Semiclassical theory of the Ehrenfest time dependence of quantum transport in ballistic quantum dots

We present a trajectory-based semiclassical calculation of the Ehrenfest-time dependence of the weak localization correction and the universal conductance fluctuations of a ballistic quantum dot with ideal point contacts. While the weak localization correction is proportional to $\exp (-{\tau }_{\mathrm{E}}∕{\tau }_{\mathrm{D}})$, where ${\tau }_{\mathrm{E}}$ and ${\tau }_{\mathrm{D}}$ are the dot’s Ehrenfest time and dwell time, respectively, the variance of the conductance is found to be independent of ${\tau }_{\mathrm{E}}$. The latter is in agreement with numerical simulations of the quantum kicked rotator [J. Tworzydlo et al., Phys. Rev. B 69, 165318 (2004) and P. Jacquod and E. V. Sukhorukov, Phys. Rev. Lett. 92, 116801 (2004)].

Figure 1

(Color online) Top: schematic picture of a pair of interfering trajectories $\alpha$ and $\beta$ that give rise to the weak localization correction to the conductance. Bottom left and bottom right: schematic picture of quadruples of interfering trajectories ${\alpha }_1$, ${\beta }_1$, ${\alpha }_2$, and ${\beta }_2$ that contribute to conductance fluctuations. The true trajectories inside the dot are piecewise straight, with specular reflections off the dot’s boundary. The small-angle (self) encounters are shown thick. In the bottom right panel, the trajectories ${\beta }_1$ and ${\alpha }_2$ have one more revolution around a periodic trajectory (shown dotted) than their partners ${\alpha }_1$ and ${\beta }_2$.

Figure 2

(Color online) If the phase space distance between the trajectories 1 and 2 is of order $\hslash$ at the point when they arrive at or depart from the periodic trajectory, their motion is correlated for a time $\sim {\tau }_{\mathrm{E}}$ while away from the periodic trajectory. The contribution of such trajectories to the conductance fluctuations is suppressed if ${\tau }_{\mathrm{E}}⪢{\tau }_{\mathrm{D}}$. After the removal of trajectories with correlated motion away from the periodic trajectory the net contribution of the trajectories shown in the bottom right panel of Fig. 1 to the conductance fluctuations zero is finite.

Figure 3

(Color online) Detail of the small-angle self-encounter and the definitions of the various times used in the calculations. The beginning and end of the encounter region are marked by solid thin lines, the location of the Poincaré surface of section is marked by a dashed line. Each trajectory passes the Poincaré surface of section twice. Left: Small-angle self-encounter fully inside the dot. Right: Small-angle self-encounter that touches the lead opening. One end of the encounter region is marked by a solid thin line, the other end is the lead opening.

Figure 4

(Color online) Two configurations of interfering trajectories that contribute to the covariance $\mathrm{cov}(R_1,R_2)$ of reflection coefficients for the left and right contacts. The encounter regions, segments of the trajectories for which the phase space distance between the trajectories 1 and 2 is less than a classical cutoff $c$, are shown thick. In the top configuration, the trajectories 1 and 2 have two consecutive well-separated encounters. In the bottom panel, the trajectories ${\alpha }_2$ and ${\beta }_1$ have one more revolution around a periodic trajectory (shown dotted) than their counterparts ${\alpha }_1$ and ${\beta }_2$.

Figure 5

(Color online) Two examples of interfering trajectories with overlapping encounters. The encounter regions are shown thick. In both panels, the trajectories ${\beta }_1$ and ${\alpha }_2$ have one more revolution around a closed periodic trajectory (shown dotted) than their partners ${\alpha }_1$ and ${\beta }_2$. The top panel shows two encounters that overlap at one end. This is the three encounter of Ref. 24. The bottom panel shows two encounters overlapping at both ends, so that the encounter region forms a closed loop. Although encounters that themselves form a closed loop do not contribute to the conductance variance if ${\tau }_{\mathrm{E}}⪡{\tau }_{\mathrm{D}}$, their contribution is essential if ${\tau }_{\mathrm{E}}⪢{\tau }_{\mathrm{D}}$.

Figure 6

(Color online) Top: Interfering trajectories of type $b$ (dashed and solid curves) and the periodic trajectory (dotted). Bottom: Short versions of the interfering trajectories together with the Poincaré surfaces of section. At each Poincaré surface of section the stable and unstable phase space coordinates $s$ and $u$ parametrize the distance to the periodic trajectory.

Figure 7

(Color online) Two successive deformations used to calculate the action difference between the pair of trajectories in the top left diagram and the pair of trajectories in the bottom left diagram.

Figure 8

(Color online) Definitions of the times appearing in Eq. (27). The top panel shows an example where the two short trajectories ${\alpha }_1$ and ${\beta }_2$ (solid curves) have strong correlations before and after arriving within the vicinity of the periodic trajectory (dotted curve). The time during which these strong classical correlations exist is ${\tau }_s$ or ${\tau }_u$. The bottom panel shows an example where there are no classical correlations except for the time the trajectories are in the vicinity of the periodic trajectory. In this case, ${\tau }_s={\tau }_u=0$. In the bottom panel, the two trajectories have different encounter times $t_{\mathrm{enc},1}$ and $t_{\mathrm{enc},2}$. In the example shown in the top panel, the two encounter times $t_{\mathrm{enc},1}$ and $t_{\mathrm{enc},2}$ are (almost) equal. The thin solid lines indicate the beginning and the end of the encounter with the periodic trajectory; the total encounter regions are marked by thick trajectories. The dashed lines indicate the positions where the Poincaré surfaces of section for the two trajectories are taken.

Figure 9

(Color online) Comparison of a part of a type $b$ encounter that fully resides inside the quantum dot (left) and an encounter that touches to lead opening (right). The relevant segments of the (short) trajectories ${\alpha }_1$ and ${\beta }_2$ are shown solid, the periodic trajectory is shown dotted.