Statistics of transmission eigenvalues in two-dimensional quantum cavities: Ballistic versus stochastic scattering

We investigate the statistical distribution of transmission eigenvalues in phase-coherent transport through quantum dots. In two-dimensional ab initio simulations for both clean and disordered two-dimensional cavities, we find markedly different quantum-to-classical crossover scenarios for these two cases. In particular, we observe the emergence of “noiseless scattering states” in clean cavities, irrespective of sharp-edged entrance and exit lead mouths. We find the onset of these “classical” states to be largely independent of the cavity’s classical chaoticity, but very sensitive with respect to bulk disorder. Our results suggest that for weakly disordered cavities, the transmission eigenvalue distribution is determined both by scattering at the disorder potential and the cavity walls. To properly account for this intermediate parameter regime, we introduce a hybrid crossover scheme, which combines previous models that are valid in the ballistic and the stochastic limit, respectively.

Figure 1

(Color online) Integrated distribution function of transmission eigenvalues, $I\left(T\right)$. Top row: Rectangular billiard with tunable shutters (see inset in Fig. 2). (a) Crossover from large to small shutter openings (at zero disorder, $V_0=0$): $I\left(T\right)$ for $w∕d=0.5$, 0.4, 0.3, and 0.2 (top to bottom). (b) Crossover from clean to disordered samples (at half opening, $w∕d=0.5$): $I\left(T\right)$ for different disorder potentials $V_0=0$, $0.03E_F$, $0.05E_F$, and $0.1E_F$ (top to bottom). The pronounced difference between (a) and (b) near $T=1$ is highlighted in the insets. Bottom row: $I\left(T\right)$ for (c) the circular- and (d) the stadium-shaped geometry for different values of $k_F$. In (c), we average over the intervals $45<N<50$, $30<N<35$, $20<N<25$, $10<N<15$, and $1<N<5$ (top to bottom), and in (d) $N=25$, $10<N<15$, and $1<N<5$ (top to bottom), where $N$ is the number of open lead modes. The RMT limit is indicated by black lines.

Figure 2

(Color online) Evolution of the crossover trajectories in the parameter space $(\alpha ,\beta )$, obtained by fitting Eq. (4) to the data displayed in Figs. 1a, 1b. Starting parameters of both trajectories: $w∕d=0.5$ and $V_0=0$. Trajectory 1 for fixed $w∕d=0.5$ (orange triangles): $V_0=0.03E_F$, $0.05E_F$, and $0.1E_F$. Trajectory 2 for fixed $V_0=0$ (blue circles): $w∕d=0.4$, 0.3, and 0.2. Inset: Rectangular cavity with tunable shutter openings and disorder strength.

Figure 3

(Color online) Circular billiard (no disorder). Top bar: Classical Poincaré surface of section [transmitted (reflected) trajectories represented by black (white) regions]. Bottom bars: Cumulative Husimi distributions $H(x,p)$ of strongly transmitted scattering states [Eq. (5)]. $H(x,p)$ is shown for specific mode intervals $N$, and black frames indicate the size of the Planck cell $h$. For $N\gtrsim 24$, the largest transmission band (see central black region in the Poincaré surface) is larger than $h$ and can be resolved by the quantum scattering process. Above this threshold, noiseless scattering states appear in $H(x,p)$ in form of pronounced density enhancements near the largest transmission bands.

Figure 4

(Color online) Rectangular billiard with tunable opening (no disorder, same color coding as in Fig. 3). Top two bars: Large shutter openings, $w∕d=0.4$. Bottom two bars: Weakly open shutters, $w∕d=0.2$. Decreasing the shutter opening below the threshold value $w∕d\approx 0.32$ (where the size of the Planck cell $h$ is equal to the largest transmission band) reduces any pronounced enhancements in the Husimi distributions $H(x,p)$.

Figure 5

(Color online) Rectangular billiard with tunable bulk disorder (fixed opening ratio $w∕d=0.3$, same color coding as in Fig. 3). Top two bars: No disorder. Bottom bar: Moderate disorder, $V_0=0.1E_F$. Bulk disorder destroys the appearance of noiseless states due to stochastic scattering.