Classical limit of transport in quantum kicked maps

We investigate the behavior of weak localization, conductance fluctuations, and shot noise of a chaotic scatterer in the semiclassical limit. Time-resolved numerical results, obtained by truncating the time evolution of a kicked quantum map after a certain number of iterations, are compared to semiclassical theory. Considering how the appearance of quantum effects is delayed as a function of the Ehrenfest time gives a new method to compare theory and numerical simulations. We find that both weak localization and shot noise agree with semiclassical theory, which predicts exponential suppression with increasing Ehrenfest time. However, conductance fluctuations exhibit different behavior, with only a slight dependence on the Ehrenfest time.

Figure 1

Schematic drawing of relevant trajectories for weak localization (a), conductance fluctuations (b), and shot noise (c). The letters “L” and “R” refer to the left and right contacts to the cavity, respectively. The trajectories shown in panel (a) are for the weak localization to the transmission $T$; the trajectories shown in panel (b) are for the covariance $\mathrm{cov}(R_{\mathrm{L}},R_{\mathrm{R}})$ of reflections from the left and right contacts. Both weak localization and conductance fluctuations require a minimal dwell time of $2{\tau }_{\mathrm{E}}$. Shot noise requires a minimal dwell time ${\tau }_{\mathrm{E}}$ only.

Figure 2

(Color online) Average conductance for the three-kick model (17) as a function of the time-reversal symmetry breaking parameter $\gamma N$. Curves shown for $N=25$, 50, 100, and 200 are offset vertically. The stochasticity parameter $K\approx 10,q=0.2$, and the dwell time ${\tau }_{\mathrm{D}}=5$. Inset: same for a larger range of $\gamma N$ values.

Figure 3

(Color online) Time-resolved difference between ensemble-averaged conductance at $\gamma N=0$ and $\gamma N=0.7$ for the three-kick model (17). The dwell time is ${\tau }_{\mathrm{D}}=5$, while $K\approx 10$ and $q=0.2$

Figure 4

Left: onset time $t_{\text{on}}$ obtained from the time-resolved difference $G_{\mathrm{LR}}\left(\gamma \right)-G_{\mathrm{LR}}\left(0\right)$. Right: weak localization correction $\delta G_{\mathrm{LR}}$, together with a fit $\propto \exp (-{\tau }_{\mathrm{E}}∕{\tau }_{\mathrm{D}})$. The stochasticity parameter $K$ uniformly chosen between 10 and 11.5; the dwell time ${\tau }_{\mathrm{D}}$ is set at ${\tau }_{\mathrm{D}}=5$ and ${\tau }_{\mathrm{D}}=10$. For ${\tau }_{\mathrm{D}}=5$, data are shown for $\gamma N=0.7$ (circles) and $\gamma N=0.35$ (squares). For ${\tau }_{\mathrm{D}}=10$, data are shown for $\gamma N=0.3$ (diamonds) and $\gamma N=0.15$ (triangles).

Figure 5

Same as Fig. 4, but for $K$ randomly chosen between 20 and 23.

Figure 6

(Color online) Time-resolved difference between ensemble-averaged conductance at $\gamma N=0$ and $\gamma N=0.2$ for the three-kick model (17), for $K\approx 10$ and ${\tau }_{\mathrm{D}}=12.8$. Left inset: onset times. Right inset: weak localization correction.

Figure 7

(Color online) Weak localization correction to reflection and transmission coefficients, obtained using Eq. (16). Data shown are for $K\approx 10$ and ${\tau }_{\mathrm{D}}=5$. The average was taken over an ensemble of 80 000 realizations, except for $N=400$, as described in the text.

Figure 8

(Color online) Weak localization correction to reflection and transmission coefficients, obtained using Eq. (16). Data shown are for $K\approx 10$ and ${\tau }_{\mathrm{D}}=10$. The number of realizations used to calculate the ensemble average is 20 000.

Figure 9

(Color online) Left: onset time $t_{\text{on}}$ obtained from the time-resolved weak localization correction $\delta G_{\mathrm{LR}}$. Right: weak localization correction $\delta G_{\mathrm{LR}}$, together with a fit $\propto \exp (-{\tau }_{\mathrm{E}}∕{\tau }_{\mathrm{D}})$. Data are taken with stochasticity parameter $K$ uniformly chosen between 10 and 11.5 (circles) and between 20 and 23 (squares) and for dwell times ${\tau }_{\mathrm{D}}=5$ (solid lines) and ${\tau }_{\mathrm{D}}=10$ (dashed lines).

Figure 10

(Color online) Time-resolved shot noise Fano factors for the open kicked rotator, with $K=10$ and ${\tau }_{\mathrm{D}}=5$. Left inset: onset times, determined from the location of the maximum of time-resolved Fano factor as a function of the truncation time $t_0$. Right inset: Fano factor compared to a fit $\propto \exp (-{\tau }_{\mathrm{E}}∕{\tau }_{\mathrm{D}})$. Results for $K\approx 10$ (circles), $K\approx 20$ (squares), ${\tau }_{\mathrm{D}}=5$ (solid lines), and ${\tau }_{\mathrm{D}}=10$ (dashed lines) are depicted in the insets.

Figure 11

(Color online) The truncation time-dependence of $\mathrm{var}{G}_{\mathrm{LL}}$, $\mathrm{cov}(G_{\mathrm{LL}},G_{\mathrm{LR}})$, and $\mathrm{cov}(G_{\mathrm{LL}},G_{\mathrm{RR}})$ for the one-kick rotator. The dwell time is ${\tau }_{\mathrm{D}}=5$, while $K$ is chosen uniformly between 10 and 11.5. The corresponding time dependence given by random matrix theory is shown as the dashed lines. The short-time behavior is magnified in the inset.

Figure 12

(Color online) Left: onset time $t_{\text{on}}$ obtained from the time-resolved conductance covariance data. Right: conductance covariance $\mathrm{cov}(G_{\mathrm{LR}},G_{\mathrm{LL}})$. Data are taken with stochasticity parameter $K$ uniformly chosen between 10 and 11.5 (squares) and between 20 and 23 (circles) and for dwell times ${\tau }_{\mathrm{D}}=5$ (solid lines) and ${\tau }_{\mathrm{D}}=10$ (dashed lines).

Figure 13

Schematic picture of the ballistic chaotic cavity of size $L$. The cavity is attached to two leads, labeled “L” and “R” and with width $d_{\mathrm{L}}$ and $d_{\mathrm{R}}$, respectively. The contours $C_{\mathrm{L}}^{\prime }$, $C_{\mathrm{L}}$, and $C_{\mathrm{R}}$ drawn in the leads are used for the calculation of transmission and reflection coefficients; see the text.