Exponential Sensitivity to Dephasing of Electrical Conduction Through a Quantum Dot

According to random-matrix theory, interference effects in the conductance of a ballistic chaotic quantum dot should vanish $\propto ({\tau }_{\varphi }/{\tau }_D{)}^p$ when the dephasing time ${\tau }_{\varphi }$ becomes small compared to the mean dwell time ${\tau }_D$. Aleiner and Larkin have predicted that the power law crosses over to an exponential suppression $\propto \exp (-{\tau }_E/{\tau }_{\varphi })$ when ${\tau }_{\varphi }$ drops below the Ehrenfest time ${\tau }_E$. We report the first observation of this crossover in a computer simulation of universal conductance fluctuations. Their theory also predicts an exponential suppression $\propto \exp (-{\tau }_E/{\tau }_D)$ in the absence of dephasing—which is not observed. We show that the effective random-matrix theory proposed previously for quantum dots without dephasing explains both observations.

Figure 1

Variance of the conductance fluctuations, normalized by the RMT value (2), as a function of the dimension $M$ of the scattering matrix of the kicked rotator with a dephasing lead. Each data set is for a fixed value of the dwell time ${\tau }_D=M/2N$ and dephasing time ${\tau }_{\varphi }={\Gamma }^{-1}(1-2N/M{)}^{-1}$. The Lyapunov exponent $\lambda =1.32$ is kept the same for all data sets. The curves show the Ehrenfest time dependence (5) predicted by the effective RMT, without any fit parameter.

Figure 2

Pictorial representation of the effective RMT of a ballistic chaotic quantum dot. The part of phase space with dwell times $>{\tau }_E$ is represented by a fictitious chaotic cavity (mean level spacing ${\delta }_{\mathrm{e}\mathrm{f}\mathrm{f}}$), connected to electron reservoirs by two long leads ($N_{\mathrm{e}\mathrm{f}\mathrm{f}}$ propagating modes, one-way delay time ${\tau }_E/2$ for each mode). The effective parameters are determined by $N_{\mathrm{e}\mathrm{f}\mathrm{f}}/N=\delta /{\delta }_{\mathrm{e}\mathrm{f}\mathrm{f}}=\exp (-{\tau }_E/{\tau }_D)$. The scattering matrix of lead plus cavity is $\exp (iE{\tau }_E/\hslash )S_{\mathrm{R}\mathrm{M}\mathrm{T}}(E)$, with $S_{\mathrm{R}\mathrm{M}\mathrm{T}}(E)$ distributed according to RMT. A finite dephasing time ${\tau }_{\varphi }$ is introduced by the substitution $E\to E+i\hslash /2{\tau }_{\varphi }$. The part of phase space with dwell times $<{\tau }_E$ has a classical scattering matrix, which does not contribute to quantum interference effects.

Figure 3

Four data sets of fixed ${\tau }_{\varphi }$, ${\tau }_D$, each consisting of a range of $M$ between ${10}^2$ and $2\times {10}^4$, plotted on a double-logarithmic scale. The solid line with slope $1/\lambda =0.76$ is the scaling predicted by Eqs. (5, 6).