By using a simple statistical model we find the distribution of scar intensities surviving the semiclassical limit. The obtained distribution is verified in a wide energy range of the quantum Bunimovich stadium billiard

Semiclassical propagation up to the Heisenberg time

By using a quantum Hamiltonian system with classically chaotic dynamics, we demonstrate that it is possible to propagate waves, at a semiclassical level, for extremely long times of the order of the Heisenberg time. We achieve this unexpected result with a new formula that evaluates the autocorrelation function of a quantum state living in the neighborhood of a short periodic orbit, the so-called resonance, in terms of the set of homoclinic orbits; this set is given by the intersection of the stable and unstable manifolds of the periodic orbit. Here we study the manifolds of the shortest periodic orbit of the hyperbola billiard (a chaotic Hamiltonian system), finding a surprisingly simple tree structure. Then, we compute a complete set consisting of the first 18 146 homoclinic orbits, and by using this data we analyze the convergence of the new formula. Finally, we compare the quantum and semiclassical autocorrelation of resonances up to the Heisenberg time, obtaining a relative error O(ℏ) in correspondence with semiclassical predictions