Resonance-assisted tunneling in deformed optical microdisks with a mixed phase space

The lifetimes of optical modes in whispering-gallery cavities depend crucially on the underlying classical ray dynamics, and they may be spoiled by the presence of classical nonlinear resonances due to resonance-assisted tunneling. Here we present an intuitive semiclassical picture that allows for an accurate prediction of decay rates of optical modes in systems with a mixed phase space. We also extend the perturbative description from near-integrable systems to systems with a mixed phase space, and we find equally good agreement. Both approaches are based on the approximation of the actual ray dynamics by an integrable Hamiltonian, which enables us to perform a semiclassical quantization of the system and to introduce a ray-based description of the decay of optical modes. The coupling between them is determined either perturbatively or semiclassically in terms of complex paths.

Figure 1

(a) Phase space of the near-integrable system and (b) phase space of the mixed system with parameters given in the text. Regular tori are depicted by lines, while dots correspond to chaotic orbits.

Figure 2

Wave numbers for whispering-gallery modes with $l=0$ (a) for the near-integrable system starting from $m=1$ and (b) for the mixed system starting from $m=4$ shown as black stars. The wave numbers of the integrable counterpart, i.e., (a) the circular cavity and (b) the elliptic cavity, are shown as gray triangles. The perturbative prediction, Eq. (9), is depicted by open magenta diamonds.

Figure 3

The intensity patterns with the respective maximum intensity normalized to unity for the modes with (a) $m=27$ and (c) $m=22$ for the near-integrable systems and (b) $m=28$ and (d) $m=23$ for the mixed system are shown. The black line represents the boundary of the respective cavity.

Figure 4

The incident Husimi representation of the modes with (a) $m=27$ and (c) $m=22$ for the near-integrable systems and (b) $m=28$ and (d) $m=23$ for the mixed system are superimposed on the classical phase space shown in gray.

Figure 5

In (a) the phase space of the mixed system (gray dots and lines) is superimposed by the adiabatically invariant curves in black in Birkhoff coordinates. The thick red line corresponds to the adiabatically invariant action $P_{a:b}$. In (b) the same orbits and adiabatically invariant curves are shown in adiabatic action-angle coordinates. The phase space of the pendulum Hamiltonian is shown by black lines in (c) on top of the actual ray dynamics depicted in gray. The same scenario for the near-integrable system is shown in (d). In (c) and (d) an example of the torus ${\tilde{P}}_{m,l}$ (thick green line) and its symmetric counterpart ${\tilde{P}}_{\mathrm{rat}}$ (thick blue line) is shown.

Figure 6

Relative errors of EBK wave numbers given by Eq. (27) for the near-integrable system (orange triangles) and for the mixed system (blue diamonds). The lines connecting the symbols are a guide to the eye.

Figure 7

Wave numbers for the near-integrable systems are shown in (a) starting from $m=1$ and in (b) for mixed systems starting from $m=4$. Numerically obtained wave numbers are depicted as black stars. The perturbative description using the imaginary parts of wave numbers in the circular and elliptic cavity is shown as red crosses, while blue circles denote the perturbative description combined with the ray-based decay rates. The semiclassical description is represented by green diamonds. The lines connecting the data points are a guide to the eye.