# The dc-Josephson effect with more than four superconducting leads

Régis Mélin  Univ. Grenoble-Alpes, CNRS, Grenoble INP††thanks: Institute of
Engineering Univ. Grenoble Alpes, Institut NEEL, 38000 Grenoble, France

###### Abstract

By definition, the $p$-terminal dc-Josephson current is sensitive to the
superconducting phase variables of $p$ terminals. In the paper, we establish
protocol for direct detection of the $p$-terminal dc-Josephson effect with
$p\geq 3$ in a device containing $N$ superconducting leads
$S_{1},\,S_{2},\,...,\,S_{N}$ having the phase variables
$\varphi_{1},\,\varphi_{2},\,...,\,\varphi_{N}$. The calculated signal
${\chi}^{(N)}$ can be probed in microwave experiments, and it corresponds to
the higher-order nonlocal inverse inductance obtained from differentiating the
current $I_{1}$ through $S_{1}$ with respect to the remaining $N-2$
independent phase differences
$\varphi_{2}-\varphi_{N},\,\varphi_{3}-\varphi_{N},\,...,\varphi_{N-1}-\varphi_{N}$.
We find that the values $p\leq N-2$ do not contribute to ${\chi}^{(N)}$, and
that ${\chi}^{(N)}\neq 0$ implies evidence for the $p=N-1$ or the
$p=N$-terminal dc-Josephson currents. For $N=4$ superconducting leads, we
demonstrate that ${\chi}^{(4)}\neq 0$ implies evidence for the $p=3$ or $p=4$
dc-Josephson effect, irrespective of the $p=2$-terminal dc-Josephson current.
Thus, we provide a way to demonstrate the dc-Josephson effect with more than
three terminals (i.e. with $p\geq 3$) in a device containing more than four
superconducting leads (i.e. with $N\geq 4$). The predicted ${\chi}^{(4)}$ is
“yes or no” answer to the $p\geq 3$ dc-Josephson effect, i.e. for $N=4$,
nonvanishingly small $\chi^{(4)}\neq 0$ implies the $p=3$ or $p=4$-terminal
dc-Josephson effect and vanishingly small $\chi^{(4)}=0$ implies absence of
the $p=3$ and $p=4$-terminal dc-Josephson effect. The paper can be viewed as
generalizing the recently considered $\varphi$-junctions in Andreev molecules
to arbitrary number $N$ of the superconducting leads, and it relies on basic
properties of the dc-Josephson effect that are not directly related to
nontrivial topology and Weyl point singularities.

##  I Introduction

The two-terminal Josephson junction Josephson  formed with the BCS
superconductors $S_{1}$ and $S_{2}$ reveals physical relevance of the gauge-
invariant difference $\varphi_{1}-\varphi_{2}$ between their macroscopic phase
variables (see figure 1a). Equilibrium dissipationless supercurrent

$I_{S}^{(2)}(\varphi_{1}-\varphi_{2})=I_{c}^{(2)}\sin\left(\varphi_{1}-\varphi_{2}\right)$
(1)

flows across the $S_{1}$-$S_{2}$ Josephson weak link Likharev , where the
superscript “$(2)$” in Eq. (1) refers to the number $N=2$ of the
superconducting leads. Anderson and Rowell Anderson  experimentally confirmed
the prediction Josephson  of the dc-Josephson effect.

Eq. (1) is valid for tunnel junctions and the current-phase relation

$I_{S}^{(2)}(\varphi_{1}-\varphi_{2})=\sum_{n}I_{c,\,n}^{(2)}\sin\left[n\left(\varphi_{1}-\varphi_{2}\right)\right]$
(2)

with $n$-Cooper pair tunneling holds more generally at arbitrary interface
transparency, see figure 1b for example of the higher-order two-Cooper pair
tunneling with $n=2$ in Eq. (2).

Current flows by Andreev reflection at a normal metal-superconductor (NS)
interface biased at the voltage $V$ smaller than the superconducting gap
$\Delta$, i.e. $|eV|<\Delta$. Spin-up electron from $N$ is Andreev reflected
as a spin-down hole while a Cooper pair is transmitted into $S$.

Nonlocality of Andreev reflection at the scale $R_{0}\approx\xi_{0}$ of the
BCS coherence length $\xi_{0}$ was predicted and experimentally probed theory-
CPBS1 ; theory-CPBS2 ; theory-CPBS3 ; theory-CPBS4 ; theory-CPBS5 ; theory-
CPBS6 ; theory-CPBS7 ; theory-CPBS8 ; theory-CPBS9 ; exp-CPBS1 ; exp-CPBS2 ;
exp-CPBS3 ; exp-CPBS4 ; exp-CPBS5 ; exp-CPBS6 ; exp-CPBS7 ; exp-CPBS8  in a
$N_{1}$-$S_{2}$-$N_{3}$ three-terminal device since the early 2000’s, see
figures 1c and 1d. The normal leads $N_{1}$ and $N_{3}$ are laterally
connected to the grounded $S_{2}$ and the voltages $V_{1}$ and $V_{3}$ are
applied on $N_{1}$ and $N_{3}$, the superconducting $S_{2}$ being grounded at
$V_{2}=0$.

Elastic cotunneling (EC) theory-CPBS6 ; theory-CPBS7 ; theory-CPBS9  over
$R_{0}\approx\xi_{0}$ transfers electrons from $N_{1}$ to $N_{3}$ across
$S_{2}$, or from $N_{3}$ to $N_{1}$, see figure 1c. Crossed Andreev reflection
(CAR) theory-CPBS5 ; theory-CPBS6 ; theory-CPBS7 ; theory-CPBS9  over
$R_{0}\approx\xi_{0}$ scatters spin-up electron from $N_{1}$ as spin-down hole
into $N_{2}$, leaving a Cooper pair in $S_{2}$, see figure 1d. Said
differently, two opposite-spin electrons from $N_{1}$ and $N_{3}$ are
cooperatively transmitted into the central $S_{2}$ and eventually join the
condensate.

EC and CAR in the above mentioned $N_{1}$-$S_{2}$-$N_{3}$ three-terminal
device were generalized to dEC Freyn  (see figure 1e) and dCAR Freyn  (see
figure 1f) in a laterally-connected $S_{1}$-$S_{2}$-$S_{3}$ three-terminal
Josephson junction with separation $R_{0}\approx\xi_{0}$ between the
$S_{1}$-$S_{2}$ and $S_{2}$-$S_{3}$ interfaces. Namely, double elastic
cotunneling (dEC) Freyn  on figure 1e transfers a Cooper pair from $S_{1}$ to
$S_{3}$ across $S_{2}$, or from $S_{3}$ to $S_{1}$. Double crossed Andreev
reflection (dCAR) Freyn  on figure 1f takes a Cooper pair from $S_{1}$ and
another one from $S_{2}$. The two pairs from $S_{1}$ and $S_{3}$ exchange
partners, yielding four-fermion state, i.e. the so-called quartet of electrons
Freyn ; Melin-EPJB ; QUARTETS1 ; Rech ; QUARTETS2 ; QUARTETS3 ; QUARTETS4 ;
QUARTETS5 ; QUARTETS6 ; QUARTETS7 . The quartet eventually dissociates as two
“outgoing” Cooper pairs joining the condensate of $S_{2}$.

Voltage biasing the $S_{1}$-$S_{2}$-$S_{3}$ three-terminal Josephson junction
Freyn ; Melin-EPJB  on figures 1e and 1f at the voltage differences
$V_{1}-V_{2}$ and $V_{3}-V_{2}$ is a possibility to reveal Lefloch ; Heiblum ;
Kim  dCAR and the quartets as emergence of the $V_{1}+V_{3}=0$ dc-Josephson
resonance line, if one of the elements of the differential conductance matrix
is plotted in the $(V_{1},V_{3})$ voltage plane while the “central” $S_{2}$ is
grounded at the reference voltage $V_{2}=0$. Other dc-Josephson resonance
lines were predictedFreyn ; Melin-EPJB  and observedLefloch ; Heiblum ; Kim ,
such as $V_{1}=V_{3}$ due to dEC. Other experiments multiterminal-exp1 ;
multiterminal-exp2 ; multiterminal-exp3 ; multiterminal-exp4 ; multiterminal-
exp5  did not report the quartet dc-Josephson anomaly, which is maybe a matter
of the materials and configurations of the devices. In addition, the recent
Ref. multiterminal-exp6,  reports progress in the fabrication of four-terminal
devices.

Figure 1: Panels a and b show the two-terminal dc-Josephson effect, with
single Cooper pair tunneling [see Eq. (1)] and two-Cooper pair tunneling [see
Eq. (2) with $n=2$] respectively. Panels c and d show double elastic
cotunneling (dEC) and double crossed Andreev reflection (dCAR) respectively,
in a three-terminal Josephson junction. Panels e and f show elastic
cotunneling (EC) and crossed Andreev reflection (CAR) respectively, in a
normal metal-superconductor-normal metal beam splitter.

Considering more generally $p_{1}$ Cooper pairs from $S_{1}$ and $p_{3}$ from
$S_{3}$ yields the energy $E_{initial}=2p_{1}V_{1}+2p_{3}V_{3}$ of the
“initial state”, and $E_{final}=2\left(p_{1}+p_{3}\right)V_{2}$ for the final
state, with $E_{final}\equiv 0$ because $V_{2}=0$ for the grounded $S_{2}$.
Energy conservation $E_{initial}=E_{final}$ implies the dc-Josephson resonance
line at

$p_{1}V_{1}+p_{3}V_{3}=0$ (3)

which is sustained by the static dc-phase variable

$\displaystyle\varphi_{p_{1},q_{1}}$ $\displaystyle=$ $\displaystyle
p_{1}\left[\varphi_{1}(t)-\varphi_{2}(t)\right]+p_{3}\left[\varphi_{3}(t)-\varphi_{2}(t)\right]$
(4) $\displaystyle=$ $\displaystyle
p_{1}\left[\varphi_{1}-\varphi_{2}\right]+p_{3}\left[\varphi_{3}-\varphi_{2}\right].$
(5)

The following Josephson relations Josephson

$\displaystyle\varphi_{1}(t)$ $\displaystyle=$
$\displaystyle\varphi_{1}+\frac{2eV_{1}}{\hbar}t$ (6)
$\displaystyle\varphi_{2}(t)$ $\displaystyle=$ $\displaystyle\varphi_{2}$ (7)
$\displaystyle\varphi_{3}(t)$ $\displaystyle=$
$\displaystyle\varphi_{3}+\frac{2eV_{3}}{\hbar}t$ (8)

for the superconducting phase variables $\varphi_{1}(t),\,\varphi_{2}(t)$ and
$\varphi_{3}(t)$ as a function of the time $t$ were combined to Eq. (3) in
order to deduce Eq. (5) from Eq. (4). Then, the multipair supercurrent
generalizing the quartets Freyn ; Melin-EPJB  takes the form

$I_{p_{1},p_{3}}^{(3)}=I_{c,p_{1},p_{3}}\sin\left[p_{1}\left(\varphi_{1}-\varphi_{2}\right)+p_{3}\left(\varphi_{3}-\varphi_{2}\right)\right].$
(9)

However, Eq. (9) also holds at equilibrium, i.e. if all superconducting leads
$S_{1}$, $S_{2}$ and $S_{3}$ are grounded at $V_{1}-V_{2}=V_{3}-V_{2}=0$, and
biased at the phase differences $\varphi_{1}-\varphi_{2}$ and
$\varphi_{3}-\varphi_{2}$. The present paper focuses on the equilibrium
multiterminal dc-Josephson effect where all leads are grounded.

Nonlocality of the dc-Josephson effect can be understood in different ways:

(i) Subgap propagation over the zero-energy BCS coherence length
$R_{0}\approx\xi_{0}$ across the “central” $S_{2}$ in a
$S_{1}$-$S_{2}$-$S_{3}$ three-terminal Josephson junction made with the
$S_{1}$-$S_{2}$ and $S_{2}$-$S_{3}$ lateral contacts separated by $R_{0}$, see
the above discussion.

(ii) Emergence of dc-Josephson current controlled by the phase of three or
more superconducting leads, see Eq. (9) with $p_{1}\neq 0$ and $p_{3}\neq 0$.

The considered devices involve $N$ superconductors connected by single-channel
weak links to the nonsuperconducting “central” region, see figure 2a. The
above item (i) for nonlocality over $R_{0}\approx\xi_{0}$ is thus not directly
relevant to the present work. According to the above item (ii), we evaluate
the contribution of the $p$-terminal Josephson effect in a device containing
$N$ superconducting leads. Namely, we evaluate the sensitivity of the current
$I_{1}$ through lead $S_{1}$ on the $p$ superconducting phase variables
$\varphi_{a_{1}},\,\varphi_{a_{2}},\,...,\varphi_{a_{p}}$ where $1\leq
a_{1}<a_{2}<...<a_{p}\leq N$. Then, we define $\chi^{(N)}$ as the partial
derivative of $I_{1}$ with respect to the $N-2$ phase differences
$\varphi_{2}-\varphi_{N}$, $\varphi_{3}-\varphi_{N}$, …,
$\varphi_{N-1}-\varphi_{N}$. We show that the $p$-terminal dc-Josephson effect
with $p\leq N-2$ does not contribute to $\chi^{(N)}$ while $p=N$ and $p=N-1$
yield nonvanishingly small contribution to $\chi^{(N)}$. Thus, microwave
experiments can reveal experimental evidence for $\chi^{(4)}\neq 0$ with
$N=4$, which implies the $p=3$ or the $p=4$ three or four-terminal dc-
Josephson effects, whatever the value of the $p=2$ dc-Josephson supercurrent.
Thus, the following paper demonstrates the possibility of directly testing the
multiterminal dc-Josephson effect with $p\geq 3$ in microwave experiments.

The paper is organized as follows. Section II establishes connection to known
results. Section III presents the model and the $p$-terminal Josephson current
in a device containing $N$ superconducting leads. Section IV provides examples
with $N=3$ and $N=4$ superconducting leads. Section V generalizes the theory
to arbitrary number $N$ of the superconducting leads. Numerical results for
$N=4$ superconducting leads are presented in section VI. Summary and final
remarks are provided in section VII.

Figure 2: The highlighted panel a shows the $N$-terminal Josephson junction
considered in the paper, where the $N$ grounded superconducting leads
$S_{1},\,\,S_{2},\,...,\,S_{N}$ with superconducting phase variables
$\varphi_{1},\,...,\,\varphi_{N}$ are connected to a small nonsuperconducting
region. The supercurrents through the leads $S_{1},\,\,S_{2},\,...,\,S_{N}$
are denoted by $I_{1},\,I_{2},\,...,\,I_{N}$, with
$\sum_{\alpha=1}^{N}I_{\alpha}=0$. Panels b-d show the devices with $N=3$ (b),
$N=4$ (c), and $N=5$ (d).

##  II Connection with recent results

We first note that the field of multiterminal Josephson junctions Freyn ;
Melin-EPJB ; QUARTETS1 ; Rech ; QUARTETS2 ; QUARTETS3 ; QUARTETS4 ; QUARTETS5
; QUARTETS6 ; QUARTETS7  has recently been enriched with the proposal of
nontrivial topology Nazarov1 ; Nazarov2 ; topo0 ; topo1 ; topo2 ; topo3 ;
topo4 ; topo5 ; Levchenko1 ; Levchenko2 ; Berry  and topology in the time-
periodic Floquet dynamics Feinberg1 ; Feinberg2 ; topo1-plus-Floquet . The
present paper is not directly related to those works on topology, in the sense
that we consider basic properties of the dc-Josephson, which generally hold in
the absence of Weyl-point singularities.

This section provides connection to known results on the multiterminal dc-
Josephson interferometers discussed in Refs. Pillet1, ; Pillet2, ; Benzoni, ;
Rech, .

The recent Refs. Pillet1, ; Pillet2, ; Benzoni,  considered an “Andreev
molecule” in the laterally-connected $S_{1}$-$S_{2}$-$S_{3}$ three-terminal
Josephson junction under equilibrium voltage biasing conditions, i.e. the
three superconducting leads $S_{1},\,S_{2}$ and $S_{3}$ are grounded at
$V_{1}-V_{2}=V_{3}-V_{2}=0$. Refs. Pillet1, ; Pillet2, ; Benzoni,
characterize the supercurrent $I_{1}(\varphi_{3})$ through lead $S_{1}$ as a
function of the phase $\varphi_{3}\neq 0$ on $S_{3}$ which is not directly
connected to $S_{1}$, while $\varphi_{2}=0$ is the reference, and it is
assumed in addition that $\varphi_{1}=0$.

This nonlocal response Pillet1 ; Pillet2 ; Benzoni  of the supercurrent
$I_{1}$ through $S_{1}$ to the phase $\varphi_{3}$ on $S_{3}$ originates from
the above mentioned dEC and dCAR Freyn . In the limit of tunnel contacts, the
dEC and dCAR Josephson currents are given by Freyn

$\displaystyle I_{dEC}^{(3)}$ $\displaystyle=$ $\displaystyle
I_{c,\,dEC}^{(3)}\sin(\varphi_{1}-\varphi_{3}),$ (10) $\displaystyle
I_{dCAR}^{(3)}$ $\displaystyle=$ $\displaystyle
I_{c,\,dCAR}^{(3)}\sin(\varphi_{1}+\varphi_{3}-2\varphi_{2}),$ (11)

where “$(3)$” in the superscript refers to the number $N=3$ of the
superconducting leads. Considering $\varphi_{1}=\varphi_{2}=0$ and
$\varphi_{3}\neq 0$ yields the dc-Josephson current $I_{1}$ through $S_{1}$:

$I_{1}^{(3)}(0)=I_{dEC}^{(3)}+I_{dCAR}^{(3)}=\left(-I_{c,\,dEC}^{(3)}+I_{c,\,dCAR}^{(3)}\right)\sin\varphi_{3}.$
(12)

In an intuitive two-terminal picture, a $\varphi$-junction is obtained Pillet1
; Pillet2 ; Benzoni : the supercurrent $I_{1}^{(3)}(\varphi_{1}-\varphi_{2})$
plotted as a function of $\varphi_{1}-\varphi_{2}$ is shifted by arbitrary
phase which is controlled by $\varphi_{3}-\varphi_{2}\neq 0$. Two limiting
cases are considered:

(a) In the tunnel limit, the dEC and dCAR current-phase relations are $0$\-
and $\pi$-shifted respectively Freyn ; Melin-EPJB . Then,
$I_{c,\,dCAR}^{(3)}<0$ and $-I_{c,\,dEC}^{(3)}<0$ are both negative, yielding
the same negative sign $-I_{c,\,dEC}^{(3)}+I_{c,\,dCAR}^{(3)}<0$ as if
$-I_{c,\,dEC}^{(3)}$ was alone. In this item (a), we assume that the contacts
have linear dimension which is large compared to the Fermi wave-length
$\lambda_{F}$ and small compared to the superconducting coherence length
$\xi_{0}$.

(b) The limit $R_{0}\ll\xi_{0}$ of small separation between the contacts
necessarily produces $I_{c,\,dCAR}^{(3)}\simeq 0$, because the assumption
$R_{0}\ll\xi_{0}$ implies equivalence to the two-terminal $S_{1}$-$S_{3}$.
Then, the sign of $-I_{c,\,dEC}^{(3)}+I_{c,\,dCAR}^{(3)}\simeq-
I_{c,\,dEC}^{(3)}<0$ is necessarily negative in the absence of dCAR.

The conditions for deducing “evidence for dCAR” from positive

$-I_{c,\,dEC}^{(3)}+I_{c,\,dCAR}^{(3)}>0$ (13)

have thus not been elucidated at present time: The above arguments show that
Eq. (13) is parameter-dependent. In the following paper, we propose another
possibility for directly probing the $p$-terminal dc-Josephson effect with
$p\geq 3$.

Considering now the $(S_{1},\,S_{2},\,S_{3},\,S_{4})$ four-terminal Josephson
junction with $N=4$, we recently proposed Rech  a superconducting quantum
interference device (SQUID SQUID ) containing two loops making four contacts
on a double quantum dot defined in a semiconducting nanowire or a carbon
nanotube. It was shown in Ref. Rech,  that dEC and dCAR can be distinguished
as different peaks in the Fourier transform of the critical current with
respect to the magnetic field. Thus, this double SQUID Rech  successfully
proposes a way to provide evidence for dCAR and for “the three-terminal dc-
Josephson effect”.

The device on figure 2c contains the four superconducting leads
$S_{1},\,S_{2},\,S_{3},\,S_{4}$ which are biased at the three independent
phases differences $\varphi_{1}-\varphi_{4}$, $\varphi_{2}-\varphi_{4}$ and
$\varphi_{3}-\varphi_{4}$. As it is mentioned above, the superconducting leads
are grounded at $V_{1}-V_{4}=0$, $V_{2}-V_{4}=0$ and $V_{3}-V_{4}=0$. In
addition, the two loops are pierced by the magnetic field fluxes $\Phi_{A}$
and $\Phi_{B}$. Compared to Ref. Rech, , we complementary propose microwave
detection of the second-order nonlocal inverse inductance

$\displaystyle{\chi}^{(4)}$ $\displaystyle\equiv$
$\displaystyle{\chi}^{(4)}\left(\varphi_{1}-\varphi_{4},\varphi_{2}-\varphi_{4},\varphi_{3}-\varphi_{4}\right)$
(14) $\displaystyle=$
$\displaystyle\frac{\partial^{2}I_{1}^{(4)}\left(\varphi_{1}-\varphi_{4},\varphi_{2}-\varphi_{4},\varphi_{3}-\varphi_{4}\right)}{\partial\left(\varphi_{2}-\varphi_{4}\right)\partial\left(\varphi_{3}-\varphi_{4}\right)},$
(15)

which is also evaluated as a function of $\varphi_{1}-\varphi_{4}$ for
$\varphi_{2}-\varphi_{4}=0$ and $\varphi_{3}-\varphi_{4}=0$:

$\displaystyle{\chi}^{(4)(0)}\equiv{\chi}^{(4)(0)}\left(\varphi_{1}-\varphi_{4}\right)$
(16) $\displaystyle=$
$\displaystyle\left.\frac{\partial^{2}I_{1}^{(4)}\left(\varphi_{1}-\varphi_{4},\varphi_{2}-\varphi_{4},\varphi_{3}-\varphi_{4}\right)}{\partial\left(\varphi_{2}-\varphi_{4}\right)\partial\left(\varphi_{3}-\varphi_{4}\right)}\right|_{\begin{array}[]{c}\varphi_{2}-\varphi_{4}=0\\\
\varphi_{3}-\varphi_{4}=0\end{array}}.$ (19)

The superscript “$(4)$” in Eqs. (14)-(15) and Eqs. (16)-(19) refers to the
number $N=4$ of the superconducting leads, see the device on figure 2c. In
experiments, Eqs. (14)-(19) can be detected with microwaves.

$N$ |  Figure number |  $p=2$ |  $p=3$ |  $p=4$ |  … |  $p=N-1$ |  $p=N$
---|---|---|---|---|---|---|---
$2$ |  |  Eqs. (1)-(2) |  |  |  … |  |
$3$ |  Fig. 2b |  Eqs. (34)-(35) |  Eq. (IV.1) |  |  … |  |
$4$ |  Fig. 2c |  Eqs. (46)-(46) |  Eqs. (46)-(46) |  Eq. (46) |  … |  |
|  |  |  Eq. (IV.2) |  Eq. (50) |  … |  |
|  |  |  Eq. (52) |  Eq. (52) |  … |  |
|  Fig. 3 |  Fig. 3-a1 |  Fig. 3-a2 |  Fig. 3-a3 |  … |  |
|  Fig. 4 |  Fig. 4-a1 |  Fig. 4-a2 |  |  … |  |
… |  … |  … |  … |  … |  … |  … |  …
$N=2q$ |  Fig. 2a |  |  |  |  … |  Eq. (V.2.2) |  Eq. (V.2.1)
$N=2q^{\prime}+1$ |  Fig. 2a |  |  |  |  … |  Eq. (V.3.2) |  Eq. (V.3.1)
Table 1: The table summarizes the values of $p$ and $N$ that are considered in
the paper.

In the paper, we demonstrate that

“${\chi}^{(4)}\left(\varphi_{1}-\varphi_{4},\Phi_{A},\Phi_{B}\right)=0$ or
${\chi}^{(4)(0)}\left(\varphi_{1}-\varphi_{4}\right)=0$”

generically implies

“Absence of the three- and four-terminal dc-Josephson effect, i.e.
$I_{c,p_{1},p_{3}}^{(3)}=0$ in Eq. (9) for all $p_{1},\,p_{3}$, and
$I_{c,p_{1},p_{2},p_{3}}^{(4)}=0$ for all $p_{1},\,p_{2},\,p_{3}$”.

The above mentioned four-terminal critical current QUARTETS6 ; Kim
$I_{c,p_{1},p_{2},p_{3}}^{(4)}$ corresponds to

$\displaystyle
I_{p_{1},p_{2},p_{3}}^{(4)}\left(\varphi_{1}-\varphi_{4},\varphi_{2}-\varphi_{4},\varphi_{3}-\varphi_{4}\right)$
$\displaystyle=$ $\displaystyle
I_{c,p_{1},p_{2},p_{3}}^{(4)}\sin\left[p_{1}\left(\varphi_{1}-\varphi_{4}\right)+p_{2}\left(\varphi_{2}-\varphi_{4}\right)\right.$
$\displaystyle\left.+p_{3}\left(\varphi_{3}-\varphi_{4}\right)\right].$

Conversely, we deduce a second logical link:

“Nonvanishingly small signal
${\chi}^{(4)}\left(\varphi_{1}-\varphi_{4},\Phi_{A},\Phi_{B}\right)\neq 0$ or
${\chi}^{(4)(0)}\left(\varphi_{1}-\varphi_{4}\right)\neq 0$”

generically implies

“Evidence for the three- or four-terminal dc-Josephson effect, i.e.
$I_{c,p_{1},p_{3}}^{(3)}\neq 0$ for some $p_{1},\,p_{3}$ or
$I_{c,p_{1},p_{2},p_{3}}^{(4)}\neq 0$ for some $p_{1},\,p_{2},\,p_{3}$”.

The two logical links are independent on the value of the two-terminal dc-
Josephson current, i.e. $I_{c,\,n}^{(2)}$ in Eq. (2) can take the arbitrary
values $I_{c,\,n}^{(2)}\neq 0$. In addition, we generalize the theory to an
arbitrary number $N$ of the superconducting leads.

##  III $p$-terminal Josephson current with $N$ superconducting leads

In this section, we establish an expression for the $p$-terminal dc-Josephson
current with $N\geq p$ superconducting leads. The Table summarizes the
different values of $p$ and $N$ that are considered in the paper. Subsection
III.1 presents the expression of the Andreev Bound States (ABS) energies.
Subsection III.2 provides the general expression of the $p$-terminal Josephson
current. Subsection III.3 presents the example of a single-level quantum dot
in the infinite-gap limit.

###  III.1 Expression of the supercurrent carried by the Andreev Bound States

In this subsection, we evaluate the Fourier transform of the ABS energies with
respect to all superconducting phase variables $\varphi_{1}$, …, $\varphi_{N}$
of the $N$ superconducting leads $S_{1},\,...,\,S_{N}$ connected to small
nonsuperconducting region.

The ABS energies
$E^{\\{\varphi\\}}_{ABS,\lambda}\left(\varphi_{1},\,...,\,\varphi_{N}\right)$
depend on all values of the $\varphi_{n}-\varphi_{N}$ (where $n=1,\,...,\,N-1$
labels the superconducting leads and $\lambda=1,\,...,\,N_{ABS}$ is used for
the negative-energy ABS). Taking the Fourier transform leads to

$\displaystyle
E^{\\{\varphi\\}}_{ABS,\lambda}\left(\varphi_{1},\,\varphi_{2},...,\,\varphi_{N}\right)$
$\displaystyle=$ $\displaystyle\sum_{m_{1}}
\sum_{m_{2}}...\sum_{m_{N}}E^{\\{m\\}}_{ABS,\lambda}\left[m_{1},\,m_{2},\,...,\,m_{N}\right]\times$
$\displaystyle\exp\left(i\left(m_{1}\varphi_{1}+m_{2}\varphi_{2}+....+m_{N}\varphi_{N}\right)\right)\delta\left(m_{1}+m_{2}+...+m_{N}\right),$

where the constraint

$m_{1}+m_{2}+...+m_{N}=0$ (22)

originates from gauge invariance, i.e. any of the superconducting phases
$\varphi_{n_{ref}}$ can be chosen as the reference, leaving $N-1$ independent
gauge-invariant phase differences $\varphi_{n}-\varphi_{n_{ref}}$ with $n\neq
n_{ref}$. For instance, choosing $n_{ref}=N$ leads to

$\displaystyle m_{1}\varphi_{1}+m_{2}\varphi_{2}+....+m_{N}\varphi_{N}$
$\displaystyle=$ $\displaystyle
m_{1}\left(\varphi_{1}-\varphi_{N}\right)+m_{2}\left(\varphi_{2}-\varphi_{N}\right)+...$
$\displaystyle+m_{N-1}\left(\varphi_{N-1}-\varphi_{N}\right),$

where Eq. (22) and Eq. (III.1) are equivalently used. The variables
$m_{1},\,m_{2},\,...,\,m_{N}$ are conjugate to the superconducting phases
$\varphi_{1},\varphi_{2},\,...,\,\varphi_{N}$, i.e. they are interpreted as
the algebraic number of Cooper pairs transmitted into
$S_{1},\,S_{2},\,...,\,S_{N}$.

The zero-temperature dc-Josephson current through the lead $S_{N_{0}}$ is
given by

$\displaystyle
I_{N_{0}}^{S,\,\left(N\right)}\left(\varphi_{1},\,...,\,\varphi_{N}\right)=-\frac{e}{\hbar}\frac{\partial}{\partial\varphi_{N_{0}}}\sum_{\lambda}E^{\\{\varphi\\}}_{ABS,\lambda}\left(\varphi_{1},\,\varphi_{2},\,...,\,\varphi_{N}\right)$
(24) $\displaystyle=$ $\displaystyle\sum_{m_{1}}
\sum_{m_{2}}...\sum_{m_{N}}I_{c,\,n_{0}}^{\left(N\right)}\left[m_{1},\,...,\,m_{N}\right]\sin\left(m_{1}\varphi_{1}+m_{2}\varphi_{2}+....+m_{N}\varphi_{N}\right)\delta\left(m_{1}+m_{2}+...+m_{N}\right),$
(25)

where

$I_{c,\,n_{0}}^{\left(N\right)}\left[m_{1},\,...,\,m_{N}\right]=-\frac{e}{\hbar}m_{N_{0}}E^{\\{m\\}}_{ABS,\lambda}\left[m_{1},\,m_{2},\,...,\,m_{N}\right].$
(26)

Figure 3: Panels a1, b1 and c1 show the “incoming” and “outgoing” Cooper pairs
for the $p=2,\,3,\,4$-terminal dc-Josephson processes respectively, with a
total of $N=4$ superconducting leads. The corresponding values of $a_{\alpha}$
and $m_{a_{\alpha}}$ are shown on panels a2, b2 and c2. The integer
$a_{\alpha}\in\\{1,\,...,\,n\\}$ is shown on the $x$-axis of panels a2, b2 and
c2 corresponding to the parameters of panels a1, b1 and c1 respectively. The
variable $m\equiv m_{\alpha}$ on the $y$-axis of panels a2, b2 and c2 is the
algebraic number of Cooper pairs transmitted in each superconducting lead.

Figure 4: Panels a1 and a2 are obtained by “dressing” the $p=2$ and
$p=3$-terminal dc-Josephson current processes on figures 3-a1 and 3-a2 by an
“excursion” in and out from one of the superconducting leads. Thus, the $p=2$
and $p=3$-terminal dc-Josephson processes on panel a1 and b1 involve three and
four terminal respectively. They receive the same $a_{\alpha}$ and
$m_{a_{\alpha}}$ values as the “bare” diagrams on figures 3-a1 and 3-a2.
Figure 5: The figure shows the distribution $P(H_{n_{1},\,\,n_{2},\,\,n_{3}})$
for a quantum dot connected to $N=3$ superconducting leads, where
$H_{n_{1},\,\,n_{2},\,\,n_{3}}$ is given by Eq. (III.3) with $N=p=3$. Panels
a, b, c and d correspond to $(n_{1},\,\,n_{2},\,\,n_{3})=(1,\,0,\,-1)$,
$(2,\,0,\,-2)$, $(1,\,1,\,-2)$ and $(2,\,1,\,-3)$ respectively. The coupling
parameters $\Gamma$s entering Eqs. (29)-(III.3) are chosen at random in a
Gaussian distribution centered at $\overline{\Gamma}=1$ and with root-mean-
square $\delta\Gamma=0.5$. The negative values of the $\Gamma$s are rejected.
Figure 6:  The figure shows the distribution
$P(H_{n_{1},\,\,n_{2},\,\,n_{3},\,\,n_{4}})$ for a quantum dot connected to
$N=4$ superconducting leads, where $H_{n_{1},\,\,n_{2},\,\,n_{3},\,\,n_{4}}$
is given by Eq. (III.3) with $N=p=4$. Panels a, b, c and d correspond to
$(n_{1},\,\,n_{2},\,\,n_{3},\,n_{4})=(1,\,-1,\,1,\,-1)$, $(3,\,1,\,-2,\,-2)$,
$(2,\,3,\,-3,\,-1)$ and $(1,\,3,\,-3,\,-1)$ respectively. The coupling
parameters $\Gamma$s entering Eqs. (29)-(III.3) are chosen at random in a
Gaussian distribution centered at $\overline{\Gamma}=1$ and with root-mean-
square $\delta\Gamma=0.5$.

###  III.2 $p$-terminal dc-Josephson current with $N$ superconducting leads

In this subsection, we provide expression for the $p$-terminal dc-Josephson
current in a device containing $N$ superconducting leads, with $p\leq N$.

We consider the $p$ integers $a_{1},\,....,\,\,a_{p}$ such that $1\leq
a_{1}<a_{2}<...<a_{p}\leq N$. They encode supercurrent flowing through the
superconducting leads $S_{a_{1}},\,S_{a_{2}},\,...,\,S_{a_{p}}$, and absence
of supercurrent through $S_{b_{1}},\,S_{b_{2}},\,...,\,S_{b_{N-p}}$, where
$a_{\alpha}\neq b_{\beta}$ for all $\alpha=1,\,2,\,...,\,p$ and
$\beta=1,\,2,\,...,\,\,n-p$. Then, $m_{a_{\alpha}}\neq 0$ if $a_{\alpha}$
belong to $\\{a_{1},\,...,\,a_{p}\\}$, and $m_{b_{\beta}}=0$ if $b_{\beta}$ is
not in $\\{a_{1},\,...,\,a_{p}\\}$, see Eqs. (24)-(25).

These statements are illustrated on figure 3 for the $N=4$ superconducting
leads $S_{1}$, $S_{2}$, $S_{3}$ and $S_{4}$. Figure 3-a1, 3-b1 and 3-c1 show
the balance in the number of “outgoing” and “incoming” Cooper pairs, for the
$p=2,\,3,\,4$-terminal dc-Josephson currents respectively, in a device
containing $N=4$ superconducting leads. Figures 3-a2, 3-b2 and 3-c2 show the
values of the variables $a_{\alpha}$ and $m_{a_{\alpha}}$ for the processes on
figures 3-a1, 3-b1 and 3-c1 with $(m_{a_{1}},\,m_{a_{2}})=(1,-1)$,
$(m_{a_{1}},\,m_{a_{2}},\,m_{a_{3}})=(3,\,-1,\,-2)$ and
$(m_{a_{1}},\,m_{a_{2}},\,m_{a_{3}},\,m_{a_{4}})=(1,\,-1,\,1,\,-1)$
respectively. We also note that dressing the processes on figures 3-a1 and
3-a2 according to figures 4-a1 and 4-a2 leaves unchanged the values of
$a_{\alpha}$ and $m_{a_{\alpha}}$.

Now, Eqs. (24)-(25) are decomposed as the following:

$I_{N_{0}}^{S,\,\left(N\right)}\left(\varphi_{1},\,...,\,\varphi_{N}\right)=\sum_{p=2}^{N}I_{N_{0}}^{S,\left(p\right)}\left(\varphi_{1},\,...,\,\varphi_{N}\right),$
(27)

where

$\displaystyle
I_{N_{0}}^{S,\,\left(N\right)\,\left(p\right)}\left(\varphi_{1},\,...,\,\varphi_{N}\right)=\sum_{a_{1}=1}^{N}\sum_{a_{2}=a_{1}+1}^{N}...\sum_{a_{p}=a_{p-1}+1}^{N}\delta\left(\prod_{\alpha=1}^{N}\left(1-\delta\left(a_{\alpha},\,n_{0}\right)\right)\right)\sum_{m_{a_{1}}\neq
0}\sum_{m_{a_{2}}\neq 0}...\sum_{m_{a_{p}}\neq 0}\times$ (28) $\displaystyle
I_{c,\,n_{0}}^{\left(N\right)\,\left(p\right)\,\left(a_{1},\,a_{2},\,...,\,a_{p}\right)}\left[m_{a_{1}},\,...,\,m_{a_{p}}\right]\sin\left(m_{a_{1}}\varphi_{a_{1}}+m_{a_{2}}\varphi_{a_{2}}+....+m_{a_{p}}\varphi_{a_{p}}\right)\delta\left(m_{a_{1}}+m_{a_{2}}+...+m_{a_{p}}\right).$

The $\delta$-function
$\delta\left(\prod_{\alpha=1}^{N}\left(1-\delta_{a_{\alpha},\,n_{0}}\right)\right)$
in Eq. (28) is used to indicate that the lead $S_{N_{0}}$ (through which the
supercurrent is evaluated) is among $S_{a_{1}},\,...,\,S_{a_{p}}$ (which
sustain the considered $p$-terminal dc-Josephson process). Namely,
$a_{\alpha_{0}}=N_{0}$ for one of the $\alpha=\alpha_{0}$ implies
$\prod_{\alpha=1}^{N}\left(1-\delta_{a_{\alpha},\,n_{0}}\right)=0$ and
$a_{\alpha}\neq N_{0}$ for all $\alpha=1,\,...,p$ implies
$\prod_{\alpha=1}^{N}\left(1-\delta_{a_{\alpha},\,n_{0}}\right)=1$.

Eqs. (24)-(25) were calculated from the contribution of the ABS to the dc-
Josephson supercurrent. The present paper is based on the general current-
phase relation given by Eq. (28), which also captures the contribution of the
continua Pillet1 ; Pillet2 ; Benzoni .

###  III.3 Single-level quantum dot

In this subsection, we illustrate Eqs. (24)-(25) for a quantum dot connected
to three or four superconducting leads. We denote by
$\Gamma_{1},\,\Gamma_{2},\,...,\,\Gamma_{N}$ the line-width broadening
associated to hopping between the dot and each of the superconducting leads
$S_{1},\,S_{2},\,...,\,S_{N}$ in the normal-state. We take the infinite-gap
limit, see for instance Refs. topo1-plus-Floquet, ; Zazunov, ; Meng, . The
$2\times 2$ Nambu Hamiltonian takes the form

${\cal
H}_{\infty}=\left(\begin{array}[]{cc}0&\sum_{\alpha=1}^{N}\Gamma_{\alpha}\exp\left(i\varphi_{\alpha}\right)\\\
\sum_{\alpha=1}^{N}\Gamma_{\alpha}\exp\left(-i\varphi_{\alpha}\right)&0\end{array}\right).$
(29)

The ABS energies

$\displaystyle
E_{ABS,\pm}\left(\varphi_{1},\,\varphi_{2},\,...,\,\varphi_{N}\right)$
$\displaystyle=$
$\displaystyle\pm\left|\sum_{\alpha=1}^{N}\Gamma_{\alpha}\exp\left(i\varphi_{\alpha}\right)\right|.$

imply nonvanishingly small Fourier coefficients
$E^{\\{m\\}}_{ABS,\pm}\left[m_{1},\,m_{2},\,...,\,m_{N}\right]$ in Eq.
(III.1). This yields the $p$-terminal Josephson effect, i.e. components of the
supercurrent which are sensitive to the superconducting phase variables of $p$
superconducting leads:

$\Psi_{p}\left[a_{1},\,a_{2},\,...,\,a_{p}\right]=m_{a_{1}}\varphi_{a_{1}}+m_{a_{2}}\varphi_{a_{2}}+...+m_{a_{p}}\varphi_{a_{p}},$
(31)

where $\\{a_{\alpha}\\}$ is a set of $p$ integers such that $1\leq
a_{1}<a_{2}<...<a_{p}\leq N$ and $m_{a_{\alpha}}\neq 0$ for all
$\alpha=1,\,...,\,p$.

Figures 5 and 6 show the distributions $P(H_{n_{1},\,n_{2},\,n_{3}})$ and
$P(H_{n_{1},\,n_{2},\,n_{3},\,n_{4}})$ for the Fourier harmonics of the three
and four-terminal dc-Josephson currents

$\displaystyle H_{n_{a_{1}},\,\,n_{a_{2}},\,...,\,n_{a_{p}}}$ $\displaystyle=$
$\displaystyle\int\frac{d\varphi_{a_{1}}}{2\pi}\int\frac{d\varphi_{a_{2}}}{2\pi}...\int\frac{d\varphi_{a_{p}}}{2\pi}\times$
$\displaystyle
I_{1}^{S(N)(p)}\left(\varphi_{a_{1}},\,\varphi_{a_{2}},\,...,\varphi_{a_{p}}\right)\times$
$\displaystyle\sin\left(m_{a_{1}}\varphi_{a_{1}}+m_{a_{2}}\varphi_{a_{2}}+...+m_{a_{p}}\varphi_{a_{p}}\right),$

where $a_{\alpha}\equiv\alpha$ on the examples of figures 5 and 6 where $p=N$.

The distribution of the $\Gamma$s in Eqs. (29)-(III.3) is taken as a Gaussian
centered at $\overline{\Gamma}=1$, with root-mean-square $\delta\Gamma=0.5$.
The negative values of the $\Gamma$s are rejected in the numerical
calculation. These parameter values imply that $H_{n_{1},\,n_{2},\,n_{3}}$ and
$H_{n_{1},\,n_{2},\,n_{3},\,n_{4}}$ are roughly of order unity on figures 5
and 6. Figure 5a shows the $(1,\,0,\,-1)$ harmonics with $N=3$, with the
expected positive sign. Figure 5b shows the expected negative sign of the
$(2,\,0,\,-2)$ harmonics. Figure 5c shows the $(1,\,1,\,-2)$ quartets with
negative sign corresponding to the negative $I_{c,\,dCAR}<0$ mentioned in
section II. Figure 5d shows the $(2,\,1,\,-3)$ harmonics. Figures 6a-6d
further illustrate the multiterminal dc-Josephson effect on the example of
$N=4$ superconducting leads, with
$(n_{1},\,\,n_{2},\,\,n_{3},\,n_{4})=(1,\,-1,\,1,\,-1)$, $(3,\,1,\,-2,\,-2)$,
$(2,\,3,\,-3,\,-1)$ and $(1,\,3,\,-3,\,-1)$ respectively.

It is concluded that the multiterminal dc-Josephson effect naturally emerges
if a quantum dot is connected to $N$ superconducting leads, in the form of the
higher-order harmonics of the dc-Josephson current-phase relation, see the
selected values of $(n_{1},\,\,n_{2},\,\,n_{3})$ and
$(n_{1},\,\,n_{2},\,\,n_{3},\,\,n_{4})$ for $N=3$ and $N=4$ superconducting
leads on figures 5 and 6 respectively.

##  IV Examples with $N=3,\,4$ superconducting leads

In this section, we present the specific examples of $N=3$ and $N=4$
superconducting leads, see the devices on figures 2b and 2c respectively. The
cases $N=3$ and $N=4$ are treated in sections IV.1 and IV.2 respectively. We
show that $\chi^{(3)}$ for $N=3$ yields the $p=2$ and the $p=3$-terminal dc-
Josephson effect, and that $\chi^{(4)}$ for $N=4$ yields the $p=3$ and the
$p=4$-terminal dc-Josephson effect, whatever the value of the $p=2$-terminal
dc-Josephson current. Thus, our proposal has the potential for directly
probing the $p=3$ or $p=4$-terminal dc-Josephson effect with $N=4$
superconducting leads in microwave experiments.

###  IV.1 Device with $N=3$ superconducting leads on figure 2b

In this subsection, we consider the device with $N=3$ superconducting leads on
figure 2b. We specifically evaluate the flux-$\Phi$ sensitivity of the
supercurrent through the lead $S_{1}$:

$\displaystyle I_{1}^{(3)}\left(\varphi_{1},\varphi_{2},\varphi_{3}\right)$
(34) $\displaystyle=$ $\displaystyle\sum_{n\neq
0}I_{c}^{{}^{\prime}\left(3\right)\,\left(2\right)\,\left(1,\,2\right)}[n]\sin\left[n\left(\varphi_{1}-\varphi_{2}\right)\right]$
$\displaystyle+$ $\displaystyle\sum_{n\neq
0}I_{c}^{{}^{\prime}\left(3\right)\,\left(2\right)\,\left(1,\,3\right)}[n]\sin\left[n\left(\varphi_{1}-\varphi_{3}\right)\right]$
(35) $\displaystyle+$ $\displaystyle\sum_{n_{1}\neq 0,\,\,n_{2}\neq
0}I_{c}^{{}^{\prime}\left(3\right)\,\left(3\right)\,\left(1,\,2,\,3\right)}[n_{1},\,\,n_{2}]\times$
$\displaystyle\sin\left[n_{1}\left(\varphi_{1}-\varphi_{3}\right)+n_{2}\left(\varphi_{2}-\varphi_{3}\right)\right].$

Taking into account that $\varphi_{3}=\varphi_{2}-\Phi$, we obtain
$\varphi_{1}-\varphi_{3}=\varphi_{1}-\varphi_{2}+\Phi$. The considered signal
for $N=3$ is defined as the following:

$\displaystyle{\chi}^{(3)}(\varphi_{1}-\varphi_{2},\Phi)=\frac{\partial
I_{1}^{(3)}(\varphi_{1}-\varphi_{3},\varphi_{2}-\varphi_{3})}{\partial\left(\varphi_{2}-\varphi_{3}\right)},$
(37)

which is analogous to $\chi^{(4)}$ in the above Eqs. (14)-(15) with $N=4$
superconducting leads. Again, $\chi^{(3)}$ in Eq. (37) can be detected in
microwave experiments.

The following is deduced from Eqs. (34)-(IV.1):

$\displaystyle{\chi}^{(3)}(\varphi_{1}-\varphi_{2},\Phi)$ (38)
$\displaystyle=$ $\displaystyle\sum_{n\neq
0}nI_{c}^{{}^{\prime}\left(3\right)\,\left(2\right)\,\left(1,\,3\right)}[n]\cos\left[n\left(\varphi_{1}-\varphi_{2}+\Phi\right)\right]$
$\displaystyle+$
$\displaystyle\sum_{n_{1},\,n_{2}}n_{2}I_{c}^{{}^{\prime}\left(3\right)\,\left(3\right)\,\left(1,\,2,\,3\right)}[n_{1},\,n_{2}]\times$
$\displaystyle\cos\left[n_{1}\left(\varphi_{1}-\varphi_{2}+\Phi\right)+n_{2}\Phi\right].$

The zero-flux limit $\Phi=0$ of Eqs. (38)-(IV.1) is the following:

$\displaystyle{\chi}^{(3)(0)}(\varphi_{1}-\varphi_{2})\equiv{\chi}^{(3)}(\varphi_{1}-\varphi_{2},0)$
(41) $\displaystyle=$ $\displaystyle\sum_{n\neq
0}n\left[I_{c}^{{}^{\prime}\left(3\right)\,\left(2\right)\,\left(1,\,3\right)}[n]+\sum_{n_{2}\neq
0}I_{c}^{{}^{\prime}\left(3\right)\,\left(3\right)\,\left(1,\,2,\,3\right)}[n,\,n_{2}]\right]\times$
$\displaystyle\cos\left[n\left(\varphi_{1}-\varphi_{2}\right)\right].$

Both $p=2$ and $p=3$ contribute to ${\chi}^{(3)}$ in Eqs. (38)-(IV.1) and
${\chi}^{(3)(0)}$ in Eqs. (41)-(41). Thus, ${\chi}^{(3)}$ or ${\chi}^{(3)(0)}$
with $N=3$ superconducting leads cannot be used to probe the $p=3$-terminal
dc-Josephson effect independently on the contribution of the $p=2$-terminal
Josephson effect, see the discussion in section II.

###  IV.2 Device with $N=4$ superconducting leads on figure 2c

In this subsection, we now consider the device with $N=4$ superconducting
leads, see figure 2c. We evaluate the sensitivity of the supercurrent
$I_{1}^{(4)}$ through the lead $S_{1}$ on the three superconducting phase
differences $\varphi_{1}-\varphi_{4}$, $\varphi_{2}-\varphi_{4}$ and
$\varphi_{3}-\varphi_{4}$. The supercurrent through the superconducting lead
$S_{1}$ takes the form

$\displaystyle I_{1}^{(4)}$ $\displaystyle=$ $\displaystyle\sum_{n\neq
0}I_{c}^{{}^{\prime}\left(4\right)\,\left(2\right)\,\left(1,\,2\right)}[n]\sin\left[n\left(\varphi_{1}-\varphi_{2}\right)\right]+\sum_{n\neq
0}I_{c}^{{}^{\prime}\left(4\right)\,\left(2\right)\,\left(1,\,3\right)}[n]\sin\left[n\left(\varphi_{1}-\varphi_{3}\right)\right]+\sum_{n\neq
0}I_{c}^{{}^{\prime}\left(4\right)\,\left(2\right)\,\left(1,\,4\right)}[n]\sin\left[n\left(\varphi_{1}-\varphi_{4}\right)\right]$
(46) $\displaystyle+\sum_{n_{1}\neq 0,\,\,n_{2}\neq
0}I_{c}^{{}^{\prime}\left(4\right)\,\left(3\right)\,\left(1,\,2,\,3\right)}[n_{1},\,\,n_{2}]\sin\left[n_{1}\left(\varphi_{1}-\varphi_{3}\right)+n_{2}\left(\varphi_{2}-\varphi_{3}\right)\right]$
$\displaystyle+\sum_{n_{1}\neq 0,\,\,n_{2}\neq
0}I_{c}^{{}^{\prime}\left(4\right)\,\left(3\right)\,\left(1,\,2,\,4\right)}[n_{1},\,\,n_{2}]\sin\left[n_{1}\left(\varphi_{1}-\varphi_{4}\right)+n_{2}\left(\varphi_{2}-\varphi_{4}\right)\right]$
$\displaystyle+\sum_{n_{1}\neq 0,\,\,n_{3}\neq
0}I_{c}^{{}^{\prime}\left(4\right)\,\left(3\right)\,\left(1,\,3,\,4\right)}[n_{1},\,\,n_{3}]\sin\left[n_{1}\left(\varphi_{1}-\varphi_{4}\right)+n_{3}\left(\varphi_{3}-\varphi_{4}\right)\right]$
$\displaystyle+\sum_{n_{1}\neq 0,\,\,n_{2}\neq 0,\,\,n_{3}\neq
0}I_{c}^{{}^{\prime}\left(4\right)\,\left(4\right)\,\left(1,\,2,\,3,\,4\right)}[n_{1},\,\,n_{2},\,\,n_{3}]\sin\left[n_{1}\left(\varphi_{1}-\varphi_{4}\right)+n_{2}\left(\varphi_{2}-\varphi_{4}\right)+n_{3}\left(\varphi_{3}-\varphi_{4}\right)\right].$

The fluxes $\Phi_{A}$ and $\Phi_{B}$ are taken into account with
$\varphi_{3}=\varphi_{2}-\Phi_{A}$ and $\varphi_{4}=\varphi_{3}-\Phi_{B}$,
which leads to

$\displaystyle\varphi_{3}-\varphi_{4}$ $\displaystyle=$
$\displaystyle\Phi_{B}$ (47) $\displaystyle\varphi_{2}-\varphi_{4}$
$\displaystyle=$ $\displaystyle\Phi_{A}+\Phi_{B}.$ (48)

Eq. (46) and Eq. (46) contribute for finite values to $\chi^{(4)}\neq 0$ in
Eqs. (14)-(15) and $\chi^{(4)(0)}\left(\varphi_{1}-\varphi_{4}\right)\neq 0$
in Eqs. (16)-(19). The other terms in Eqs. (46)-(46) do not contribute to
$\chi^{(4)}$ and $\chi^{(4)(0)}\left(\varphi_{1}-\varphi_{N}\right)$ after the
partial derivatives with respect to $\varphi_{2}-\varphi_{4}$ and
$\varphi_{3}-\varphi_{4}$ have been taken:

$\displaystyle{\chi}^{(4)}(\varphi_{1}-\varphi_{4},\Phi_{A},\Phi_{B})$
$\displaystyle=$ $\displaystyle\sum_{n_{1}\neq 0,\,\,n_{2}\neq
0}n_{2}\left(n_{1}+n_{2}\right)I_{c}^{{}^{\prime}\left(4\right)\,\left(3\right)\,\left(1,\,2,\,3\right)}[n_{1},\,\,n_{2}]\times$
$\displaystyle\sin\left[n_{1}\left(\varphi_{1}-\varphi_{4}\right)+n_{2}\left(\Phi_{A}+\Phi_{B}\right)-\left(n_{1}+n_{2}\right)\Phi_{B}\right]$
$\displaystyle-$ $\displaystyle\sum_{n_{1}\neq 0,\,\,n_{2}\neq 0,\,\,n_{3}\neq
0}n_{2}n_{3}I_{c}^{{}^{\prime}\left(4\right)\,\left(4\right)\,\left(1,\,2,\,3,\,4\right)}[n_{1},\,\,n_{2},\,\,n_{3}]\sin\left[n_{1}\left(\varphi_{1}-\varphi_{4}\right)+n_{2}\left(\Phi_{A}+\Phi_{B}\right)+n_{3}\Phi_{B}\right],$
(50)

where we used Eqs. (46)-(48).

Assuming zero flux $\Phi_{A}=\Phi_{B}=0$ (or, equivalently,
$\varphi_{2}-\varphi_{4}=0$ and $\varphi_{3}-\varphi_{4}=0$), Eqs. (IV.2)-(50)
simplify as

$\displaystyle{\chi}^{(4)(0)}(\varphi_{1}-\varphi_{4})$ $\displaystyle\equiv$
$\displaystyle{\chi}^{(4)}(\varphi_{1}-\varphi_{4},0,0)$ (51) $\displaystyle=$
$\displaystyle\sum_{n_{1}\neq 0,\,\,n_{2}\neq
0}n_{2}\left[\left(n_{1}+n_{2}\right)I_{c}^{{}^{\prime}\left(4\right)\,\left(3\right)\,\left(1,\,2,\,3\right)}[n_{1},\,\,n_{2}]-\sum_{n_{3}\neq
0}n_{3}I_{c}^{{}^{\prime}\left(4\right)\,\left(4\right)\,\left(1,\,2,\,3,\,4\right)}[n_{1},\,\,n_{2},\,\,n_{3}]\right]\sin\left[n_{1}\left(\varphi_{1}-\varphi_{4}\right)\right].$
(52)

Eqs. (IV.2)-(52) contain the $p=3$ and the $p=4$-terminal contributions
$I_{c}^{{}^{\prime}\left(4\right)\,\left(3\right)\,\left(1,\,2,\,3\right)}$
and
$I_{c}^{{}^{\prime}\left(4\right)\,\left(4\right)\,\left(1,\,2,\,3,\,4\right)}$.
The $p=2$ two-terminal
$I_{c}^{{}^{\prime}\left(4\right)\,\left(2\right)\,\left(1,\,2\right)}$,
$I_{c}^{{}^{\prime}\left(4\right)\,\left(2\right)\,\left(1,\,3\right)}$,
$I_{c}^{{}^{\prime}\left(4\right)\,\left(2\right)\,\left(1,\,4\right)}$ do not
appear in Eqs. (IV.2)-(52). Thus, experimental detection of $\chi^{(4)}\neq 0$
or $\chi^{(4)}(0)\left(\varphi_{1}-\varphi_{4}\right)\neq 0$ implies evidence
for the $p=3$ or $p=4$ dc-Josephson effects. Specifically, we demonstrated the
following for $N=4$ superconducting leads (see figure 2c):

“Nonvanishingly small
${\chi}^{(4)}\left(\varphi_{1}-\varphi_{4},\Phi_{A},\Phi_{B}\right)\neq 0$ or
${\chi}^{(4)(0)}\left(\varphi_{1}-\varphi_{4}\right)\neq 0$ with $N=4$
superconducting leads”

generically implies

“Evidence for the $p=3$ or $p=4$-terminal Josephson effect”.

Now, we generalize this statement to a device containing an arbitrary number
$N$ of the superconducting leads, see figures 2b, 2c and 2d for $N=3,\,4,\,5$
respectively, and figure 7 for $N=6$.

##  V $N$ superconducting leads

We demonstrated in the previous section IV that $\chi^{(3)}\neq 0$ or
$\chi^{(3)(0)}(\varphi_{1}-\varphi_{3})\neq 0$ for $N=3$, and that
$\chi^{(4)}\neq 0$ or $\chi^{(4)(0)}(\varphi_{1}-\varphi_{4})\neq 0$ for $N=4$
imply evidence for the $p=2,\,3$ and the $p=3,\,4$ dc-Josephson effects
respectively.

Now, we generalize this statement to arbitrary values of the number $N$ of the
superconducting leads. Subsection V.1 provides expression of the higher-order
nonlocal inverse inductance $\chi^{(N)}$ at arbitrary $N$. Subsections V.2 and
V.3 present the calculations for even and odd values of $N$ respectively.

###  V.1 Higher-order nonlocal inverse inductance

In this subsection, we discuss how $N=3,\,4$ in the previous sections IV.1 and
IV.2 respectively can be generalized to arbitrary number $N$ of
superconducting leads.

Eqs. (27)-(28) for the dc-Josephson current are written as

$\displaystyle
I^{{}^{\prime}S\,\left(N\right)}_{1}\left(\varphi_{1}-\varphi_{N},\,\varphi_{2}-\varphi_{N},\,...,\varphi_{N-1}-\varphi_{N}\right)=\sum_{p=2}^{N-1}\sum_{a^{\prime}_{1}=1}^{N-1}\sum_{a^{\prime}_{2}=a^{\prime}_{1}+1}^{N-1}...\sum_{a^{\prime}_{p}=a^{\prime}_{p-1}+1}^{N-1}$
(53)
$\displaystyle\delta\left(\prod_{\alpha=1}^{N-1}\left(1-\delta\left(a^{\prime}_{\alpha},1\right)\right)\right)\sum_{m^{\prime}_{a^{\prime}_{1}}\neq
0}\sum_{m^{\prime}_{a^{\prime}_{2}}\neq
0}...\sum_{m^{\prime}_{a^{\prime}_{p}}\neq
0}I_{c}^{{}^{\prime}\left(N\right)\,\left(1\right)\,\left(p\right)\,\left(a^{\prime}_{1},\,a^{\prime}_{2},\,...,\,a^{\prime}_{p}\right)}[m^{\prime}_{a^{\prime}_{1}},\,...,\,m^{\prime}_{a^{\prime}_{p}}]\sin\left(\sum_{\alpha=1}^{p}m^{\prime}_{a^{\prime}_{\alpha}}\left(\varphi_{a^{\prime}_{\alpha}}-\varphi_{N}\right)\right),$

where we evaluated the supercurrent through the lead labeled by $N_{0}=1$, and
used the phase $\varphi_{N}$ of the superconducting lead $S_{N}$ as a
reference. The variable $a^{\prime}_{\alpha}$ in Eq. (53) is defined in the
interval $1\leq a^{\prime}_{\alpha}\leq N-1$ while $a_{\alpha}$ in Eq. (28) is
such that $1\leq a_{\alpha}\leq N$.

We define the order-$\left(N-2\right)$ nonlocal inductance as the signal which
is proposed to be detected in microwave experiments:

$\displaystyle{\chi}^{\left(N\right)}\equiv\left[L^{-1}_{N-2}\right]^{\prime}\left(\varphi_{1}-\varphi_{N},\,\varphi_{2}-\varphi_{N},\,...,\varphi_{N-1}-\varphi_{N}\right)$
(54) $\displaystyle=$
$\displaystyle\frac{\partial^{N-2}}{\partial\left(\varphi_{2}-\varphi_{N}\right)\partial\left(\varphi_{3}-\varphi_{N}\right)...\partial\left(\varphi_{N-1}-\varphi_{N}\right)}\times$
$\displaystyle
I^{{}^{\prime}S}_{1}\left(\varphi_{1}-\varphi_{N},\,\varphi_{2}-\varphi_{N},\,...,\varphi_{N-1}-\varphi_{N}\right).$

We note that the lead $S_{N}$ is singled out in Eq. (54): the superconducting
phase variable $\varphi_{N}$ is the reference for evaluating the partial
derivatives of $I^{{}^{\prime}S}_{1}$ with respect to all of the
$\varphi_{n}-\varphi_{N}$, with $n=2,\,...,\,n-1$. Now, we provide a
calculation of Eq. (54).

Figure 7: The figure shows a $N=6$ dc-Josephson junction (a). Panels b, c and
d, e show the $p=3,\,6,\,5,\,5$-terminal dc-Josephson processes in a device
containing $N=6$ superconducting leads respectively. The processes on the
green-highlighted panels c and e contribute for a nonvanishingly small
$\left[\tilde{L}^{-1}_{N-2}\right]^{{}^{\prime}\left(N\right)}\neq 0$ in Eq.
(54), on the condition of nonvanishingly small corresponding critical
currents.

###  V.2 Even number $N=2q$ of superconducting leads

In this subsection, we assume that $N=2q$ is even. In addition, we assume in
subsection V.2.1 that
$\sum_{\alpha=1}^{2q-1}m^{\prime}_{b^{\prime}_{\alpha}}\neq 0$ (with
$m^{\prime}_{b^{\prime}_{\beta}}\neq 0$ for all $\beta=1,\,...,2q-1$).
Conversely, subsection V.2.2 deals with
$\sum_{\alpha=1}^{2q-1}m^{\prime}_{b^{\prime}_{\alpha}}=0$.

####  V.2.1 $p=2q$-terminal Josephson processes with $N=2q$ superconducting
leads

In this subsection, we assume that $N=2q$ is even and
$\sum_{\alpha=1}^{2q-1}m^{\prime}_{b^{\prime}_{\alpha}}\neq 0$ (with
$m^{\prime}_{b^{\prime}_{\beta}}\neq 0$ for all $\beta=1,\,...,2q-1$).

This results in the $p=2q$-terminal dc-Josephson contribution
$\left[L^{-1}_{2q-2}\right]^{{}^{\prime}(2q)}$ to
${\chi}^{(2q)}=\left[L^{-1}_{2q-2}\right]^{\prime}$ in Eq. (54). Taking into
account that $b^{\prime}_{\alpha}=\alpha$ for a $p=2q=N$-terminal dc-Josephson
process, we find

$\displaystyle\left[L^{-1}_{2q-2}\right]^{{}^{\prime}(2q)}$ $\displaystyle=$
$\displaystyle\left(-\right)^{q+1}\sum_{m^{\prime}_{1}\neq
0}\sum_{m^{\prime}_{2}\neq 0}...\sum_{m^{\prime}_{2q-1}\neq
0}I_{c,1}^{{}^{\prime}\left(2q\right)\,\left({2q}\right)\,\left(1,2,\,...,{{2q-1}}\right)}[m^{\prime}_{1},\,...,\,m^{\prime}_{{{2q}-1}}]\times$
$\displaystyle\left(\prod_{\alpha=2}^{{2q}-1}m^{\prime}_{\alpha}\right)\sin\left(\sum_{\alpha=1}^{{2q}-1}m^{\prime}_{{\alpha}}\left(\varphi_{{\alpha}}-\varphi_{{2q}}\right)\right).$

Specializing to $\varphi_{2}-\varphi_{2q}=...=\varphi_{{2q}-1}-\varphi_{2q}=0$
and $\varphi_{1}-\varphi_{2q}\neq 0$ leads to

$\displaystyle\left[\tilde{L}^{-1}_{{2q}-2}\right]^{{}^{\prime}(2q)(0)}$
$\displaystyle=$ $\displaystyle\left(-\right)^{q+1}\sum_{m^{\prime}_{1}\neq
0}\sum_{m^{\prime}_{2}\neq 0}...\sum_{m^{\prime}_{{{2q}-1}}\neq
0}I_{c}^{{}^{\prime}\left(1\right)\,\left({2q}\right)\,\left(1,2,\,...,{{2q-1}}\right)}[m^{\prime}_{1},\,...,\,m^{\prime}_{{{2q}-1}}]\times$
$\displaystyle\left(\prod_{\alpha=2}^{{2q}-1}m^{\prime}_{\alpha}\right)\sin\left(m^{\prime}_{1}\left(\varphi_{1}-\varphi_{{2q}}\right)\right).$

####  V.2.2 $p=\left(2q-1\right)$-terminal Josephson processes with $N=2q$
superconducting leads

Now, we assume in this subsection that $N=2q$ is even and

$\sum_{\alpha=1}^{{2q}-1}m^{\prime}_{c^{\prime}_{\alpha}}=0,$ (57)

with $c^{\prime}_{\alpha}=\alpha$ for all $\alpha=1,\,...,\,2q-1$. Then, we
make use of the identities

$\displaystyle\sin\left(\sum_{\alpha=1}^{2q-1}m^{\prime}_{{\alpha}}\left(\varphi_{\alpha}-\varphi_{2q}\right)\right)=\sin\left(\sum_{\alpha=1}^{2q-1}m^{\prime}_{\alpha}\varphi_{\alpha}\right)=\sin\left(\sum_{\alpha=2}^{2q-1}m^{\prime}_{\alpha}\left(\varphi_{\alpha}-\varphi_{1}\right)\right)$
$\displaystyle=$
$\displaystyle\sin\left(\sum_{\alpha=2}^{2q-1}m^{\prime}_{\alpha}\left(\varphi_{\alpha}-\varphi_{2q}\right)-\sum_{\alpha=2}^{{2q}-1}m^{\prime}_{\alpha}\left(\varphi_{{2q}}-\varphi_{1}\right)\right)=\sin\left(\sum_{\alpha=2}^{{2q}-1}m^{\prime}_{\alpha}\left(\varphi_{\alpha}-\varphi_{2q}\right)+m^{\prime}_{1}\left(\varphi_{{2q}}-\varphi_{1}\right)\right).$

Specializing to $\varphi_{1}-\varphi_{2q}\neq 0$ and
$\varphi_{2}-\varphi_{2q}=...=\varphi_{{{2q}-1}}-\varphi_{2q}=0$, we deduce
the contribution $\left[\tilde{L}^{-1}_{{2q}-2}\right]^{{}^{\prime}(2q-1)}$ of
the $p=2q-1$-terminal dc-Josephson processes to
$\left[\tilde{L}^{-1}_{{2q}-2}\right]^{\prime}$ in the presence of $N=2q$
superconducting leads:

$\displaystyle\left[\tilde{L}^{-1}_{{2q}-2}\right]^{(2q-1)(0)}$
$\displaystyle=$ $\displaystyle\left(-\right)^{q}\sum_{m^{\prime}_{2}\neq
0}\sum_{m^{\prime}_{3}\neq 0}...\sum_{m^{\prime}_{{{2q}-2}}\neq
0}I_{c,1}^{{}^{\prime}\left(2q\right)\,\left(1\right)\,\left({2q-1}\right)\,\left(1,2,\,...,{{2q}-1}\right)}\left[-\left(\sum_{\alpha=2}^{2q-1}m^{\prime}_{\alpha}\right),\,m^{\prime}_{2},\,...,\,m^{\prime}_{{{2q}-1}}\right]\times$
$\displaystyle\left(\prod_{\alpha=2}^{{2q}-1}m^{\prime}_{\alpha}\right)\sin\left(\left(\sum_{\alpha=2}^{2q-1}m^{\prime}_{\alpha}\right)\left(\varphi_{1}-\varphi_{2q}\right)\right),$

where we used $m^{\prime}_{{2q-1}}=-\sum_{\alpha=1}^{2q-2}m^{\prime}_{\alpha}$
deduced from Eq. (57).

####  V.2.3 Summary

The zero-flux limit of the predicted experimental signal with $N=2q$
generalizes the above Eq. (52) with $q=2$:

${\chi}^{(2q)(0)}\left(\varphi_{1}-\varphi_{2q}\right)=\left(\ref{eq:IprimeS-2-even-2-A}\right)+\left(\ref{eq:IprimeS-2-even-2-B}\right).$
(60)

###  V.3 Odd number $N=2q^{\prime}+1$ of superconducting leads

In this subsection, we assume now that $N=2q^{\prime}+1$ is odd. We assume in
subsection V.3.1 that
$\sum_{\alpha=1}^{2q-1}m^{\prime}_{b^{\prime}_{\alpha}}\neq 0$ (with
$m^{\prime}_{b^{\prime}_{\beta}}\neq 0$ for all $\beta=1,\,...,2q-1$) and we
assume in subsection V.3.2 that
$\sum_{\alpha=1}^{2q-1}m^{\prime}_{b^{\prime}_{\alpha}}=0$.

####  V.3.1 $p=\left(2q^{\prime}+1\right)$-terminal Josephson processes with
$N=2q^{\prime}+1$ superconducting leads

In this subsection, we assume that $N=2q^{\prime}+1$ is odd and
$\sum_{\alpha=1}^{2q^{\prime}}m^{\prime}_{b^{\prime}_{\alpha}}\neq 0$ (with
$m^{\prime}_{b^{\prime}_{\beta}}\neq 0$ for all $\beta=1,\,...,2q^{\prime}$).

We find

$\displaystyle\left[L^{-1}_{2q^{\prime}-1}\right]^{{}^{\prime}\left(2q^{\prime}+1\right)}$
$\displaystyle=$
$\displaystyle\left(-\right)^{q^{\prime}+1}\sum_{m^{\prime}_{1}\neq
0}\sum_{m^{\prime}_{2}\neq 0}...\sum_{m^{\prime}_{2q^{\prime}}\neq
0}I_{c,1}^{{}^{\prime}\left(2q^{\prime}+1\right)\,\left(2q^{\prime}\right)\,\left(1,2,\,...,2q^{\prime}\right)}[m^{\prime}_{1},\,...,\,m^{\prime}_{2q^{\prime}}]\times$
(62)
$\displaystyle\left(\prod_{\alpha=2}^{2q^{\prime}}m^{\prime}_{\alpha}\right)\cos\left(\sum_{\alpha=1}^{2q^{\prime}}m^{\prime}_{\alpha}\left(\varphi_{\alpha}-\varphi_{2q^{\prime}+1}\right)\right).$

Specializing Eq. (62) to
$\varphi_{2}-\varphi_{2q^{\prime}+1}=...=\varphi_{2q^{\prime}}-\varphi_{2q^{\prime}+1}=0$
and $\varphi_{1}-\varphi_{2q^{\prime}+1}\neq 0$ leads to

$\displaystyle\left[\tilde{L}^{-1}_{2q^{\prime}-1}\right]^{{}^{\prime}\left(2q^{\prime}+1\right)(0)}$
$\displaystyle=$
$\displaystyle\left(-\right)^{q^{\prime}+1}\sum_{m^{\prime}_{1}\neq
0}\sum_{m^{\prime}_{2}\neq 0}...\sum_{m^{\prime}_{2q^{\prime}}\neq
0}I_{c,1}^{{}^{\prime}\left(2q^{\prime}+1\right)\,\left(2q^{\prime}\right)\,\left(1,2,\,...,2q^{\prime}\right)}[m^{\prime}_{1},\,...,\,m^{\prime}_{2q^{\prime}}]\times$
$\displaystyle\left(\prod_{\alpha=2}^{2q^{\prime}}m^{\prime}_{\alpha}\right)\cos\left(m^{\prime}_{1}\left(\varphi_{1}-\varphi_{2q^{\prime}+1}\right)\right).$

####  V.3.2 $p=2q^{\prime}$-terminal Josephson processes with
$N=2q^{\prime}+1$ superconducting leads

In this subsection, we assume that $N=2q^{\prime}+1$ is odd and
$\sum_{\alpha=1}^{2q^{\prime}}m^{\prime}_{b^{\prime}_{\alpha}}=0$ (with
$m^{\prime}_{b^{\prime}_{\beta}}\neq 0$ for all $\beta=1,\,...,2q^{\prime}$).

We specialize to
$\varphi_{2}-\varphi_{2q^{\prime}+1}=...=\varphi_{{2q^{\prime}}}-\varphi_{2q^{\prime}+1}=0$
and $\varphi_{1}-\varphi_{2q^{\prime}+1}\neq 0$, we deduce

$\displaystyle\left[\tilde{L}^{-1}_{{2q^{\prime}}-1}\right]^{(2q^{\prime})(0)}$
$\displaystyle=$
$\displaystyle\left(-\right)^{q^{\prime}+1}\sum_{m^{\prime}_{2}\neq
0}\sum_{m^{\prime}_{3}\neq 0}...\sum_{m^{\prime}_{{{2q^{\prime}}}}\neq
0}I_{c,1}^{{}^{\prime}\left(2q^{\prime}+1\right)\,\left({2q^{\prime}}\right)\,\left(1,2,\,...,{{2q^{\prime}}}\right)}\left[-\left(\sum_{\alpha=2}^{2q^{\prime}}m^{\prime}_{\alpha}\right),\,m^{\prime}_{2},\,...,\,m^{\prime}_{{{2q^{\prime}}}}\right]\times$
$\displaystyle\left(\prod_{\alpha=2}^{{2q^{\prime}}}m^{\prime}_{\alpha}\right)\cos\left(\left(\sum_{\alpha=2}^{2q^{\prime}}m^{\prime}_{\alpha}\right)\left(\varphi_{1}-\varphi_{2q^{\prime}+1}\right)\right).$

####  V.3.3 Summary

The zero-flux limit of the predicted experimental signal with
$N=2q^{\prime}+1$ generalizes the above Eqs. (41)-(41):

${\chi}^{(2q^{\prime}+1)(0)}\left(\varphi_{1}-\varphi_{2q}\right)=\left(\ref{eq:IprimeS-2-odd-2-A}\right)+\left(\ref{eq:IprimeS-2-odd-2-B-x}\right).$
(65)

###  V.4 Discussion

To summarize, the $p=N$-terminal and some of the $p=N-1$-terminal dc-Josephson
processes contribute for finite values to
${\chi}^{\left(N\right)}\equiv\left[L^{-1}_{N-2}\right]^{\prime}$ given by Eq.
(54):

${\chi}^{\left(N\right)}=\left[L^{-1}_{N-2}\right]^{\prime}=\left[L^{-1}_{N-2}\right]^{{}^{\prime}\left(N\right)}+\left[L^{-1}_{N-2}\right]^{{}^{\prime}(N-1)}.$
(66)

In general,

$\displaystyle\left[L^{-1}_{N-2}\right]^{{}^{\prime}\left(N\right)}\neq 0$
(67) $\displaystyle\left[L^{-1}_{N-2}\right]^{{}^{\prime}(N-1)}\neq 0$ (68)

if the corresponding critical currents are nonvanishingly small. We find:

$\left[L^{-1}_{N-2}\right]^{{}^{\prime}(p)}=0\mbox{ if }p\leq N-2.$ (69)

In addition, the $\left(N-2\right)$-th order derivative of
$\sin\left(m_{a^{\prime}_{1}}\varphi_{a^{\prime}_{1}}+m_{a^{\prime}_{2}}\varphi_{a^{\prime}_{2}}+....+m_{a^{\prime}_{p}}\varphi_{a^{\prime}_{p}}\right)$
in Eq. (54) produces $\sin$ or $\cos$ according to the parity of $N$.

The Appendix details examples with $N=6$ superconducting leads, see figure 7.

Figure 8: The figure shows $\chi^{(4)(0)}(\varphi_{1}-\varphi_{4})$ as a
function of $\varphi_{1}-\varphi_{4}$, for $\alpha_{2T}=1$, $\alpha_{4T}=0$.
The values of $\alpha_{3T}$ between $0$ and $1$ are shown on the figure.

##  VI Numerical results

Now, we present a numerical illustration that
$\chi^{(4)(0)}(\varphi_{1}-\varphi_{4})$ defined in Eq. (16) is vanishingly
small if the $p=3$ and the $p=4$-terminal critical currents are also
vanishingly small i.e.
$I_{c}^{{}^{\prime}\left(4\right)\,\left(3\right)\,\left(1,\,3,\,4\right)}=0$
and
$I_{c}^{{}^{\prime}\left(4\right)\,\left(4\right)\,\left(1,\,2,\,3,\,4\right)}=0$
in Eqs. (46)-(46).

This implies that “measurement of $\chi^{(4)(0)}(\varphi_{1}-\varphi_{4})\neq
0$” is evidence for the $p=3$ or $p=4$-terminal dc-Josephson critical currents
$I_{c}^{{}^{\prime}\left(4\right)\,\left(3\right)\,\left(1,\,3,\,4\right)}\neq
0$ or
$I_{c}^{{}^{\prime}\left(4\right)\,\left(4\right)\,\left(1,\,2,\,3,\,4\right)}\neq
0$.

We use the following phenomenological modeling of the ABS energies:

$\displaystyle E_{ABS}(\varphi_{1},\varphi_{2},\varphi_{3},\varphi_{4})$
$\displaystyle=$ $\displaystyle\alpha_{0}$ $\displaystyle+$
$\displaystyle\alpha_{2T}\left\\{\cos(\varphi_{1}-\varphi_{2})+\cos(\varphi_{1}-\varphi_{3})+\cos(\varphi_{1}-\varphi_{4})+\cos(\varphi_{2}-\varphi_{3})+\cos(\varphi_{2}-\varphi_{4})+\cos(\varphi_{3}-\varphi_{4})\right\\}$
$\displaystyle+$
$\displaystyle\alpha_{3T}\left\\{\cos(\varphi_{1}+\varphi_{2}-2\varphi_{3})+\cos(\varphi_{1}+\varphi_{2}-2\varphi_{4})+\cos(\varphi_{2}+\varphi_{3}-2\varphi_{1})+\cos(\varphi_{2}+\varphi_{3}-2\varphi_{4})\right.$
$\displaystyle+\left.\cos(\varphi_{1}+\varphi_{3}-2\varphi_{2})+\cos(\varphi_{1}+\varphi_{3}-2\varphi_{4})+\cos(\varphi_{1}+\varphi_{4}-2\varphi_{2})+\cos(\varphi_{1}+\varphi_{4}-2\varphi_{3})\right.$
$\displaystyle+\left.\cos(\varphi_{2}+\varphi_{3}-2\varphi_{1})+\cos(\varphi_{2}+\varphi_{3}-2\varphi_{4})+\cos(\varphi_{2}+\varphi_{4}-2\varphi_{1})+\cos(\varphi_{2}+\varphi_{4}-2\varphi_{3})\right.$
$\displaystyle+\left.\cos(\varphi_{3}+\varphi_{4}-2\varphi_{1})+\cos(\varphi_{3}+\varphi_{4}-2\varphi_{2})\right.$
$\displaystyle+$
$\displaystyle\alpha_{4T}\left\\{\cos(\varphi_{1}-\varphi_{2}-\varphi_{3}+\varphi_{4})+\cos(\varphi_{1}-\varphi_{2}+\varphi_{3}-\varphi_{4})+\cos(\varphi_{1}+\varphi_{2}-\varphi_{3}-\varphi_{4})\right\\},$
(71)

where $\alpha_{0}$ is a constant energy shift, and $\alpha_{2T}$,
$\alpha_{3T}$ and $\alpha_{4T}$ parameterize the $p=2,\,3,\,4$ dc-Josephson
effect.

The current $I_{1}$ through the lead $S_{1}$ is deduced from Eq. (24), and
$\chi^{(4)(0)}(\varphi_{1}-\varphi_{4})$ is obtained from Eqs. (16)-(19).

Figure 8 shows $\chi^{(4)(0)}(\varphi_{1}-\varphi_{4})$ for $\alpha_{2T}=1$,
$\alpha_{4T}=0$ and $\alpha_{3T}$ between $0$ and $1$. It is concluded that
$\alpha_{3T}=\alpha_{4T}=0$ implies
$\chi^{(4)(0)}(\varphi_{1}-\varphi_{4})=0$. Conversely, $\alpha_{3T}\neq 0$
with $\alpha_{4T}=0$ implies nonvanishingly small
$\chi^{(4)(0)}(\varphi_{1}-\varphi_{4})\neq 0$, whatever $\alpha_{2T}$. Thus,
Figure 8 confirms that $\chi^{(4)(0)}(\varphi_{1}-\varphi_{4})\neq 0$ is
evidence for the $p=3$ or $p=4$ dc-Josephson effect in a device with $N=4$
superconducting leads. The effect is “yes or no” answer to the $p\geq 3$
multiterminal dc-Josephson effect.

##  VII Conclusions

Now, we present summary of the main results and final remarks.

We considered multiterminal dc-Josephson interferometers that were
investigated in devices containing $N=4$ (or with $N\geq 4$) superconducting
leads, in a way which is different from the previous Refs. Pillet1, ; Pillet2,
; Benzoni, ; Rech, .

Specifically, we proposed to detect the $p\geq 3$-terminal Josephson effect in
a device containing $N=4$ superconducting leads, see figure 2c. The
experimental signal ${\chi}^{(4)}$ or
${\chi}^{(4)(0)}\left(\varphi_{1}-\varphi_{4}\right)$ is given by Eqs.
(14)-(15) or Eqs. (16)-(19) respectively. We obtained the following result:

“Nonvanishingly small second-order nonlocal inverse inductance ${\chi}^{(4)}$
or ${\chi}^{(4)(0)}\left(\varphi_{1}-\varphi_{4}\right)$”

generically implies

“Evidence for the three or four-terminal dc-Josephson effect”.

We generalized this statement to a device containing arbitrary number $N\geq
4$ of the superconducting leads, thus with $N-1$ independent phase
differences. One of those phase differences (e.g. $\varphi_{1}-\varphi_{N}$)
is reserved for controlling the current-phase relation of the overall two-
terminal device. The remaining $N-2$ independent superconducting phase
differences are used in the definition Eq. (54) of the higher-order nonlocal
inverse inductance ${\chi}^{(N)}$ and they are related to the $N-2$
independent fluxes.

Practically, the partial derivative of the supercurrent $I_{1}$ through the
superconducting lead $S_{1}$ with respect to $\varphi_{2}-\varphi_{N}$,
$\varphi_{3}-\varphi_{N}$, …, $\varphi_{N-1}-\varphi_{N}$ [see Eqs. (14)-(15),
(16)-(19) and Eq. (54)] can be evaluated by changing variables from the $N-1$
independent phase differences $\varphi_{1}-\varphi_{N}$,
$\varphi_{2}-\varphi_{N}$, …, $\varphi_{N-1}-\varphi_{N}$ to
$\varphi_{1}-\varphi_{N}$ and the $N-2$ fluxes $\Phi_{A_{1}}$, $\Phi_{A_{2}}$,
… , $\Phi_{A_{N-2}}$.

We demonstrated that finite values for ${\chi}^{(N)}\neq 0$ or
${\chi}^{(N)(0)}\left(\varphi_{1}-\varphi_{N}\right)\neq 0$ necessarily
originate from the $p=N-1$ and $p=N$-terminal dc-Josephson effects: the
$p$-terminal dc-Josephson effect does not contribute to ${\chi}^{(N)}$ or
${\chi}^{(N)(0)}\left(\varphi_{1}-\varphi_{N}\right)$ if $p\leq N-2$.

Finally, a device with number $N=4$ of the superconducting leads and two loops
seems to be already quite interesting from the point of view of possible
experiments, see figure 2c. Microwave detection of ${\chi}^{(N)}\neq 0$ or
${\chi}^{(N)(0)}\left(\varphi_{1}-\varphi_{N}\right)\neq 0$ is promising, see
also the recent Refs. Feinberg1, ; Feinberg2, ; topo1-plus-Floquet, .

## Acknowledgements

The idea of the paper came from reading the manuscript of V. Bezoni’s PhD
thesis Benzoni . The author wishes to thank Ç.Ö. Girit for useful discussions
and J.D. Pillet for sharing their results Pillet1 ; Pillet2  prior to making
their preprints public. The author wishes to thank Romain Danneau for useful
comments on a previous version of the manuscript. The author thanks the
Infrastructure de Calcul Intensif et de Données (GRICAD) for use of the
resources of the Mésocentre de Calcul Intensif de l’Université Grenoble-Alpes
(CIMENT). The author acknowledges support from the French National Research
Agency (ANR) in the framework of the Graphmon project (ANR-19-CE47-0007).

## Appendix: Examples with $N=6$ superconducting leads

In this Appendix, we detail the examples shown on figure 7 with $N=6$
superconducting leads.

Figure 7b shows example of a $p=3$-terminal dc-Josephson process with $N=6$.
Figure 7c shows a $p=6$-terminal process with $N=6$. Figures 7d and 7e show
$p=5$ with $N=6$. Not all values of $p$ contribute to
${\chi}^{(N)}=\left[L^{-1}_{N-2}\right]^{\prime}$ in Eq. (54):

(i) The $p=3$-terminal dc-Josephson current
$i_{1}(\varphi_{3}-\varphi_{1},\,\varphi_{5}-\varphi_{1})$ shown on figure 7b
depends on the two independent phase differences $\varphi_{3}-\varphi_{1}$ and
$\varphi_{5}-\varphi_{1}$, thus it cannot contribute to
${\chi}^{(6)}=\left[L^{-1}_{4}\right]^{\prime}$ in Eq. (54) where
nonvanishingly small value for
${\chi}^{(6)}=\left[L^{-1}_{4}\right]^{\prime}\neq 0$ requires additional
sensitivity of the supercurrent through the lead $S_{1}$ on both the
superconducting phase variables $\varphi_{2}$ and $\varphi_{4}$.

(ii) The $p=6$-terminal dc-Josephson current $i_{2}$ shown on figure 7c is
sensitive to all phase differences between $S_{n}$ and $S_{m}$ with
$n,\,m=1,\,...,6$, thus it contributes for a finite value to
${\chi}^{(N)}=\left[L^{-1}_{N-2}\right]^{\prime}$ in Eq. (54), see below.

(iii) The $p=5$-terminal dc-Josephson current
$i_{3}(\varphi_{2}-\varphi_{1},\,\varphi_{4}-\varphi_{1},\varphi_{5}-\varphi_{1},\,\varphi_{6}-\varphi_{1})$
shown on figure 7d depends on $N-2=4$ phase differences. However, this dc-
Josephson process produces ${\chi}^{(6)}=\left[L^{-1}_{4}\right]^{\prime}=0$
because $i_{3}$ on figure 7d is independent on $\varphi_{3}$.

(iv) The $p=5$-terminal dc-Josephson current
$i_{4}(\varphi_{2}-\varphi_{1},\,\varphi_{3}-\varphi_{1},\,\varphi_{4}-\varphi_{1},\varphi_{5}-\varphi_{1})$
shown on figure 7e is sensitive on all the phase variables
$\\{\varphi_{1},\,\varphi_{2},\,\varphi_{3},\,\varphi_{4},\,\varphi_{5}\\}$.
Then, $i_{4}\equiv
i^{\prime}_{4}(\varphi_{1}-\varphi_{6},\,\varphi_{2}-\varphi_{6},\,\varphi_{3}-\varphi_{6},\,\varphi_{4}-\varphi_{6},\,\varphi_{5}-\varphi_{6})$
contributes for a finite value to
${\chi}^{(6)}=\left[L^{-1}_{4}\right]^{\prime}$, see below.

Now, we detail the above items (ii) and (iv) corresponding to figures 7c and
7e.

First, concerning item (ii) and figure 7c, the lowest-order supercurrent
$i_{2}$ through the lead $S_{1}$ is given by

$\displaystyle
i_{2}(\varphi_{1},\varphi_{2},\varphi_{3},\varphi_{4},\varphi_{5},\varphi_{6})$
$\displaystyle=$ $\displaystyle
j_{2}^{(0)}\sin\left[\left(\varphi_{2}-\varphi_{1}\right)+\left(\varphi_{3}-\varphi_{1}\right)+\left(\varphi_{4}-\varphi_{1}\right)+\left(\varphi_{5}-\varphi_{1}\right)+\left(\varphi_{6}-\varphi_{1}\right)\right]$
(72) $\displaystyle=$ $\displaystyle
j_{2}^{(0)}\sin\left[-5\varphi_{1}+\varphi_{2}+\varphi_{3}+\varphi_{4}+\varphi_{5}+\varphi_{6}\right]$
(73) $\displaystyle=$ $\displaystyle
j_{2}^{(0)}\sin\left[-5\left(\varphi_{1}-\varphi_{6}\right)+\left(\varphi_{2}-\varphi_{6}\right)+\left(\varphi_{3}-\varphi_{6}\right)+\left(\varphi_{4}-\varphi_{6}\right)+\left(\varphi_{5}-\varphi_{6}\right)\right].$
(74)

Thus,

$\displaystyle\frac{\partial^{4}i_{2}(\varphi_{1},\varphi_{2},\varphi_{3},\varphi_{4},\varphi_{5},\varphi_{6})}{\partial\left(\varphi_{2}-\varphi_{6}\right)\partial\left(\varphi_{3}-\varphi_{6}\right)\partial\left(\varphi_{4}-\varphi_{6}\right)\partial\left(\varphi_{5}-\varphi_{6}\right)}=i_{2}(\varphi_{1},\varphi_{2},\varphi_{3},\varphi_{4},\varphi_{5},\varphi_{6})$
(75)

and we find:

$\displaystyle\left.\frac{\partial^{4}i_{2}(\varphi_{1},\varphi_{2},\varphi_{3},\varphi_{4},\varphi_{5},\varphi_{6})}{\partial\left(\varphi_{2}-\varphi_{6}\right)\partial\left(\varphi_{3}-\varphi_{6}\right)\partial\left(\varphi_{4}-\varphi_{6}\right)\partial\left(\varphi_{5}-\varphi_{6}\right)}\right|_{\begin{array}[]{c}\varphi_{2}-\varphi_{6}=0,\,\varphi_{3}-\varphi_{6}=0\\\
\varphi_{4}-\varphi_{6}=0,\,\varphi_{5}-\varphi_{6}=0\end{array}}=-j^{(0)}_{2}\sin\left[5\left(\varphi_{1}-\varphi_{6}\right)\right].$
(78)

Second, we consider item (iv) and figure 7d:

$\displaystyle
i_{4}(\varphi_{1},\varphi_{2},\varphi_{3},\varphi_{4},\varphi_{5},\varphi_{6})$
$\displaystyle=$ $\displaystyle
j_{4}^{(0)}\sin\left[\left(\varphi_{2}-\varphi_{1}\right)+\left(\varphi_{4}-\varphi_{1}\right)+\left(\varphi_{5}-\varphi_{1}\right)+\left(\varphi_{6}-\varphi_{1}\right)\right]$
(79) $\displaystyle=$ $\displaystyle
j_{4}^{(0)}\sin\left[-4\varphi_{1}+\varphi_{2}+\varphi_{4}+\varphi_{5}+\varphi_{6}\right]$
(80) $\displaystyle=$ $\displaystyle
j_{4}^{(0)}\sin\left[-4\left(\varphi_{1}-\varphi_{6}\right)+\left(\varphi_{2}-\varphi_{6}\right)+\left(\varphi_{4}-\varphi_{6}\right)+\left(\varphi_{5}-\varphi_{6}\right)\right],$
(81)

which implies that the following is vanishingly small:

$\frac{\partial^{4}i_{4}(\varphi_{1},\varphi_{2},\varphi_{3},\varphi_{4},\varphi_{5},\varphi_{6})}{\partial\left(\varphi_{2}-\varphi_{6}\right)\partial\left(\varphi_{3}-\varphi_{6}\right)\partial\left(\varphi_{4}-\varphi_{6}\right)\partial\left(\varphi_{5}-\varphi_{6}\right)}=0.$
(82)

The above Eqs. (75) and (82) confirm the general theory presented in the
paper.

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