Decoherence and full counting statistics in a Mach-Zehnder interferometer

We investigate the full counting statistics of an electrical Mach-Zehnder interferometer penetrated by an Aharonov-Bohm flux, and in the presence of a classical fluctuating potential. Of interest is the suppression of the Aharonov-Bohm oscillations in the distribution function of the transmitted charge. For a Gaussian fluctuating field we calculate the first three cumulants. The fluctuating potential causes a modulation of the conductance leading in the third cumulant to a term cubic in voltage and to a contribution correlating modulation of current and noise. In the high voltage regime we present an approximation of the generating function.

Figure 1

(Color online) The Mach-Zehnder interferometer, a four-terminal $AB$ ring, threaded by a magnetic flux. The interferometer arms are coupled to classical fluctuating potentials modeling the effect of dephasing. The influence of dephasing on current fluctuations in lead 3 is investigated.

Figure 2

(Color online) (a) The distribution function at zero temperature (logarithmic scale, $V{t}_0∕2\pi =40$). Left, 3D plot of the binomial distribution, which is periodic in flux. Right, the same in contour representation, dark areas correspond to low probability. (b) The distribution function at high temperature $k_{B}T=10\mathrm{V}$ (for a long observation time $t_0{k}_{B}T=100$). The flux dependence is washed out, and the distribution tends towards a Gaussian.

Figure 3

(Color online) Noise (a), $(k_{B}T=0)$ and third cumulant (c), $(k_{B}T=0.1)$ as a function of decoherence parameter $z$ and flux $\Phi$ (for $V\tau =1$): the $AB$ oscillations are suppressed for decreasing $z$. (b) and (d) display the dependence on voltage and temperature for $z=0.3$. The third cumulant changes little with temperature, but the noise shows linear thermal noise for high temperature.

Figure 4

(Color online) Processes contributing to the third cumulant at zero temperature. The black points correspond to channels $\alpha$, $\beta$, $\gamma$ at time $t+\tau$, and the crosses to channels $\rho$, $\mu$, $\nu$ at time $t$. In $C_I^3$ (a) three one-particle processes are connected by correlations of the fluctuating potential (circled in grey/green). In (b) the correlation of the modulated current and the shot noise are displayed, which contributes to $C_{I\Sigma }^3$. $C_{3p}^3$ (c) is the three particle contribution, averaged over excitations.

Figure 5

(Color online) The third cumulant and its three contributions as a function of voltage. At high $V\tau$ the cubic term dominates. The dashed line $C_{\text{slow}}^3$ shows the approximation in the high voltage regime.

Figure 6

(Color online) The distribution function (logarithmic scale) at zero temperature under influence of decoherence $(N_{\tau }=10,t_0=\tau )$. (a) Weak dephasing, $z=e^{-0.32}$; (b) strong dephasing, $z=e^{-2}$. The flux dependence is suppressed, and the distribution tends towards a symmetric function for $\mathcal{R}=0.5$.