Quantum-to-classical crossover for Andreev billiards in a magnetic field

We extend the existing quasiclassical theory for the superconducting proximity effect in a chaotic quantum dot, to include a time-reversal-symmetry breaking magnetic field. Random-matrix theory (RMT) breaks down once the Ehrenfest time ${\tau }_E$ becomes longer than the mean time ${\tau }_D$ between Andreev reflections. As a consequence, the critical field at which the excitation gap closes drops below the RMT prediction as ${\tau }_E∕{\tau }_D$ is increased. Our quasiclassical results are supported by comparison with a fully quantum mechanical simulation of a stroboscopic model (the Andreev kicked rotator).

Figure 1

Classical trajectory in an Andreev billiard. Particles are deflected by the potential $V=[{(r∕L)}^2-1]V_0$ for $r<L,V=[-4{(r∕L)}^2+10(r∕L)-6]V_0$ for $r>L$, with $r^2=x^2+y^2$ (the dotted lines are equipotentials). At the insulating boundaries (solid lines) there is specular reflection, while the particles are Andreev reflected at the superconductor ($y=0$, dashed line). Shown is the trajectory of an electron at the Fermi level $(E=0)$, for $B=0$ and $E_F=0.84{\mathrm{eV}}_0$. The Andreev reflected hole will retrace this path.

Figure 2

Andreev reflection at a NS boundary (dashed line) of an electron to a hole. The left panel shows the case of perfect retroreflection (zero excitation energy $E$ and zero magnetic field $B$). The middle and right panels show that the hole does not precisely retrace the path of the electron if $E$ or $B$ are nonzero.

Figure 3

(Color online) Poincaré map for the Andreev billiard of Fig. 1. Each dot marks the position $x$ and tangential velocity ${\upsilon }_x$ of an electron at the NS boundary. Subsequent dots are obtained by following the electron trajectory for $E,B\to 0$ at fixed ratio $B∕E=1∕3\sqrt{m∕V_0{L}^2{e}^3}$. The inset shows the full surface of section of the Andreev billiard, while the main plot is an enlargement of the central region. The drift is along closed contours defined by $\mathcal{K}=$constant [see Eq. (4)]. The value of the adiabatic invariant $\mathcal{K}$ (in units of $\sqrt{{\mathrm{mL}}^2∕{\mathrm{eV}}_0}$) is indicated for several contours. All contours are closed loops, but for some contours the opening of the loop is not visible in the figure.

Figure 4

Directed area for a classical trajectory, consisting of the area enclosed by the trajectory after joining begin and end points along the NS boundary (dashed line). Different parts of the enclosed area have different signs because the boundary is circulated in a different direction.

Figure 5

Illustration of a bunch of trajectories within a single scattering band in the billiard defined in Fig. 1. All trajectories in this figure have starting conditions in the band containing the contour with $\mathcal{K}=11$ of Fig. 3. Both $T$ and $A$ vary only slightly from one trajectory to the other, so that the whole band can be characterized by a single $\overline{T}$ and $\overline{A}$, being the average of $T$ and $A$ over the scattering band.

Figure 6

Pictorial representation of the effective RMT of an Andreev billiard. The part of phase space with long trajectories (return time $>{\tau }_E$) is represented by a chaotic cavity with level spacing ${\delta }_{\mathrm{eff}}$, connected to the superconductor via a fictitious ballistic lead with $N_{\mathrm{eff}}$ channels. The lead introduces a channel-independent delay time ${\tau }_E∕2$ and a channel-dependent phase shift ${\varphi }_n$, which is different from the distribution of phase shifts in a real lead.

Figure 7

(Color online) Ensemble averaged density of states $\rho \left(E\right)$ of the Andreev kicked rotator. The dark (red) curves show the numerical results from the fully quantum mechanical model, while the light (green) curves are obtained from Eq. (43) with input from the classical limit of the model. The energy is scaled by the Thouless energy $E_T=\hslash ∕2{\tau }_D$ and the density is scaled by the level spacing $\delta$ of the isolated billiard. The parameters of the kicked rotator are $M=2048,N=204,q=0.2,K=200$ in panel (a) and $M=16384,N=3246,q=0.2,K=14$ in panels (b), (c), (d). The three-peak structure indicated by the arrow in panels (b), (c), (d) is explained in Fig. 8.

Figure 8

Conditional distribution $P(A\mid T)$ of directed areas $A$ enclosed by classical trajectories with $T=2{\tau }_0$, for $K=14,q=0.2$, and ${\tau }_D=5{\tau }_0$. The distribution was obtained from the classical map (A11) at $\gamma =0$. Trajectories with $T=2{\tau }_0$ give rise to a peak in the density of states centered around $E∕E_T=(m+1∕2)\pi \hslash ∕2{\tau }_0$, cf. Eq. (14). On the energy scale of Fig. 7 only the peak with $m=0$ can be seen, at $E∕E_T=2.5\pi \approx 7.9$. In a magnetic field this peak broadens and it obtains the side peaks of $P(A\mid 2{\tau }_0)$.

Figure 9

Critical magnetic field $B_c$ of the Andreev kicked rotator as a function of the Ehrenfest time. The Ehrenfest time ${\tau }_E={\lambda }^{-1}\ln (N^2∕M)$ is changed by varying $M$ and $N$ while keeping $q=0.2$ and ${\tau }_D∕{\tau }_0=M∕N=5$ constant. For the closed circles the kicking strength $K=14$, while for the squares from left to right $K=4000$, 1000, 400, 200, 100, 50. The solid curve is the quasiclassical prediction (47). The open circles are obtained from the closed circles by the transformation $\lambda {\tau }_E\to \lambda {\tau }_E+1.75$, allowed by the terms of order unity in Eq. (48).

Figure 10

(Color online) Ensemble averaged density of states of the Andreev kicked rotator for fully broken TRS. The histogram shows the numerical results, while the curve is the theoretical prediction (50) of the Laguerre unitary ensemble. Both the energy and the density of states are scaled by the level spacing $\delta$ of the isolated billiard. The parameters of the kicked rotator are $M=2048,N=204,q=0.2$, while $K$ was varied between 200 and 250 to obtain an ensemble average.