Diffractive paths for weak localization in quantum billiards

We study the weak-localization effect in quantum transport through a clean ballistic cavity with regular classical dynamics. We address the question which paths account for the suppression of conductance through a system where disorder and chaos are absent. By exploiting both quantum and semiclassical methods, we unambiguously identify paths that are diffractively backscattered into the cavity (when approaching the lead mouths from the cavity interior) to play a key role. Diffractive scattering couples transmitted and reflected paths and is thus essential to reproduce the weak-localization peak in reflection and the corresponding antipeak in transmission. A comparison of semiclassical calculations featuring these diffractive paths yields good agreement with full quantum calculations and experimental data. Our theory provides system-specific predictions for the quantum regime of few open lead modes and can be expected to be relevant also for mixed as well as chaotic systems.

Figure 1

(a) Geometry of circular quantum billiard with radius $\rho =\sqrt{1+4∕\pi }$ (in scaled units) and width of the leads $d=0.25$ $(\approx 0.16\rho )$ with one pair of time-reversal symmetric paths $q$ and $\overline{q}$ depicted. (b) Internal diffractive backscattering at the lead described by diffraction amplitude $v({\theta }_2^{\prime },{\theta }_2,k)$ (plotted is $\mathrm{Re}\left[v({\theta }_2^{\prime },{\theta }_2=\frac{\pi }{4},k)\right]\cos \left(kr\right)$, with $r$ as the radial distance from the center of the lead and $k=\frac{2.5\pi }{d}$).

Figure 2

(Color online) Two-dimensional path length-area spectrum of the exact quantum amplitude ${\mid {\tilde{S}}_{22}(L,a)\mid }^2$ [Eq. (5)] with integration ranges $k∊[2.2\pi ∕d,3.45\pi ∕d]$ and $B∕c∊[-3,3]$ (a) for reflection ${\mid {\tilde{r}}_{22}(L,a)\mid }^2$ and (b) for transmission ${\mid {\tilde{t}}_{22}(L,a)\mid }^2$. Strength given by intensity of color/grayscale, the dots mark classical paths. The pairing of classical and peudopaths contributing to WL is visualized [insets in (b)] and examples of diffractive contributions by pseudopaths are highlighted (red/gray markings).

Figure 3

Weak localization (WL) in reflection and transmission for the circular quantum dot (see Fig. 1) with two open modes and wave-number average $[2.2\pi ∕d⩽k⩽2.8\pi ∕d]$. (a) Quantum calculation of WL with truncated set of quantum paths $(L⩽L_{\max }=40)$ [see Eq. (5)]. (b) WL calculated within the PSCA and the same truncation (No. of paths $\approx 7\times {10}^6$). (c) WL calculated within SCA (only classical paths) and $L⩽L_{\max }$. Note that a dip in $T$ is completely absent. (d) Full quantum calculation (no truncation).