Dynamical Scheme for Interferometric Measurements of Full-Counting Statistics

We propose a dynamical scheme for measuring the full-counting statistics in a mesoscopic conductor using an electronic Mach-Zehnder interferometer. The conductor couples capacitively to one arm of the interferometer and causes a phase shift which is proportional to the number of transferred charges. Importantly, the full-counting statistics can be obtained from average current measurements at the outputs of the interferometer. The counting field can be controlled by varying the time delay between two separate voltage signals applied to the conductor and the interferometer, respectively. As a specific application, we consider measuring the entanglement entropy generated by partitioning electrons on a quantum point contact. Our scheme is robust against moderate environmental dephasing and may be realized thanks to recent advances in gigahertz quantum electronics.

Figure 1

Interferometric measurements of FCS. Single electrons are injected into a Mach-Zehnder interferometer enclosing the magnetic flux $\varphi$. Separate voltage signals are applied to the interferometer $V(t)$ and a nearby conductor $V_c(t)=V(t-\tau )$ with a time delay $\tau$. The average current $⟨I⟩$ measured at an output is sensitive to a phase shift caused by the capacitive coupling to the conductor in the interaction region of length $a$. The phase shift is proportional to the number of transferred electrons in the conductor. By varying the magnetic flux and the time delay $\tau$, the FCS of the conductor can be obtained from average current measurements only.

Figure 2

Interferometric measurement of the full-counting statistics in a QPC. (a) Cumulants of the current as functions of the QPC transmission $T$. We show results for different pulse widths $\mathrm{\Gamma }$ in terms of the length $a$ of the interaction region. The exact results for a binomial process are $⟨⟨I^2⟩⟩=T(1-T)$, $⟨⟨I^3⟩⟩=T(1-T)(1-2T)$, and $⟨⟨I^4⟩⟩=T(1-T)(1-6T+6T^2)$, having set $e=1$ and $\mathcal{T}=1$ here and in the figure. (b) Full distribution of the current $I=en/N$ with $T=0.4$ and $N=40$. For a large number of periods $N\gg 1$, the distribution takes on the large-deviation form $\ln [P(I)]/N=\mathcal{G}(I)$ with the rate function $\mathcal{G}(I)$ being independent of $N$. For a binomial process, we find $\ln [P(I)]/N=\ln [(1-T)/(1-I)]+I\{\ln [T/(1-T)]-\ln [I/(1-I)]\}+\mathcal{O}(N^{-1})$.

Figure 3

The entanglement entropy generated per period obtained from Eq. (17). The exact result for the entanglement entropy reads $\mathcal{S}=(T-1)\ln (1-T)-T\ln T$. The maximum value $\mathcal{S}=\ln 2$ is obtained for $T=1/2$.