Quantum Hall Valley Splitters and a Tunable Mach-Zehnder Interferometer in Graphene

Graphene is a very promising test bed for the field of electron quantum optics. However, a fully tunable and coherent electronic beam splitter is still missing. We report the demonstration of electronic beam splitters in graphene that couple quantum Hall edge channels having opposite valley polarizations. The electronic transmission of our beam splitters can be tuned from zero to near unity. By independently setting the beam splitters at the two corners of a graphene $p\text{−}n$ junction to intermediate transmissions, we realize a fully tunable electronic Mach-Zehnder interferometer. This tunability allows us to unambiguously identify the quantum interferences due to the Mach-Zehnder interferometer, and to study their dependence with the beam-splitter transmission and the interferometer bias voltage. The comparison with conventional semiconductor interferometers points toward universal processes driving the quantum decoherence in those two different 2D systems, with graphene being much more robust to their effect.

Figure 1

Quantum Hall valley splitter. Schematic representation of the $p\text{−}n$ junction. The $n$ region is depicted in blue, the $p$ one in pink. Electrons are injected from the upper right Ohmic contact (defining an injected current $I_0$) and transmitted current $I_T$ is measured at the lower left contact.

Figure 2

Oscillations of the quantum Hall valley splitter. (a) Measured transmission as a function of the top side gate voltage $V_1$, for ${\nu }_2=0$ below the bottom side gate. For ${\nu }_1=0$ below the top side gate (dashed arrow), the injected current is fully reflected. For ${\nu }_1<0$, current can be transmitted. (b) Measured transmission $T_1$ of the top beam splitter as a function of $V_1$ for ${\nu }_1\le -1$ and ${\nu }_2=0$. Right panel: sketch of the corresponding edge states configuration. (c) $T_1$ calculated with kwant as a function of the position $x_1$ of the intersection between the $p\text{−}n$ interface and the top sample edge. Top panel: edge configuration used in the calculations. Bottom panel: resulting calculated transmission.

Figure 3

Oscillations of the Mach-Zehnder interferometer. (a) Measured $T_1$ as a function of $V_1$ and the magnetic field $B$, with the filling factor below the bottom side gate set to ${\nu }_2=0$ ($V_2=0.45\mathrm{V}$, see sample configuration in inset). (b) Measured device transmission $T_{\mathrm{MZ}}$ in the same range of $V_1$ and $B$, with ${\nu }_2\le -1$, forming a Mach-Zehnder interferometer (inset: sample configuration). (c) $T_{\mathrm{MZ}}$ as a function of $B$ and $V_1$ with $T_1=T_2\sim 1/2$.

Figure 4

Transmission and energy dependence of the Mach-Zehnder visibility. (a) Interference visibility as a function of the normalized ${\overline{T}}_1$ (see Supplemental Material [13]). The visibility (yellow dots) of the experimentally measured oscillation of $T_{\mathrm{MZ}}$ and the curve of $\alpha \sqrt{\overline{T_1}(1-\overline{T_1})}$ where $\alpha$ is an adjustable parameter show good agreement. (b) $T_{\mathrm{MZ}}$ as a function of the dc bias $V_{\mathrm{dc}}$ (applied to the upper right Ohmic contact for the injection of $I_0$) and the magnetic field. (c) Measured visibility (yellow dots) as a function of $V_{\mathrm{dc}}$. Computed visibility (black solid line) based on Gaussian phase fluctuations is in agreement with the experimental data.