Electron Waiting Times in Mesoscopic Conductors

Electron transport in mesoscopic conductors has traditionally involved investigations of the mean current and the fluctuations of the current. A complementary view on charge transport is provided by the distribution of waiting times between charge carriers, but a proper theoretical framework for coherent electronic systems has so far been lacking. Here we develop a quantum theory of electron waiting times in mesoscopic conductors expressed by a compact determinant formula. We illustrate our methodology by calculating the waiting time distribution for a quantum point contact and find a crossover from Wigner-Dyson statistics at full transmission to Poisson statistics close to pinch-off. Even when the low-frequency transport is noiseless, the electrons are not equally spaced in time due to their inherent wave nature. We discuss the implications for renewal theory in mesoscopic systems and point out several analogies with level spacing statistics and random matrix theory.

Figure 1

QPC connected to voltage-biased electrodes. The QPC transmission is denoted as $T$, the Fermi energy is $E_F$, and $V$ is the voltage applied to the source electrode. The average time separation of the in-coming electrons is $\overline{\tau }=h/\mathrm{eV}$. The distribution of waiting times $\tau$ between transmitted electrons is determined by the many-body state of the in-coming electrons as well as the QPC which may cause electrons to reflect back. Reflected (missing) electrons are indicated by dashed lines.

Figure 2

Electron WTDs for a QPC. (a) WTDs for different transmission probabilities $T$ of the QPC. The waiting time $\tau$ is given in units of $\overline{\tau }=h/\mathrm{eV}$ or $⟨\tau ⟩=\overline{\tau }/T$ (inset). For a fully open QPC, $T=1$, the WTD is well approximated by a Wigner-Dyson distribution, Eq. (3). Close to pinch-off, $T\simeq 0$, the WTD approaches Poisson statistics given by the exponential distribution $\mathcal{W}(\tau )\simeq e^{-\tau /⟨\tau ⟩}/⟨\tau ⟩$. (b) WTD in the low-transmission regime, $T\simeq 0.1$. The short-time behavior given by Eq. (4) is indicated with a dashed line. Thin lines show expansions of the WTD to first, second, and third order in $T$. Arrows indicate small oscillations with period $\overline{\tau }$.

Figure 3

Fano factors as functions of QPC transmission $T$. The exact results $F_2=1-T$ and $F_3=(1-T)(1-2T)$ are shown together with predictions based on renewal theory $F_2^r=《{\tau }^2》/《\tau {》}^2$ and $F_3^r=3《{\tau }^2{》}^2/《\tau {》}^4-《{\tau }^3》/《\tau {》}^3$, where $《{\tau }^m》$ are the cumulants of the waiting time $\tau$. Renewal theory only holds in the low-transmission regime (shaded area), where $F_m\simeq F_m^r$.