Avoided-level-crossing statistics in open chaotic billiards

We investigate a two-level model with a large number of open decay channels in order to describe avoided level crossing statistics in open chaotic billiards. This model allows us to describe the fundamental changes in the probability distribution of the avoided level crossings compared with the closed case. Explicit expressions are derived for systems with preserved and broken time-reversal symmetry. We find that the decay process induces a modification at small spacings of the probability distribution of the avoided level crossings due to an attraction of the resonances. The theoretical predictions are in complete agreement with the recent experimental results of Dietz et al. [Phys. Rev. E 73, 035201 (2006)].

Figure 1

(Color online) Probability distributions of nonzero avoided level crossings for GOE (left side) and for GUE (right side). The number of open channels is $M=1$, 3, 5, 10, 20, and the coupling strength $\lambda ={\sigma }^2/\left(2\Delta \right)$ equals $\lambda =0.250$, 0.083, 0.05, 0.025, 0.013 from top to bottom. The histograms show the numerical simulations, the analytic distributions are shown as straight lines. The dashed curves result from a fit of Eq. (11) with $\lambda$ as a parameter to the numerical distributions, resulting in the effective coupling strengths ${\lambda }_{eff}=0.020$, 0.038, 0.036, 0.028, 0.016. For all curves, $\alpha$ is chosen such that $⟨c^{\prime }⟩=1$.

Figure 2

(Color online) In boxes the experimental distribution of avoided crossings [23]. The continuous line shows the analytical prediction with $\lambda =0.058$ (obtained through a least-square procedure). The vertical bars represent the numerically obtained distribution with $M=3$ and $\lambda =0.02$ [23]. For all curves, the average is chosen such that $⟨c^{\prime }⟩=1$.