Quantum-Classical Correspondence in the Wave Functions of Andreev Billiards

We present a classical and quantum mechanical study of an Andreev billiard with a chaotic normal dot. We demonstrate that the nonexact velocity reversal and the diffraction at the edges of the normal-superconductor contact render the classical dynamics of these systems mixed indicating the limitations of a widely used retracing approximation. We point out the close relation between the mixed classical phase space and the properties of the quantum states of Andreev billiards, including periodic orbit scarring and localization of the wave function onto other classical phase space objects such as intermittent regions and quantized tori.

Figure 1

The Poincaré section of the SA billiard for $h=0.58$ (a) and $h=0$ (b). (For other parameters see Ref. 22.) Each dot represents a starting point of an electron trajectory; the diamonds (◇) (red online) and boxes (□) (blue online) correspond to very elongated regular islands. The torus shown with diamonds in (a) is projected onto real space (c).

Figure 2

The probability densities of the electron ($|u{|}^2$) and hole ($|v{|}^2$) components of a chaotic eigenstate ($\epsilon /{\Delta }_0=0.060395$) of the SA billiard (a). The corresponding smoothed projections of the Wigner function onto the PS (b) in units of $y/W$ and $p_y/p_e$, $p_y/p_h$ for the electron and hole components, respectively. Here $p_e=m{v}_e$ and $p_h(\epsilon )=p_e(-\epsilon )$. Eight equally spaced positive contours [dark gray (red online) lines] and five negative contours [light gray (green online) lines] are plotted. The distribution of the scaled probabilities $z_e$ [dark gray (red online) line], $z_h$ [light gray (green online) line] and the Porter-Thomas distribution [black (blue online) line] (c).

Figure 3

The probability densities of the electron ($|u{|}^2$) and hole ($|v{|}^2$) components of a scarred eigenstate ($\epsilon /{\Delta }_0=0.88323$) of the SA billiard (a) and the corresponding unstable periodic orbit (b). The probability densities of a decoupled state at $\epsilon /{\Delta }_0=0.42901$ (c) and the marginally stable periodic orbit family over which the hole component shows enhancement (d).

Figure 4

The probability density of the electron ($|u{|}^2$) and hole ($|v{|}^2$) components of an intermittent eigenstate ($\epsilon /{\Delta }_0=0.44652$) of the SA billiard (a). One of the slowly drifting electron-hole orbits during the intermittent-like motion (b) which form the regular strip around $v_y/v_e\approx -0.2$ in Fig. 1a. The orbit comprises electron [dark gray (red online)] and hole [light gray (green)] trajectories. The smoothed projections ${\mathcal{W}}_{\mathrm{P}}$ for the electron and the hole components of the eigenstate (c). Eight equally spaced positive contours [dark gray (red online) lines] and one negative contour [light gray (green online) line] are plotted. The black (blue online) curve marks the location of that region of the PS, in which the drifting electron-hole orbits, like the one in (b), stay longer than ${\tau }_{\mathrm{H}}$ (see text). The electron component of a wave function localized onto a quantized torus when $h=0$ (d). The hole component is almost identical and thus not shown here.