Non-universal suppression of the excitation gap in chaotic Andreev billiards: Superconducting terminals as sensitive probes for scarred states

When a quantum-chaotic normal conductor is coupled to a superconductor, random-matrix theory predicts that a gap opens up in the excitation spectrum which is of universal size $E_g^{\mathrm{R}\mathrm{M}\mathrm{T}}\approx 0.3\mathrm{\hslash }\mathrm{/}{t}_D$, where $t_D$ is the mean scattering time between Andreev reflections. We show that a scarred state of long lifetime $t_S\gg t_D$ suppresses the excitation gap over a window $\mathrm{\Delta }E\approx 2E_g^{\mathrm{R}\mathrm{M}\mathrm{T}}$ which can be much larger than the narrow resonance width ${\mathrm{\Gamma }}_S=\mathrm{\hslash }\mathrm{/}{t}_S$ of the scar in the normal system. The minimal value of the excitation gap within this window is given by ${\mathrm{\Gamma }}_S\mathrm{/}2\ll E_g^{\mathrm{R}\mathrm{M}\mathrm{T}}$. Hence the scarred state can be detected over a much larger energy range than it is the case when the superconducting terminal is replaced by a normal one.