Superconducting Terminals as Sensitive Probes for Scarred States

When a quantum-chaotic normal conductor is coupled to a superconductor, the random-matrix theory (RMT) predicts that a gap opens up in the excitation spectrum which is of universal size $E_g^{\mathrm{RMT}}\approx 0.3\hslash /t_D$, 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 $\Delta E\approx 2E_g^{\mathrm{RMT}}$ which can be much larger than the narrow resonance width ${\Gamma }_S=\hslash /t_S$ of the scar in the normal system. The minimal value of the excitation gap within this window is given by ${\Gamma }_S/2\ll E_g^{\mathrm{RMT}}$. Via this suppression of the gap to a nonuniversal value, the scarred state can be detected over a much larger energy range than it is in the case when the superconducting terminal is replaced by a normal one.

Figure 1

(a) Excitation spectrum as a function of the Fermi energy $E_F$, obtained by numerical computations for the Andreev billiard shown in panel (b). The lowest excitation energy (triangles) determines the excitation gap. The second, third, forth, and fifth excitation energy are shown as solid circles. The energy of the scarred eigenstate is highlighted by open squares. The dashed line shows the random-matrix prediction $E_g^{\mathrm{RMT}}$ for the gap. Other lines are guides for the eye. (b) The Andreev billiard is a segment of a Sinai billiard with $W=0.21a$, $h=0.58a$, and $R=0.8a$. The pair potential in the superconductor is ${\Delta }_0=0.015E_F$. In the energy range of (a), there are $N=25$ open channels in the superconducting lead.

Figure 2

The square moduli $|u{|}^2$ of the electron and $|v{|}^2$ of the hole components of the wave function for $E_F/E_{\mathrm{Th}}=707.434<E_{F,\min }$ (a), $E_F=E_{F,\min }$ (b) and $E_F/E_{\mathrm{Th}}=707.695>E_{F,\min }$ (c).

Figure 3

Scattering wave function of the open normal dot at energies $E=E_{F,\min }$ (a) and $E=707.657E_{\mathrm{Th}}$ (b). The states are excited by an incoming wave in the third transverse mode of the lead.

Figure 4

(a) The phase $\Theta (E)=\mathrm{Im}\ln \det [S(E)]$ of the scattering matrix of the open normal billiard [Fig. 1b with the superconducting terminal replaced by a normal lead] is plotted as a function of energy (circles). The solid line shows a fit of the function $2\arctan \frac{{\Gamma }_S}{2(E-E_S)}+CE+{\Theta }_0$ to $\Theta (E)$. (b) Quantitative comparison of the excitation energies of the scarred Andreev bound state (squares) with Eq. (5) (solid line), using the parameters $E_S$ and ${\Gamma }_S$ of the scar in the normal system.