# Nonlocal supercurrent of quartets in a three-terminal Josephson junction

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Edited by Eduardo Fradkin, University of Illinois at Urbana–Champaign, Urbana, IL, and approved May 21, 2018 (received for review January 2, 2018)

## Significance

In this work we present detailed studies of a nondissipative current, called “quartet supercurrent,” where two distinct Cooper pairs, originating in different terminals, recombine into a four-electron quasiparticle: a quartet. Employing conductance measurements and highly sensitive cross-correlation of current fluctuation, we identified the existence of such a coherent nonlocal state.

## Abstract

A novel nonlocal supercurrent, carried by quartets, each consisting of four electrons, is expected to appear in a voltage-biased three-terminal Josephson junction. This supercurrent results from a nonlocal Andreev bound state (ABS), formed among three superconducting terminals. While in a two-terminal Josephson junction the usual ABS, and thus the dc Josephson current, exists only in equilibrium, the ABS, which gives rise to the quartet supercurrent, persists in the nonlinear regime. In this work, we report such resonance in a highly coherent three-terminal Josephson junction made in an InAs nanowire in proximity to an aluminum superconductor. In addition to nonlocal conductance measurements, cross-correlation measurements of current fluctuations provided a distinctive signature of the quartet supercurrent. Multiple device geometries had been tested, allowing us to rule out competing mechanisms and to establish the underlying microscopic origin of this coherent nondissipative current.

Superconductivity is an emblematical example of modern condensed matter physics as it manifests a macroscopic phenomenon governed by quantum mechanics, stressing the significance of the phase of a “macroscopic” wavefunction (1). Most striking is the dc Josephson effect (2). An established phase difference Δφ between two superconductors (SCs) leads to a nondissipative supercurrent flow carried by Andreev bound states (ABS, Fig. 1). Biasing the two SCs junctions leads to time evolution of Δφ, resulting in an oscillatory supercurrent: the ac Josephson effect.

In recent years, multiterminal Josephson junctions (MTJs) have been considered as generalizations of the ubiquitous two-terminal junction (3⇓⇓⇓⇓⇓⇓⇓⇓–12). The MTJs are expected to show a wealth of new phenomena thanks to the existence of several independent phase variables. For instance, in equilibrium, dc supercurrent of Cooper pairs can flow from any terminal to another one, with the underlying ABS spectrum possessing topological features, such as robust zero-energy states (9⇓⇓–12). On the other hand, when biasing a three-terminal Josephson junction (3TJ), a new type of “multipair” dc supercurrent may emerge, involving all three terminals (3⇓⇓–6). The simplest multipair quasiparticles appear when *V*_{L} *=* −*V*_{R}, with both voltages applied to the *S*_{L} and *S*_{R} terminals, respectively, relative to the grounded terminal, *S*_{M} (Fig. 1*B*). Under this condition, two Cooper pairs, one emerging from *S*_{L} and one from *S*_{R}, are transferred to *S _{M}* through a quartet, which is composed of four electrons (3⇓⇓–6). As shown in Fig. 1

*B*, this can happen only if

*L*< ξ, with

*L*the size of

*S*

_{M}and ξ is the superconducting coherence length. The quartet forms via “crossed-Andreev reflection” (CAR) (13⇓⇓–16). Evidently, the reversed process also takes place: two Cooper pairs in

*S*

_{M}split (each via a CAR process), and exchange electron partners that recombine to form two spatially separated Cooper pairs in terminals

*S*

_{L}and

*S*

_{R}. An ABS, involving four Andreev reflections within the 3TJ, is formed (Fig. 1

*D*), carrying a nonlocal supercurrent where the current, say, from

*S*

_{M}to

*S*

_{R}, depends on the phase of

*S*

_{L}.

A previous study of the conductance in a diffusive metallic 3TJ already provided a signature of the quartet current (6); however, several alternative models for that current could not be ruled out. Here, we verify an emergent coherent quartet supercurrent in a 3TJ, which is formed in a proximitized semiconducting nanowire. Care was exercised to rule out other possible mechanisms such as multiple Andreev reflections (MAR) (7, 8, 17) and circuit electromagnetic coupling mechanisms (18, 19).

## Quartet Supercurrent

The microscopic mechanism leading to the supercurrent in a short two-terminal superconductor–normal-superconductor Josephson junction (JJ) is described in Fig. 1*C*. An electron impinging on a superconductor with energy smaller than the single-particle superconducting energy gap may enter the superconductor as a Cooper pair while reflecting back a hole via Andreev reflection (AR). When two superconductors are placed in close proximity, an ABS forms, allowing a flow of an equilibrium supercurrent. The magnitude of the supercurrent obeys the energy-phase relation, *E*_{ABS} the energy of the ABS and

In a similar fashion the microscopic mechanism of the quartet supercurrent in a 3TJ is shown in Fig. 1*D*. Due to CAR processes in the small terminal *S*_{M}, an outgoing hole, in response to an incoming electron from one side, propagates toward the opposite terminal on the other side. A unique ABS is formed, with all three superconducting terminals participating via four ARs (3). At equilibrium (*V*_{L} = *V*_{R} = 0), the ABS energy *V* = *V*_{L} = −*V*_{R} (Fig. 1*D*), using *e* in the prefactor is the signature that current is carried by two Cooper pairs, namely, a quartet.

The nonlocal ABS depends on a single-phase *SI Appendix*). Detailed calculations (4, 5, 8) show that the quartet current components in both branches *S*_{L} − *S*_{M} and *S*_{R} *− S*_{M} are equal even if the resistances of the two branches differ. This relates to their nondissipative, energy-conserving, processes. However, accompanying dissipative MAR processes of quasiparticles may not be equal in the two branches.

The existence of quartets relies on mediating CAR processes through the middle contact, *S*_{M}. Considering the geometry shown in Fig. 1*B*, the probability amplitude for a CAR process is expected to be large if

Quantum (nonequilibrium) noise is a powerful probe of quantum correlations in transport. A few comments on noise in Cooper pairs current are due. First, the equilibrium dc two-terminal Josephson current is, in principle, noiseless; namely, the ABSs support only fluctuations free dc current. However, thermal transitions between the two branches of the ABS, with each branch carrying an oppositely propagating current, evidently lead to current fluctuations (23, 24). Similarly, in a 3TJ device, the quartet current contains fluctuations (Fig. 1*E* and *SI Appendix*) (25). The fluctuations in both branches are expected to be positively correlated (25). Note that positive cross-correlations previously provided evidence for CAR in a normal-superconductor–normal Cooper splitting device (16).

## Experimental Setup

Two different configurations of the three-terminal JJ were realized by coupling an aluminum superconducting contact to an InAs nanowire: device type d1––with the central contact *S*_{M} smaller than the superconductor coherence length (Fig. 2*A*); and device type d2––with the central contact *S*_{M} much larger than the superconductor coherence length (Fig. 2*B*).

The InAs nanowires were grown by the gold assisted molecular beam epitaxy process, using the well-established vapor–liquid–solid (VLS) growth technique. Growth was initiated on an unpatterned (100) InAs substrate, where single wires and Y-shaped intersections were simultaneously grown (26). The nanowires were spread on an oxidized P^{+}- doped Si wafer (with 150-nm-thick SiO_{2}), with superconducting contacts and local gates made by depositing 5-nm/120-nm Ti/Al on the wires. The setup allowed measuring the differential conductance and the “zero-frequency” cross-correlation (CC) of the current fluctuations in *S*_{L} and *S*_{R} (*SI Appendix*, section S2). We define the conductance *I*_{L/R} is the current in *S*_{L} or *S*_{R}, and *dV*_{M} is a small ac signal applied to the central contact relative to ground. The dc bias was applied to *S*_{L} and *S*_{R} across two grounded 5Ω resistors.

The induced superconducting energy gap in the nanowire was found to be

## Results and Discussion

### Differential Conductance Measurement.

A color representation of *G*_{L} as a function of *V*_{L} and *V*_{R} in device type d1 is plotted in Fig. 2*C* (an equivalent plot of *G*_{R} is shown in *SI Appendix*, Fig. S2*B*). The equilibrium dc Josephson current is manifested as a wide “horizontal” structure (*V*_{L} = 0) in the “*G*_{L} plot.” In the same plot the Josephson current in the right junction is observed as an attenuated “vertical” structure (being transconductance across the central contact). Similar results are obtained when plotting *G*_{R}.

Most importantly, a pronounced, Josephson-type, high-conductance peak is observed at *V*_{L} = −*V*_{R}, agreeing with the expected signature of the quartets. Other nondissipative processes are manifested by conductance peaks with different slopes, for example the “sextet” line at *V*_{R} = −2*V*_{L} (and *V*_{L} = −2*V*_{R}), which represents a six-electron state (three Cooper pairs) (3⇓–5, 8). In sharp contrast, the plot in Fig. 2*D*, representing the conductance in device type d2, has no sign of nonlocal effects, since CARs are not possible in the large *S*_{M}.

Fig. 2*E* shows traces of *G*_{L} and *G*_{R} as a function of *V*_{L} with *V*_{R} = −16 *µ*V, with the sharp quartets conductance peaks appearing at *V*_{L} = +16 *µ*V. Fig. 2*E* (*Inset*) shows a zoom into the quartet conductance peak in *G*_{R}, demonstrating the characteristic shape of a Josephson-like conductance peak, engulfed in two symmetric conductance dips (28, 29). The quartet supercurrent signature, which results from a coherent ABS, shared by all three terminals, is similar in both *G*_{L} and *G*_{R}, showing that a single microscopic mechanism takes place (Fig. 4*A*). In contrast, one can observe broader conductance peaks, appearing only in *G*_{L}. This results from MAR between *S*_{L} and *S*_{M}, with a width dictated by the interterminal transparencies and the quasiparticle density of states at the superconducting leads. The vertical displacement between *G*_{L} and *G*_{R} (Fig. 4*A*) is explained in *SI Appendix* using a simple resistively shunted junction model (*SI Appendix*, section S1*C*).

Tuning the transparency of each junction by the back-gate voltage, the differential conductance of *S*_{L} and *S*_{R} to ground is plotted in Fig. 3. The mutual correspondence between the left and right quartet currents is clearly demonstrated. Pinching the right-hand junction (with negative *V*_{GR}) quenches the quartet anomaly on both sides, hence confirming the observed correlation of quartet currents.

### CC of Current Fluctuations: Experiment and Theory.

To further verify the quartet anomaly we utilize a highly sensitive measurement of CC of current fluctuations in the left and right segments of 3TJ. In Fig. 4*A* we plot line-cuts of the differential conductance *G*_{L} and *G*_{R} as well as the corresponding CC as a function of *V*_{L} for *V*_{R} *= −*15 *µ*V. A pronounced positive CC peak coincides with a peak in the differential conductance (Fig. 4*B*). This coincidence indicates that the current fluctuations in both sides of the device are positively correlated, as indicated by the currents anomaly, and thus cannot be a result of MAR processes. Moreover, the latter, involving fermionic quasiparticles dressed by Cooper pairs, would rather lead to a negative CC signal (30) (*SI Appendix*, section S*1D*). The evolution of the CC signal along the quartet conductance line, *V = V*_{L} *=* −*V*_{R} (Fig. 4*C*, *Upper*), agrees qualitatively with numerical calculations based on nonequilibrium Green’s functions (25) (*SI Appendix*). These calculations demonstrate that the quantum noise and the CC are both expected to be nonmonotonic with the voltage *V*, and thus reflect the nonadiabatic transitions between the two branches of the ABS (when a multiple of 2 eV matches the ABS spacing that varies with *A*) is ascribed to MAR processes (7, 8, 17, 31, 32).

### Can an “Extrinsic” Effect Mimic the Quartet Current?

An argument that relates the quartet conductance peak to an “extrinsic cause” should be addressed. Under quartet biasing condition, *V* = *V*_{L} = −*V*_{R}, the nanowire system is expected to generate two oscillating Josephson currents, with matching frequencies, *d1* and d2*,* having similar electromagnetic environments (identical circuit coupling mechanisms), should have displayed the same anomalies, which they do not.

One may also argue that coupling between the two junctions is possible via the common resistance in the middle branch. This scenario was experimentally tested in coupled junctions at temperature close to *Tc* by Jillie et al. (18), and further discussed by Likharev (19). Testing the effect, a common resistance was performed by measuring the CC in the normal state. No CC signal was found, suggesting that the common resistance is much smaller than that of the individual junctions (*SI Appendix*).

## Summary

We presented a detailed study of a nonlocal, coherent, Josephson current, under strong nonequilibrium conditions in a 3TJ. The dc nondissipative supercurrent was measured in an InAs nanowire in proximity to an aluminum superconductor. CAR processes led to a many-body quantum state, involving quartets, each composed of two Cooper pairs. Measurements of nonlocal conductance and CC of current fluctuations, performed on two types of 3TJ devices, provided a definite signature of the quartet supercurrent. Alternative mechanisms that may have produced similar effects are disproved. We provide theoretical support and estimates that agree qualitatively with the measured quantities.

## Acknowledgments

D.F. and R.M. acknowledge support from L’Agence Nationale de la Recherche Nanoquartets 12-BS-10-007-04 and the Centre Régional Informatique et d’Applications Numériques de Normandie computing center. M.H. acknowledges the partial support of the Israeli Science Foundation (ISF), the Minerva Foundation, and the European Research Council (ERC) under the European Community’s Seventh Framework Program (FP7/2007-2013)/ERC Grant Agreement 339070. H.S. acknowledges partial support by ISF Grant 532/12, and Israel Ministry of Science and Technology (IMOST) Grants 0321-4801 and 3-8668. H.S. is incumbent of the Henry and Gertrude F. Rothschild Research Fellow Chair.

## Footnotes

↵

^{1}Y.C. and Y.R. contributed equally to this work.- ↵
^{2}To whom correspondence should be addressed. Email: Moty.Heiblum{at}weizmann.ac.il.

Author contributions: Y.C., Y.R., M.H., and H.S. designed research; Y.C., Y.R., J.-H.K., M.H., D.F., R.M., and H.S. performed research; Y.C., Y.R., M.H., D.F., and R.M. contributed new reagents/analytic tools; Y.C., Y.R., D.F., and R.M. analyzed data; and Y.C., Y.R., M.H., D.F., R.M., and H.S. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1800044115/-/DCSupplemental.

Published under the PNAS license.

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