Dephasing in a Mach-Zehnder Interferometer by an Ohmic Contact

We study dephasing in an electronic Mach-Zehnder (MZ) interferometer based on quantum Hall edge states by a micrometer-sized Ohmic contact embedded in one of its arms. We find that at the filling factor $\nu =1$, as well as in the case where an Ohmic contact is connected to a MZ interferometer by a quantum point contact that transmits only one electron channel, the phase coherence may not be fully suppressed. Namely, if the voltage bias $\mathrm{\Delta }\mu$ and the temperature $T$ are small compared to the charging energy of the Ohmic contact $E_C$, the free fermion picture is manifested, and the visibility saturates at its maximum value. At large biases, $\mathrm{\Delta }\mu \gg E_C$, the visibility decays in a power-law manner.

Figure 1

Schematics of the electronic Mach-Zehnder interferometer based on a Corbino disk shaped QH system at filling factor $\nu =2$ (shown by the gray shaded area), so that two chiral edge states are formed (shown by arrowhead lines). The incoming outer edge state originates from the source, biased with the voltage $\mathrm{\Delta }\mu$, and is split by ${\mathrm{QPC}}_L$ (quantum point contact) in two paths, which propagate in the upper and lower parts of the mesa and recombine at the ${\mathrm{QPC}}_R$. Thus, outer edge states enclose an $AB$ flux, which leads to $AB$ oscillations in the charge current, measured at the drain 2. An Ohmic contact is coupled to the edge state in the upper arm of the interferometer. It is a piece of metal, which absorbs incoming electron edge states and turns them into neutral electron-hole excitations. Then, it emits equilibrium neutral excitations as well as the charge current into the outgoing edge states. The number of channels transmitted to the Ohmic contact is controlled by the upper QPC. We assume, that at filling factor 2, the outer channel is fully transmitted to the Ohmic contact, while the inner channel can be both fully transmitted or fully reflected.

Figure 2

The simplified scheme of the MZI is shown. It is obtained by cutting the interferometer in Fig. 1 at the drain 2, unfolding edge states, and leaving only outer edge state, since it contributes to the $AB$ oscillations. The tunneling amplitudes at the left and right QPCs are denoted by ${\tau }_L$ and ${\tau }_R$, respectively. $L_u$ and $L_d$ are the lengths of the upper and lower paths of interferometer. The voltage bias $\mathrm{\Delta }\mu$ is applied to the lower channel. The current $I$ measured at the drain 2, and the differential conductance $G=dI/d\mathrm{\Delta }\mu$ oscillate as a function of the $AB$ phase ${\varphi }_{AB}$.

Figure 3

An equivalent representation of the Ohmic contact at filling factor $\nu =1$. Edge states are described by two bosonic fields ${\varphi }_{+}(x)$ and ${\varphi }_{-}(x)$ of opposite chiralities. The region inside the Ohmic contact, where the capacitive interaction is assumed, is shown by the light blue color.

Figure 4

The normalized visibility $V/V_0$ is plotted versus the dimensionless temperature $T/E_C$ in log-log scale (here $T\ne 0$, $\mathrm{\Delta }\mu \to 0$).

Figure 5

The normalized visibility $V/V_0$ is plotted versus the dimensionless bias $\pi \mathrm{\Delta }\mu /E_C$ in log-log scale (here $T\to 0$, $\mathrm{\Delta }\mu \ne 0$).