Electron waiting times in coherent conductors are correlated

We evaluate the joint distributions of electron waiting times in coherent conductors described by scattering theory. Successive electron waiting times in a single-channel conductor are found to be correlated due to the fermionic statistics encoded in the many-body state. Our formalism allows us also to investigate the waiting times between charge transfer events in different outgoing channels. As an application we consider a quantum point contact in a chiral setup with one or both input channels biased by either a static or a time-dependent periodic voltage described by Floquet theory. The theoretical framework developed here can be applied to a variety of scattering problems and can in a straightforward manner be extended to joint distributions of several electron waiting times.

Figure 1

Joint WTD for a fully open conduction channel. The two side panels show the joint WTD along the cuts indicated with colored dashed lines in the main panel. The joint WTD is well captured by the generalized Wigner-Dyson distribution (dashed lines) in Eq. (3). The dotted lines in the side panels show the waiting time distributions if there were no correlations between subsequent waiting times. Exact results have been obtained using Eq. (51).

Figure 2

Chiral setup with electrons in two incoming channels being partitioned on a QPC. A (time-dependent) voltage can be applied to the contacts of the incoming channels. The QPC has transmission $T$ and reflection $R$. The transmitted and reflected electrons are detected in the outgoing channels at different positions $x=t_{A/B}^{s/e}$ with $v_F=1$.

Figure 3

Idle time probability (ITP), first passage time distribution, and waiting time distribution (WTD). (a) Results for a static voltage applied to contact 1, $V_1\left(t\right)=V$, and no voltage applied to contact 2, $V_2\left(t\right)=0$. The QPC is tuned to $T=0.3$. We consider the electrons that are transmitted into outgoing channel $B$. The time is given in units of $\overline{\tau }=h/\left(\mathrm{eV}\right)$. (b) Results for a series of Lorentzian-shaped voltage pulses of unit charge applied to contact 1; see Eq. (27). The width of the pulses is $\mathrm{\Gamma }=0.02\mathcal{T}$. The time is given in units of the period of the pulses $\mathcal{T}$.

Figure 4

Comparison with renewal theory. Exact results (solid lines) for two different transmissions of the QPC are shown together with the approximation in Eq. (39) (dashed lines) based on a renewal assumption of uncorrelated waiting times. (a) Results for a static voltage applied to contact 1, $V_1\left(t\right)=V$, and no voltage applied to contact 2, $V_2\left(t\right)=0$. The QPC is tuned to $T=0.3$ (green lines) or $T=0.7$ (blue lines). We consider the electrons that are transmitted into outgoing channel $B$. The time is given in units of $\langle \tau \rangle =h/\left(T\mathrm{eV}\right)$. (b) Results for a series of Lorentzian-shaped voltage pulses of unit charge applied to contact 1. The width of the pulses is $\mathrm{\Gamma }=0.02\mathcal{T}$. The time is given in units of the period of the pulses $\mathcal{T}$.

Figure 5

Joint waiting time distributions for a static voltage. (a) Results for the voltage $V_1\left(t\right)=V$ applied to contact 1 and no voltage applied to contact 2, $V_2\left(t\right)=0$. The QPC is fully transmitting $T=1$ and we consider the electrons that are transmitted into outgoing channel $B$. The time is given in units of $\langle \tau \rangle =h/\left(T\mathrm{eV}\right)$. (b) Difference $\mathrm{\Delta }\mathcal{W}({\tau }_1,{\tau }_2)=\mathcal{W}({\tau }_1,{\tau }_2)-\mathcal{W}\left({\tau }_1\right)\mathcal{W}\left({\tau }_2\right)$ between exact results and results for uncorrelated waiting times. Positive correlations are indicated with red, while areas of negative correlations are blue. (c) and (d) Similar results for a half-transmitting QPC, $T=1/2$.

Figure 6

Joint waiting time distribution for levitons. (a) Results for a series of Lorentzian-shaped voltage pulses of unit charge applied to contact 1. The width of the pulses is $\mathrm{\Gamma }=0.05\mathcal{T}$ and the QPC is tuned to $T=1/2$. The time is given in units of the period of the pulses $\mathcal{T}$. (b) Difference $\mathrm{\Delta }\mathcal{W}({\tau }_1,{\tau }_2)=\mathcal{W}({\tau }_1,{\tau }_2)-\mathcal{W}\left({\tau }_1\right)\mathcal{W}\left({\tau }_2\right)$ between exact results and results for uncorrelated waiting times. Positive correlations are indicated with red, while areas of negative correlations are blue.

Figure 7

Distributions of waiting times for a Hong-Ou-Mandel experiment with levitons. Two sources emit levitons of pulse width $\mathrm{\Gamma }=0.05\mathcal{T}$ toward a QPC with transmission $T=1/2$. The tunable time delay between the sources is denoted as $\mathrm{\Delta }\tau$. (a) WTD for waiting times between events occurring in either of the two outgoing channels for different values of the time delay. (b) WTD for waiting times between detection events in different channels. The left inset shows the WTD for just a single channel. The right inset shows the WTD at $\tau =0$ as a function of the time delay $\mathrm{\Delta }\tau$ between the sources.