\section*{Introduction}

The superconducting proximity effect describes the processes in which correlations are induced in a normal metal by a superconducting electrode~\cite{klapwijk_proximity_2004}. The microscopic origin of the proximity effect lies in the Andreev reflections which couple the electron and hole states at the border of a normal metal and a superconductor (SC). In the past few years, interest in Andreev processes has experienced a renaissance, driven by the promise of producing exotic states and excitations, such as Majorana zero modes and chiral Majorana fermions~\cite{BeenakkerRMP2015}, which may be used for topological quantum computing~\cite{Stern_topological_2013}. Many concrete implementations of Majorana modes have been proposed, relying on superconductors proximitizing either materials with spin-orbit coupling~\cite{lutchyn_majorana_2018} or various quantum Hall (QH) systems~\cite{mong_universal_2014,qi_chiral_2010}. 

In this work, we directly probe chiral Andreev edge states (CAES), which result from inducing superconducting correlations in the integer QH edge states. Semiclassically, CAES result from skipping orbit trajectories, in which an electron turns into a hole and back to an electron upon successive Andreev reflections from a SC interface. Quantum mechanically, this combination of Andreev reflections with the QH edge states yields fermionic modes in which the electron and hole states are hybridized and propagate chirally along the QH-SC interface~\cite{takagaki_transport_1998,hoppe_andreev_2000,Khaymovich_2010}. Under certain conditions, CAES are predicted to be self-conjugate, becoming chiral Majorana fermions~\cite{lian_edge-state-induced_2016,gamayun_two-terminal_2017}. 

The early search for CAES in \Rom{3}-\Rom{5} semiconductor devices focused on magneto-conductance oscillations in the QH regime~\cite{eroms_andreev_2005,batov_andreev_2007}. Later, graphene samples in the QH regime were shown to have enhanced conductance between superconducting contacts~\cite{Komatsu2012,rickhaus_quantum_2012}.
Recent progress in making transparent type \Rom{2} superconducting contacts to both GaAs~\cite{wan_induced_2015} and encapsulated graphene~\cite{BenShalom_quantum_2015,Calado_ballistic_2015} has enabled the observation of several new phenomena, including edge state-mediated supercurrent~\cite{amet_supercurrent_2016} and crossed Andreev conversion~\cite{lee_inducing_2017}.

Nevertheless, despite some recent attempts \cite{park_propagation_2017,kozuka_andreev_2018,matsuo_equal-spin_2018}, direct evidence for CAES remains elusive. To conclusively identify the CAES one must demonstrate their propagation along the superconducting contact. Naively, one may expect that any electrical signal spreading along the contact will be shunted by the superconductor. Here, we demonstrate that this is not the case. The main mechanism allowing us to detect the CAES in this work is their interference, which can be described as follows: An incoming electron approaching the superconducting contact is decomposed into a linear combination of CAES~\cite{Khaymovich_2010} propagating along the QH-SC interface with different wavevectors. The accumulated phase difference between these modes can result in the original electron turning into a hole as it exits the opposite end of the interface~\cite{van_ostaay_spin-triplet_2011,lian_edge-state-induced_2016}. The appearance of the hole can be then detected by measuring the voltage on a normal contact located downstream from the grounded SC contact~\cite{lee_inducing_2017}. We observe the beating signal which proves that the CAES are formed by a coherent superposition of the electron and hole amplitudes. By further analyzing the interference between the CAES, we show that these modes are on average neutral -- their electron and hole components have a roughly equal weight. These results demonstrate that transport measurements can detect the presence of CAES despite their charge neutrality. Our approach opens the door for detecting the chiral Majorana modes in topological superconductors. 

\section*{Results}

\begin{figure}
\includegraphics[width=1\textwidth]{AES_figs/Fig1/fig1.pdf}

\noindent {\bf Fig.\,1.} {\bf Andreev reflection in the QH regime.} (\textbf{A}) Optical image of the sample. Superconducting contacts (gray) are placed between normal contacts (gold). The magnetic field is applied into the plane, resulting in counterclockwise travel of electron and hole when the graphene (blue) is n-doped. We use a four-terminal scheme to measure the Hall voltage $V_{xy}$ simultaneously with the SC downstream longitudinal voltage $\widetilde{V}_{xx}$. The sign of $\widetilde{V}_{xx}$ is carefully defined as the voltage of contact \textbf{\textit{d}} minus the voltage at contact \textbf{\textit{c}}. (\textbf{B}) Fan diagram of the zero-bias Hall conductance $G_{xy}$. The filling factors are labeled on the plateaus. Degeneracy of the LLs starts to lift around $B=1$ T, suggesting the high quality of the graphene region. (\textbf{C}) Temperature dependence of zero-bias SC downstream longitudinal resistance $\widetilde{R}_{xx}$ around $\nu=2$ at $B=3$ T together with the simultaneously measured Hall conductance $G_{xy}$. The oscillations of $\widetilde{R}_{xx}$ on $\nu=2$ plateau disappears at 1.8 K. (\textbf{D}) At a certain $V_G$, an electron injected into a pair of CAES exits as a hole. The bottom panel shows the electron (red) and hole (blue) wavefunction densities from a tight-binding calculation in the lowest Landau-level.
\end{figure}

Our samples are made from graphene encapsulated in hexagonal boron nitride (hBN). The graphene-hBN heterostructure is deposited on a doped Si wafer capped with a 280 nm SiO$_2$ layer, which serves as a back gate. One-dimensional contacts are made to the heterostructure~\cite{wang_one-dimensional_2013} using both normal and superconducting metal electrodes as shown in Fig.\,1A. The SC electrodes are sputtered molybdenum-rhenium alloy (MoRe), a type \Rom{2} superconductor with an upper critical field $H_{c2} \sim 10$ T, a critical temperature $T_c \sim 10$ K, and a superconducting gap $\Delta_0 \sim 1.3$ meV. We have previously demonstrated that the interface between MoRe and graphene is highly transparent~\cite{borzenets_ballistic_2016}. In addition to the SC electrodes, the sample has several normal contacts made of thermally evaporated Cr/Au. 

The lengths of the top and bottom SC interfaces $L$ are 150 nm and 600 nm, respectively. We focus on the bottom contact (\textbf{\textit{c}} in Fig.\,1A). The $L=600$ nm length is chosen so that it falls in the range between the induced superconducting coherence length $\xi_s=\hbar v/\pi \Delta_0 \sim160$ nm, and the phase coherence length of QH edge states $\xi_{\varphi}=\hbar v/2\pi k_B T$~\cite{qhcoherence_2008}, which is $\sim 12 \mu$m at $T=0.1$ K. In this estimates, the velocity of the edge states, $v$, is taken to be equal to the Fermi velocity of graphene, $v_F\sim10^6$ m/s. While this is typically true only for sharp vacuum edges, our simulation (see \cite{sup} section S6) suggests that the CAES velocity is likely lower but comparable to $v_F$, which places $L=600$ nm comfortably between $\xi_{\varphi}$ and $\xi_s$. This condition ensures both that the propagation of the CAES along the contact is quantum-mechanically coherent, and that the crossed Andreev conversion~\cite{lee_inducing_2017} is suppressed. 

Throughout these measurements, we apply a current from the normal contact labeled \textbf{\textit{a}} while keeping the bottom superconducting contact \textbf{\textit{c}} grounded (Fig.\,1A). The current is comprised of a variable DC component, $I$, and a small AC excitation of 10 nA which allows us to measure the differential resistance. To probe CAES propagation along the SC contact, we study the longitudinal resistance, $\widetilde{R}_{xx}=d\widetilde{V}_{xx}/dI$, where $\widetilde{V}_{xx}$ is measured between the normal contact \textbf{\textit{d}} and the adjacent grounded superconducting contact \textbf{\textit{c}}. We refer to this quantity as the ``downstream resistance'' to reflect the quantum Hall intuition that the edge states propagate along the chiral direction and thus contact \textbf{\textit{d}} is located downstream from contact \textbf{\textit{c}}. In conventional devices with normal contacts, $\widetilde{R}_{xx}$ would correspond to the longitudinal resistance $R_{xx}$, which equals zero on QH plateaus and is positive between them (fig.\,S1).  


We also simultaneously measure the differential Hall conductance, $G_{xy}=dI/dV_{xy}$, where the transverse voltage is measured between the normal contacts \textbf{\textit{b}} and \textbf{\textit{d}} in Fig.\,1A. This quantity remains well-quantized through the relevant range of gate voltages and temperatures below 3 K. This ensures that the bulk of the sample is gapped and that transport occurs only through the edge states. In the data presented, the sample is held at the base temperature below 100 mK unless otherwise specified, and the MoRe remains superconducting at all the magnetic fields ($B<6$ T) and temperatures studied. 


Fig.\,1B shows the map of Hall conductance $G_{xy}$ measured vs. $B$ and gate voltage $V_{G}$. The valley and spin degeneracies in this sample start to lift at $B \sim 1$ T, leading to the appearance of all integer filling factors, $\nu$. The broken symmetry states have smaller activation gaps than the main sequence of filling factors $\nu=4 (n+1/2)$. In order to ensure that the bulk of the sample remains insulating when we vary the current bias and temperature, we focus on the robust filling factor $\nu=2$ (filled lowest Landau level) which has an energy gap on the order of tens of meV. In Fig.\,1C, we plot the simultaneous measurement of $\widetilde{R}_{xx}$ and $G_{xy}$ versus $V_{G}$ in the range corresponding to the Landau level filling factor $\nu=2$ at $B=3$ T. The Hall conductance is well-quantized, despite the fact that the current flows through a superconducting drain contact. This observation is in agreement with the Landauer-B\"uttiker (LB) formula in \cite{sup} section S3, which shows that $G_{xy}$ measured between the normal contacts \textbf{\textit{b}} and \textbf{\textit{d}} should be quantized regardless of the properties of the drain contact. 

We focus next on the range of gate voltages ($5$ V$<V_G<5.7$ V) in which $G_{xy}$ remains well-quantized at $2\ e^2/h$ even at higher temperature ($T < 3$ K), ruling out any possible contribution from the bulk. Remarkably, in this range the downstream resistance $\widetilde{R}_{xx}$ shows clear deviations from the zero signal usually expected in the QH regime (see \cite{sup} section S1). As the temperature is increased, $\widetilde{R}_{xx}$ gradually flattens and approaches zero, and eventually a conventional QH behavior of zero longitudinal resistance is recovered around $2$ K. Note that this temperature is still very small compared to the QH gap and indeed $G_{xy}$ remains well-quantized. We conclude that the deviations from zero in the downstream resistance are due to superconductivity of contact \textbf{\textit{c}}, whose influence is suppressed by raising the temperature. Incidentally, the vanishing of $\widetilde{R}_{xx}$ observed around $2$ K suggests that contact \textbf{\textit{c}} would be fully transparent in its normal state. 

A particularly intriguing aspect of $\widetilde{R}_{xx}$ is the fact that it becomes negative at some gate voltages -- this means that contact \textbf{\textit{d}} acquires a chemical potential lower than the chemical potential of the grounded contact \textbf{\textit{c}}! We attribute this behavior to the following process: An electron approaching the superconductor turns into a linear combination of CAES. For each electron state, a pair of CAES is formed when the proximity effect couples the electron edge state with the hole edge state at the same energy~\cite{Khaymovich_2010} (see \cite{sup} section S2\&S6). Since their wavevectors are different, the two CAES acquire a phase difference while propagating along the superconducting interface, resulting in a beating pattern between the electron and hole components of the wavefunction~\cite{lian_edge-state-induced_2016}. If the CAES interference produces a hole at the end of the SC-graphene interface, the hole will flow to contact \textbf{\textit{d}} and lower its chemical potential. Note that in contrast to Ref.~\cite{lee_inducing_2017}, which studies crossed Andreev conversion \emph{across} the superconductor, the negative signal observed here is due to the interference of CAES propagating \emph{along} the contact. As a result, $\widetilde{R}_{xx}$ is sensitive to the phase accumulated along the interface, which makes it dependent on the gate voltage.
 
Fig.\,1D shows the results of a tight-binding calculation of an electron injected in a quantum Hall edge state toward an interface with a superconductor (See \cite{sup} section S6). Note the clear beating pattern between the electron and hole probabilities. The value of the chemical potential is chosen such that the outgoing state is almost purely a hole. We can alternatively obtain an outgoing electron, or any superposition of electron and hole, by changing the chemical potential in our simulation, which corresponds to varying the gate voltage in the experiment. As a result, the calculated probabilities of the outgoing state being an electron, $P_e$, or a hole, $P_h$, show pronounced oscillations (fig.\,S11). 

Experimentally, the beating pattern between the two CAES is likely to be affected by multiple parameters, such as the interface roughness, disorder potential, electron density profile near the contact, and even positions of vortices in the SC contact (fig.\,S3 and S4). As a result, the downstream resistance measured as a function of the gate voltage acquires a pattern of random but highly reproducible fluctuations (Fig.\,1C), in which the signal is positive or negative depending on whether the superconductor emits predominantly an electron or a hole. 

\begin{figure}[h]
\includegraphics[width=1\textwidth]{AES_figs/Fig2/fig2.pdf}

\noindent {\bf Fig.\,2.} {\bf The interference of CAES on various QH plateaus and its magnetic field dependence.} (\textbf{A}) Zero-bias $\widetilde{R}_{xx}$ plotted versus gate voltage $V_G$ at $B=3$ T together with the simultaneously measured $G_{xy}$. $\widetilde{R}_{xx}$ oscillates around zero on the well quantized plateaus of $\nu=2, 3, 5, 6, 7$ (shaded regions). (\textbf{B}) Zero-bias $\widetilde{R}_{xx}$ plotted versus gate voltage $V_G$ and magnetic field $B$. Filling factors are labeled based on the $G_{xy}$ fan diagram in Fig.\,1C, with the  boundaries for $\nu=2$ and $6$ drawn as black lines. The oscillations in the well-quantized region gradually die out with increasing magnetic field. (\textbf{C}) Peak values of $\widetilde{R}_{xx}$. The maximum and minimum of $\widetilde{R}_{xx}(V_G)$ on the $\nu=2$ plateau are plotted vs. $B$. On average, $\widetilde{R}_{xx}$ varies around zero, indicating that the CAES are roughly equal superpositions of electrons and holes. 
\end{figure}

The behavior observed in Fig.\,1C is generic. In Fig.\,2A, we plot $\widetilde{R}_{xx}$ and $G_{xy}$ at $3$ T in a wider range of gate voltages. Clearly, $\widetilde{R}_{xx}$ oscillates around zero as a function of the gate voltage for a range of integer filling factors. We could also expect mesoscopic fluctuations in the downstream resistance to be induced by changing the magnetic field, $B$, which changes the magnetic length. Indeed, our simulations indicate that $P_e-P_h$ oscillates with magnetic field (fig.\,S10).

To explore this dependence experimentally, we plot $\widetilde{R}_{xx}$ as a function of both $B$ and $V_G$ in Fig.\,2B. The overall pattern is reminiscent of the traditional Landau fan diagram of the longitudinal resistance, with the exception that the downstream resistance $\widetilde{R}_{xx}$ is not equal to zero on the plateaus. Mesoscopic fluctuations that deviate from zero resistance (white) appear inside the QH plateaus as blue (negative) and red (positive) pockets. A prominent feature of this data is the frequent abrupt changes of the $\widetilde{R}_{xx}(V_G)$ pattern while sweeping the magnetic field.  Although the field sweeps are stable and reproducible in a very small range of $B$, changing the field by several mT completely changes the $\widetilde{R}_{xx}(V_G)$ curves (see fig.\,S3B for more details). This stochastic switching complicates the analysis of the map.

We note that when switching of the $\widetilde{R}_{xx}(V_G)$ curve occurs, the $G_{xy}$ stays unchanged (compare Fig.\,1B and 2B). This can also be noticed in fig.\,S3A in the regions slightly outside the quantized plateau, where $G_{xy}$ develops a recognizable mesoscopic pattern. This observation indicates that the switching events do not involve the normal contacts \textbf{\textit{b}} and \textbf{\textit{d}} between which $G_{xy}$ is measured, nor the bulk of the sample. Instead, they must originate in the superconducting contact \textbf{\textit{c}}. We surmise that the switching of the $\widetilde{R}_{xx}$ pattern is caused by the rearrangement of vortices inside the type II superconducting contact. Indeed, we have routinely observed similar switching events in the interference pattern of supercurrent in Josephson junctions fabricated with similar contacts. In \cite{sup} section S4, we show that we can hysteretically switch between two distinct patterns of $\widetilde{R}_{xx}$ multiple times, indicating that the vortices can be controllably added to and removed from the superconductor.

To explain the observed sensitivity of $\widetilde{R}_{xx}$ to the vortex configuration, we note that adding a vortex close to the interface should change the phase of the order parameter along the QH-SC interface $\theta (x)$. As a result of this change, a pure electron or a pure hole state would only acquire an overall phase, which would not change $\widetilde{R}_{xx}$. However, the change of $\theta (x)$ is expected to change the relative phase shift between the two interfering CAES. This idea is supported by our simulations, which show that the $P_e-P_h$ curve is strongly affected by adding a single vortex, modeled as a kink of magnitude $\pi$ in $\theta (x)$ (fig.\,S11). 

To extract information otherwise buried in the stochastic switching, we analyze the impact of $B$ on the gate-dependent oscillations on the $\nu=2$ plateau. The plateau region used for this analysis is selected such that $G_{xy}$ is within 1\% off the quantized values, as indicated by the black lines in Fig.\,2B. We then find the minimum and maximum $\widetilde{R}_{xx}(V_G)$ for a given field and plot the resulting $\min\widetilde{R}_{xx}(V_G)$ and $\max \widetilde{R}_{xx}(V_G)$ as a function of $B$ in Fig.\,2C. First, we find that the amplitude of the fluctuations decreases with $B$. Most likely, this suppression is explained by the loss of individual carriers that may be absorbed by the superconductor. Evidently, this process becomes more effective at higher $B$. 
Second, the typical amplitudes of the positive and negative signals are very close. We argue that this observation indicates that CAES are on average neutral (see \cite{sup} section S2\&S6). Indeed, if the two CAES $\psi_{1,2}$ had predominantly electron-like and hole-like characters, the incoming electron would couple primarily to $\psi_1$. This would in turn result in a greater likelihood of electrons being emitted downstream, and $\widetilde{R}_{xx}$ would mostly stay positive, i.e. $\max\widetilde{R}_{xx}(V_G) > |\min \widetilde{R}_{xx}(V_G)|$, contrary to our observations. 

Our numerical simulations support this argument: the eigenmodes $\psi_{1,2}$ are given by coherent superpositions of electron and hole amplitudes, which have distinct patterns in space. Nevertheless, the integral of the probability of the electron and hole components is close to $1/2$ (fig.\,S9), resulting in the overall approximately neutral character of the CAES. Due to particle-hole symmetry of the model, $\psi_2$ at zero energy is the charge conjugate partner of $\psi_1$, meaning that the pattern of the electron and hole amplitudes is interchanged. 

\begin{figure}
\includegraphics[width=1\textwidth]{AES_figs/Fig3/fig3.pdf}

\noindent {\bf Fig.\,3.} {\bf The bias-dependence of the interference effect.} (\textbf{A}) The SC downstream longitudinal resistance $\widetilde{R}_{xx}$ plotted versus the DC bias current $I$ and gate voltage $V_G$ on a well-quantized $\nu=2$ plateau at $B=$ 3 T. Gate-dependent oscillations centered at zero-bias are observed, indicating interference of the CAES. (\textbf{B-D}) Bias-dependent oscillations of $\widetilde{R}_{xx}$ at the gate voltages $V_G=5.08$ (\textbf{B}), $5.19$ (\textbf{C}) and $5.29$ (\textbf{D}) V marked by the arrows in (\textbf{A}). Oscillations die out with increasing temperature. (\textbf{E}) The dispersion relation of a pair of CAES at a QH-SC interface calculated from the tight-binding simulation. The momentum difference between the two modes varies with energy, causing oscillations of $\widetilde{R}_{xx}$ in bias.
\end{figure}

Finally, we address the non-linearity of the CAES energy-momentum dispersion. In Fig.\,3A, we plot the $\widetilde{R}_{xx}$ map measured as a function of $I$ and $V_{G}$. This data corresponds to the range nominally identical to Fig.\,1C; however it was measured following a sweep of magnetic field, so the individual mesoscopic features have changed. The dependence of this map on temperature and additional maps for $\nu=6$ are shown in section S5 of \cite{sup}.

Most notably, we find that $\widetilde{R}_{xx}$ in Fig.\,3A oscillates not only with the gate voltage, but also as a function of the DC current bias. These oscillations are revealed in the vertical cross-sections of the map, taken at $V_G=5.08, 5.19$ and $5.29$ V and plotted in Fig.\,3B-D. Quite unusually, the signal can even oscillate several times as a function of bias, as shown in Fig.\,3C. To interpret these oscillations, we note that the applied current tunes the energy of the injected electrons with respect to the grounded superconducting contact, $E=eI/G_{xy}$. The wavevector difference $\delta k$ between the two CAES depends on their energy, as demonstrated in our model calculations in Fig.\,3E (see also \cite{sup} section S6). The phase difference accumulated by the CAES along the interface, $\delta k L$, thus produces the observed bias oscillations of $\widetilde{R}_{xx}$. Eventually, $\widetilde{R}_{xx}$ goes to zero when the applied voltage becomes comparable to the superconducting gap, $I/G_{xy} \sim \Delta/e$. At that point, the incoming electrons have high probability to enter the superconductor as quasi-particles, and no downstream signal is expected. 

\section*{Discussion}

We have demonstrated robust coupling of QH edge states to a superconductor via Andreev reflections, resulting in the formation of chiral Andreev edge states which are coherent superpositions of electrons and holes. The resistance measured downstream from the superconducting contact, $\widetilde{R}_{xx}$, oscillates as a function of gate voltage and bias, providing strong evidence of the interference between the CAES. Numerical simulations qualitatively support the observed behavior by reproducing a pair of charge-conjugate CAES states which have momentum difference that depends on the Fermi energy and injection energy. 

The phase difference between the CAES also depends on the vortex configuration, which results in the observed switching behavior as a function of magnetic field. The presence of the vortices is typically neglected in theoretical studies, but we find that they have a dramatic effect on the beating pattern of the CAES -- even one vortex entering the SC can change the downstream resistance.

Our analysis shows that the CAES are on average charge neutral. Nevertheless, we find that they can be detected by a DC transport measurement. Our results therefore open the door for future experiments involving neutral fermionic states, such as manipulating, detecting, and braiding chiral Majorana fermions.


