Semiclassical theory of the Ehrenfest-time dependence of quantum transport

In ballistic conductors, there is a low-time threshold for the appearance of quantum effects in transport coefficients. This low-time threshold is the Ehrenfest time ${\tau }_E$. Most previous studies of the ${\tau }_E$ dependence of quantum transport assumed ergodic electron dynamics, so that they could be applied to ballistic quantum dots only. In this paper, we present a theory of the ${\tau }_E$ dependence of three signatures of quantum transport—the Fano factor for the shot noise power, the weak localization correction to the conductance, and the conductance fluctuations—for arbitrary ballistic conductors.

Figure 1

Schematic picture of a ballistic quantum dot (left) and a Lorentz gas (right). In a ballistic quantum dot, the conductance is dominated by the conductances of the two contacts that connect the quantum dot to the electron reservoirs. A Lorentz gas is a ballistic conductor with scattering from circular disks.

Figure 2

Hierarchy of time scales in two-dimensional ballistic conductors in the semiclassical limit $\hslash \to 0$. At finite temperatures or for time-dependent transport, the role of ${\tau }_D$ is taken over by the minimum of ${\tau }_D$, the dephasing time ${\tau }_{\varphi }$, or the inverse frequency ${\omega }^{-1}$.

Figure 3

(Color online) Left: Schematic drawing of a set of four trajectories $\alpha$, $\beta$, $\gamma$, and $\delta$ that contribute to the Fano factor $f$. Right: Detail of the encounter between the trajectories $\alpha$ and $\gamma$ together with definitions of the phase-space points, coordinates, and partial encounter durations used in the main text. The true trajectories may involve specular reflection at the boundaries or at scatterers inside the sample. The separation between the trajectories is magnified for clarity. The true encounter regions involve trajectories separated by a submacroscopic distance which cannot be resolved in a figure.

Figure 4

Ehrenfest-time dependence of the Fano factor $f$ in a quasi-one-dimensional Lorentz gas.

Figure 5

(Color online) Schematic drawing of sets of four trajectories that contribute to the conductance fluctuations. In panels (b) and (c), the periodic reference orbit $\gamma$ is shown thick. The period ${\tau }_{\gamma }$ of $\gamma$ is $t$.

Figure 6

Schematic drawing of two of the interfering trajectories (one of each pair) for all possible contributions to $K\left(t\right)$. The figure also contains the definitions of the phase-space points (represented by solid dots), Poincaré surfaces of section (thick arrows), and time intervals used in the main text (thin arrows).

Figure 7

(Color online) The three contributions to the conductance autocorrelation function $K\left(t\right)$ for ${\tau }_E∕{\tau }_D=0.2$ (top) and ${\tau }_E∕{\tau }_D=2$ (bottom). The dashed curves show $K\left(t\right)$ for the limit ${\tau }_E∕{\tau }_D\to 0$.

Figure 8

Conductance variance versus Ehrenfest time for a quasi-one-dimensional Lorentz gas. The figure shows $\mathrm{var}g$ in the absence of a time-reversal symmetry. In the presence of a time-reversal symmetry, $\mathrm{var}g$ is a factor of 2 larger.