Interplay between Josephson and Aharonov-Bohm effects in Andreev interferometers
Pavel E. Dolgirev Mikhail S. Kalenkov Andrei D. Zaikin





Skolkovo Institute of Science and Technology, Skolkovo Innovation Center, 3 Nobel St., 143026 Moscow, Russia

I.E. Tamm Department of Theoretical Physics, P.N. Lebedev Physical Institute, 119991 Moscow, Russia
Moscow Institute of Physics and Technology, Dolgoprudny, 141700 Moscow region, Russia


Institut für Nanotechnologie, Karlsruher Institut für Technologie (KIT), 76021 Karlsruhe, Germany
National Research University Higher School of Economics, 101000 Moscow, Russia





Proximity induced quantum coherence of electrons in multi-terminal voltage-driven hybrid normal-superconducting nanostructures  may result in a non-trivial interplay between topology-dependent Josephson and Aharonov-Bohm effects. We elucidate a trade-off between stimulation of the voltage-dependent Josephson current due to non-equilibrium effects and quantum dephasing of quasiparticles causing reduction of both Josephson and Aharonov-Bohm currents. We also predict phase-shifted quantum coherent oscillations of the induced electrostatic potential as a function of the externally applied magnetic flux. Our results may be employed for engineering superconducting nanocircuits with controlled quantum properties.


Interplay between Josephson and Aharonov-Bohm effects in Andreev interferometers
    Andrei D. Zaikin
    December 30, 2023
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§ INTRODUCTION


Long-range quantum coherence under non-equilibrium conditions in normal-superconducting (NS) heterostructures manifests itself in a large number of interesting and non-trivial phenomena <cit.>. These phenomena become particularly pronounced in the low temperature limit since in this case proximity-induced quantum coherence of electrons in a normal metal may persist
even far away from a superconductor being limited only by dephasing due to electron-electron interactions <cit.>.

In multi-terminal hybrid NS nanostructures (also called Andreev interferometers) one can easily drive electrons out of equilibrium by applying an external voltage bias to (some of) the normal terminals. In three-terminal NSN systems long-range quantum coherence of electrons results in conductance anomalies associated with non-local Andreev reflection <cit.>. Non-trivial phenomena also occur in  cross-like structures
 with two normal and two superconducting terminals interconnected by normal wires <cit.>. Biasing the normal terminals by some voltage V, one can control both the magnitude and the phase dependence of the supercurrent between the two superconducting terminals  demonstrating switching between 0- and π-junction states <cit.>. In other words, in this case the dc Josephson current I_J between the two S-terminals is determined not only by the superconducting phase difference ϕ but also by the bias voltage V, i.e. I_J=I_J(V,ϕ).

Likewise, dissipative currents in multi-terminal hybrid superconducting circuits can also be controlled both by external voltage and
the superconducting phase <cit.> further emphasizing a non-trivial interplay between quantum coherence and non-equilibrium effects.
In ring-shaped geometries one can conveniently fix the phase difference ϕ by inserting an external magnetic flux Φ inside the
ring and investigate proximity-enhanced Aharonov-Bohm current oscillations <cit.> I_AB(V,ϕ), where ϕ=2πΦ/Φ_0 and Φ_0 is the superconducting flux quantum.

Thus, in NS hybrid nanostructures there exist two physically different contributions to the current – I_J(V,ϕ) and I_AB(V,ϕ) –
sensitive to both proximity-induced quantum coherence and non-equilibrium conditions. Until recently these two currents had been investigated
separately from each other. For instance, no Josephson current can possibly occur in ring-shaped NS structures <cit.>
where Aharonov-Bohm oscillations of the current I_AB(V,ϕ) have been demonstrated <cit.>. And vice versa, no
Aharonov-Bohm effect can emerge in symmetric cross-like four-terminal setups <cit.> where the voltage-controlled dc Josephson current
has been observed <cit.>.

Recently we argued <cit.> that by slightly modifying the topology of a four-terminal Andreev interferometer – e.g., just by making the cross-like
geometry <cit.> asymmetric – one can induce non-vanishing Aharonov-Bohm currents, thus being able to directly observe a trade-off between Josephson and Aharonov-Bohm effects in the same setup. The competition between the two 2π-periodic in ϕ terms I_J(V,ϕ) and I_AB(V,ϕ) – respectively odd and even functions of ϕ – yields novel features such as, e.g., the (I_0,ϕ_0)-junction state <cit.> for which the current-phase relation turns out to be
phase-shifted by the value ϕ_0 controlled by an external voltage bias V. Interestingly enough, applying a temperature gradient to the system one can induce the thermoelectric voltage signal which
also demonstrates coherent phase-shifted oscillations as a function of Φ. Such oscillations turn out to be quite similar <cit.> (although not exactly identical) to those of the electric current.

In this work we will further investigate a non-trivial interplay between dissipative (Aharonov-Bohm) and
non-dissipative (Josephson) contributions to the current in multiterminal Andreev interferometers at low temperatures and under non-equilibrium conditions. In particular, we will demonstrate that providing extra low energy quasiparticles in a voltage biased setup (e.g., by attaching an extra normal terminal to the system) yields a trade-off between effective dephasing of quasiparticles (causing reduction of both Josephson and Aharonov-Bohm currents) and stimulation of the (voltage-dependent) Josephson current. The combination of these two effects may result in a substantial modification of the current-phase relation in Andreev interferometers and to further interesting topology-dependent phenomena like, e.g., coherent oscillations of the voltage induced at the normal terminal isolated from external leads.

The structure of our paper is as follows. In Sec. II we define our model and briefly
describe the quasiclassical formalism employed in our calculations. In Sec. III we discuss phase-shifted coherent oscillations of the current in four-terminal Andreev interferometers. Sec. IV is devoted to the analysis of novel non-equilibrium effects which occur in the five-terminal configurations. In Sec. <ref>, we briefly summarize our main findings and provide further discussion. Some details of our calculation are relegated to Appendix.






§ THE MODEL AND BASIC FORMALISM



Below in this work we will mainly focus our attention on a five-terminal hybrid NS structure consisting of two superconducting (S_1 and S_2) and three normal (N_1,2,3) terminals interconnected by normal diffusive wires of equal cross section A and different lengths as it is illustrated in Fig. <ref>.
The superconducting order parameter in the two S-terminals has the form Δ e^± iϕ/2, implying that the phase difference between these terminals equals to ϕ. The two normal terminals N_1 and N_2 are attached to an external voltage source thus fixing the voltage difference between these terminals V_2-V_1=V. The third terminal N_3 is kept isolated from any external circuit. Nevertheless, depending on the system topology a non-zero electric potential V_N=V_N(V,ϕ) may be generated at this terminal.



In what follows we will assume that the effective distance between the two superconducting terminals L = l_S,1 + l_c + l_S,2 strongly exceeds the superconducting coherence length ξ. Then the corresponding Thouless energy of our device ℰ_ Th = D/L^2 (with D being the wire diffusion constant) remains  well below the superconducting gap, i.e. we have ℰ_ Th≪Δ.

Our further analysis will be based on the well established quasiclassical formalism of the superconductivity theory <cit.>. Employing the so-called θ-parameterisation <cit.> we express the 2× 2 matrix in the Nambu space representing the retarded quasiclassical Green function Ĝ^R in the form

    Ĝ^R = [ G_11 F_12; F_21 G_22 ] = [       coshθ   e^iχsinhθ; -e^-iχsinhθ      -coshθ ],

where θ and χ are two complex functions obeying the spectral Usadel equations

    D Δθ    =     -2 i ϵsinhθ + 1/2 D (∇χ)^2 sinh 2θ, 
    ∇ j_E     =     0, j_E = sinh^2 θ·∇χ.


The quantum kinetic equations read <cit.>

    ∇ j_L = 0 , j_L =  D_L ∇ f_L - 𝒴∇ f_T + j_s f_T, 
       ∇ j_T = 0 , j_T =  D_T ∇ f_T + 𝒴∇ f_L + j_s f_L,

where f_L(T)(ϵ) is symmetric (antisymmetric) in energy part of the electron distribution function.
In Eqs. (<ref>)–(<ref>) we also introduced the kinetic coefficients

    D_L/T = 1/2 (1 +  | coshθ |^2 ∓ |sinhθ|^2 cosh ( 2χ )),
        𝒴 = 1/2|sinhθ|^2sinh ( 2χ ),  j_s = j_E.

It is worth pointing out that the function Y accounts for electron-hole asymmetry in our structure.

The electric current density j in our system is expressed in terms of the energy-integrated j_T-component of the spectral current as

    j = -σ_N/2e∫ j_T (ϵ) d ϵ ,

where σ_N is the Drude conductivity of a normal metal. By solving the above Usadel equations one can also determine the distribution of the electrostatic potential in our structure by means of the formula

    e V(x)=  ∫_0^∞ d ϵ f_T (x,ϵ) ν_ϵ (x) ,

where ν_ϵ (x) = {coshθ (x,ϵ)} is the coordinate-dependent electron density of states.

Eqs. (<ref>)–(<ref>) should be solved separately in each of the metals and the corresponding solutions should be matched with the aid of proper boundary conditions at all interfaces of our structure. Here we will assume that all inter-metallic interfaces are fully transparent implying that at all wire nodes (a) the functions θ, χ, f_L and f_T are continuous and (b) the spectral currents A∇θ  ,  A∇χ,   A j_L and A j_T remain conserved. In addition, at the boundaries between the wires and the N-terminals the functions θ, χ, f_L and f_T are continuously matched with their bulk values deep inside these terminals. At the NS interfaces we have j_L = 0 and f_T = 0 at energies | ϵ | < Δ. The latter condition just means that charge imbalance (possibly existing inside normal wires) disappears at the NS interfaces.









§ FOUR-TERMINAL INTERFEROMETER


To begin with, let us somewhat simplify our system and disconnect the N_3 terminal from the rest of the structure, thus reducing our five-terminal system depicted in Fig. <ref> to a four-terminal one <cit.>. It is instructive to first discuss the behavior of this reduced structure
since it will help us to elucidate all essential physics and to make our subsequent analysis of the five-terminal setup of Fig. <ref> a lot easier. In addition, for simplicity we set l_S(N),1 = l_S(N), 2 = l_S(N), in which case one has V_1/2 = ∓ V/2.

As usually, in order to proceed, one first solves the spectral part of the problem and finds the retarded and advanced Green functions for the structure under consideration. This task can easily be accomplished: At energies |ϵ| ≪ℰ_ Th and |ϵ| ≫ℰ_ Th an analytic solution can be obtained <cit.> (see also Appendix), while for |ϵ| ∼ℰ_ Th it is in general necessary to resort to numerics <cit.>.

The next step is to resolve the kinetic equations. The corresponding solution can also be obtained analytically
provided we <cit.> (i) disregard terms containing 𝒴 and (ii) resolve the kinetic equations in the first order in j_s. Strictly speaking, the approximations (i) and (ii) are fully justified only in the vicinity of the phase values ϕ≈π n. Fortunately, the exact numerical analysis of the problem <cit.> verifies that the above approximations work sufficiently well allowing to capture all essential physics even far away from ϕ≈π n.



Assuming that the superconducting order parameter Δ strongly exceeds any other energy scale in our problem, in the leading order in j_s we find

    f_L(x) ≡ f_L^N (V/2),
       f_L/T^N (V) = 1/2[ tanhϵ + eV/2T±tanhϵ - eV /2T],

and j_L = 0 in every wire of the structure.

Turning now to electric currents flowing in our system, for the wire l_N,1 we may write

    j^N_T = D_T f_T' ⇒ j_T^N = (f_T^c,1 - f_T^N(V_1)) [∫_l_N dx/D_T]^-1,

where f_T^c,1 is evaluated at the crossing point c_1 of the wires l_S,1, l_N,1 and l_c.
Similarly, for the wires l_S,1 and l_c we get:

    j_T^S = j_s^S f_L^N (V/2) + f_T^c,1[∫_l_Sdx/D_T]^-1,
       j_T^c =  j_s^c f_L^N (V/2) - 2f_T^c,1[∫_l_c dx/D_T]^-1.

Making use of the conservation of the spectral charge current, j_T^c  = j_T^S  + j_T^N, we eventually recover the expression for the spectral current I_S(ϵ) = σ_N j_T^S  A/(2e), which, after energy integration, determines the current I_S flowing between the two superconducting terminals S_1 and S_2:

    I_S = ∫ dϵ[σ_N f_L^N (V/2)j_s  A/(2e) - f_T^N(V/2)  R_c^T/ N],

where we denoted N =  R_c^T ( R_S^T +  R_N^T) + 2  R_S^T  R_N^T and
R^T_i = ( Aσ_N)^-1∫_l_i dx/D_T,i.

Making use of Eq. (<ref>) we obtain <cit.>

    I_S = I_0(V) + I_J(V,ϕ) + I_AB(V,ϕ).

The first term in the right-hand side of this formula represents the averaged over ϕ current value I_0(V)=⟨ I_S⟩_ϕ, while two other – sensitive to the phase – terms are respectively the Josephson (odd in ϕ) and the Aharonov-Bohm (even in ϕ) contributions to the current. In the interesting for us limit of sufficiently large bias voltages eV ≫ℰ_ Th we find <cit.>

    I_J(V,ϕ) ≃ I_C^(4)(V) sinϕ,     I_AB(V,ϕ)≃ I_ m^(4)(V) cosϕ

with (see also Appendix)

    I_C^(4) (V) ≃128 (1+v^-1)/9(3+2√(2))V/R_L e^-vsin(v + v^-1),
       I_ m^(4)≃0.18 ℰ_ Th/e R_L.

In Eq. (<ref>)  we introduced the dimensionless parameter v = √(eV/2ℰ_ Th)≫ 1.

Note that both results (<ref>) and (<ref>) hold only in the low temperature limit. In particular, Eq. (<ref>) is valid for T ≪√(e|V|ℰ_ Th), while low temperature asymptotics (<ref>) is correct even for a wider temperature range.  The full voltage dependence for both I_C^(4) and I_ m^(4) is illustrated in Fig. <ref>. We observe that at low voltages the Josephson critical current I_C^(4) shows the π-junction feature <cit.> and  dominates over the Aharonov-Bohm contribution I_ m^(4), whereas at high voltages |I_C^(4)| decays exponentially with increasing V in accordance with Eq. (<ref>). The Aharonov-Bohm  current shows just the opposite trend: I_m^(4) increases with V and saturates to the value in Eq. (<ref>) at eV ≳ 100 ℰ_ Th.


For completeness, let us also point out that the currents I_C and I_ m are described by very different temperature dependencies: The Josephson term decays exponentially with increasing temperature being completely suppressed already at T≃ 20 ℰ_ Th, while the Aharonov-Bohm current decays much slower, typically as a power-law <cit.>.














§ FIVE-TERMINAL INTERFEROMETER


Let us now go back to our initial five-terminal configuration schematically depicted in Fig. <ref>.
In other words, as compared to the situation considered in the previous section we now attach an extra reservoir of normal electrons N_3 to the central wire l_c. At the first glance, an immediate and
obvious consequence of this modification could only be a reduction of superconducting correlations in our system and, hence, partial suppression of both Josephson and Aharonov-Bohm contributions to the current I_S(V). This is because a certain fraction of “phase-coherent electrons” propagating in the normal wires connecting the two S-terminals can now make a “detour” into N_3 being replaced by electrons from the latter terminal which carry no information about the phase ϕ. As it is demonstrated in Appendix, at T → 0 and eV ≫ D/l_c^2 this decoherence mechanism yields a reduction of the Josephson critical current for the five-terminal setup as

    I_C,0^(5) (V)=2/3 I_C^(4) (V),

where I_C^(4) (V) is defined in Eq. (<ref>).

Likewise, the Aharonov-Bohm current component of I_S(V) in the five-terminal setup gets reduced as compared to
that in the four-terminal one. As it is illustrated in Fig. <ref>b, for the symmetric case (see below) we have:

    I_ m,0^(5) (V)≈1/2 I_ m^(4)(V),

where I_ m^(4)(V) is specified in Eq. (<ref>).

In what follows, we will demonstrate that along with the above decoherence scenario, there is yet another effect which, on the contrary, may yield a significant enhancement of the Josephson current. On top of that, by applying an external voltage bias V,  we, in general, induce a non-zero electric potential V_N at the terminal N_3. Below we will observe that the voltage V_N(V, ϕ) is also sensitive to proximity-induced quantum coherence effects and, hence, V_N exhibits the (phase shifted) coherent oscillations as a function of the superconducting phase ϕ.





 §.§ Symmetric setup


We start by considering a fully symmetric configuration, in which case the terminal N_3 is connected by the wire l_N,3 to the central point of the wire l_c. As before, we also set l_S(N),1 = l_S(N), 2 = l_S(N). Then by symmetry we have V_N ≡ 0 and V_1/2≡∓ V/2, i.e. no further evaluation of V_N(V, ϕ) would be necessary in this case. Adopting the same set of approximations and employing the same analysis as in the previous section, we evaluate the spectral Josephson current between the two superconducting terminals with the result

    I_J(ϵ)    ≃    [ (1-κ) f_L^N(V/2) + κ f_L^N(0) ] σ_N j_s  A/2e,
    κ   =   2 R_N,1/R_c + 2 R_N,1 + 4 R_N,3.

The first term in the right-hand side of Eq. (<ref>) has exactly the same origin as the corresponding contribution in Eq. (<ref>) controlled by the voltage V between the normal terminals N_1 and N_2. In contrast, the last term is new. It emerges here only due to the presence of the terminal N_3 not considered in the previous section. Since the voltage V_N = 0, the latter term turns out to be independent of the bias voltage V. Then, in the interesting limit T ≪√(e |V| ℰ_ Th)≪ e|V| we obtain:

    I_J ≃ (1-κ) I_C,0^(5)(V) sinϕ + κ I_ eq^(5)(ϕ ),

where I_ eq^(5)(ϕ ) is the equilibrium Josephson current for the five-terminal setup of Fig. <ref> at T → 0. This current differs from that for an SNS junction <cit.> only by a geometry-dependent numerical prefactor smaller than unity.


Equations (<ref>)–(<ref>) represent an important result: We observe that, while the first – voltage controlled – term in the right-hand side of Eq. (<ref>) decays exponentially with increasing eV ≫ℰ_ Th, the second term remains nonzero being equal to a voltage-independent constant, cf. also Fig. <ref>a. In other words, under these non-equilibrium conditions the maximum value of the Josephson current

    I_C^(5)(V) ≃κ  max_ϕ [I_ eq^(5)(ϕ )] ≃ 3.2 κℰ_ Th/(e R_L)

may strongly exceed I_C,0^(5)(V) in Eq. (<ref>). The physical reason for this enhancement effect is transparent: The terminal N_3 supplies extra quasiparticles with energies |ϵ |∼ T –  well below both eV and
ℰ_ Th – to the wires connecting the two superconducting terminals. Accordingly, the Josephson
current acquires an extra contribution, which is not exponentially suppressed at low enough temperatures no matter how large the external bias V is. Nevertheless, this non-equilibrium effect may be considered curious because the supercurrent enhancement is provided by the normal terminal N_3, which “knows nothing” about superconductivity at all.

It is also interesting that, unlike for I_J, no such enhancement effect is observed for the Aharonov-Bohm contribution I_AB, here the only effect of the terminal N_3 is the current suppression (<ref>), see also Fig. <ref>b. This tendency is also understandable since, unlike in the case of the supercurrent, low energy quasiparticles mainly contribute to the Aharonov-Bohm current even at high voltages <cit.>. Accordingly, no significant impact of the terminal N_3 on I_AB (apart from that accounted for by Eq. (<ref>)) could be expected. We can also add that with increasing temperature above
ℰ_ Th both current components I_C^(5) and I_ m^(5) decay (respectively exponentially and as a power-law) similarly to the case of a four-terminal setup. This behavior is illustrated in Fig. <ref> for the case of an asymmetric setup to be addressed below.









 §.§ Asymmetric setup


Let us now consider an asymmetic setup, in which case the terminal N_3 is attached to the wire l_c in a non-symmetric fashion, just as it is shown in Fig. <ref>. Then the problem gets somewhat more involved since the conditions V_1/2 = ∓ V/2 and V_N = 0 no longer apply. In other words, the voltages V_1/2 and V_N should now be evaluated self-consistently by solving the Usadel equations combined with Eq. (<ref>). Treating this problem numerically, bearing in mind that (i) no current can flow into the normal terminal N_3, (ii) I_N,1 = I_N,2 and (iii) V= V_2 - V_1, we arrive at the results for V_N which contains an oscillating in ϕ part displayed in Fig. <ref>.





We observe that this oscillating part of the voltage V_N depends on both ϕ and V: it is an odd-like 2π-periodic function of the phase at smaller voltages eV ≲ 20 E_ Th and shows an even-like behavior at higher voltage values eV ≳ 80 E_ Th. Thus, the value V_N(ϕ, V) turns out to be sensitive to the proximity-induced long-range quantum coherence of the electrons in normal wires, and it essentially originates from an interplay between the Aharonov-Bohm and Josephson effects.

Without loss of generality the function V_N(ϕ, V) can be decomposed into even and odd terms as

    V_N(ϕ, V) = V_ even(ϕ, V) + V_ odd(ϕ, V).

In order to estimate the even part, one can solve the kinetic equations analytically by setting D_L/T≈ 1 and neglecting both j_s and Y. Then one finds:

    V_ even≈R_c_1 - R_c_28(R_c_1 + R_c_2) V + V_ AB,

where R_c_i= l_i/(σ_N𝒜) is the normal state resistance of the wire segment of length l_i.  Hence, the even in ϕ part of the voltage V_N equals to the sum of Ohmic and Aharonov-Bohm terms, where at large enough V the latter saturates to the value

    V_ AB≈ 0.29 ( E_ Th/e)cosϕ.


As far as the odd in ϕ term V_ odd is concerned, our numerical analysis demonstrates that, being important at smaller voltage values V, this term becomes strongly suppressed in the large voltage limit. This behavior is reminiscent of that for the currents I_C^(4)(V) and I_C,0^(5)(V), thereby indicating that the presence of the odd in ϕ contribution V_ odd may be associated with the Josephson-like effect. At the same time, one should keep in mind that in the asymmetric setup one has Y≠ 0 in the wire l_N,3. Hence, electron-hole asymmetry <cit.> induced in the kinetic equations by the Y-term  should also be taken into account while evaluating the contribution V_ odd. More detailed description of the electron-hole asymmetry effects is beyond the scope of the present paper and will be presented elsewhere.

Turning now to the analysis of the current-phase relation, we note that in the leading in j_s order one has f_L(x) ≈ f_L^c,1 inside the wire attached to the first superconducting terminal. Observing that (a) the kinetic equations are linear and (b) f_L is an odd function of energy, we conclude that the function f_L^c,1 can be expressed in terms of some linear combination of the functions f_L^N(V_1), f_L^N(V_2) and f_L^N(V_N). Since the value |V_N| remains smaller than V/2, it follows immediately that I_C^(5)(V) in the asymmetric five-terminals setup becomes suppressed at higher voltages as compared to that for the four-terminals setup. This observation is supported by the results of our numerical analysis displayed in Fig. <ref>.  We observe that the Josephson contribution evaluated for an asymmetric five-terminals setup survives up to the highest voltage values employed in the calculation (Fig. <ref>a). In this case for the same voltage range the Aharonov-Bohm current I_m shows no sign of saturation, as it is indicated in Fig. <ref>b.

In Fig. <ref> we further compare the full current-phase relations in both four- and five-terminal geometries at different bias voltages and T → 0. In the four-terminals case – in accordance with Eqs. (<ref>)–(<ref>) – we observe a clear crossover from the odd-like behavior of I_ osc(ϕ) at lower voltages to the even-like one at higher values of V. By contrast, in five-terminal configurations the odd (Josephson-like) component remains dominant up to very high voltages. With increasing temperature, however, this component gets suppressed much stronger than I_ m, as it is illustrated in Fig. <ref>.









§ CONCLUDING REMARKS


In this work we investigated proximity-induced long-range quantum coherent effects in multi-terminal Andreev interferometers under non-equilibrium conditions.  We demonstrated that at low enough temperatures the current flowing between two superconducting terminals results from a non-trivial interplay between Josephson-like and Aharonov-Bohm-like effects. The corresponding contributions to the current I_J and I_AB are controlled both by the magnetic flux Φ threading the system and the external bias voltage V. As functions of the magnetic flux both currents I_J and I_AB exhibit coherent oscillations with the period Φ_0 being respectively odd and even functions of Φ. The magnitudes of these two current components demonstrate very different dependencies on both voltage bias and temperature, thus offering a unique opportunity to at will engineer the current-phase relation in Andreev interferometers.

The system topology is yet another important factor that may strongly affect its non-equilibrium behavior at low enough T. Here we demonstrated that by attaching an extra normal reservoir of electrons or just by changing the symmetry of our multi-terminal hybrid structure one can further modify both currents I_J and I_AB in a non-trivial manner. For instance, in the presence of the normal terminal N_3 (see Fig. 1)
some “superconducting” (i.e. phase-coherent) electrons propagating in the central normal wire get absorbed by this terminal being replaced by “normal” (i.e. insensitive to proximity-induced quantum coherence) electrons
from N_3. This process results in two (in part competing) effects: (i) quantum decoherence that yields partial suppression of both currents I_J and I_AB and (ii) modification in the electron distribution function that may produce significant enhancement of the Josephson component I_J but has (almost) no extra effect on I_AB. We also discussed topology-dependent coherent oscillations of the voltage induced at the normal terminal isolated from the external leads.

Our predictions can be directly verified in modern experiments and may be used for designing
superconducting hybrid nanocircuits with controlled quantum properties.







Acknowledgements

This work was supported in part by RFBR Grant No. 18-02-00586. P.E.D. acknowledges support by Skoltech NGP Program (Skoltech-MIT joint project).





§ 





Consider first an SNS junction with a normal-metal wire of length L connecting two superconducting terminals. At sufficiently high energies ϵ≫ E_ Th the solution of the Usadel equation in the normal wire can be expressed as a superposition of the two independent anomalous propagators <cit.>:

    F_12(x)    =  F_ SN(L/2 + x) e^iϕ/2 +  F_ SN(L/2 - x) e^-iϕ/2,
     F_21 (x)    = - F_ SN(L/2 + x) e^-iϕ/2 -  F_ SN(L/2 - x) e^iϕ/2

where

    F_ SN(x) = -4 q (1 + q^2)/(1-q^2)^2, q(x) = i/1+√(2) e^x√(-2iϵ/D)

and x is the coordinate along the wire (-L/2 ≤ x ≤ L/2). Combining the above expressions with that for the spectral supercurrent

    j_s     =   1/4τ̂_3 ( Ĝ^R ∂_x Ĝ^R - Ĝ^A ∂_x Ĝ^A) =
       =   1/4 [  F_12∂_x  F_21 -   F_21∂_x  F_12 + c.c.]

we readily find

    j_s = 16/3 + 2√(2)sinϕ[ i √(-2iϵ/D) e^√(-2iϵ/D) + c.c.].

Having established this relation, we can now turn to the symmetric cross-like geometry <cit.> with two superconducting and two normal terminals biased by the voltage V, cf. also Fig. <ref> with l_c=0 and disconnected terminal N_3. Integrating Eq. (<ref>) over energies and bearing in mind that at T → 0 only states with energies |ϵ | ≥ eV/2 contribute to this integral, in the limit eV ≫ℰ_ Th we obtain

    I_C^SNS (V) ≃32 (1+v^-1)/3+2√(2)V/R_L e^-vsin(v + v^-1),

where the parameter v was defined above in Eq. (<ref>).






Note that this simple analysis totally neglects partial suppression of superconducting correlations inside the normal wires due to the presence of two extra normal terminals. The magnitude of this effect depends on the system topology and can be accounted for in a straightforward manner by formulating the matching conditions at the wire branching points. In a general form such conditions have already been discussed, e.g., in Refs. <cit.>. In essence, they are equivalent to the current conservation law interpreted as a “Kirchoff law for the Green functions” – see, e.g., Eq. (29) in Ref. <cit.>.



Here we derive a simplified version of these matching conditions sufficient for our present purposes. To do so, we note that for energies |ϵ| ≫ℰ_ Th and at distances from a superconductor exceeding the length scale ∼√(D/|ϵ |) the Usadel equation can be linearized, thereby reducing to a simple wave-like equation:

    D d^2 F/dx^2 + 2 i ϵ F = 0.

As this equation is linear, the effect from the superconducting terminals can be treated independently, cf., e.g., Eqs. (<ref>)–(<ref>).


To this end let us consider a setup schematically depicted in Fig. <ref>a. The setup consists of
one superconducting terminal attached to a normal wire which, in turn, is connected to N other normal wires at some branching point located sufficiently far from the superconductor. By solving the wave-like equation (<ref>), we obtain the following matching conditions

    F_i(x) = α F_ SN(x), α =2 A_0 / (∑_j = 0^N A_j ), i = 1,…, N,

where A_i denotes the cross section of the i-th normal wire and x is a coordinate along the corresponding wire. This condition implies that the anomalous propagator F_i(x) inside the i-th wire is suppressed by the factor α as compared to the propagator F_ SN(x) in the absence of the branching point (N =1).

Applying the matching condition (<ref>) to the setup of Fig. <ref> and bearing in mind that all wire cross sections are assumed to be equal A_i= A,
for the anomalous propagator F_12(x) inside the wire l_c we obtain

    F_12(x) = α F_ SN(L/2 + x) e^iϕ/2 + α̃ F_ SN(L/2 - x) e^-iϕ/2,

with α=α̃=2/3 in the absence of the terminal N_3 (four-terminal configuration) and
α =2/3, α̃= 4/9 (or vice versa depending on whether x is located to the left or to the right with respect to the wire l_N,3) for the five-terminal setup in Fig. <ref>. Accordingly, in these two cases the currents I_C^(4) (V) and I_C,0^(5) (V) get reduced respectively by the factors 4/9 and 8/27 compared to I_C^SNS (V) in Eq. (<ref>).

Note that, strictly speaking, the latter results for the prefactors  α and α̃ apply only provided different branching points in the setup of Fig. <ref> are located sufficiently far from each other. In order to address a more general situation let us consider the configuration depicted in Fig. <ref>b with the wire length l_c varying from zero to large values.  According to Eq. (<ref>),
for l_c → 0 we have α (0) = 1/2 (we again assume that all wires have the same cross section), while in the limit of large l_c one finds α = (2/3)^2 = 4/9. Making use of the linearity of Eq. (<ref>) and the conservation of the spectral currents, we recover the complete dependence α(l_c):

    α(l_c) ≈  4( 9 - exp(2 l_c √(-2iϵ/D))  )^-1.

This result demonstrates that in a general case the function α(l_c) depends on energy.

Bearing in mind the above matching conditions, for the setup in Fig. <ref> we obtain

    I_C^(4) (V)    ≃    I_C^SNS (V) 4/9    l_c^2 ≫ D/(eV) 
    1/2    l_c^2 ≪ D/(eV)  ,
    
    I_C,0^(5) (V)    ≃    I_C^SNS (V) 8/27    l_c^2 ≫ D/(eV) 
    2/5    l_c^2 ≪ D/(eV) .








99
belzig1999quasiclassical W. Belzig, F.K. Wilhelm, C. Bruder, G. Schön and A.D. Zaikin,
Superlatt. Microstruct. 25, 1251 (1999).
deph1 A.G. Semenov, A.D. Zaikin, and L.S. Kuzmin, Phys. Rev. B 86, 144529 (2012).
deph2 A.G. Semenov and A.D. Zaikin, Phys. Rev. B 91, 024505 (2015).
Pino G. Falci, D. Feinberg, and F.W.J. Hekking, Europhys. Lett. 54,
255 (2001).
BeckmannCAR D. Beckmann, H.B. Weber, and H. v. Löhneysen,
Phys. Rev. Lett. 93, 197003 (2004).
TeunCAR S. Russo,  M. Kroug, T.M. Klapwijk, and A.F. Morpurgo,
Phys. Rev. Lett. 95, 027002 (2005).
VenkatCAR P. Cadden-Zimansky and V. Chandrasekhar,
Phys. Rev. Lett. 97, 237003 (2006).
KZCAR M.S. Kalenkov and A.D. Zaikin, Phys. Rev. B 76, 224506 (2007).
GKZ D.S. Golubev, M.S. Kalenkov, and A.D. Zaikin, Phys. Rev. Lett. 103, 067006 (2009).
V A.F. Volkov, Phys. Rev. Lett. 74, 4730 (1995).
WSZ F.K. Wilhelm, G. Schön, and A.D. Zaikin, Phys. Rev. Lett. 81, 1682 (1998).
Yip S. Yip, Phys. Rev. B 58, 5803 (1998).
Teun J.J.A. Baselmans, A.F. Morpurgo, B.J. van Wees and T. M. Klapwijk, Nature 397, 43 (1999).
nakano1991quasiparticle H. Nakano and H. Takayanagi, Solid State Commun. 80, 997 (1991).
Petr V.T. Petrashov, V.N. Antonov, P. Delsing, and T. Claeson, Phys. Rev. Lett. 70, 347 (1993); 74, 5268 (1995).
Zaitsev A.V. Zaitsev, Physica B 203, 274 (1994).
SN96 T.H. Stoof and Yu.V. Nazarov, Phys. Rev. B 54, R772 (1996).
Grenoble H. Courtois, P. Gandit, D. Mailly, and B. Pannetier, Phys. Rev. Lett. 76, 130 (1996).
GWZ97 A.A. Golubov, F.K. Wilhelm, and A.D. Zaikin, Phys. Rev. B 55, 1123 (1997).
PD18 P.E. Dolgirev, M.S. Kalenkov and A.D. Zaikin, Phys. Rev. B 97, 054521 (2018).
PD18Th P.E. Dolgirev, M.S. Kalenkov and A.D. Zaikin, arXiv:1806.06882.
ZZh A.D. Zaikin and G.F. Zharkov, Fiz. Nizk. Temp. 7, 375
  (1981) [Sov. J. Low Temp. Phys. 7, 181 (1981)].
GreKa P. Dubos, H. Courtois, B. Pannetier, F.K. Wilhelm, A.D. Zaikin, and G. Schön,
Phys. Rev. B 63,  064502 (2001).
VH P. Virtanen and T.T. Heikkilä, Phys. Rev. Lett. 92, 177004 (2004);
J. Low Temp. Phys. 136, 401 (2004).
KZ17 M.S. Kalenkov and A.D. Zaikin, Phys. Rev. B 95, 024518 (2017).


