Quantum Andreev Map: A Paradigm of Quantum Chaos in Superconductivity

We introduce quantum maps with particle-hole conversion (Andreev reflection) and particle-hole symmetry, which exhibit the same excitation gap as quantum dots in the proximity to a superconductor. Computationally, the Andreev maps are much more efficient than billiard models of quantum dots. This makes it possible to test analytical predictions of random-matrix theory and semiclassical chaos that were previously out of reach of computer simulations. We have observed the universal distribution of the excitation gap for a large Lyapunov exponent and the logarithmic reduction of the gap when the Ehrenfest time becomes comparable to the quasiparticle dwell time.

Figure 1

Main plot: Gap distribution for the Andreev kicked rotator with $M=8192$, $K=45$, and $M/N={\tau }_{\mathrm{d}\mathrm{w}\mathrm{e}\mathrm{l}\mathrm{l}}=10$ (diamonds), 20 (circles), 40 (+), and 50 (×). The solid line gives the RMT prediction [13]. Inset: Density of states for the same system. The solid line is the RMT prediction [9]. The dashed line is a numerical result in the mixed regime ($M=8192$, $K=1.2$, $M/N=10$).

Figure 2

Density of states for the Andreev kicked rotator with $M=131072$, ${\tau }_{\mathrm{d}\mathrm{w}\mathrm{e}\mathrm{l}\mathrm{l}}=5$, and $K=14$ (solid line), compared with the Bohr-Sommerfeld calculation (histogram), and the RMT prediction (dashed line).

Figure 3

Main plot: Dependence of the mean gap on the system size $M$, for ${\tau }_{\mathrm{d}\mathrm{w}\mathrm{e}\mathrm{l}\mathrm{l}}=M/N=5$ and $K=14$. Averages have been calculated with 400 (for $M=512$) to 40 (for $M>5\times {10}^5$) different positions of the contacts to the superconductor. The error bars represent the root mean square of ${\epsilon }_0$. The dashed line is the RMT prediction and the solid line is a linear fit to the data points. Inset: Dependence of the mean gap on ${\tau }_{\mathrm{d}\mathrm{w}\mathrm{e}\mathrm{l}\mathrm{l}}=M/N$ for $K=14$ and $M=524288$. The dashed line is the RMT prediction and the solid and dotted curves are given by Eqs. (8, 9), respectively, with coefficients extracted from the linear fit in the main plot.