Proposal for detecting the $\pi-$shifted Cooper quartet supercurrent

The multiterminal Josephson effect aroused considerable interest recently, in connection with theoretical and experimental evidence for correlations among Cooper pairs, that is, the so-called Cooper quartets. It was further predicted that the spectrum of Andreev bound states in such devices could host Weyl-point singularities. However, the relative phase between the Cooper pair and quartet supercurrents has not yet been addressed experimentally. Here, we propose an experiment involving four-terminal Josephson junctions with two independent orthogonal supercurrents, and calculate the critical current contours (CCCs) from a multiterminal Josephson junction circuit theory. We predict a generically $\pi$-shifted contribution of both the local or nonlocal second-order Josephson harmonics. Furthermore, we show that these lead to marked nonconvex shapes for the CCCs in zero magnetic field, where the dissipative state reenters into the superconducting one. Eventually, we discuss distinctive features of the non-local Josephson processes in the CCCs. The experimental observation of the latter could allow providing firm evidence of the $\pi$-shifted Cooper quartet current-phase relation.

In spite of intense experimental efforts for observing signatures of the quartet state and its new physics beyond the standard Resistively Shunted Josephson Junction model [34][35][36], novel schemes are necessary for ascertaining the Cooper quartets. When driving current between pairs of contacts in a multiterminal Josephson junction with an even number 2n of superconducting leads, n equations of current conservation are imposed by the external circuit. Those n constraints (for a total of 2n phase variables) allow for supercurrent inside a region in phase space parameterized by 2n − n ≡ n independent variables. With four terminals, a DC supercurrent is thus established within a two-dimensional region in the * regis.melin@neel.cnrs.fr plane of the bias currents, separated from the resistive state by a one-dimensional critical current contour (CCC). In a recent work, Pankratova et al. [39] reported nonconvex shapes in the CCCs of four-terminal semiconductor-superconductor Josephson junctions. However, these nontrivial features appeared only at rather high magnetic fields, corresponding to about half a flux quantum threading the central part of the device. The observation of nonconvex CCCs was interpreted using Random Matrix Theory, assuming time-reversal symmetry breaking, either due to an applied magnetic field or preexisting in the normal state [39].
Here, we demonstrate that in the presence of at least one contact with an intermediate transmission, another mechanism for the emergence of nonconvex CCCs is possible, which does not require a magnetic field. Namely, we find correspondence between the quartet physics and the emergence of nonconvex sharp-angled points in the CCCs at zero magnetic field. This distinctive signature stems from the interference between symmetric quartet channels, which are dephased by a transverse supercurrent (see Fig. 1). In other words, we demonstrate that macroscopic critical current measurements can probe the microscopic internal structure of entangled split Cooper pairs [2][3][4][5][6][7][8][9].
The article is organized as follows. The π-shifted quartets are introduced in Sec. II. The device and the model are presented in Sec. III. The numerical results, analytical and numerical, are presented and discussed in Sec. IV. Concluding remarks are provided in Sec. V.

II. π-SHIFTED COOPER QUARTETS
In this section, we provide physical arguments supporting the π-shifted Cooper quartet current-phase relation. The key underlying concept can readily be understood starting from a three-terminal configuration of Josephson junctions in the DC superconducting state, connecting the leads S with respective indices i, j, k [28,54], and biased with respective phases ϕ. The corresponding spin-singlet wave-function of a split FIG. 1. Sketch of the superconducting four-terminal device with either one current and one phase bias (a), or two orthogonal current biases (b). The superconductors S L , S R , S T and S B are connected to the normal metallic region N 0 . The four N 0 -S i junctions consist of tunable quantum point contacts, where the transmission of the N 0 -S T interface is reduced by a scaling factor τ T . Panels c-e represent the lowest-order Josephson processes occurring in a simplified toymodel. Panel c shows the two-terminal DC-Josephson effect from S B to S T , which is insensitive to the horizontal contacts. Panels d and e show the Cooper quartet processes, which take two Cooper pairs from S B , exchange partners and transmit the outgoing pairs into (S T , S L ) and (S T , S R ), respectively. In presence of a horizontal phase drop, these two processes pick up opposite phases, as shown in panels f and g. This leads to interfering quartet supercurrent components within this simplified model, without and with horizontal phase drop, respectively. Due to the π-shift, the critical current along the vertical direction in panel f is reduced by the two quartet processes. On panel g, a phase drop along the horizontal direction dephases the negative contribution of both processes, resulting in an increased critical current and thus a nonconvex CCC.
Cooper pair for instance between S i and S j takes the form where c + i,σ creates a spin-σ fermion in S i . The splitting event in Eq. (1) can come along with a second one. The resulting composite four-fermion transient state, i.e. a Cooper quartet [28,[34][35][36][37], ends up as two Cooper pairs transmitted into S i and S j , respectively, and described by where ⟨ ... ⟩ is a quantum mechanical expectation value (details can be found in Appendix A). By probing the internal structure of (double) split Cooper pairs, we mean providing experimental evidence for the negative sign in Eq. (2), which is a direct consequence of both quantum mechanical exchange and the split Cooper pair structure of Eq. (1). Consequently, the relation between the quartet supercurrent I q and the quartet phase ϕ q is inverted: where ϕ a , ϕ b and ϕ c are the superconducting phase variables of the leads S a , S b and S c respectively. Eq. (3) can be rewritten as I q (ϕ q ) = |I c,q | sin(ϕ q + π) and this π-shift is a macroscopic signature for the specific internal structure of single split Cooper pairs, see Eq. (1). Another simple perspective on the π-shift of the quartets readily follows from considering a single two-terminal superconducting weak link with normal-state transmission α.
Here, the energy-phase relation can be Fourier-expanded as E J (ϕ) = E J 0 + E J 2e cos ϕ + E J 4e cos 2ϕ + .... The cos ϕ term represents the Josephson Cooper-pair energy, and is dominant in the limit of small transparency, while the cos 2ϕ one describes correlated tunneling of two Cooper pairs. We find E J 4e /E J 2e ≈ −α/16 in the small-α limit and more generally E J 4e /E J 2e < 0 for all α < 1, see Appendix B. This negative sign echoes the above current-phase relation of the quartets. More generally, our work proposes a method to directly reveal these π-shifted second-order Josephson harmonics, using a multiterminal configuration.

III. THE DEVICE AND MULTITERMINAL JOSEPHSON CIRCUIT THEORY
In this section, we present the two types of devices and the approximations sustaining multiterminal Josephson circuit theory. The proposed device consists of four BCS superconducting leads S L , S R , S B and S T , with the respective superconducting phase variables ϕ L , ϕ R , ϕ B and ϕ T , and connected via a square-shaped normal conductor N 0 as shown in Fig. 1.
The external circuit imposes current in orthogonal directions, that is, a vertical current I v ≡ I T = −I B and a horizontal one I h ≡ I R = −I L . The absence of coupling between I v and I h produces a square or rectangular CCC, while rounded CCCs are indicative of coupling.
Our main result is that assuming a single or two contacts with transparency smaller than the others, nonconvex CCCs emerge in the (I v , I h ) plane already under zero applied magnetic field. We thus find reentrance of the dissipative state into the superconducting region as a distinctive signature of the π-shifted contribution of second-order Josephson harmonics. Furthermore, we show that the π-shifted Cooper quartet supercurrent produces distinctive sharp reentrant sharp-angled points in the CCCs.
The four-terminal geometry is found by a straightforward generalization of Josephson circuits, where now the I v and I h supercurrents result from an interference between multipair processes involving the phases of more than two terminals [28,31]. For instance, in a two-terminal Josephson junction, the terms corresponding to Cooper pairs transmitted from S i to S j couple to the difference δ i, j = ϕ i − ϕ j . Similarly, with four terminals, the relevant phase variables are then given by gauge-invariant combinations such as δ i, j + δ k,l [31], which reduces to Eq. (4) for three terminals [28].
In our multiterminal Josephson circuit model, we assume tunable contacts with a few transmission modes connecting the four superconductors to a central normal metal island (see Fig. 1), as was recently demonstrated in bilayer graphenebased two-terminal Josephson devices [81] and in multiterminal semiconducting-superconducting quantum point contacts [37]. Considering intermediate contact transparencies, although the DC-Josephson effect is dominant, the next-order Cooper quartets still yield a sizable contribution, while the even higher-order terms are smaller. This hierarchy justifies the approach of the Letter, considering within a single fourterminal device all the Josephson processes involving two, three and then four terminals. The calculation involves two steps: our starting points are the approximate analytical expressions of the current-phase relations discussed above, with sign and amplitude as free parameters. This allows comparing the CCCs with respectively positive or negative Cooper quartet contributions. From this we will arrive to the conclusion that nonconvex CCCs in zero field carry the unique signature of the microscopic π-shifted Cooper quartet current-phase relation, and would be absent with a 0-shift.
We consider intermediate transparency interfaces, with hopping amplitudes J L , J R , J B and J T connecting respectively the four superconducting leads S L , S R , S B and S T to a normal tight-binding lattice N 0 . The DC-Josephson supercurrent of Cooper pairs from lead S i to lead S j is written as I P = I c,P i, j sin δ i, j . The nonlocal DC-Josephson supercurrent of the Cooper quartets involves, at the lowest order in tunneling, the following three terms: Here, I c,q i, j,(k) for instance represents the critical quartet current of two pairs emitted by S k and recombining into S i and S j . We introduce the individual channel transmissions τ i such that all J i = √ τ i J (0) , with J (0) a constant smaller than the band-width W . The critical currents scale as follows: (i), j,k for the Cooper quartets, where the I c(0) s do not scale with the transmissions.

A. Polarization with one current and one phase bias
In this subsection we present analytical results for the device polarized with one current and one phase bias, see 1a. An external source drives a supercurrent from S B to S T and an external loop fixes the phase difference between S L and S R . We additionally assume that the N 0 -S T link has a tunneling amplitude J T small compared to J L = J R = J B ≡ J (0) , i.e. τ T ≲ τ L , τ R , τ B ≲ 1. Then, we make a perturbation expansion in tunneling of the Josephson circuit to the dominant order τ T , neglecting the processes of order τ 2 T (see Appendix C). In absence of the quartets, we find two types of processes: (i) The direct two-terminal DC-Josephson effect of the Cooper pairs from S B to S T (see Fig. 1e), and (ii) The two-terminal DC-Josephson processes of the Cooper pairs involving the lateral superconductors S L and S R . Adding now the quartets, we include all possible processes appearing at the orders τ 0 T and τ T .
The cartoon shown in Fig. 1 illustrates the case where, at the order τ T , the critical current I c v from S B to S T results from an interference between the amplitudes of the two-terminal DC Josephson effect and both Cooper quartets (see Figs. 1f,g). Taking an opposite relative sign of the two-and three-terminal contributions, respectively, leads to a reduction of I c v upon including the Cooper quartets. Notably, because each quartet process picks up an opposite phase ϕ L = −ϕ R ≡ −ϕ/2, their respective contributions are dephased and the value of I c v is restored upon applying a supercurrent I h (or a phase gradient) in the transverse direction, as shown in Figs. 1f and g. Now, we evaluate the full set of microscopic two-and three-terminal processes at the relevant orders (details in Appendix C). Using the notations ϕ L = −ϕ R = −ϕ/2, we demonstrate in Appendix C that, at small τ T and quartet Josephson energy E q = (h/2e)I c q , the critical current I c v from S B to S T can be approximated as where I c P and I c q are proportional to the critical currents of the two-and three-terminal Cooper pair and Cooper quartet processes respectively. Eventually, Eq. (6) predicts nonconvex CCCs if the condition is fulfilled. In this case, the dissipative state reenters into the superconducting one, as a result of the π-shifted Cooper quartet current-phase relation coming from the spin-singlet minus signs in Eqs. (1) and (2). We rewrite Eq. (6) as I c v (ϕ) = τ T I c P F(ϕ/2π), with The variations of F(ϕ/2π) are shown in Fig. 2, confirming emergence of nonconvex or convex CCC if I c q < −I c P /14 or I c q > −I c P /14 respectively.

B. Polarization with two orthogonal current biases
In this subsection, we numerically solve a related model where we impose current biases in both horizontal and vertical directions, such that I v = I T = −I B and I h = I R = −I L (see Fig. 1b). The four superconducting phase variables adjust accordingly. The numerical calculations are based on evaluating convergence of the steepest descent algorithm for a multiterminal Josephson junction. A dichotomic search was implemented, in order to locate the CCCs to high accuracy. We use I c k,l ≡ I c P and I c k,l,(m) ≡ I c q for the critical currents of the processes coupling to two and three superconducting phase variables, respectively. Fig. 3 shows the CCCs of a fourterminal device with the transmission coefficient scaling factors τ B = τ L = τ R = 1 and different values of τ T . For positive values of I c q /I c P , the CCCs have the shape of nested rounded rectangles. For sufficiently negative I c q /I c P however, the CCCs evolve from diamond-like to a shape presenting nonconvex sharp-angled points when lowering τ T . Notably, the CCCs with nonconvex sharp-angled points are only obtained for a sufficiently negative Cooper quartet critical current (here I c q /I c P = −0.2), which is in agreement with the preceding analytical solution.
In Fig. 4a, we further implement two weak links with τ T , τ L ≤ 1, while maintaining τ B = τ R = 1, and we use a negative Cooper quartet critical current I c q /I c P = −0.2. Focusing on the panels on the diagonal, i.e. τ T = τ L = 1/4, 1/2, 1, we obtain an evolution from diamond-like to square-like CCCs, as τ T = τ L decreases. Since a rectangular CCC is indicative of independent currents in orthogonal directions, this evolution demonstrates a loss of quantum mechanical coupling between I v and I h as the contact transmission coefficient scaling factor decreases. The intermediate value τ T = τ L = 1/2 yields reentrance on both supercurrent axes, which originates from the underlying diagonal mirror symmetry in the device. Considering now the off-diagonal panels in Fig. 4a, we obtain shapes with nonconvex sharp-angled points on the I c v axis if τ T = 1/4, 1/2 and τ L = 1, and the same on the I c h axis if τ T = 1 and τ L = 1/4, 1/2. This is again in qualitative agreement with the analytical model calculations presented in the above Sec. IV A.
In Fig. 4b, we introduce all possible higher-order twoterminal I ′ c 2T sin(2(ϕ i − ϕ j )) coupling terms, in addition to the Cooper quartets. We observe the robustness of the reentrant sharp-angled points with respect to addition of these. Qualitatively, this can be interpreted as due to the fact that a smooth feature on top of a sharp cusp does not alter the latter.

V. CONCLUSIONS
To conclude, it follows from basic theoretical arguments that the quartet supercurrent contribution must be π-shifted with respect to the lowest order Josephson Cooper pair supercurrent. We demonstrated that the nonconvex twodimensional critical current contours (CCCs) of a currentbiased four-terminal Josephson junction are generically due to a relative π-shift of the higher-order terms in the currentphase relation. These can either originate simply from the two-terminal Josephson current-phase relation, or, more interestingly, from the Cooper quartets. Finally, we demonstrated that nonconvex sharp-angled points in the CCCs are a distinctive signature of negative Cooper critical current contributions. However, we note that too small negative quartet critical currents will restore convex CCC, which sets constraints on the transmissions for the observation of the characteristic reentrance. A recent experiment [39] reported the appearance of nonconvex CCCs only under applied magnetic field. However, in contrast to our assumptions, all contacts had large transparencies. Conclusive evidence for the πshifted quartet term could be realized with bilayer grapheneor semiconducting-quantum point contacts [37,81] with tunable contact transparencies. In this Appendix, we detail how to deduce Eq. (2) from Eq. (1). Namely, we square the wave-function of a Cooper pair split between S i and S j : Evaluating quantum mechanical expectation values in the final state leads to where we used ⟨ c + i,↑ 2 ⟩ = 0 because of spin conservation. The above Eq. (A5) matches the above Eq. (2).

Appendix B: Details on a single superconducting weak link
In this Appendix, we evaluate the first-and second-order harmonics of the Josephson current-phase relation for a single-channel superconducting weak link having hopping amplitude J 0 , and connecting the "left" and "right" superconducting leads S L and S R with superconducting phases ϕ L and ϕ R respectively. The Andreev Bound State (ABS) energies take the following form: where the dimensionless normal-state transmission coefficient α between 0 and 1 is given by The ABS energies are expressed as Expanding to second order, we obtain Using cos 2 (ϕ R − ϕ L ) = [1 + cos (2(ϕ R − ϕ L ))] /2, we obtain in the small-α limit, where E J 2e > 0 is positive and E J 4e < 0 is negative. Now, we present supplemental numerical calculations for the amplitudes H 1 (α) and H 2 (α) of the first and second Josephson harmonics as a function of the dimensionless normal-state transmission coefficient α: (B9) Figure 6 shows that H 2 (α)/H 1 (α) < 0 is negative for all values of α < 1. The limiting behavior H 2 (α)/H 1 (α) = −α/16 is obtained at small α, in agreement with the above Eq. (B7). Appendix C: Details on the device controlled with one current and one phase bias In this Appendix, we consider the multiterminal Josephson circuit shown on the above figure 1a, consisting of the four superconducting leads S L , S R , S B and S T with the phases ϕ L , ϕ R , ϕ B and ϕ T . The phase difference ϕ R − ϕ L = Φ is controlled by the flux Φ in the loop, and supercurrent I v = I B = −I T is forced to flow from the "bottom" to the "top" superconducting leads. We use the notation ϕ T = ψ. Given those constraints, the supercurrent transmitted into S T is parameterized by a single phase variable, for instance by the phase variable ψ, and the critical current from bottom to top is obtained by maximizing the supercurrent I v (ψ) over ψ.
The superconductor S T is assumed to be connected to the normal conductor N 0 by hopping amplitude that is weaker than the others. A reduction factor τ T is applied to each Cooper pair crossing the N 0 − S T interface. First considering vanishing small quartet Josephson energy I c q = 0, we obtain the following expression of the four-terminal Josephson junction energy E (0) : where E P is the Josephson energy associated to transferring Cooper pairs between the leads. Each term entering Eq. (C1) is schematically shown in figure 7. Using ϕ L = −ϕ R = −ϕ/2 and ϕ T = ψ, we obtain Then: The current source imposes which leads to the self-consistent ϕ B = ϕ * B , with showing that ϕ * B is of order τ T . Then, at the order τ T , we obtain Taking the maximum over ψ and expanding in small ϕ leads to following expansion of the critical current flowing from bottom to top at small ϕ, at the order τ T : leading to convex CCC in absence of the quartets. Eq. (C10) is rewritten as where I c P = 2eE P /h is related to the Cooper pair critical current. Now, we include finite but small quartet Josephson energy E q . Those processes are shown in figure 8, and, at the order τ T , they yield the following correction δ E (0) to the energy E (0) in Eq. (C1): Then, we find + sin (ϕ T + ϕ L − 2ϕ R ) + sin (ϕ T + ϕ B − 2ϕ R )} = τ T E q sin − ϕ 2 + ψ − 2ϕ B + sin ϕ 2 + ψ − 2ϕ B + sin ψ + 3ϕ 2 + sin (ψ + ϕ B + ϕ) (C14) Since ϕ B is of order τ T [see the above Eq. (C8)], we replace ϕ B by ϕ B = 0 in the above Eqs. (C13)-(C14) to obtain the order-τ T contribution to −∂ δ E (0) /∂ ϕ T : − ∂ δ E (0) ∂ ϕ T ≃ τ T E q sin − ϕ 2 + ψ + sin ϕ 2 + ψ + sin ψ + 3ϕ 2 + sin (ψ + ϕ) + sin ψ − 3ϕ 2 + sin (ψ − ϕ) (C15) = 2τ T E q cos ϕ 2 + cos ϕ + cos 3ϕ 2 sin ψ.
Considering now −∂ δ E (0) /∂ ϕ B , we find The phase variable ϕ B turns out to be linear in τ T at E q = 0 [see the above Eq. (C8)]. At small E q , both −∂ δ E (0) /∂ ϕ T and −∂ δ E (0) /∂ ϕ B are of order E q τ T . Then, ϕ * B is linear in τ T in the presence of small E q , as it was the case for E q = 0.
The supercurrent transmitted into the superconducting lead S T is then given by the sum of Eqs. (C9) and (C16): Taking the maximum over ψ leads to the following expression of the critical current: where I c P = 2eE P /h and I c q = 2eE q /h are related to the Cooper pair and Cooper quartet critical currents. This concludes the demonstration of the above Eq. (6). Eq. (C23) goes to Eq. (C11) in the I c q → 0 limit of vanishingly small quartet energy. As a consequence of Eq. (C23), the CCC is nonconvex if thus necessarily being negative.