Coherent transport through a Majorana island in an Aharonov-Bohm interferometer

TL;DR

This study demonstrates phase-coherent single-electron transport through a Majorana island embedded in an Aharonov-Bohm interferometer, providing evidence for electron teleportation via Majorana modes in a topological superconductor system.

Contribution

First experimental observation of flux-periodic conductance oscillations indicating coherent electron transport through a Majorana island in an interferometer.

Findings

Persistent 1e-periodic Coulomb blockade peaks at zero energy

h/e flux period oscillations in conductance

Evidence of electron teleportation via Majorana modes

Abstract

Majorana zero modes are leading candidates for topological quantum computation due to non-local qubit encoding and non-abelian exchange statistics. Spatially separated Majorana modes are expected to allow phase-coherent single-electron transport through a topological superconducting island via a mechanism referred to as teleportation. Here we experimentally investigate such a system by patterning an elongated epitaxial InAs-Al island embedded in an Aharonov-Bohm interferometer. With increasing parallel magnetic field, a discrete sub-gap state in the island is lowered to zero energy yielding persistent 1e-periodic Coulomb blockade conductance peaks (e is the elementary charge). In this condition, conductance through the interferometer is observed to oscillate in a perpendicular magnetic field with a flux period of h/e (h is Planck's constant), indicating coherent transport of single electrons through the islands, a signature of electron teleportation via Majorana modes, could also be observed, suggesting additional non-Majorana mechanisms for 1e transport through these moderately short wires.

Full text

Coherent transport through a Majorana island

in an Aharonov-Bohm interferometer

A. M. Whiticar

These authors contributed equally to this work.

Center for Quantum Devices, Niels Bohr Institute, University of Copenhagen and Microsoft Quantum Lab Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark

  

A. Fornieri

These authors contributed equally to this work.

Center for Quantum Devices, Niels Bohr Institute, University of Copenhagen and Microsoft Quantum Lab Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark

  

E. C. T. O’Farrell

Center for Quantum Devices, Niels Bohr Institute, University of Copenhagen and Microsoft Quantum Lab Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark

  

A. C. C. Drachmann

Center for Quantum Devices, Niels Bohr Institute, University of Copenhagen and Microsoft Quantum Lab Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark

  

T. Wang

Department of Physics and Astronomy and Microsoft Quantum Lab Purdue, Purdue University, West Lafayette, Indiana 47907 USA

Birck Nanotechnology Center, Purdue University, West Lafayette, Indiana 47907 USA

  

C. Thomas

Department of Physics and Astronomy and Microsoft Quantum Lab Purdue, Purdue University, West Lafayette, Indiana 47907 USA

Birck Nanotechnology Center, Purdue University, West Lafayette, Indiana 47907 USA

  

S. Gronin

Department of Physics and Astronomy and Microsoft Quantum Lab Purdue, Purdue University, West Lafayette, Indiana 47907 USA

Birck Nanotechnology Center, Purdue University, West Lafayette, Indiana 47907 USA

  

R. Kallaher

Department of Physics and Astronomy and Microsoft Quantum Lab Purdue, Purdue University, West Lafayette, Indiana 47907 USA

Birck Nanotechnology Center, Purdue University, West Lafayette, Indiana 47907 USA

  

G. C. Gardner

Department of Physics and Astronomy and Microsoft Quantum Lab Purdue, Purdue University, West Lafayette, Indiana 47907 USA

Birck Nanotechnology Center, Purdue University, West Lafayette, Indiana 47907 USA

  

M. J. Manfra

Department of Physics and Astronomy and Microsoft Quantum Lab Purdue, Purdue University, West Lafayette, Indiana 47907 USA

Birck Nanotechnology Center, Purdue University, West Lafayette, Indiana 47907 USA

School of Materials Engineering, Purdue University, West Lafayette, Indiana 47907 USA

School of Electrical and Computer Engineering, Purdue University, West Lafayette, Indiana 47907 USA

  

C. M. Marcus

marcus@nbi.ku.dk

Center for Quantum Devices, Niels Bohr Institute, University of Copenhagen and Microsoft Quantum Lab Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark

  

F. Nichele

fni@ibm.zurich.com

Present address: IBM Research - Zurich, Sumerstrasse 4, 8803 Rschlikon, Switzerland.

Center for Quantum Devices, Niels Bohr Institute, University of Copenhagen and Microsoft Quantum Lab Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark

Majorana zero modes are leading candidates for topological quantum computation due to non-local qubit encoding and non-abelian exchange statistics. Spatially separated Majorana modes are expected to allow phase-coherent single-electron transport through a topological superconducting islands via a mechanism referred to as teleportation. Here we experimentally investigate such a system by patterning an elongated epitaxial InAs-Al island embedded in an Aharonov-Bohm interferometer. With increasing parallel magnetic field, a discrete sub-gap state in the island is lowered to zero energy yielding persistent 1$e$-periodic Coulomb blockade conductance peaks ($e$ is the elementary charge). In this condition, conductance through the interferometer is observed to oscillate in a perpendicular magnetic field with a flux period of $h/e$ ($h$ is Planck’s constant), indicating coherent transport of single electrons through the islands, a signature of electron teleportation via Majorana modes.

I Introduction

Initial experiments reporting signatures of Majorana zero modes (MZMs) in hybrid superconductor-semiconductor nanowires focussed on zero-bias conductance peaks (ZBPs) using local tunnelling spectroscopy Mourik2012b ; Deng2016 ; Nichele2017 ; Zhang2018 . Subsequently, Majorana islands provided additional evidence of MZMs based on nearly 1$e$-spaced Coulomb blockade (CB) peaks Albrecht2016b , and indicated a Rashba-like spin orbit coupling with the spin-orbit field lying in-plane, perpendicular to the wire axis  OFarrell2018 . Under some circumstances, these signatures can be mimicked by trivial modes Chiu2017 ; Moore2018d ; prada2019andreev , motivating a new generation of experiments that explicitly probe non-local properties, which are more difficult to mimic. For instance, non-locality of MZMs was recently investigated by measuring the energy splitting induced by the interaction of a quantum dot and a zero-energy state in a hybrid nanowire Deng2018 .

Non-locality can also be accessed by interferometric measurements of a Majorana island, where CB couples separated MZMs and fixes fermion parity Fu2010 ; Sau2015a ; Vijay2016 ; Hell2018b ; Drukier2018 . In the topological regime, a Majorana island can coherently transfer a single-electron between its two ends through MZMs Fu2010 ; Sau2015a . To demonstrate the effect, a Majorana island can be embedded in the arm of an Aharonov-Bohm (AB) interferometer. If single-electron transport in both the reference arm and the Majorana island is coherent, conductance through the interferometer is expected to show oscillations with a flux period $h/e$ Aharonov1959 ; Fu2010 . In addition, interferometry offers a way to distinguish between localized trivial modes and MZMs Sau2015a ; Hell2018b . This technique was used to investigate coherent transport in semiconductor quantum dots Yacoby1996 ; Schuster1997a ; Avinun-Kalish2005 ; Edlbauer2017 .

In this work, we study coherent single-electron transport through a Majorana island with an AB interferometer. At high magnetic fields, we observe $1e$ periodic CB peaks due to a discrete zero-energy state. In this case, we observe the conductance through the interferometer to oscillate periodically for a flux $h/e$ piercing the AB ring, while oscillations are suppressed when the island shows a $2e$ periodicity or is in the normal state. The observation of conductance oscillations in the $1e$ regime indicates coherent single-electron transport through the Majorana island, as predicted for electron teleportation mediated by MZMs Fu2010 .

II Results

II.1 Majorana island interferometer.

Devices were fabricated using an InAs two-dimensional electron gas (2DEG) heterostructure covered by 8 nm of epitaxially grown Al Shabani2016a . The bare 2DEG (without Al) showed a phase coherence length of $l_{\phi}\sim 4~{}\mu\rm{m}$ (see Supplementary Figure 1). Figure 1a shows a micrograph of device 1 with a 1.2 $\mu$m long and 0.1 $\mu\rm{m}$ wide superconducting Al stripe formed by wet etching. Ti/Au top-gates were evaporated on top of a 25 nm HfO2 dielectric grown by atomic layer deposition. We studied two lithographically similar interferometers with circumferences $L_{\rm loop}$ of 5.6 $\mu$m for device 1 and 5 $\mu$m for device 2.

Applying a negative voltage, $V_{\rm{pg}}$, to the central gate serves two purposes. It depletes the 2DEG surrounding the Al wire to form both the Majorana island and the AB ring center and also adjusts the chemical potential and charge occupancy of the island. Energizing all exterior gates confines the 2DEG into an AB interferometer by connecting the Majorana island to a normal conducting reference arm. The resistance of the reference arm was independently tuned by a negative gate voltage $V_{\rm{c}}$. A source-drain bias voltage ($V_{\rm{sd}}$) was applied to one lead and the resulting current and four-terminal voltage was recorded. The in-plane magnetic fields $B_{\parallel}$ and $B_{\rm{t}}$, and perpendicular field, $B_{\perp}$, were controlled by a three-axis vector magnet.

At low temperatures, tunneling of single electrons onto a Majorana island with a superconducting gap $\Delta$ is suppressed by CB, except at charge degeneracies. When the lowest sub-gap state energy, $E_{\rm{0}}$, exceeds the charging energy $E_{\rm{c}}$, ground-state degeneracies only occur between even-occupied states, resulting in 2$e$-periodic CB conductance peaks Hekking1993a . Odd-occupied ground states are lowered into the accessible spectrum by a Zeeman field, resulting in even-odd CB peak spacing when $0<E_{\rm{0}}<E_{\rm{c}}$. The difference in peak spacings between even and odd states, $S=S_{\rm{e}}-S_{\rm{o}}$, is proportional to $E_{\rm{0}}$ Albrecht2016b (see Fig. 1b). For well-separated MZMs, $E_{\rm{0}}$ tends exponentially toward zero, yielding 1$e$ periodic CB peaks with a discrete zero-bias state at consecutive charge degeneracy point VanHeck2016 ; Albrecht2016b . Both observations are necessary for a MZM interpretation. When MZMs are not widely separated, CB peak spacings oscillate with field and chemical potential OFarrell2018 ; Albrecht2016b ; Chiu2017 .

II.2 Coulomb blockade spectroscopy.

We first investigated the Majorana island without interferometry by depleting a segment of the reference arm (see Fig. 1a). Figure 1b shows zero-bias differential conductance $G=dI/dV$ of the island as a function of parallel magnetic field $B_{\parallel}$ and gate voltage $V_{\rm{pg}}$, which controls the electron occupancy and chemical potential of the island. CB peaks are 2$e$ periodic at zero field and split around 2 T, becoming 1$e$ periodic as the sub-gap state moves toward zero energy (see Fig. 3a for peak spacing analysis). Performing CB spectroscopy, that is, measuring $G$ as a function of both source-drain bias $V_{\rm{sd}}$ and $V_{\rm{pg}}$ reveals Coulomb diamonds (Fig. 1c-f). At low $B_{\parallel}$, diamonds are 2$e$ periodic with distinct negative differential conductance (Fig. 1c), which transition to an even-odd peak spacing difference at moderate fields (Fig. 1d), similar to previous work on superconducting Coulomb islands Tuominen1992 ; Eiles1993 ; Hekking1993a ; Higginbotham2015b ; Albrecht2016b ; Shen2018 ; OFarrell2018 . At high fields, the 1$e$ periodic diamonds show a discrete ZBP for consecutive charge degeneracy points that is well separated from the superconducting gap (Fig. 1e). This sub-gap feature remained at zero bias until the superconducting gap closure, and persists for 3 mV in $V_{\rm{pg}}$, corresponding to an energy range of 0.8 meV. For systems with strong spin-orbit interaction, a helical gap of $g\mu_{\rm B}B_{\parallel}$ is expected, where $\mu_{\rm B}$ and $g$ are the Bohr magneton and electron g-factor, respectively. We estimate a helical gap of $\sim 0.6$ meV at $B_{\parallel}=2.5$ T for a g-factor of 4, comparable to the span of the ZBP in $V_{\rm{pg}}$. The stability of the zero-bias state in both magnetic field and in chemical potential is consistent with the MZM picture (see Supplementary Figure 2) Albrecht2016b , however, the observation of coherent single-electron transport is needed to draw conclusions about non-locality. Below, we additionally show that the zero-bias state is sensitive to rotations of the in-plane field. The magnitude of $B_{\parallel}$ where 1$e$ periodicity is observed is in agreement with ZBPs measured in tunneling spectroscopy in InAs 2DEGs Nichele2017 . In contrast, as a function of $B_{\rm{t}}$ the peak spacing showed an oscillating even-odd periodicity, and no consecutive ZBPs were observed (see Supplementary Figure 3d-f), as expected for extended modes in the wire OFarrell2018 ; prada2019andreev ; stanescu2013dimensional . The normal state 1$e$ regime of the Majorana island appears above $B_{\parallel}\sim 3$ T with $E_{\rm{c}}=80~{}\mu$eV (Fig. 1f), where no discrete ZBPs are observed (see Supplementary Figure 2 and 3a-c).

II.3 Interferometry and coherent single-electron transport.

The reference arm of the AB interferometer was connected to the Majorana island by tuning $V_{\rm{c}}$ from -8 V to -3 V while $V_{\rm{pg}}$ was compensated. This lifted the overall conductance by opening a path through the reference arm (see Fig. 2a-d). Figure 2e-h shows the conductance $\Delta G$ through the full interferometer (with smooth background subtracted; see methods) as a function of $B_{\perp}$ and gate voltage $V_{\rm{pg}}$, which control the flux in the interferometer and occupancy of the island, respectively. Figure 2e shows small oscillations in $\Delta G(B_{\perp})$ at $B_{\parallel}=0~{}\rm{T}$ for the 2$e$ periodic peaks. For $B_{\parallel}=2.2$ T, where the peak spacing is even-odd (Fig. 2f), the conductance showed moderate oscillations with a period $\Delta B_{\perp}=1.5$ mT. This periodicity corresponds to a single flux quantum $h/e$ threading an area of $\sim 2.7~{}\mu\rm{m}^{2}$, consistent with the ring center area defined by $V_{\rm{pg}}$. This indicates coherent $1e$ transport through both the reference arm and the Majorana island. At $B_{\parallel}=2.7$ T, the CB peak spacing is uniformly 1$e$, and oscillation amplitude is maximal (see Figs. 2g). When the Majorana island is driven normal, $B_{\parallel}>3$ T, conductance oscillations are reduced, becoming comparable to the low field oscillations (Fig. 2h). The appearance of strong $h/e$ periodic conductance oscillations in the 1$e$ regime of the island is a key experimental signature of electron teleportation.

The strength and periodicity of the oscillations are examined more quantitatively using Fourier power spectrum (PS) analysis (see methods). In Figs. 2i-l, the PS of $\Delta G(B_{\perp})$ are shown. Increasing $B_{\parallel}$ led to a peak appearing around $f=0.65~{}\rm{mT}^{-1}$, the frequency expected for AB interference. The PS is maximized in the 1$e$ regime. To quantify the oscillations amplitude, $\langle\tilde{A}\rangle$, we average the integrated PS (see methods).

We next correlate the $B_{\parallel}$ dependence of the oscillations amplitude, $\langle\tilde{A}\rangle(B_{\parallel})$, with the $B_{\parallel}$ dependence of the lowest sub-gap state, $E_{\rm{0}}(B_{\parallel})$, of the island. The sub-gap energy is found from the difference between even and odd CB peak spacings, averaged separately, $\langle S\rangle=\langle S_{\rm{e}}\rangle-\langle S_{\rm{o}}\rangle$ (see Fig. 1b). In Figure 3a, $\langle S\rangle$ remains constant as a function of $B_{\parallel}$ (indicating 2$e$ transport) until a sub-gap state moves below $E_{\rm{c}}$, reaching zero at 2.2 T without overshoot (as expected for well separated MZMs in a long wire VanHeck2016 ; Albrecht2016b ). At low fields, where the CB periodicity is 2$e$, the oscillation amplitude $\langle\tilde{A}\rangle$ is small (Fig. 3b). When $\langle S\rangle$ approaches zero at high fields ($B_{\parallel}>2$ T), $\langle\tilde{A}\rangle$ exhibits a sharp increase that coincides with the 2$e$ to 1$e$ transition. Above 3 T, the device is in the normal state and $\langle\tilde{A}\rangle$ returned to the low value found in the 2$e$ regime. This comparison shows that the oscillation amplitude is correlated with the energy of the lowest subgap state, and is maximal in the 1$e$ superconducting regime, as expected for electron teleportation.

Figure 3c,d shows a similar study for device 2. In Fig. 3c, $\langle S\rangle$ shows strong even-odd below 1 T, fluctuates around $\langle S\rangle=0$ between 1-2 T, then settles to 1$e$ ($\langle S\rangle=0$) above 2 T. CB spectroscopy reveals a discrete state that oscillates around zero bias in both $B_{\parallel}$ and $V_{\rm{pg}}$ without forming a stable $1e$ periodic zero-bias peak (see Supplementary Figure 5). These oscillations about zero energy are compatible with hybridized Majorana modes due to wavefunction overlap resulting from the finite island length prada2019andreev ; stanescu2013dimensional . This overlap causes an energy splitting that oscillates both in field and chemical potential OFarrell2018 ; Albrecht2016b ; Shen2018 . Figure 3d shows that phase coherent transport first appears above 1 T and $\langle\tilde{A}\rangle$ gradually increases until reaching a maximum amplitude for 1$e$ peak spacing at 2.1 T, before diminishing in the normal state. In comparison to device 1, phase coherent transport appears when $\langle S\rangle$ oscillations about zero, suggesting that hybridized Majorana modes characterized by an extended wavefunction may also contribute to coherent transport. The absence of energy oscillations may distinguish non-local MZMs from hybridized Majorana modes (see Fig. 3a,c). We attribute the reduced oscillation amplitude in device 1 to result from the longer loop length $L_{\rm loop}>l_{\phi}$, leading to increased dephasing in the reference arm.

We observe that conductance oscillations measured on opposite sides of a CB peak in the $1e$ regime are out of phase (see yellow ticks in the insets of Fig. 3) indicating a transmission phase shift of $\pi$ is acquired when the parity of the island is flipped. This demonstrates interferometric detection of island parity, which offers a way to detect MZM parity as proposed by several recent topological qubit schemes  Karzig2017b ; Plugge2017 ; Vijay2016 . In some cases, we found that the the phase shift is restored through the CB valley, such that the same sides of adjacent CB peaks have the same phase. What determines whether there is phase recovery in the CB valley is not currently understood (see Supplementary Figure 6) Drukier2018 ; Schuster1997a ; Avinun-Kalish2005 ; Edlbauer2017 .

The angular dependence of the in-plane magnetic field is investigated by fixing the field magnitude $B_{\rm{r}}=2.5~{}\rm{T}$ and rotating the field by an angle $\alpha$ (see Fig. 1). Theoretically, a rotation of the in-plane field towards the Rashba field direction is expected to close the topological gap Osca2014 . Figure 4a shows 1$e$ periodic Coulomb diamonds at $B_{\parallel}=2.5$ T with a discrete ZBP at each charge degeneracy point (similar to Fig. 1e). Rotating by an angle $\alpha=5^{\circ}$ lifted the discrete state from zero energy, leading to even-odd peak spacing; at $\alpha=10^{\circ}$, $1e$ periodicity is recovered, though without a discrete ZBP. The observed sensitivity of the zero-energy state to in-plane field rotation is consistent with MZMs Osca2014 .

Small rotations ($|\alpha|<7.5^{\circ}$) reduced the oscillation amplitude, $\langle\tilde{A}\rangle$, as expected for even-odd periodicity (see Fig. 3). However, at larger angles ($|\alpha|>10^{\circ}$) where the discrete ZBP is absent, a strong interference signal is observed (Fig. 4d). The observation of coherent transport in the absence of a discrete zero-energy state suggests trivial extended modes are also phase coherent over the length of the island. Therefore, the additional information provided by bias spectroscopy is needed to distinguish teleportation from other coherent transport mechanisms, as shown in Figs. 4a-c.

We further studied the effect of different magnetic field direction. The results are shown in Supplementary Figure 7. In summary, all three axes showed coherent transport, with oscillation amplitude first increasing as $\langle S\rangle$ approached zero. This shows that the oscillation amplitude is dictated by the energy $E_{\rm{0}}$ in all field directions and indicates that interference is not unique to a parallel magnetic field.

Finally, we comment on the physical mechanism that correlates the oscillation amplitude to the energy of $E_{\rm{0}}$. At low fields, the Majorana island favours an even parity where transport of electrons occurs as two sequential tunneling events on either end of the island Hekking1993a ; Eiles1993 . The two electrons acquire the condensate phase when forming a Cooper pair, which suppresses single-electron coherence. At moderate fields, a discrete sub-gap state is brought below $E_{\rm{c}}$ and a single-electron transport channel is opened, allowing coherent resonant tunneling through the Majorana island. When the discrete state is brought to zero-energy, the contribution of coherent transport is increased due to electron teleportation. Finally, in the normal state, we interpret the reduction in interference signal to reflect the short coherence length in the diffusive Al wire.

In conclusion, we report signatures of single-electron teleportation via non-local MZMs using AB interference in combination with spectroscopy of a discrete zero-energy state. Our results also reveal that coherent transport by topologically trivial modes extending over the full length of the Majorana island are allowed. These extended trivial modes may be precursors of topological states in a finite length system that could transition into non-local MZMs by adjusting experimental parameters stanescu2013dimensional . We have shown that interferometry accompanying bias spectroscopy revealing stable 1$e$ periodic CB in magnetic field and chemical potential can discriminate non-local MZMs from extended modes (that display characteristic energy oscillations). Increasing the wire length to greatly exceed the diffusive coherence length $\xi=\sqrt{\xi_{0}~{}l_{e}}\sim 1~{}\mu$m (for $\Delta=75~{}\mu$eV at $B_{\parallel}=2.5$ T), where $\xi_{0}$ is the clean coherence length and $l_{e}\sim~{}300$ nm is the semiconducting mean free path will suppress $1e$ transport via trivial extended modes  tinkham2004introduction . The observation of coherent transport through the island rules out localized ABS at the ends of the wire as the source of the studied zero-bias state. Indeed, transport flips the parity of localized modes and suppresses interference, while transport through MZMs preserves island parity and coherent transport Sau2015a ; Hell2018b . These localized modes could have been a possible interpretation of the previously observed ZBPs in single-end tunneling experiments Moore2018d ; prada2019andreev .

These results suggest that InAs-Al 2DEGs are a promising route towards more complex experiments related to the braiding or fusion of MZMs. We have established coherent transport and parity readout from the transmission phase shifts in Majorana islands, two key results for future topological qubit networks Karzig2017b ; Plugge2017 ; Vijay2016 . Future devices will take advantage of improved material quality to allow for increased wire lengths to suppress coherent trivial quasiparticle transport, allowing MZM contributions to be better separated from other potential contributions.

Methods

II.4 Wafer structure.

The devices were fabricated on wafers grown by molecular beam epitaxy on a InP substrate. The wafer stack consists of a 1 $\rm{\mu m}$ graded $\rm{In_{1-x}Al_{x}As}$ insulating buffer, a $4~{}\rm{nm}$ $\rm{In_{0.81}Ga_{0.19}As}$ bottom barrier, a $5~{}\rm{nm}$ InAs quantum well, and a top barrier consisting of $5~{}\rm{nm}$ $\rm{In_{0.9}Al_{0.1}As}$ for device 1 and $10~{}\rm{nm}$ $\rm{In_{0.81}Ga_{0.19}As}$ for device 2. A $7~{}\rm{nm}$ film of epitaxial Al was then grown in-situ without breaking the vacuum of the chamber. The InAs 2DEGs were characterized with a Hall bar geometry (Al removed), which showed a peak mobility of $\mu=17,000~{}\rm{cm^{2}V^{-1}s^{-1}}$ for an electron density of $n=1.7\times 10^{12}~{}\rm{cm^{-2}}$ and $n=7.5\times 10^{11}~{}\rm{cm^{-2}}$ for device 1 and device 2, respectively.

II.5 Device fabrication.

Devices were fabricated using standard electron beam lithography techniques. The devices were electrically isolated by etching mesa structures by first removing the top Al film with Al etchant Transene D, followed by a deep III-V chemical wet etch $\rm{H_{2}O:C_{6}H_{8}O_{7}:H_{3}PO_{4}:H_{2}O_{2}}$ (220:55:3:3). Next, the Al film on the mesa was selectively etched with Al etchant Transene D to produce the Al strip. A $25~{}\rm{nm}$ thick layer of insulating $\rm{HfO_{2}}$ was grown by atomic layer deposition at a temperature of $90^{\circ}\mathrm{C}$ over the entire sample. Top gates of Ti/Au (5/25$~{}\rm{nm}$) were then evaporated and connected to bonding pads with leads of Ti/Au (5/250$~{}\rm{nm}$).

II.6 Measurements.

Electrical measurements were performed by standard lock-in techniques at 166 Hz by applying the sum of a variable dc bias voltage $V_{\rm{sd}}$ and an ac excitation voltage of 3 to 10 $\mu\rm{V}$ applied to one of the top ohmic contacts as shown in Fig. 1a. The resulting current across the device was recorded by grounding a bottom ohmic via a low-impedance current-to-voltage converter, and the four terminal voltage was measured by an ac voltage amplifier with an input impedance of $500~{}\rm{M\Omega}$. All measurements were taken in a dilution refrigerator with a base temperature of $20~{}\rm{mK}$ and an electron temperature of $40~{}\rm{mK}$ estimated by the temperature dependence saturation of ZBP conductance Nichele2017 .

II.7 Data analysis.

To highlight the oscillating components of the differential conductance $G(B_{\perp})$, a smooth background was subtracted with a low-degree polynomial Savitzky-Golay filter resulting in $\Delta G$  Savitzky1964a . Analysis of the oscillations was performed by first performing a fast Fourier transform $F(f)$ of $\Delta G(B_{\perp})$ using a Hanning window then calculating the power spectral density $PS(f)=|F(f)|^{2}$. The oscillation amplitude $\langle\tilde{A}\rangle$ was obtained by averaging integrated power spectra. The integration was limited to a band in frequency between $f_{1}=0.55~{}\rm{mT}^{-1}$ and $f_{2}=0.75~{}\rm{mT}^{-1}$, spanning the range of a single flux quantum $\Phi_{0}=h/e$ through the area $A$ defined by either the central gate $(A_{1}=2.25~{}\mu\rm{m}^{2})$ or the exterior gates $(A_{2}=3.1~{}\mu\rm{m}^{2})$, where $f=A/\Phi_{0}$ (see Figs. 2i-l).

III Supplementary Information