Towards a semiclassical justification of the effective random matrix theory for transport through ballistic chaotic quantum dots

The scattering matrix $S$ of a ballistic chaotic cavity is the direct sum of a classical and a quantum part, which describe the scattering of channels with typical dwell time smaller and larger than the Ehrenfest time, respectively. According to the effective random matrix theory of Silvestrov, Goorden, and Beenakker [Phys. Rev. Lett. 90, 116801 (2003)], statistical averages involving the quantum-mechanical scattering matrix are given by random matrix theory. While this effective random matrix theory is known not to be applicable for quantum interference corrections to transport, which appear to subleading order in the number of scattering channels $N$, it is believed to correctly describe quantum transport to leading order in $N$. We here partially verify this belief, by comparing the predictions of the effective random matrix theory for the ensemble averages of polynomial functions of $S$ and $S^{†}$ of degree 2, 4, and 6 to a semiclassical calculation.

Figure 1

Schematic drawing of one of the contacts to the quantum dot. The coordinate $y$ as well as the perpendicular component of the momentum $p_{\perp }$ refers to the direction perpendicular to the lead axis. In the sum (11) over classical trajectories there is no restriction on $y$, but only discrete values of $p_{\perp }$ are considered.

Figure 2

(Color online) Left-hand side: Four classical trajectories that give the random matrix contribution to the average $Q_4$. These trajectories have a small-angle encounter which fully resides inside the quantum dot. Right-hand side: Schematic drawing of the encounter together with the definitions of the various times used in the text. The encounter region is shown thick.

Figure 3

(Color online) Schematic drawing of four classical trajectories that give the random matrix contribution with one Kronecker delta to the average $Q_4$. These trajectories have a small-angle encounter that touches one of the lead openings. The Kronecker delta appears for the entrance lead index in (a) and for the exit lead index in (b).

Figure 4

(Color online) Schematic drawing of four classical trajectories with a small-angle encounter that touches both lead openings. Such configurations of classical trajectories give the classical contribution to $Q_4$.

Figure 5

(Color online) Schematic drawing of six classical trajectories that contribute to $Q_6$ without encounters that touch the lead openings. The configuration shown in (a) consists of two separate two-encounters. There are two more configurations of this type, which are obtained by the cyclic permutations $(1,2)\to (3,4)\to (5,6)$. The configuration shown in (b) consists of one three-encounter.

Figure 6

(Color online) Schematic drawing of six classical trajectories that provide the remaining contributions to $Q_6$. The configuration shown in (a) consists of two separate two-encounters, where one two-encounter touches a lead opening. The configuration shown in (b) consists of one three-encounter, which partially touches one lead opening. The configuration of (c) consists of a three-encounter that fully touches one of the lead openings. The configuration of (d) has two two-encounters that each touch lead openings. The configuration of (e) has a three-encounter that partially touches two lead openings. The configuration of (f) has a three-encounter that partially touches one, and fully touches the other lead opening. Finally, the configuration of (g) fully touches both lead openings. The contributions from (a) and (b) have one Kronecker delta involving the lead indices, the contributions from (c), (d), and (e) have two Kronecker deltas, the contribution of (f) is zero, and the contribution of (g) has four Kronecker deltas.

Figure 7

Poincaré surface of section of lead opening, together with the standard coordinates $y$, $p_{\perp }$, as well as the phase space coordinates $s$, $u$, taken along the stable and unstable directions in phase space. The trajectory pair ${\alpha }_1$, ${\alpha }_2$, which have equal perpendicular momentum components $p_{\perp ,1}=p_{\perp ,2}$ are mapped to a trajectory pair ${\tilde{\alpha }}_1$, ${\tilde{\alpha }}_2$ with equal stable phase space coordinates, ${\tilde{s}}_1={\tilde{s}}_2$. The unstable phase space coordinates do not change in the mapping.

Figure 8

Definition of the four measures of the encounter duration used in the text: $t_{\mathrm{enc}}$ is the total duration of the three-encounter, $t_{\mathrm{enc}}^{\prime }$ is the duration of the part of the encounter that all trajectories involved in the three-encounter are within a phase-space distance $c$, and $t_s$ and $t_u$ are the durations of the encounter stretches where one pair of trajectories has already diverged from the remaining two pairs.

Figure 9

Definition of the four measures of the encounter duration used in the text: $t_{\mathrm{enc}}$ is the total duration of the three-encounter, $t_{\mathrm{enc}}^{\prime }$ is the duration of the part of the encounter that all trajectories involved in the three-encounter are within a phase-space distance $c$, and $t_s$ and $t^{\prime }$ are the durations of the encounter stretches where one pair of trajectories has already diverged from the remaining two pairs. The right end of the encounter is at the lead opening.