Noise Dephasing in Edge States of the Integer Quantum Hall Regime

An electronic Mach-Zehnder interferometer is used in the integer quantum Hall regime at a filling factor 2 to study the dephasing of the interferences. This is found to be induced by the electrical noise existing in the edge states capacitively coupled to each other. Electrical shot noise created in one channel leads to phase randomization in the other, which destroys the interference pattern. These findings are extended to the dephasing induced by thermal noise instead of shot noise: it explains the underlying mechanism responsible for the finite temperature coherence time ${\tau }_{\phi }(T)$ of the edge states at filling factor 2, measured in a recent experiment. Finally, we present here a theory of the dephasing based on Gaussian noise, which is found to be in excellent agreement with our experimental results.

Figure 1

(a) Tilted SEM view of the device, with schematic representation of the edge states. $G1$ and $G2$ are Quantum Point Contact (QPC) which define the two beam splitters of the Mach-Zehnder interferometer. They are set to transmission ${\mathcal{T}}_1\sim {\mathcal{T}}_2\sim 1/2$ for the OES, while fully reflecting the IES. The two arms (a) and (b) are $L=11.3\mu \mathrm{m}$ long defining an area $S$ of $34\mu {\mathrm{m}}^2$. The small inner Ohmic contact is connected to the ground via an Au metallic bridge. SG is a side gate. $G0$ is an additional beam splitter which makes it possible to bias the IES by $V_2$, while the other is biased by $V_1$. $G0$ is tuned such that the OES is fully reflected, while the IES is transmitted with a probability ${\mathcal{T}}_0$. (b) Schematic representation of the edge states coupled by a geometrical capacitance $C$. (c) Low frequency equivalent circuit with $V_1$ set to 0 V, $C_Q=\tau /R_Q$.

Figure 2

(a) Phase sweeping by varying the side gate voltage $V_{\mathrm{SG}}$. (b) Phase sweeping by varying $V_2$ with ${\mathcal{T}}_0=1$ for two different magnetic fields. The periodicity $V_0$ depends on the magnetic field as shown in Fig. 4.

Figure 3

(a) Visibility decrease of the interferometer as a function of $V_2$ at ${\mathcal{T}}_0=1/2$ for two different magnetic fields 4.7 and 3.9 T. The solid lines are fit to the data $\mathcal{V}={\mathcal{V}}_0{e}^{-2{\pi }^2\Delta S_{22}\Delta \nu /V_0^2}$ with an electronic temperature of 25 mK (for a base temperature of 20 mK) and ${\mathcal{T}}_0=1/2$. The high bias fit of the exponential decrease $\mathcal{V}={\mathcal{V}}_0\exp [-{\mathcal{T}}_0(1-{\mathcal{T}}_0)V_2/V_{\phi }]$ allows us to determine $V_{\phi }$ which is found to depends on the magnetic field. (b) Visibility decrease of the interferometer as a function of ${\mathcal{T}}_0$ for $V_2=0$, 21, 31, 42, 53, and $63\mu \mathrm{V}$ from top to bottom. The solid lines are fits to the data using Eq. (1) with ${\mathcal{V}}_0=0.45$, $V_{\phi }=7.2\mu \mathrm{V}$, and $T=25\mathrm{mK}$.

Figure 4

$V_0$ and $V_{\phi }$ as a function of the magnetic field. The dashed line is the general behavior of $4k_B{T}_{\phi }/e$ (right scale) measured in Ref. 3, on the same sample.