Suppression of weak localization and enhancement of noise by tunneling in semiclassical chaotic transport

We add simple tunneling effects and ray splitting into the recent trajectory-based semiclassical theory of quantum chaotic transport. We use this to derive the weak-localization correction to conductance and the shot noise for a quantum chaotic cavity (billiard) coupled to $n$ leads via tunnel barriers. We derive results for arbitrary tunneling rates and arbitrary (positive) Ehrenfest time ${\tau }_E$. For all Ehrenfest times, we show that the shot noise is enhanced by the tunneling, while the weak localization is suppressed. In the opaque barrier limit (small tunneling rates with large lead widths, such that the Drude conductance remains finite), the weak localization goes to zero linearly with the tunneling rate, while the Fano factor of the shot noise remains finite but becomes independent of the Ehrenfest time. The crossover from random matrix theory behavior $({\tau }_E=0)$ to classical behavior $({\tau }_E=\infty )$ goes exponentially with the ratio of the Ehrenfest time to the paired-path survival time. The paired-path survival time varies between the dwell time (in the transparent barrier limit) and half the dwell time (in the opaque barrier limit). Finally, our method enables us to see the physical origin of the suppression of weak localization; it is due to the fact that tunnel barriers “smear” the coherent-backscattering peak over reflection and transmission modes.

Figure 1

(Color online) A chaotic cavity (billiard) with $n$ leads. The $m\text{th}$ lead has width $W_m$ and is coupled to the cavity through a tunnel barrier with transmission probability ${\rho }_m$. We assume that all the tunnel barriers are very high and thin, so that this transmission probability is independent of the lead mode number.

Figure 2

(Color online) Schematic of the calculation of Green’s functions in the vicinity of a tunnel barrier. The evolution from ${\mathbf{r}}^{\prime }\to {\mathbf{r}}^{″}$ and ${\mathbf{r}}^{\prime }\to {\mathbf{r}}^{‴}$ is treated exactly, while semiclassics is used for the evolution from ${\mathbf{r}}_0\to {\mathbf{r}}^{\prime }$ and ${\mathbf{r}}^{″}\to \mathbf{r}$. We calculate the contribution for the part of the incoming Green’s function due to paths in the vicinity of the classical path $\alpha$ by matching the semiclassical and exact solutions of the Green's function equation on cross section A, which is just to the left of the tunnel barrier. Just to the right of the tunnel barrier, on cross section B, we couple to the lead modes.

Figure 3

(Color online) In all figures of contributions in this paper, we mark the regions where paths are correlated with vertical cross hatching. The encounter is at the center of those regions. The diagonal approximation is good everywhere except at the encounter; thus, for simplicity, we show only two of the four paths (or path segments) in all other figures in this paper. Here, we show what the cross-hatched regions in the other figures mean in terms of all four paths that enter them. In the cross-hatched region the two paths marked by solid lines are paired with each other. This region is macroscopic in size, the length of the region in time is ${\tau }_E^{\mathrm{op}}$ (which we assume to be of similar magnitude to the classical time scales such as ${\tau }_{D1}$ and ${\tau }_{D2}$), and the distance between the paths at either end of this region is of order of the lead width. In contrast, the encounter (where both paths marked by dashed lines swap from one path marked by a solid line to the other) is microscopic in size, the distance between the noncrossing paths is typically ${\left({\lambda }_{F}L\right)}^{1∕2}$ which goes to zero in the classical limit, and the time which the encounter takes is a few times the inverse of the Lyapunov exponent.

Figure 4

(Color online) Weak-localization contributions to conductance (contributions to conductance of order $N^0$) when there are tunnel barriers on the leads. All contributions except the second are for any $m_0$ and $m$, while the second (cbs0) is only for $m=m_0$. The vertical cross hatching indicates the regions in which the paths are correlated with each other, so their survival probability is given by Eq. (13) and not by Eq. (10). The encounter occurs at the center of the correlated region, as shown in Fig. 3. Details of the correlated regions for successful and failed coherent backscatterings are shown in Fig. 5.

Figure 5

(Color online) Details of the paths that contribute to successful and failed coherent backscatterings (cbs0, cbs1, and cbs2). Path $\gamma 1$ (solid black line) start on the cross section of lead $m_0$ at position $y_0$ with momentum angle ${\theta }_0$ and returns at $y$ with momentum angle $\theta$. For the successful contribution, it then tunnels into the lead; for the failed contribution, it is reflected off the barrier and remains in the chaotic system. All paths are drawn in the basis parallel and perpendicular to $\gamma 1$ at injection; the initial position and momentum of path $\gamma 1$ at exit are $r_{0\perp }=(y_0-y)\cos {\theta }_0$, $r_{0\Vert }=(y_0-y)\sin {\theta }_0$, and $p_{0\perp }\simeq -p_F(\theta -{\theta }_0)$.

Figure 6

(Color online) The set of contributions to $\mathrm{Tr}\left[{\mathbb{S}}_{m{m}_0}^{†}{\mathbb{S}}_{m{m}_0}{\mathbb{S}}_{m{m}_0}^{†}{\mathbb{S}}_{m{m}_0}\right]$ which we call $D_{1\mathrm{a}},\dots ,D_{1\mathrm{i}}$. These are the contributions which do not vanish for infinite Ehrenfest time. Here, we show only the tunnel barriers on leads $m_0$ and $m$ as shaded rectangles; a path which crosses the barrier on lead $m$ has succeeded in tunneling out of the cavity into the lead. The contributions are made up of four classical paths; here, we show only two of the paths (1 and 3). The other two paths (2 and 4) look the same as the paths shown, except that they cross at the center of the correlated region (indicated by the vertical cross hatching). In the correlated region, the paths have the same topology as shown in Fig. 3 if we replace the labels in the manner given in Eq. (58). Thus, path 4 is paired with path 1 at lead $m_0$ but paired with path 3 at lead $m$ (and vice versa for path 2). The noise in these contribution is purely due to the stochastic nature of scattering at the tunnel barriers; if they were absent, these contributions would be noiseless. The calculation will show that all these contributions go like $(1-e^{-{\tau }_E^{\mathrm{op}}∕{\tau }_{D2}})$.

Figure 7

(Color online) The set of contributions to $\mathrm{Tr}\left[{\mathbb{S}}_{m{m}_0}^{†}{\mathbb{S}}_{m{m}_0}{\mathbb{S}}_{m{m}_0}^{†}{\mathbb{S}}_{m{m}_0}\right]$ which we call $D_{2\mathrm{a}},\dots ,D_{2\mathrm{g}}$. These are the contributions which vanish for infinite Ehrenfest time. The contributions are drawn in the same manner as Fig. 6, with only paths 1 and 3 shown. As before, paths 2 and 4 are equivalent to 1 and 3 except that they cross at the center of the correlated region (indicated by vertical cross hatching). For finite Ehrenfest times, the paths must escape the correlated region (vertical cross hatching) before either tunnels out of the cavity (on at least one side of the encounter). The calculation will show that all these contributions go like $e^{-{\tau }_E^{\mathrm{op}}∕{\tau }_{D2}}$.

Figure 8

(Color online) Here, we list all the contributions to ${\sum }_{m^{\prime }\ne m}\mathrm{Tr}\left[{\mathbb{S}}_{m^{\prime }{m}_0}^{†}{\mathbb{S}}_{m^{\prime }{m}_0}{\mathbb{S}}_{m{m}_0}^{†}{\mathbb{S}}_{m{m}_0}\right]$ which remain finite in the limit of infinite Ehrenfest time. The contributions are drawn in the same manner as Fig. 6, with only paths 1 and 3 shown. The noise in these contribution is purely due to the stochastic nature of scattering at the tunnel barriers; if they were absent, these contributions would be noiseless. The calculation will show that (a) and (b) are independent of ${\tau }_E^{\mathrm{op}}$, while all other contributions go like $(1-e^{-{\tau }_E^{\mathrm{op}}∕{\tau }_{D2}})$.

Figure 9

(Color online) Here, we list all the contributions to ${\sum }_{m^{\prime }\ne m}\mathrm{Tr}\left[{\mathbb{S}}_{m^{\prime }{m}_0}^{†}{\mathbb{S}}_{m^{\prime }{m}_0}{\mathbb{S}}_{m{m}_0}^{†}{\mathbb{S}}_{m{m}_0}\right]$ which vanish in the limit of infinite Ehrenfest time. The contributions are drawn in the same manner as Fig. 6, with only paths 1 and 3 shown. The calculation will show that all these contributions go like $e^{-{\tau }_E^{\mathrm{op}}∕{\tau }_{D2}}$. Thus, for ${\tau }_E^{\mathrm{op}}∕{\tau }_{D2}\to 0$, the noise is given by the sum of these contributions and (a) and (b).