Observation of nonlocal Josephson effect on double InAs nanowires

Short-range coherent coupling of two Josephson junctions (JJs) are predicted to generate a supercurrent in one JJ nonlocally modulated by the phase difference in the other. We report on observation of the nonlocal Josephson effect on double InAs nanowires as experimental evidence of the coherent coupling. We measured one JJ sharing one superconducting electrode with the other JJ and observed switching current oscillation as a control of the nonlocal phase difference. Our result is an important step toward engineering of novel superconducting phenomena with the short-range coherent coupling.

Short-range coherent coupling of two Josephson junctions (JJs) are predicted to generate a supercurrent in one JJ nonlocally modulated by the phase difference in the other.We report on observation of the nonlocal Josephson effect on double InAs nanowires as experimental evidence of the coherent coupling.We measured one JJ sharing one superconducting electrode with the other JJ and observed switching current oscillation as a control of the nonlocal phase difference.Our result is an important step toward engineering of novel superconducting phenomena with the short-range coherent coupling.
The development of Josephson junction (JJ) physics [1] is of significance for exploiting novel coherent macroscopic quantum phenomena and superconducting (SC) device applications in various quantum information technologies [2].In particular, coherent coupling between two JJs is the key for SC circuit designs to engineer qubitqubit couplings and realise a novel SC phase in multiple JJ arrays.Recently, the Andreev molecular state (AMS), a new concept for producing short-range coherent coupling between two JJs on nanowires has been introduced [3][4][5].In the two adjacent JJs, upper (JJU) and lower (JJL), on the nanowires sharing one SC electrode, as described in Fig. 1(a), the respective Andreev bound states (ABSs) [6,7] are hybridised because of the penetration of the ABS wave functions into the shared SC electrode, forming an AMS as the bonding and antibonding states.ABSs have been utilised for the Andreev qubit [8][9][10], and AMS physics holds the possibility of producing short-range coherent coupling between qubits.In such a device, the ABS energy and supercurrent in JJU depend on not only δ U , the phase difference on JJU, but also on the ABS energy and the phase difference, δ L , of JJL.Consequently, the supercurrent in JJU can be controlled nonlocally by manipulation of JJL, which is referred to as the nonlocal Josephson effect [3].The AMS has been observed in a double dot coupled with SCs [11]; however, experimental studies of superconducting transport related to the AMS, such as the nonlocal Josephson effect, have not been reported.
When two nanowires are coherently coupled through an SC electrode, the coupling forms a nonlocal SC correlation on the double nanowire.Such nonlocal SC correlation is an essential ingredient for engineering timereversal invariant topological SCs and can be applied in topological quantum computing with Majorana fermions or parafermions in SC-semiconductor hybrid systems [12][13][14].The nonlocal SC correlation on double nanowires has been addressed in electrically tunable devices [11,15,16].
Experimental study of the nonlocal Josephson effect is an important step to address the phase control of the nonlocal SC correlation; it paves the way for realising time-reversal invariant topological SC devices.
Considering that multiple phase differences dominate the junction properties, the coupled JJs in Fig. 1(a) resemble multiterminal JJs [17][18][19][20][21][22][23][24], providing a platform for topological physics [25][26][27][28].In this sense, establishing the physics of the coupling between JJs will also help arXiv:2112.12960v1[cond-mat.mes-hall]24 Dec 2021 to develop multiterminal JJ physics, as studies on not only devices of a single normal conductor with multiple SC electrodes but also those of two or more coupled JJs on various independently controllable materials may be conducted.
Here, we report experimental evidence of the nonlocal Josephson effect using gate-tunable InAs nanowires to demonstrate the coherent coupling between two JJs on a double nanowire.Our results provide a means to nonlocally control the D.C. Josephson effect.Our strategy for this demonstration is to determine whether the SC correlation between any two SC electrodes exists and then measure the switching current in JJU dependent on δ L .
We prepared two coupled JJ devices, named samples A and B. We chose an InAs double-nanowire structure, created using the selective area growth (SAG) method as this method allows the preparation of two spatially separated parallel nanowires with high yield compared to using self-assembled nanowires [15,16,29].Additionally, aluminium is epitaxially grown on InAs nanowires, which provides a highly transparent interface [30].The two SAG nanowires were spatially separated by 60 nm.JJU and JJL were formed on InAs nanowires U and L, respectively.The SC electrode on the left side is shared by the two JJs, but the two right SC electrodes contact nanowire U and nanowire L separately.Sample A was fabricated into a 3-terminal JJ device (left shared SC, right upper SC, and right lower SC electrodes) on nanowires U and L. A scanning electron microscope (SEM) image of the same structure is shown in Fig. 1(b).For sample B, JJL is embedded in an SC loop to change δ L by a magnetic field (B) penetrating the SC loop.The two gate electrodes were used to electrically control JJU and JJL with gate voltages of V gU and V gL , respectively.The SEM image of sample B is shown in Fig. 1(c).The junction of sample B has the same structure as that of sample A, but the shared and lower SC electrodes are connected to an SC loop.Sample A was used to confirm the SC correlation between any two SC electrodes and sample B to demonstrate the nonlocal control of the D.C. Josephson effect.
First, we measured sample A at 10 mK to study the correlation between the SC electrodes.For this purpose, we simultaneously measured V U and V L , the voltages of the upper and lower SC electrodes with the shared SC grounded by sweeping the bias currents I U and I L through JJU and JJL, respectively (see Fig.   From the supercurrent region, the three additional regions where the differential resistance is reduced are extended, as labelled with SC US , SC LS , andSC UL in Figs.2(b) and (c).These diagonal features have also been reported in multiterminal Josephson junctions [22][23][24].In these regions, the supercurrent flowing between any two contacts remains within these contacts, whereas the dissipative current flows between the other contacts.For example, in SC US , there is considerable reduction, as seen in Fig. 2(b), but not in Fig. 2(c).In addition, SC US is strongly dependent on I U but is slightly sensitive to I L .This means that the supercurrent between the upper and shared SC electrodes remains in SC US with the dissipative current at JJL because |I L | > 60 nA.For similar reasons, SC LS can be assigned to the region where the supercurrent between the lower and shared SC electrodes remains.
As for SC US and SC LS regions, the figures indicate that the differential resistance in the remained supercurrent regions is finite.For example dV U /dI U in SC US is around 300 Ohm at I L ∼ ±400 nA (see supplemental material (SM)).This can be assigned to the Joule heating effect derived from the normal transport in JJL.The heating increases the electron temperature in JJU, resulting in the phase diffusion which produces the finite differential resistance even in the supercurrent region [42].Additionally, SC US and SC LS have finite tilts on the V U V L plane.These tilts can be derived from the cotunneling of the quasiparticles.For example, when a finite potential difference on JJU is present, the current in JJU flows from the upper SC to the shared SC.Additionally, due to the cotunneling through the shared SC, the finite current also flows to the lower SC.This cotunneling current generates the finite tilt of SC LS .
We focus on the diagonal feature SC UL in which both dV U /dI U and dV L /dI L are slightly reduced, as shown in Figs.2(b) and (c).A supercurrent can exist in three pairs of upper, lower, and shared SC electrodes.The supercurrent in the upper and shared SC pair is assigned to SC US , and that in the lower and shared SC pair is assigned to SC LS .Therefore, SC UL can be assigned to the supercurrent between the upper and lower SC electrodes.This result indicates that a nontrivial SC correlation exists between the upper and lower SC electrodes, although the shared SC electrode only intermediates two nanowires; no other material connects the two SC electrodes.This is a signature of the nonlocal SC correlation between the two Josephson junctions.However, the SC UL signal is small and vague because the dissipative current and nonequilibrium quasiparticles coexist with the supercurrent.Therefore, a detailed discussion of the nonlocal SC correlation from these 3-terminal results is difficult.
Then, we use another strategy to control the phase difference δ L in sample B. We measured V U by changing I U and B at several V gU and V gL , the gate voltages on JJU and JJL respectively (see Fig. 3(a)).Fig. 3(b) represents V U vs. I U at B = 0 mT and several V gU .At I U = 105 nA for V gU = 0 V, the supercurrent region is switched to the normal region.The switching current I sw decreases as V gU decreases, indicating that JJU can be controlled electrically.V gU = −5 V provides about half of I sw at V gU = 0 V. Figure 3(c) indicates V U vs. I U at B = 1.9 mT and several V gL .V gL = −4.4V provides about 0.9 of I sw at V gL = 0 V, meaning that I sw of JJU depends weakly on V gL in the present range.
We measured V U vs. I U at various magnetic field strengths.The obtained values of V U as a function of I U and B are shown in Fig. 3(d).As illustrated, the switching current around 100 nA, corresponding to the boundary between the blue and red regions, oscillates with B. The oscillation period is 0.22 mT, which reasonably agrees with the calculated value of 0.30 mT, derived from the superconducting loop area (6.88 µm 2 ).The magnetic field changes only δ L and the phase difference of JJL in the SC loop.Therefore, B would not affect the I sw of JJU if there were no nonlocal SC correlation between JJU and JJL.The obtained switching current oscillation was reproduced in different devices on a single or double nanowire (see SM). Furthermore, we confirmed that this oscillation disappeared as the distance between the two junctions increased, as expected from the theory that the distance should be shorter than the coherence length of SC to form the coupling.Disappearance of the oscillation with the long distance implies that the single JJ on the single nanowire produces no oscillation of the switching current (see SM). Consequently, we present the observation of the nonlocal Josephson effect derived from the coupling between JJU and JJL through the shared SC electrode.Additionally, our observation of JJU switching current in sample B proves coherence of the coupling because the nonlocal phase modulation is observed on the supercurrent which is one of the phase coherent phenomena.
To clarify that the I sw oscillation originates not only from the JJU, but also from the properties of JJL, we studied the gate voltage control of the nonlocal Josephson effect signal.We evaluated the peak-to-peak value and the average of I sw vs. B at the respective gate voltages.First, we indicate the V gU dependence of I sw vs. B with V gL = 0 V, indicating local gate control.Both the peak-to-peak value and the average of I sw change with V gU , as shown in Fig. 4(a).Then, we move onto the V gL dependence with V gU = 0 V; this refers to nonlocal gate control.As shown in Fig. 4(b), the average of I sw is almost constant, whereas the peak-to-peak value decreases as V gL decreases (see SM).This indicates that the average is affected only by the local gate voltage; the oscillation is affected by both the local and nonlocal gate voltages.Therefore, the oscillation originates from the coherent coupling between the two JJs.Finally, we discuss the microscopic origin of the observed coherent coupling between JJU and JJL.In the literature, the coupling in the ballistic junctions is dominantly formed by the double elastic cotunneling through the shared SC.The observed tilt of SC LS in Fig. 2 provides a ratio of the cotunneling current to the current in JJU as around 0.14.This value is similar to a ratio of the peak to peak value to the average of I sw found in Fig. 4. Therefore, the similarity between the cotunneling ratio and the oscillation ratio seems to imply that the coherent coupling in our devices is dominantly originated from the cotunneling process as theoretically expected in the ballistic JJs.
Although our experiments clearly provide evidence of existence of the nonlocal Josephson effect, we only addressed the nonlocal control of the switching current.In order to unveil the local and nonlocal transport on the coupled JJs, it is significant to evaluate the current phase relation, which gives the supercurrent as a function of local and nonlocal phase differences.In addition, microscopic mechanism to produce the coherent coupling should be revealed experimentally.Further studies on the coupled JJs are necessary.
In summary, we confirmed the nonlocal superconducting correlation between two JJs in a 3-terminal device and observed the nonlocal Josephson effect in a double JJ device with a superconducting loop.These demonstrated that the short-range coherent coupling between the two different Josephson junctions can be formed through a superconductor.The observed coherent coupling is not just for the semiconductor-superconductor junctions but expected in various kinds of Josephson junctions coupled through superconductors.Our results pave the way to engineering short-range coherent coupling between superconducting qubits, such as the Andreev qubit [8][9][10], and provide a new building block for designing novel SC devices consisting of the two JJs on the different materials.
When the AMS is formed on the two JJs, the correlation is expected to vanish when the separation between two JJs is greater than the coherence length of the superconductivity between the two JJs.To measure whether the oscillations vanish in a device with a long separation between JJs, we fabricated four devices with the same structure as sample F, but with different separations of 0.2, 0.4, 0.6, and 1.0 µm, on a single SAG nanowire.The SEM image is shown in Fig. S6.The observed I sw values of the coupled JJ devices with different separations L are shown in Fig. S7(a).We note that the quality of the SAG nanowire is inhomogeneous even in a single nanowire; therefore we cannot ensure that all JJs have the same properties.However, the results indicate that a longer distance between two JJs produces less or no oscillation of the switching current.We executed the additional measurement on the single nanowire device with 150 nm separation on the different wafer as shown in Fig. S7(b).As a result, we found bigger amplitude of the oscillation whose ratio to the Isw average is about 0.2.These results indicate that the longer separation device tends to generate less oscillation.
The oscillation vanishes in devices of length 0.6 and 1.0 µm.In this study of the length dependence, we used the single nanowire samples.Therefore, the two JJs are coupled through the proximity region beneath the shared SC.The coherence length is modified as the formula of hv F /π∆ where v F and ∆ indicate the Fermi velocity and the superconducting energy, respectively.Since we cannot pinch off the nanowires due to the interfacial problem, we are not sure how large the carrier density and the Fermi velocity are.However, around 600 nm in the similar InAs/Al systems was reported previously [32,33].This length is comparable to the shortest distance between the two JJs in our devices which did not show the oscillation.Then this length dependence is qualitatively consistent with AMS physics that the nonlocal SC correlation disappears when two JJs are at a larger distance than the coherence length.
We note that the selective area growth nanowires are not uniform and the Josephson junction quality highly depends on the nanowires and also the junction location on the single nanowire.In case the grain boundary or impurities happen to appear in the junction, the junction quality can easily become small.Therefore, we assume that the general trend of decreasing modulation amplitude with increasing distance between nanowires is consistent between the prediction and experiment but only qualitatively.VgU (V)

FIG. 1 .
FIG. 1. Concept of this study and device structure.(a) AMS formation is shown.The ABSs of JJU and JJL penetrate the SC electrodes, as illustrated in the middle panel.When the separation of the two JJs is short, the ABSs in the respective nanowires are hybridised to form an AMS.(b) SEM image of the 3-terminal JJ on a double nanowire.The two JJs share the left SC electrode, but on the right side, the two SC electrodes were placed on the respective nanowires.(c) SEM image of sample B. The shared and lower SC electrodes in (b) are connected by a superconductor loop.

2
(a)).Figs.2(b) and (c) show the measured differential resistances dV U /dI U and dV L /dI L as functions of I U and I L , respectively.In the square region of -30 nA< I U < 30 nA and -60 nA< I L < 60 nA, dV U /dI U and dV L /dI L almost vanish in both figures, indicating that the supercurrent flows between the upper and the shared SC and between the lower and the shared SC.This means that the switching current is almost 30 nA for JJU and 60 nA for JJL.

FIG. 2 .
FIG. 2. 3 terminal measurement of the two coupled Josephson junctions on the double nanowire.(a) shows a schematic of the electrical circuit used for measurement of sample A. (b) and (c) show the differential resistances of JJU and JJL, respectively, as a function of IU and IL, measured simultaneously.The supercurrent flows in both JJs in the central blue regions of -30 nA < IU < 30 nA and -60 nA < IL < 60 nA.In the regions arranged with SCUS, SCLS, and SCUL extending outward from the central blue supercurrent region, the local supercurrent remains between a pair of the two SC electrodes with the dissipative current in the other pairs.

FIG. 3 .
FIG. 3. Switching current oscillation indicating a coherent coupling between two Josephson junctions.(a) shows a schematic of the measurement circuit for sample B .(b) illustrates the IU − VU curves obtained at several VgU points, indicating that the supercurrent in JJU can be tuned by VgU locally.(c) indicates VU vs. IU at B = 1.9 mT and several VgL.The supercurrent in JJU is little controllable by VgL.(d) shows VU as a function of IU and B. The boundary between the red and blue regions corresponds to the switching current in JJU as a function of B. The image clearly indicates the oscillation of the switching current in the JJU.

FIG. 5 .
FIG. 5. (a) shows the conductance as a function of VgL and VgU of sample B measured at 4 K.The line profiles at various VgLs appear in (b).The small measured VgL dependence can be attributed to the cross-capacitance contribution.(c) Resistance as a function of B at 4 K.There appear small fluctuations but no periodic oscillation.

FIG. 6 .FIG. 8 .FIG. 9 .
FIG. 6. VU as a function of IU and B obtained from sample C. A clear oscillation, as seen in Fig. 3 (c), was measured.
Local and nonlocal gate control of the switching current oscillation in JJU.(a) and (b) show the peakto-peak value in red and the average in blue of the switching current in JJU as a function of VgU and VgL, respectively.In the local gate control case in (a), both the amplitude and average change with VgU.However, only the peak-to-peak value largely varies in the non-local gate control case in (b).This means that the switching current oscillation is derived from the hybridisation between JJU and JJL.