Controlled localization of anyons in a graphene quantum Hall interferometer

Abstract

Exchange statistics are a fundamental principle of quantum mechanics, dictating the symmetry of identical particle wavefunctions and thereby enabling emergent phenomena of many-body quantum states. The exchange-induced unitary transformation of both abelian and non-abelian anyonic wavefunctions can be probed using electronic fractional quantum Hall (FQH) interferometers, where quasiparticles propagating along the interfering FQH edge braid with those localized within the interferometer. Here, we add a gate-controlled dot/anti-dot in the center of a bilayer graphene FQH interferometer cavity to tune the number of enclosed anyons. We observe hundreds of controlled phase slips in the diagonal conductance across the interferometer for both abelian and non-abelian states, consistent with discrete changes in the localized quasiparticle population. For abelian anyons, the observed phase slips agree with the theoretically expected value. At half filling, our results suggest the interfering edge carries charge $|e^*/e| = 1/2$ abelian excitations, whereas charge $|e^*/e| = 1/4$ putative non-abelian anyons remain localized in the interferometer cavity. Controlling the population of localized $e/4$ anyons in an interferometer marks a significant milestone towards observing their non-local exchange statistics and building a fault tolerant topological qubit based on non-abelian anyon manipulation.

Figures (15)

• Figure 1: Fabry-Pérot fractional quantum Hall interferometry in bilayer graphene.a, Schematic of device structure showing overhanging bridge gate, lithographically patterned top gates, graphene, and bottom gate. b, False-color scanning electron microscopy image of a similar device to the one measured. The 4-probe measurement setup across the interferometry cavity is illustrated in green to measure diagonal voltage $V_D$. Scale bar is $1\, \mu m$ on large scan, and $100\, nm$ on zoom-in. c, Hall conductivity $\sigma_{xy}$ and longitudinal conductivity $\sigma_{xx}$ measured with the contacts on the left reservoir of the device, showing states at $\nu = -1/2, -2/5, -1/3, 1+1/3, 1+1/2$. d, Two-dimensional map of the conductance across the interferometer $\Delta G$ as a function of magnetic field $B$ and plunger gate $V_{PG}$ showing clear Aharonov-Bohm oscillations in the five fractional states observed in (c). e, 2D FFT analysis of each panel in (d) indicating the magnetic field periodicity $\Phi_0/\Delta B$ and the plunger gate periodicity $1/\Delta V_{PG}$. f, Magnetic field periodicity $\Phi_0/\Delta B$ in each of the five fractional states used to extract the charge carried by quasiparticles on the interfering edge. The line fit yields a cavity area $A=0.56\, \mu m^2$, close to the lithographic area of $0.7\, \mu m^2$. All measurements were performed at $T=20\, mK$ and $B=9.95\, T$ unless otherwise specified.
• Figure 1: Device contacts, gates, and measurement setup.a, Optical image of measured device, with a scale bar of 10 $\mu\text{m}$. b, Schematic of the gate setup shown in Fig. \ref{['FIG:1']}. Each gate is contacted via a metallic bridge. The top gate and bottom gate set the density in the graphene, while the split gates and plunger gate define the interferometry cavity. The bridge gate controls the AD in the center of the cavity. c, Schematic of the measurement of quantum Hall transport reported in Fig. \ref{['FIG:1']}c and Extended Fig. \ref{['fig:BG_states']}. The metallic contacts connect to the BLG in regions without top and bottom gates, with the silicon back gate doping this region. The voltage probes are labeled $V_{xx}$ and $V_{xy}$ for the longitudinal and Hall measurements, respectively. The source contact is indicated by the AC bias and the drain current is labeled $I_d$. d, Schematic of the measurement for diagonal conductance $G_D = I_d/V_D$ reported throughout the paper. The diagonal voltage drop across the interferometry cavity measures the conductance through the QPCs.
• Figure 1: Density profile sketches of one side of the AD for $\nu_{cav}=1+1/3$ and $-1<\nu_{AD}<2$. In the upper panel, we sketch the various possible density profiles on one side of the AD for the experimental range of $\nu_{cav}=1+1/3$ (blue end) and $-1<\nu_{AD}<2$ (green end). In the middle panel, we then pick the density profiles with the least total variation from the upper panel for each $\nu_{AD}$, as strong variations in charge density are penalized electrostatically. In the lower panels, we then note that we observe jumps experimentally only in $\nu_{AD} \approx -1/3, 2/3, 1+1/3$, which corresponds exclusively to the fillings where there is a 1/3 edge both inside the AD and around it, in the cavity.
• Figure 2: Observation of discrete phase slips with the bridge gate tuning. a, Conductance $\Delta G_D$ across the interferometer as a function of $V_{TG}$ and $V_{brg}$ showing Aharonov-Bohm (AB) oscillations at a fixed $V_{BG} = 0.2425\, V$. In the center of the scan, the AB oscillations turn into rapidly changing wiggles. Absolute value of the change in conductance with respect to $V_{brg}$ shows low magnitude when the oscillations change slowly, and larger magnitude when the oscillations become non-uniform. We identify two regions: one with high $|\delta G_D/\delta V_{brg}|$ in a green colored background, and the other with low $|\delta G_D/\delta V_{brg}|$ in a white background. b, High resolution scan of a section of the green region in (a), showing discrete phase slips. This scan is taken around the same anti-dot (AD) density as in (a) indicated in pink at fixed $V_{BG} = 0.2706\, V$. c, Top: linecut from (b) showing phase slips are induced as the bridge gate is swept. Bottom: linecut from (d) showing smooth AB oscillation, which is the zoomed in region marked by purple window in (a). This smooth background oscillation indicates that that it arises from a change in area, where the bridge gate overhangs the trenches defining the interferometry cavity. d, High resolution scan of a section of the white region in (a), showing no phase slips. This scan is taken around the same AD density as in (a) indicated in purple at fixed $V_{BG} = 0.3133\, V$. All measurements were performed at $T=20\, mK$ and $B=9.95\, T$.
• Figure 2: BLG FQH states.a, Longitudinal resistance $R_{xx}$ of the filling $\nu$ with respect to the displacement field $D$ for $-1<\nu<2$. The five states measured in this experiment $\nu = -1/2, -2/5, -2/3, 1+1/3, \text{ and } 1+1/2$ are persistent throughout large ranges of $D$. The obscuring of the hole-like states ($\nu<0$) for $D<0$ is attributed to issues with our metallic contacts to the BLG layer. b, Quantum Hall plateaus in $\sigma_{xy}$ in the same range. c, Schematic of states present in the measurement, with colors indicating the eight possible ground states of $K0\sigma, K'0\sigma, K1\sigma, K'1\sigma$ for the spin index $\sigma=\pm$, the valley index $K,K'$ and the orbital index $N=0,1$. The five states discussed in this work are indicated by thick black lines. Note that our experiments were carried out only in the cavity states with orbital index $N=1$ (yellow and orange), as these states assume a more complex wavefunction necessary for even-denominator states to develop. All measurements were performed at $T=20\, mK$. and $B=9.95\, T$.