Finite bias visibility of the electronic Mach-Zehnder interferometer

We present an original statistical method to measure the visibility of interferences in an electronic Mach-Zehnder interferometer in the presence of low frequency fluctuations. The visibility presents a single side lobe structure shown to result from a Gaussian phase averaging whose variance is quadratic with the bias. To reinforce our approach and validate our statistical method, the same experiment is also realized with a stable sample. It exhibits the same visibility behavior as the fluctuating one, indicating the intrinsic character of finite bias phase averaging. In both samples, the dilution of the impinging current reduces the variance of the Gaussian distribution.

Figure 1

Scanning electron microscope view of the electronic Mach-Zehnder interferometer with a schematic representation of the edge state. G0, G1, G2 are quantum point contacts which mimic beam splitters. The pairs of split gates defining a QPC are electrically connected via an Au metallic bridge deposited on an insulator (SU8). G0 allows a dilution of the impinging current, G1 and G2 are the two beam splitters of the Mach-Zehnder interferometer. SG is a side gate which allows a variation of the length of the lower path (b).

Figure 2

Sample 1: (a) Transmission $\mathcal{T}=d{I}_T∕d{I}_0$ as a function of the gate voltages $V1$ and $V2$ applied on G1 and G2. (엯) $\mathcal{T}={\mathcal{T}}_1$ vs $V1$. (●) $\mathcal{T}={\mathcal{T}}_2$ vs $V2$. The solid line is the transmission $\mathcal{T}$ obtained with ${\mathcal{T}}_1$ fixed to $1∕2$ while sweeping $V2$: transmission fluctuations due to interferences with low frequency phase noise appear. (b) Stack histogram on 6000 successive transmission measurements as a function of the normalized deviation from the mean value. The solid line is the distribution of transmission expected for a uniform distribution of phases. (c) Visibility of interferences as a function of the transmission ${\mathcal{T}}_2$ when ${\mathcal{T}}_1=1∕2$. The solid line is the $\sqrt{{\mathcal{T}}_2(1-{\mathcal{T}}_2)}$ dependence predicted by the theory.

Figure 3

(Color online) Sample 1: Visibility of the interferences as a function of the drain-source voltage $I_{0}h∕e^2$ for three different values of ${\mathcal{T}}_0$. The curves are shifted for clarity. The energy width of the lobe structure is modified by the dilution whereas the maximum visibility at zero bias is not modified. Solid lines are fits using Eq. (1). From top to bottom, ${\mathcal{T}}_0=0.02$ and $V_0=31\mu \mathrm{V}$, ${\mathcal{T}}_0=0.14$ and $V_0=22\mu \mathrm{V}$, ${\mathcal{T}}_0=1$ and $V_0=11.4\mu \mathrm{V}$.

Figure 4

(Color online) Sample 2: (a) Gray plot of the transmission $\mathcal{T}$ as a function of the bias voltage $V$ and the side gate voltage $V_{\mathrm{SG}}$. Note the $\pi$ shift of the phase when the visibility reaches 0. (b), (c) $\mathcal{T}$ as a function of the side gate voltage for two different values of the drain source voltage corresponding to the dashed line of (a) (0 and $16\mu \mathrm{V}$, respectively). (d) Lobe structure of the visibility fitted using Eq. (1) for a diluted and an undiluted impinging current.

Figure 5

(Color online) $V_0$ obtained by fitting the visibility with Eq. (1), normalized to $V_0$ at ${\mathcal{T}}_0=1$, as a function of ${\mathcal{T}}_0$ at zero bias.