Numerical and experimental tests of the semiclassical

The universal part of the correlation function has been tested either directly or by means of its statistical implications in several works [37]. In the

meso-scopic realm, however, such experiments are very difficult to realize and the available results typically required an extra spatial average, washing away the non-universal contributions we want to study.

Another option is to use an exact mapping between the Schr¨odinger equa-tion for billiard systems and the Helmotz equaequa-tion in the case of microwave cavities [50]. In the microwave case the measurement of high-lying eigen-states can be done with great accuracy and this sort of experiments have become very popular to check the predictions of theoretical models. The experimental results show an impressive agreement with the Bessel-like cor-relation function but, as mentioned before, this is to be expected when spatial averages are taken, as it is almost invariably the case.

Numerical tests of the theory at the universal level are also available, confirming the correctness of the Bessel result when both energy and spatial averages are used [51]. To our knowledge the only systematic study of the structure of the correlation function beyond the universal regime when only spectral averages are considered is presented here [49].

The specific system we study is the so called Africa billiard, obtained by a conformal deformation of the unit circle [52]. The reason to use such billiard is that it is easy to handle numerically and it can be modeled experimentally [50]. The exact correlation functions are obtained by explicit calculation of the eigenfunctions up to the 300-th energy level and use of the definition Eq. (3.1) with an energy window satisfying the semiclasical requirements and such thatτW is of the order of the traveling time trough the system (so-called Thouless time). Due to the damping function Γ, this allow us to keep in the semiclassical two-point correlation function only classical paths with at most one bounce with the boundary.

The semiclassical calculations, when naively applied, gave very bad results except in a region very close to the boundary. After examination, it turned out that this system has a very problematic optical structure, namely, much of the billiard’s domain is affected by effects beyond the ray-description which is the optical analogue of the semiclassical approximation. Such deviations from the ray-picture are called in general diffraction effects [53], and almost all of them are present in our system (penumbra effects, caustics, focal points, etc).

To overcome this problem, we include diffraction effects in the semiclas-sical expression for the two-point correlation function with paths up to one bounce to the boundary. This is achieved by calculating the full diffraction integral in the vicinity of each classical trajectory involved in Eq. (3.6). This

is done by means of the expression [54] wheresparameterizes a small segment around the bouncing point of theγi,j -th classical trajectory joining~ri with~rj after just one reflection, and n(s) is the normal at point ~r(s). We mention that uniform approximations can be done in order to render this expression into the usual semiclassical structure sum over paths of some classical prefactor times cosine of some action, but for our purposes this is not necessary.

In figure (3.1) we present our findings for the two-point correlation func-tion R^sc+dif(~r1, ~r2). Since this is a function of four variables, we present our results by keeping ~r1 fix , while moving ~r2 along the line indicated in the insets. The symbols are the exact numerical results, the dashed lines are the predictions of the isotropic contribution to the correlation function com-ing from the direct classical path (givcom-ing both the isotropic Random Wave and Nonlinear Sigma Model results). The continuous line is the sum of the isotropic and first non-isotropic contribution, the later calculated as a sum over all paths with one bounce including diffraction effects.

The comparison with the universal Bessel correlation shows how system dependent effects are essential, particularly close to the boundaries. This is to be expected since the isotropic result obviously misses any non-universal effects. On the other hand, the robustness of the Bessel result is indeed remarkable. We note that the non-zero curvature of the boundary makes the application of the non-isotropic Random Wave Models impossible.

The most spectacular evidence of non-universal effects beyond the Ran-dom Wave and Nonlinear Sigma Models is the behavior of the average in-tensity R(~r, ~r). The universal prediction in this approaches gives a constant value 1/A with A = π in our case, while the semiclassical plus diffraction theory predicts a much richer structure.

The results for the average intensity are presented in figure (3.2) for the position~rmoving along the directions indicated in the inset. The horizontal line without structure is the isotropic result (which is the prediction of the isotropic Random Wave and the Sigma Models), the symbols are the exact quantum mechanical calculations and the continuous line the semiclassical prediction using Eq. (3.10) including diffraction effects.

Figure 3.1: Two-point correlation functionR(~r1, ~r2) for~r2 pointing along the lines indicated in the Africa billiard (inset). The symbols mark numerical quantum results for R, Eq. (3.1), the thin lines depict the prediction em-ploying Eq. (3.10) where the Green function is approximated by a sum over paths, including diffraction effects, with at most one reflection at the bound-ary. The dashed lines shows the result from the isotropic Random Wave and Nonlinear Sigma Model (3.9).

Figure 3.2: Average intensity R(~r, ~r) for~r pointing along the lines indicated in the Africa billiard (inset). The symbols mark numerical quantum results forR(~r, ~r), Eq. (3.1), the thin lines depict the prediction employing Eq. (3.10) where the Green function is approximated by a sum over paths, including diffraction effects, with at most one reflection at the boundary. The hori-zontal line shows the isotropic Random Wave and Nonlinear Sigma Model results (1/π).

Even when the agreement is not perfect our results clearly indicate the im-portance of the non-universal effects beyond the isotropic predictions, which are very well described by the semiclassical approach.

On the other hand, since we have systematically used the Gutzwiller version of the semiclassical Green function, and we know this object has serious problems of analyticity, we expect that some effects are not correctly taken into account by the semiclassical two-point correlation presented here, even when supplemented with diffraction effects. This is indeed the case as we discuss now.