Controllable odd-frequency Cooper pairs in multi-superconductor Josephson junctions

We consider Josephson junctions formed by multiple superconductors with distinct phases and explore the formation of nonlocal or inter-superconductor pair correlations. We find that the multiple superconductor nature offers an additional degree of freedom that broadens the classification of pair symmetries, enabling nonlocal even- and odd-frequency pairings that can be highly controlled by the superconducting phases and the energy of the superconductors. Specially, when the phase difference between two superconductors is $\pi$, their associated nonlocal odd-frequency pairing is the only type of inter-superconductor pair correlations. Finally, we show that these nonlocal odd-frequency Cooper pairs dominate the nonlocal conductance via crossed Andreev reflections, which constitutes a direct evidence of odd-frequency pairing.


I. INTRODUCTION
Superconductivity is caused by electrons binding together into Cooper pairs below a critical temperature and has attracted great interest due to its properties for quantum technologies [1].The applications of superconductors are thus intimately linked to the Cooper pairs, specially to the symmetries or their wavefunction or pair amplitude.Due to the fermionic nature of electrons, the pair amplitude is antisymmetric under the exchange of all the quantum numbers describing the paired electron states plus the exchange of their relative time coordinates.Of particular interest is that the antisymmetry enables the formation of odd-frequency Cooper pairs, where the pair amplitude is odd in the relative time, or frequency ω, of the paired electrons [2][3][4][5][6][7].As a result, the odd-ω Cooper pairs characterize a unique type of superconducting pairing that is intrinsically dynamic [8][9][10][11][12][13].
In this work we demonstrate the generation, control, and direct detection of spin-singlet odd-ω Cooper pairs in Josephson junctions (JJs) formed by multiple superconductors [Fig.1].In particular, we exploit the degree of freedom offered by the multi superconductor nature of the setup and find that inter-superconductor even-and odd-ω Cooper pairs naturally arise and can be controlled by the superconducting phases and onsite energies of the superconductors.Interestingly, for a JJ with two superconductors, the even-ω amplitude vanishes either when the superconducting phase difference is π or at zero onsite energy, leaving only odd-ω pairing.This behaviour remains when the number of superconductors increases but only at weak couplings between superconductors.Furthermore, we discover that crossed Andreev reflections (CARs) directly probe odd-ω Cooper pairs and can be controlled by the superconducting phases.Our work thus puts forward multi-superconductor JJs as a powerful and entirely different route for odd-ω Cooper pairs.The remainder of this article is organized as follows.In Sec.II, we introduce the multi-superconductor JJs stud-ied in this work, while in Sec.III we show how to obtain the emerging pair amplitudes.In Sec.IV we present the obtained even-and odd-ω pair amplitudes and discuss their tunability by the superconducting phases.In Sec.V we demonstrate how the nonlocal odd-ω pair amplitude is detected via CAR processes.Finally, in Sec.VI we present our conclusions.

II. MULTI-SUPERCONDUCTOR JJS
We consider JJs as shown in Fig. 1, where n conventional spin-singlet s-wave superconductors are coupled directly.For the sake of simplicity, we model these JJs by only considering the contact regions, with a Hamiltonian given by where the first two terms describe the superconductor S j , where c jσ (c † jσ ) annihilates (creates) an electronic state with spin σ at site j with onsite energy ϵ j , phase ϕ j , and induced pair potential ∆ from a parent spinsinglet s-wave superconductor with order parameter ∆ sc .Moreover, H T = t 0 n j=1 c † jσ c j+1σ + h.c.represents the coupling between superconductors with equal strength t 0 and c n+1 = c 1 .Away from the bulk gap edges, ∆ is determined as ∆ = τ 2 /∆ sc [110][111][112][113] where τ is the coupling between S j and the bulk superconductor.Below we choose τ = 0.7 and ∆ = 0.5 such that ∆ sc = 1 is larger than the induced gap and fix it as our energy unit.We also drop the spin index for simplicity but keep in mind that the superconductors in Eq. ( 1) are spin-singlet.Despite the simplicity of our model, it captures the main effects we aim to explore in this work, namely, the multi superconductor nature and the distinct superconducting phases.Systems involving multiple JJs have been studied before but in the context of topological phases [114][115][116][117][118][119][120][121][122][123][124][125].Here, we expand the playground of these multi-superconductor JJs for realizing controllable odd-ω Cooper pairs.

III. SUPERCONDUCTING PAIR AMPLITUDES
We are interested in inter-superconductor pair correlations which we also refer to as nonlocal pair correlations as they reside between superconductors.Pair correlations are described by the anomalous Green's function )⟩ where T is the time ordering operator, c n annihilates an electronic state with quantum numbers n at time and position 1 = (x 1 , t 1 ) [126,127].The fermionic nature of electrons dictates the antisymmetry condition F nm (1, 1 ′ ) = −F mn (1 ′ , 1), which enables the classification of superconducting pair correlations based on all the quantum numbers, including time and space coordinates [8][9][10][11][12][13].Thus, this con- dition enables even-and odd-ω pair correlations when F nm (ω) = ±F nm (−ω), with F nm (ω) being the Fourier transform of F nm (1, 1 ′ ) into frequency domain.In the case of multi-superconductor junctions, the multiple superconductor nature introduces an additional quantum number n, the superconductor index, that broadens the classification of pair symmetries in a similar way as the band index in multiband superconductors [12].In Table I we present all the allowed pair symmetry classes that respect the antisymmetry condition in JJs with spin-singlet and spin-triplet superconductors: four classes correspond to odd-ω pair correlations which are the four bottom classes in Table I, see Supplementary Material [128] for details.It is evident that the superconductor index (sup.index) plays a crucial role for broadening the allowed pair symmetries.
In the JJs with spin-singlet s-wave superconductors considered here, the symmetric and antisymmetric combination F +(−) nm = (F nm ± F mn )/2 become even-and odd-ω pair symmetry classes, respectively [128].These two pair symmetry classes correspond to the ESEE and OSOE classes indicated in orange in Table I.In practice, the pair correlations F nm are obtained from the electron-hole component of the Nambu Green's function, whose equation of motion in frequency space reads [ω − H nJJ ]G(ω) = I, where H nJJ is the Nambu Hamiltonian of the JJ with n superconductors described by Eqs.(1)

IV. INTER-SUPERCONDUCTOR PAIR AMPLITUDES IN JJS
To begin, we focus on the pair correlations in a JJ with two superconductors coupled directly.This system is modelled by H 2JJ with n = 2 in Eq. (1).As described in the previous section, the pair correlations are obtained from electron-hole components of the Green's function associated to the Nambu Hamiltonian in the basis Ψ = (c 1 , c † 1 , c 2 , c † 2 ) T .Without loss of generality, we assume a phase difference ϕ 2 − ϕ 1 = ϕ.Then, considering ϵ 1,2 ≡ ϵ, the symmetric and antisymmetric pair amplitudes in superconductor index are given by [128] where ω represents complex frequencies unless otherwise stated and First, both pair amplitudes in Eqs. ( 2) have the same denominator which is an even function of ω and reveals the formation of Andreev bound states (ABSs) when P +2∆ 2 t 2 0 cos(ϕ) = 0.This is seen in the bright regions of Fig. 2, where we plot the absolute value of the symmetric and antisymmetric amplitudes as a function of the phase difference ϕ.Second, the numerators of both F + 12 and F − 12 have different functional dependences, oscillating with the phase difference ϕ in an alternate fashion as cos(ϕ/2) and sin(ϕ/2), respectively [129].While the numerator of the symmetric term is an even function of ω with a linear dependence on ϵ, the antisymmetric component is interestingly linear in ω and, therefore, an odd function of frequency.The symmetric even-ω part vanishes either when ϵ = 0 or ϕ = π, while the antisymmetric odd-ω pair amplitude remains remarkably finite at these points and even acquiring large values.The surprising features of the nonlocal pair amplitudes can be seen by comparing the panels of Fig. 2(a,b,d), where the vanishing values of the even-ω part is indicated by white arrows in Fig. 2(a).The vanishing values of the even-ω pairing can be better seen in Fig. 2(c) where we plot the ratio between the two 12 is an odd function of ω and thus vanishes at ω = 0, R ± has a clear interpretation only for ω ̸ = 0.In sum, JJs with two superconductors exhibit highly tunable odd-ω pairing that is the only type of inter-superconductor pair correlations.
For JJs with more superconductors n > 2, the expressions for the nonlocal pair amplitudes become lengthy, but still capturing the formation of ABSs in the denominator and with numerators that strongly depend on all ϕ i [128].We find that the symmetric and antisymmetric pair amplitudes between nearest neighbour superconductors develop even-and odd-ω symmetries, respectively.While the odd-ω part is proportional to ∼ (e iϕj+1 − e iϕj ), the even-ω term to ∼ (e iϕj+1 + e iϕj ) + P (ϕ 1,••• ,n ), where P is a function of all the system parameters [128].Thus, the odd-ω term depends on the sine of the phase difference of the involved superconductors as in JJs with two superconductors discussed above.However, the even-ω part has a cosine part as for JJs with two superconductors, but also an additional contribution due to the rest of the system.Nevertheless, both pair amplitudes exhibit a high degree of tunability by means of the superconducting phases.To visualize this fact, in Fig. 3 we plot the even-ω and odd-ω pair amplitudes for a JJ with three superconductors as a function of ϕ 2 and ϕ 3 at ϕ 1 = 0.The main feature of this figure is that the behaviour of both pair amplitudes is highly controllable by the superconducting phases.Interestingly, there are regions where the even-ω component acquires vanishing small values while the odd-ω remains sizeable large, see dark and bright regions in Fig. 3(a,c) and Fig. 3(b,d), respectively.
The vanishing and finite values of the even-and odd-ω pair amplitudes can be further visualized in a simpler regime.Specially, for very weak couplings between superconductors t 0 and for superconductors with the same onsite energy ϵ, the nearest neighbour nonlocal pair amplitudes up to linear order in t 0 are given by [128] where j = 1, • • • , n and ϕ n+1 = ϕ 1 .Strikingly, only the pair amplitudes between nearest neighbour superconductors remain finite at leading order in t 0 [130].
(3) exhibit evenand odd-ω spin-singlet symmetries, respectively.Interestingly, both pair amplitudes acquire the same form as their counterparts in JJs with two superconductors, see Eqs. (2).In this regime, the even-ω pairing thus vanishes either at ϵ = 0 or when e iϕj+1 + e iϕj = 0 which needs a phase difference of ϕ j+1 − ϕ j = π between superconductors.However, the odd-ω component remains always finite in this regime, exhibiting high tunability by ϕ j .We have verified that this behaviour remains even in JJs with finite superconductors and also in JJs with superconductors coupled via a normal region [128].Hence, multi-superconductor JJs represent a rich platform for the generation and control of nonlocal odd-ω pair correlations that do not require magnetic elements.Before closing this part, we highlight that the odd-ω pair amplitudes presented here are a proximity-induced superconducting effect bound to the device, exhibiting wide controllability by the superconducting phases and with important impact on physical observables as we discuss next.

V. CAR DETECTION OF ODD-ω PAIRING
Having established the emergence of intersuperconductor odd-ω pairs in multi-superconductor JJs, now we inspect a direct detection protocol.Due to the nonlocal character of the pair correlations found here, it is natural to explore nonlocal transport of Cooper pairs [28,36,75,131].Without loss of generality, we focus on JJs formed by two superconductors and aim at detecting the odd-ω pairs obtained in Eqs. 2. Hence, we attach two normal leads at the left and the right of the system as in Fig. 1 and include them in our model via retarded selfenergies Σ r L(R) , such that the system's retarded Green's function is Here, H 2JJ describes the JJ described by Eq. ( 1) with n = 2 and ω now represents real frequencies.In the wide-band limit, Σ r j = −iΓ j /2, where Γ j = π|τ | 2 ρ j characterizes the coupling to lead j with surface density of states ρ j , and τ the hopping between leads and superconductors.
At weak Γ j , the JJ can be probed by nonlocal transport.Specially, the transport of Cooper pairs is characterized by nonlocal Andreev reflection or crossed Andreev reflection (T CAR ), which competes with electron tunneling (T ET ) to determine the nonlocal conductance ∼ (T CAR − T ET ) [128].These CAR and ET processes involve electron-hole (hole-electron) and electron-electron (hole-hole) transfers, T CAR = T eh + T he and T ET = T ee + T hh , which can be obtained from G r as [75] where g r 12 (ḡ r 12 ) and F r 12 ( F r 12 ) are the normal and anomalous (or pair amplitude) components of the intersuperconductor retarded Green's function, obtained from G r [128].Interestingly, the CAR processes T eh(he) are directly determined by the squared modulus of the intersuperconductor pair amplitudes F r 12 .We note that, while the pair amplitudes F r 12 and F r 12 are not directly measurable, their modulo respectively determines the finite value of the nonlocal probabilities T eh and T he , thus facilitating the detection of these emergent pairings.
Under general circumstances, F r 12 includes both symmetric even-ω and antisymmetric odd-ω terms, the symmetric part vanishes at ϵ = 0 for any ϕ, see Eqs. 2. Thus, the CAR amplitudes have the potential to directly probe the antisymmetric inter-superconductor odd-ω pairing.However, as shown above, the CAR processes T eh(he) are always accompanied by electron tunnelings T ee(hh) .Therefore, even if T eh(he) directly probes odd-ω pairs, their total effect in the non-local conductance can be masked if T ee(hh) are larger.For this reason, to directly detect inter-superconductor odd-ω pairing, a regime where T ee(hh) ≪ T eh(he) is needed.Even though this regime might sound challenging to find, we now demonstrate that it is in fact possible.To show this, we consider ϕ 1 = −ϕ/2, ϕ 2 = ϕ/2 and assume symmetric couplings to the leads Γ j = Γ.Then, for ϵ = 0, g r 12 and the antisymmetric pair amplitude F r,− 12 are given by [128] where , and ḡr 12 (ϕ) = −g r 12 (−ϕ), and F r 12 (ϕ) = F r 12 (ϕ).We note that F r 12 can be obtained from Eqs. (2) by replacing ω → ω + i0 + + iΓ/2.Now, we can exploit the fact that the energy of the ABSs at ϵ = 0 and ϕ = π is given by |ω ± | = |t 0 − ∆|, which clearly vanishes for t 0 = ∆.In this regime we have |g r 12 |/|F r,− 12 | ≈ ω/(2∆) ≪ 1 for low frequencies.Thus, it is possible to obtain a regime where the antisymmetric pair amplitude is larger than the normal contribution.Hence, in this regime T eh(he) are expected to be larger than T ee(hh) and constitute the main contribution to the non-local conductance, whose finite value indicates a direct evidence of inter-superconductor odd-ω pairing.To visualize the above argument, in Fig. 4 we plot ET and CAR processes as a function of ϕ and ω at ϵ = 0.The most important feature is that at high frequencies, ET processes T ee(hh) acquire large values near ϕ = 0, 2π but are vanishing small at low ω near ϕ = π, in line with the discussion presented above.Interestingly, the CAR processes T eh(he) acquire large values around ϕ = π at low frequencies but smaller values at higher frequencies.The finite values of these CAR processes directly probe the formation of induced odd-ω pairs.Of particular relevance here are the values around ϕ = π and low ω, because, at such points, CAR dominates over ET and it thus determines the nonlocal conductance.We have verified that this behaviour also holds for JJs with more than two superconductors but in the weak tunneling regime, thus supporting the direct detection of proximity-induced inter-superconductor odd-ω pairing in a nonlocal transport measurement.Hence, despite being an induced effect, the nonlocal odd-ω pairs determine CAR processes by simply tuning the superconducting phases in multisuperconductor JJs.

VI. CONCLUSIONS
In conclusion, we have studied multi-superconductor Josephson junctions and found that inter-superconductor even-and odd-ω Cooper pairs can be generated, controlled, and detected by virtue of the superconducting phases.We found that even-ω pairing vanishes when the phase differences between two superconductors is π, thus leaving only odd-ω pairing as the only type of intersuperconductor pair correlations.While this finding is exact for Josephson junctions with two superconductors, it is only valid at weak couplings between superconductors in junctions with more than two superconductors.Due to the vanishing of even-ω pairing, only odd-ω pairs contribute to CAR processes, whose finite values directly probe the presence of odd-ω Cooper pairs.Given the advances in the fabrication of superconducting heterostructures, including a promising tunability of CAR processes [133], we expect that the physics discussed here could be soon realized in multi-terminal Josephson junctions [117,122,[134][135][136] and in superconducting quantum dots [137][138][139][140][141][142][143][144][145][146].Of particular relevance are Refs.[117,122,[134][135][136] because they have already demonstrated the fabrication of multi-superconductor Josephson junctions and the control of several superconducting phases.In this regard, our work offers an entirely unexplored route for the generation, control, and detection of odd-ω Cooper pairs that might be even possible to explore using already existent experimental techniques.
Before carrying out any calculation in a specific system, here we present all the allowed superconducting pair symmetries in multi-superconductor Josephson junctions with quantum numbers that involve frequency (ω), spins (σ, σ ), superconductor indices (n, m), and spatial coordinates (x, x ).For this purpose, we remind that the antisymmetry condition dictates that the pair amplitudes F σ,σ n,m (ω; x, x ) must be antisymmetric under the total exchange of quantum numbers, namely, where ω represents complex frequencies unless otherwise stated.Thus, the allowed pair symmetries should fulfil Eq. (S1) under the total exchange of quantum numbers.However, F σ,σ n,m (ω; x, x ) can be even or odd under the individual exchange of quantum numbers as long as Eq.(S1) holds.Thus, for instance, the pair amplitude can be even (odd) under the exchange of frequency ω → −ω and pick up a plus (minus) sign, which translates as Moreover, as stated at the beginning of this part, F σ,σ n,m (ω; x, x ) can be even (odd) under the individual exchange of the rest of the quantum numbers.Thus, F σ,σ n,m (ω; x, x ) can be even (odd) under the exchange of spins, superconducting indices, or spatial coordinates, respectively, when Therefore, the allowed pair amplitudes can be obtained by performing all the possible combinations of the individual exchanges of quantum numbers (Eqs.(S2) and ( S3)) that fulfil the antisymmetry condition Eq. (S1).By doing this, we find eight allowed pair symmetry classes that respect the antisymmetry condition, where 4 correspond to odd-frequency correlations, see Table S1.Of particular relevance is that the superconducting index (sup.index) plays a crucial role for broadening the allowed pair symmetries.Table S1 is presented as Table 1 in the section on "Inter-superconductor pair amplitudes in JJs" of the main text.
In the absence of any spin-mixing field, the spin symmetry of the emergent pair correlations is the same as the spin symmetry of the parent superconductor.Thus, the pair symmetry classes allowed in our study, where no spin-mixing field is present, are ESEE and OSOE pair symmetry classes: they correspond to a pair amplitude with even-frequency (odd-frequency) spin-singlet even (odd) in superconductor indices, even parity.By including a spin-mixing field, it is possible to obtain odd-frequency spin-triplet pair amplitudes which correspond to OTEE and OTOO pair symmetry classes in Table S1, which could be used as sources of spins highly controllable by the superconducting phases and thus promising for superconducting spintronics.Since we do not have spin-mixing fields in the results presented in the main text, the pair symmetries therein exhibit the spin-symmetry of the parent superconductor, namely, spin singlet.This is specially discussed in the section on "Inter-superconductor pair amplitudes in JJs" of the main text.

GREEN'S FUNCTIONS OF JOSEPHSON JUNCTIONS WITH SUPERCONDUCTORS COUPLED DIRECTLY
To obtain the Green's functions of the Josephson junctions studied in the main text, we first write their model Hamiltonian in Nambu (electron-hole) space and then obtain them from the equation of motion [ω − H nJJ ]G(ω) = I, where H nJJ is the Hamiltonian of the phase-biased Josephson junctions and G(ω) its associated Green's function.A Josephson junction with n superconductors coupled directly is modeled by Eq. ( 1) in the main text.In Nambu space, the Hamiltonian of each superconductor S j is given by where j represents the onsite energy of the superconductor, ∆ is the spin singlet s-wave pair potential and φ j its superconducting phase.Similarly, the coupling between superconductors is described by where t j,j+1 represents the coupling strength between nearest superconductors S j and S j+1 .For simplicity, we consider that all such couplings are the same t j,j+1 = t 0 and thus drop the indices in the coupling matrix (V j,j+1 = V ).Below, we discuss Josephson junctions with distinct superconductors and obtain expressions for their associated Green's functions.

Josephson junctions with two superconductors
The Hamiltonian of a Josephson junction between two superconductors is The eigenvalues of this Hamiltonian are given by which, for 1,2 = , reduce to Then, Therefore, the gap closes at φ = π if t 0 = √ ∆ 2 + 2 ; for = 0, the gap closes when t 0 = ∆.The associated Green's function has the following structure The entries of G 2JJ correspond to the Green's functions inside the superconductors (G 11 (22) ) or between the superconductors (G 12 (21) ).We thus term these components as intra-and inter-superconductor Green's functions, respectively.The Nambu structure of each G ij is given by where the diagonal elements allow us to obtain the density of states, while the off-diagonal ones give the pair amplitudes.Thus, for the pair amplitudes we obtain For the normal components we get extract the pair amplitudes between the last site of the left superconductor and the first site of the right superconductor and label them just by L and R indices denoting that they represent pair correlations between left (L) and right (R) superconductors.These nonlocal pair amplitudes F LR are then decomposed into their symmetric and antisymmetric components under the exchange of L and R, which are then denoted by F + LR and F − LR , respectively.Moreover, we note that since there are no spin-mixing fields, these nonlocal pair amplitudes have spin-singlet symmetry which implies that F ± LR corresponds to even-ω (odd-ω), spin-singlet, even (odd) in superconductor indices.Interestingly, these nonlocal pair amplitudes correspond to the even-and odd-ω nonlocal pair amplitudes discussed in the main part of our manuscript.In Fig. S3(a-d) and Fig. S4(a,b,d,e) we present the magnitude of these pair amplitudes as a function of frequency ω and phase difference φ for distinct lengths of the superconductors.In Fig. S4(c,f) we also show the ratio between even-and odd-ω pair amplitudes for several realistic lengths of the superconductors.
At zero phase φ = 0 only the even-ω component is finite for frequencies within the gap (and also outside) [Figs.S3(a,c)], while at φ = π it is the odd-ω pair amplitude the only pair amplitude that remains finite [Figs.S3(b,d)].
Of course that at zero frequency the odd-ω pairing vanishes as expected for any odd-ω function.Having odd-ω pairing as the only type of nonlocal superconducting pairing at φ = π remains even when the length of the superconductors L S increase, supporting the discovery reported in the first part of our manuscript, specially Fig. 2. The intriguing and interesting behaviour of these pair correlations can be further seen in Fig. S4, where we clearly observe that the even-ω part completely vanishes at φ = π, while the odd-ω pairing becomes the only finite nonlocal pair amplitude, seen by comparing Fig. S4(a,d) with Fig. S4(b,e).Furthermore, by inspecting the ratio between even-and odd-ω pair correlations, we note that such ratio vanishes at φ = π due to the vanishing of even-ω pairing, see Fig. S4(c,f).As a result, we conclude that odd-ω pairing becomes the only type of nonlocal superconducting pairing at φ = π in Josephson junctions with realistic superconductors.These results are in line with our findings presented in the first part of the main text where, however, Josephson junctions are modelled by single site superconductors.The agreement between the results presented in this section and those shown in the main text clearly demonstrate that our findings about odd-ω pairing being the only type of nonlocal superconducting pairing at φ = π are robust and very likely to appear even in realistic Josephson junctions.The reason for this agreement is because our simple model in the main text Eq. (1) already captures the tunnelling processes that permits us to explore Josephson transport in multi-superconductor Josephson junctions.As a result, having odd-ω pairing as the only type of nonlocal pairing implies that it is the main effect for enabling CAR at φ = π exactly in the same way as discussed in the section on "CAR detection of odd-ω pairing" of the main text.

FIG. 1 .
FIG. 1. JJs formed by coupling superconductors Si with distinct phases ϕi, and same induced pair potential ∆.In each Si local pairs are depicted in gray ellipses containing two electrons (black filled circles), referred to as intra superconductor (local) pairs.Due to the tunneling between superconductors, inter-superconductor (nonlocal) pair correlations emerge (cyan) which can be controlled by ϕi.Normal leads (green) are attached to two Si for exploring nonlocal transport and detecting inter-superconductor Cooper pairs.

TABLE S1 .
Allowed superconducting pair symmetries in multi-superconductor Josephson junctions under the presence of spin-mixing fields.The classes ESEE and OSOE, which are spin-singlet, correspond to the pair correlations reported in the main text of this work.