Formation of Fabry-Perot resonances in double-barrier chaotic billiards

We study wave transport through a chaotic quantum billiard attached to two waveguides via barriers of arbitrary transparencies in the semiclassical limit of a large number of open scattering channels. We focus attention on the ergodic regime, which is described by using a random-matrix approach to chaotic resonance scattering together with an extended version of Nazarov’s circuit theory. By varying the relative strength of the barriers’ transparencies a reorganization of the relevant resonances in the energy interval where transport takes place leads to a full suppression of high transmission modes. We provide a detailed quantitative description of the process by means of both numerical and analytical evaluations of the average density of transmission eigenvalues. We show that the density of Fabry-Perot modes can be used as a kind of order parameter for this quantum transition. A diagram is presented as a function of the transparencies of the barriers exhibiting the transport regimes and the transition lines.

Figure 1

Level density $\nu \left(x\right)$ for $T_1=1$ and $T_2=1$ (circle), $T_2=0.8$ (square), $T_2=0.5$ (triangle), and $T_2=0.15$ (inverted triangle). The full lines are exact numerical solutions of Eq. (74). Note the transition point at $T_2=0.5$.

Figure 2

Density of Fabry-Perot resonances, $\nu \left(0\right)$, as a function of the transmission coefficients $T_1$ and $T_2$.

Figure 3

Diagram illustrating the transport regimes as a function of the tunnel probabilities. The solid lines represent the transition lines $\zeta ={\zeta }_0$ [case (iv)] and $\zeta =1$ [case (ii)]. The region above, between and below these solid lines stand for cases (v), (iii), and (i), respectively. The dashed line represents the particular case $T_1=T_2$, whereas the dotted lines are guides to the eye at $T_1=0.5$ and $T_2=0.5$.