# Nonlocal Josephson effect in Andreev molecules

J.-D. Pillet  [  V. Benzoni  J. Griesmar  J.-L. Smirr  Ç. Ö. Girit
<EMAIL_ADDRESS>[

###### Abstract

We propose the “Andreev molecule,” an artificial quantum system composed of
two closely spaced Josephson junctions. The coupling between Josephson
junctions in an Andreev molecule occurs through the overlap and hybridization
of the junction’s “atomic” orbitals, Andreev Bound States. A striking
consequence is that the supercurrent flowing through one junction depends on
the superconducting phase difference across the other junction. Using the
Bogolubiov-de-Gennes formalism, we derive the energy spectrum and non-local
current-phase relation for arbitrary separation. We demonstrate the
possibility of creating a $\varphi$-junction and propose experiments to verify
our predictions. Andreev molecules may have potential applications in quantum
information, metrology, sensing, and molecular simulation.

###### keywords:

superconductivity, Josephson junction, Andreev bound states, superconducting
circuits, quantum information

Collège de France] $\Phi_{0}$, JEIP, USR 3573 CNRS, Collège de France, PSL
University, 11, place Marcelin Berthelot, 75231 Paris Cedex 05, France [École
Polytechnique] LSI, CEA/DRF/IRAMIS, Ecole polytechnique, CNRS, Institut
Polytechnique de Paris, F-91128 Palaiseau, France Collège de France]
$\Phi_{0}$, JEIP, USR 3573 CNRS, Collège de France, PSL University, 11, place
Marcelin Berthelot, 75231 Paris Cedex 05, France

Understanding and exploiting the interaction between Josephson junctions is
paramount for superconducting device applications in quantum information 1,
magnetometry 2, metrology 3, and quantum simulation 4. In typical
superconducting circuits, junctions interact indirectly via electromagnetic
coupling to inductors, capacitors, transmission lines, and microwave
resonators. In addition to this well understood long-range interaction 5,
there is a short range interaction via quasiparticle diffusion which can
modify superconducting energy gaps and critical currents, but is only
important close to $T_{c}$, the superconducting transition temperature, or at
large bias voltages 6.

A second short-range interaction, mediated by Cooper pairs, is relevant to the
majority of applications where characteristic energies are much smaller than
the gap, but is still poorly understood. It becomes significant when the
distance between Josephson junctions is comparable to $\xi_{0}$, the
superconducting coherence length, and can modify the electrical properties in
a dramatic way.

Initially, minor effects resulting from this “order-parameter interaction”
were calculated for temperatures near $T_{c}$ using the Ginzburg-Landau
equations 7. More recently, theorists have investigated this problem at
arbitrary temperature using Green’s function techniques. In the two-electrode
geometry, where it is not possible to independently apply a phase difference
to each junction, the overall current-phase relation and dc current were
obtained 8, 9. For the more relevant three-electrode geometry, non-local out-
of-equilibrium supercurrents were calculated and the existence of $\pi$ shifts
in the current-phase relation were demonstrated 10, 11, 12, 13, 14, 15. A
remarkable phase-locking similar to Shapiro steps was predicted and
subsequently measured experimentally in superconducting bi-junctions biased
with commensurate voltages16, 17. The authors attribute these phenomena to the
formation of entangled Cooper pairs called “quartets.”

Although this interpretation is intriguing, many open questions remain: what
is the microscopic mechanism involved? How does the interaction depend on the
distance between the junctions? Have all non-local effects been revealed? How
can one detect these effects and exploit them in devices?

In this work we answer these questions and show that the phenomena can be
explained by a simple analogy to the formation of a hydrogen molecule,
complete with orbital hybridization and level splitting. The analog of the
hydrogen atom’s atomic orbitals are Andreev Bound States (ABS), localized
electronic states that are the microscopic basis for the Josephson effect 18,
19. The equivalent of the Bohr radius is $\xi_{0}$, the characteristic length
scale for the spread of ABS wavefunctions. In our superconducting “Andreev
molecule,” where two Josephson junctions are separated by a distance
$l\lesssim\xi_{0}$, the ABS wavefunctions localized at each junction overlap
and hybridize. This hybridization modifies the total energy spectrum and can
result in a $\varphi$-junction where the supercurrent flowing through one
junction depends on the superconducting phase difference across the other
junction.

Figure 1:  Two single-channel short weak links (gray) of transmissions
$\tau_{L}=\tau_{R}\approx 0.94$ are connected to three superconducting regions
(colored), of which the central one (pink) has length $l=10\xi_{0}$ (a), or in
the case of the Andreev molecule, $l=\xi_{0}$ (b). The ground connection
allows applying phase differences $\delta_{L}$ and $\delta_{R}$ independently.
(c) The Andreev Bound State (ABS) spectrum as a function of $\delta_{L}$ for
fixed $\delta_{R}=3\pi/5$ for $l=10\xi_{0}$ and the corresponding
wavefunctions at the two weak links (d). The degeneracies at
$\delta_{L}=\pm\delta_{R}$ are lifted in the spectrum of the Andreev molecule
(e) and the wavefunctions hybridize (f). Since the phases are $2\pi$ periodic,
the $\delta_{L}=-\delta_{R}$ degeneracy is shown at
$\delta_{L}=2\pi-\delta_{R}=7\pi/5$. Colors indicate relative localization of
the wavefunction on the weak links (left: blue, right: red).

The simplest Josephson junction is composed of two superconductors connected
by a short, one-dimensional quantum conductor with a single electronic
channel. Experimentally, this has been realized with superconducting atomic
contacts 18, 20 and in semiconductor-superconductor nanowires 21, 22, 23, 24,
25. These experimental results are well described by the Bogolubiov-de-Gennes
(BdG) formalism and motivates us to use the same formalism to calculate the
junction interaction. We consider a series connection of two ideal short weak
links as shown in Figure 1(a) for large separation ($l\gg\xi_{0}$) and in
Figure 1(b) for an Andreev molecule ($l=\xi_{0}$). By connecting a ground
electrode to the central conductor one can flow current independently through
each junction or apply individual superconducting phase differences
$\delta_{L\left(R\right)}$ using magnetic fields. Experimentally, achieving
small separations $l\lesssim\xi_{0}$ and making an electrical connection to
the central superconductor is feasible with microfabrication techniques since
the effective superconducting coherence length is approximately 100 nm for a
disordered thin film of aluminum, a typical superconductor.

Within the BdG formalism, electrons in such a circuit can be described by a
$2\times 2$ Hamiltonian $H$ in Nambu space,

$\displaystyle H$
$\displaystyle=\left(\begin{array}[]{cc}H_{0}+H_{\mathit{WL}}&\Delta\left(x\right)\\\
\Delta^{*}\left(x\right)&-H_{0}-H_{\mathit{WL}}\end{array}\right),$ (3)
$\displaystyle\Delta(x)$ $\displaystyle=\begin{cases}\Delta
e^{i\delta_{L}}&\text{if }x<-l/2\\\ \Delta&\text{if }|x|<l/2\\\ \Delta
e^{i\delta_{R}}&\text{if }x>l/2\end{cases},$ (4)

where $H_{0}=\frac{-\hbar^{2}}{2m}\partial_{x}^{2}-\mu$ is the single particle
energy ($m$ is the electron mass, $\mu$ the chemical potential) and
$H_{\mathit{WL}}=U_{L}\delta\left(x+l/2\right)+U_{R}\delta\left(x-l/2\right)$
models scattering at the weak links at $x=\pm l/2$ with amplitudes
$U_{L\left(R\right)}$. For simplicity we use Dirac $\delta$-functions for the
scatterers, appropriate for weak links of length shorter than $\xi_{0}$, but
expect the effects to persist in longer weak links, diminishing on a length
scale given by $\xi_{0}$. In the following, we consider symmetric scattering
$U_{L}=U_{R}=U_{0}$ corresponding to both single-channel junctions having
transmission $\tau\approx 0.94$, comparable to experimental values 24, 25.

The off-diagonal terms of the Hamiltonian, $\Delta(x)$ (Eq. 3), describe
electron pairing in each superconductor. The superconducting gap amplitude
$\Delta$ is constant along the whole device and by gauge invariance we choose
the phase of the central superconductor to be zero. In the one- to few-channel
limit considered here the weak link supercurrents are much smaller than the
critical currents of the superconducting regions and it is justified to ignore
additional spatial variations in the order-parameter amplitude or phase.

Diagonalization of the BdG equation $H\psi=E\psi$ using an evanescent plane-
wave basis for the wavefunctions $\psi$ gives a discrete spectrum of Andreev
Bound States with energies smaller than the superconducting gap $|E|<\Delta$
and a continuum of states for $|E|>\Delta$. For the plane-waves we make the
standard approximation $\xi_{0}\gg\lambda_{F}$ appropriate for typical
superconductors, where $\lambda_{F}$ is the Fermi wavelength. Analytic
expressions for the spectrum can be obtained at arbitrary separation for
opaque junctions ($\tau\ll 1$) as well as for arbitrary transmission at large
($l/\xi_{0}\gg 1$) or small separation ($l/\xi_{0}\ll 1$). However the results
shown here in the experimentally accessible intermediate case are calculated
numerically.

Figure 1(c) shows the ABS spectrum for large separation, $l\gg\xi_{0}$, as a
function of the left weak link’s phase $\delta_{L}$ for fixed
$\delta_{R}=3\pi/5$ across the right weak link. As expected there is no
interaction and the four levels are determined by the single junction
expression
$E^{\pm}_{A}(\delta,\tau_{0})=\pm\Delta\sqrt{1-\tau_{0}\sin^{2}\frac{\delta}{2}}$
by substituting $\delta=\delta_{L(R)}$ and $\tau_{0}=\tau$. As a consequence
of setting $\delta_{R}=3\pi/5$, there are two dispersionless ABS (red lines)
at energies $E^{\pm}_{A}(3\pi/5,\tau)$. Since the transmissions of both
junctions are equal, $\tau_{L}=\tau_{R}=\tau$, the energy levels cross at
$\delta_{L}=\pm\delta_{R}=\pm 3\pi/5$. The degeneracy at
$\delta_{L}=-\delta_{R}$ is actually plotted at
$\delta_{L}=2\pi-\delta_{R}=7\pi/5$ because the phases are $2\pi$ periodic. We
confirm that the spectra correspond to distinct ABS localized at each weak
link by plotting the ground-state wavefunctions in Figure 1(d) at the
degeneracy point $\delta_{L}=\delta_{R}$.

The spectrum of the Andreev molecule Figure 1(e) changes drastically as a
result of interactions between the junctions at small separation $l=\xi_{0}$.
Not only do gaps open and lift the preceding degeneracies, but all spectral
lines now depend on $\delta_{L}$, despite the fact that $\delta_{R}$ remains
fixed. Additional spectra for separation $l=2\xi_{0},l=0.5\xi_{0}$, and
$l\approx 0$ can be found in the Supporting Information.

The breaking of symmetry about $\delta_{L}=\pi$, absent in the spectra of the
isolated junctions, is permitted by time-reversal invariance, which only
requires $E(\delta_{L},\delta_{R})=E(-\delta_{L},-\delta_{R})$ and not
$E(\delta_{L},\delta_{R})=E(-\delta_{L},\delta_{R})$. Although the continuum
states are not plotted, states near the gap edge also develop a dispersion in
the regions where the bound states approach the gap.

The opening of gaps is reminiscent of a coupling which hybridizes previously
orthogonal states, and this picture is confirmed by a plot of the ABS
wavefunctions in Figure 1(f) for $\delta_{L}=\pm\delta_{R}$. Only the
wavefunction of the negative-energy state closest to zero energy is plotted in
each case. The rapid oscillations here and in Figure 1(d) are due to the
propagating part of the plane waves, whose wavelength $\lambda_{F}=\xi_{0}/5$
is chosen for visibility. The ABS wavefunctions are spread over both junctions
with a significant weight in the central superconductor. Due to the off-
diagonal term $\Delta(x)$ (Eq. 4) the parity symmetry of the Hamiltonian is
not the same for $\delta_{L}=\delta_{R}$ and $\delta_{L}=-\delta_{R}$. This
results in the asymmetric spectrum as well as the differences in symmetry
between the two wavefunctions, similar to bonding and anti-bonding states of a
molecule.

In the context of quartets, the distinct hybridization mechanisms at
$\delta_{L}=\delta_{R}$ and $\delta_{L}=-\delta_{R}$ are called double crossed
Andreev reflection (dCAR) and double elastic cotunneling (dEC), respectively
10, 13. Microscopically, dCAR corresponds to the transfer of Cooper pairs from
the inner superconducting electrode to the outer ones while dEC corresponds to
the transfer of Cooper pairs directly from the left to right electrode. One
can link this Cooper pair transfer picture to our formalism via the scattering
matrix approach as well as obtain analytical expressions for the dCAR and dEC
currents in special limiting cases (see Supporting Information), but it is not
necessary to invoke entanglement as with the quartet interpretation.

Figure 2:  (a) Positive Andreev Bound State spectrum at symmetry points
$\delta_{L}=-\delta_{R}$ (solid lines) and $\delta_{L}=\delta_{R}$ (dashed
lines) as a function of $l/\xi_{0}$ for fixed $\delta_{R}=3\pi/5$. Due to
hybridization the energy levels split as the separation is reduced and some
states are pushed into the continuum. (b) The level splitting, as well as the
number of bound states, can also be controlled by adjusting $\delta_{R}$ for
fixed separation, $l/\xi_{0}=0.5$. Oscillations on a scale $\lambda_{F}$ are
avoided by choosing $k_{F}l=\pi/2$ (mod $2\pi$) with $k_{F}l\gg 1$.

The transition from two independent junctions ($l\gg\xi_{0}$) to a single one
($l=0$) with an intermediate molecular state is visualized in Figure 2(a)
where the positive spectral lines for $\delta_{L}=\pm\delta_{R}$ are plotted
as a function of $l/\xi_{0}$ for $\delta_{R}=3\pi/5$ and the same transmission
as before. At large separation $l/\xi_{0}\gg 1$ all positive ABS converge to
the same energy, $E^{+}_{A}(\delta=3\pi/5,\tau)$, that of the upper red line
in Figure 1(c).

For the hybridization mechanism at $\delta_{L}=-\delta_{R}$ (dEC, solid
lines), with decreasing $l/\xi_{0}$ the two degenerate ABS of the left and
right junction gradually split to form bonding and anti-bonding molecular
states until the higher-energy ABS escapes into the continuum. The ABS pushed
into the continuum is referred to as a “leaky” Andreev state 26 but unlike in-
gap ABS are delocalized plane waves 27. Beyond that point the device hosts
only one pair of ABS, symmetric in energy, such that it behaves as a single
artificial atom rather than a molecule. Incidentally, around this transition
from molecule to atom, the lower ABS reaches a minimum resulting in an overall
shape which suggests the interatomic potentials used to describe the formation
of molecules. Physically, this minimum occurs due to competition between
hybridization which pushes the lower ABS toward zero energy and scattering
which pushes it away (see Supporting Information).

At $l/\xi_{0}=0$, in contrast to a real interatomic potential, the energy of
this lower state does not diverge and is less than the energy at large
separation, $l/\xi_{0}\gg 1$. The full spectrum for $l/\xi_{0}=0$ can be found
in the Supporting Information.

For the dashed $\delta_{L}=\delta_{R}$ (dCAR) spectral lines, the level
splitting is weaker than for dEC (solid lines) and both states are pushed into
the continuum as $l/\xi_{0}$ approaches zero. As with the level splitting, the
depth of the potential well for the lower dashed line is smaller than for dEC
(lower solid line). This is because the dCAR process is in general weaker than
dEC for high transmissions.

A more experimentally feasible possibility to tune the hybridization strength
is adjusting $\delta_{R}$ at fixed separation, as shown in Figure 2(b) for
$l/\xi_{0}=0.5$. Different regimes can be identified by the number of bound
states. As $\delta_{R}$ increases from $0$ to $\pi$, states gradually detach
from the continuum and form bound states. The size of the level splitting of
the $\delta_{L}=\delta_{R}$ line (dCAR, dashed) is maximal at $\delta_{R}=\pi$
whereas for $\delta_{L}=-\delta_{R}$ (dEC, solid line) this maximum occurs at
an intermediate value. By changing $\delta_{R}$ one can control the degree of
hybridization and tune the molecular nature of the system.

As a function of the transmission $\tau$, the general tendency is for the
energy splitting at $\delta_{L}=-\delta_{R}$ to monotonically increase with
$\tau$ whereas the splitting at $\delta_{L}=\delta_{R}$ has a peak and goes to
zero at $\tau=1$.

The Andreev molecule, as a tunable multi-level qubit, may be useful for the
preparation and manipulation of quantum states. In an Andreev molecule with
moderate transmission and separation, one could envision a protocol where the
system, initially in the ground state, is tuned via the phases $\delta_{L,R}$
to a level splitting in the gigahertz range. Microwave pulses could then
prepare an excited state which would be protected by subsequently detuning the
phases away from the splitting where transitions are less likely. Although
coherence times of Andreev-based qubits 25, 28 are currently an order of
magnitude lower than that of conventional superconducting qubits, the
decoherence mechanisms are not understood and may well be mitigated in the
future.

The remarkable features of the spectrum of the Andreev molecule, in particular
the asymmetry and additional energy dispersion in Figure 1(e), dramatically
modifies the Josephson effect and gives rise to non-local Josephson
supercurrents. These supercurrents are calculated from the eigenstates of the
BdG equation 29, 30, 31 in a manner analogous to calculating probability
currents. Both Andreev bound states and continuum states contribute
significantly to the total supercurrent, unlike the case for isolated
junctions where the continuum plays no role 30.

Figure 3: Andreev molecule current-phase relation and $\varphi$-junction
behavior. (a) The current-phase relation of the left weak link,
$I_{L}(\delta_{L},\delta_{R}=3\pi/5)$, is shifted and distorted for small
separation ($l=0.5\xi_{0}$, solid line) as compared to the isolated junction
($l\gg\xi_{0}$, dashed line). (b) The zero-phase supercurrent
$I_{L}(\delta_{L}=0)$ (black vertical arrow) is plotted as a function of
$\delta_{R}$, showing a tunable $\varphi$-junction effect. Currents are
normalized to $\Delta/\varphi_{0}$, with $\varphi_{0}=\hbar/2e$ the reduced
flux quantum.

Figure 3(a) shows the supercurrent-phase relation of the left junction
$I_{L}\left(\delta_{L},\delta_{R}\right)$ for a fixed superconducting phase
difference $\delta_{R}=3\pi/5$ across the right junction. Plots are given for
an isolated junction (dashed line) and for an Andreev molecule with
$l=0.5\xi_{0}$ (solid line). The curve for small separation is shifted with
respect to that for large separation, and increases as the separation
$l/\xi_{0}$ is reduced. In addition there is a distortion to the curve which
breaks the odd symmetry in $\delta_{L}$ about the zeroes in the current.
Furthermore, the positive and negative critical currents are not equal, and
although not shown here, have magnitudes which depend on $\delta_{R}$. These
modifications, resulting from the dependence of $I_{L}$ on $\delta_{R}$ for
$l\lesssim\xi_{0}$, are non-local effects in that the phase drop $\delta_{R}$
occurs away from the left weak link, at a distance on the order of $\xi_{0}$,
which can be hundreds of nanometers depending on the superconductor.

For an isolated, time-reversal invariant Josephson junction, Figure 1(a), the
supercurrent must be zero at a phase difference of $0$ and $\pi$, Figure 3(a)
(dashed line) 9. But for the Andreev molecule where $I_{L}$ depends on
$\delta_{R}$, since time-reversal symmetry only imposes
$I_{L}\left(-\delta_{L},-\delta_{R}\right)=-I_{L}\left(\delta_{L},\delta_{R}\right)$,
the current $I_{L}$ can be non-zero at $\delta_{L}=0$. A non-zero supercurrent
at zero phase difference is the hallmark of a $\varphi$-junction, in which the
current crosses zero at a phase $\varphi$ instead of zero 32.

In Figure 3(b) the zero-phase supercurrent
$I_{L}^{0}=I_{L}(\delta_{L}=0,\delta_{R})$ is plotted as a function of
$\delta_{R}$. The sign of $I_{L}^{0}$ for small $\delta_{R}$ is negative,
indicating that the current flows in the same direction as the phase
difference $\delta_{R}$, from left to central electrode in Figure 1(b). This
is akin to a “ferromagnetic” interaction in that the current $I_{L}^{0}$ and
phase difference $\delta_{R}$ are aligned. This tendency is explained by the
fact that for vanishing separation $l$, there is only one junction with an
overall phase drop given by $\delta_{L}-\delta_{R}$ and therefore
$I_{L}(\delta_{L},\delta_{R})=I_{L}(\delta_{L}-\delta_{R})$. In other words,
the $\varphi$-junction phase offset approaches $\varphi=-\delta_{R}$ in the
limit $l/\xi_{0}\rightarrow 0$.

Figure 4: Experimental proposals. (a) The Josephson tunnel junction biased at
$V_{J}$ (boxed cross) acts as a spectrometer for transitions of the Andreev
molecule (colored crosses). (b) Setup for switching-current measurements of
the current-phase relation of an Andreev molecule (described in text). An
overall magnetic field $B$ and control current $I_{d}$ allow independent
control of $\delta_{L(R)}$ via the loop fluxes $\Phi,\Phi_{L(R)}$.

The two main signatures of Andreev molecules at equilibrium are the spectral
avoided crossings (Figure 2) and the non-local current-phase relation (Figure
3). For large effects, the Andreev molecule should have small separation
($l\lesssim\xi_{0}$) and high transmission ($\tau\lesssim 1$), realistic
conditions for epitaxial semiconductor-superconductor devices 24, 25.
Additional calculations have shown that these effects are robust to disorder
and persist if there are multiple conduction channels. Andreev molecules can
be detected with experiments at zero voltage bias, as described below. In non-
equilibrium dc transport experiments, where junctions are voltage biased 10,
16, 17, these signatures are not present and dynamic processes such as mutual
phase locking 5 complicate interpretation.

In Figure 4(a), we propose a setup to perform spectroscopy of an Andreev
molecule in a similar manner to the spectroscopy of an Andreev “atom” in a
superconducting atomic contact 18. A large tunnel Josephson junction (yellow)
is simultaneously used as an on-chip microwave source and detector. When a
voltage bias $V_{J}$ is applied across this spectroscopy junction, it emits
microwave current at frequency $\nu=2eV_{J}/h$ via the AC Josephson effect.
This high-frequency current is coupled via the capacitor to the Andreev
molecule, and due to inelastic Cooper-pair tunneling, will give peaks in the
DC current-voltage characteristic of the spectrometer junction,
$I_{J}(V_{J})$, at voltages $h\nu/2e=E_{T}/2e$ corresponding to the
transitions in the Andreev molecule spectrum, Figure 1(e), of energy $E_{T}$.
The magnetic field $B$ and gradiometric control current $I_{d}$ allow
independently tuning the phases $\delta_{L(R)}$ via the magnetic fluxes
$\Phi_{L(R)}$, assuming that loop inductances are negligible compared to the
Josephson inductance of the weak links. A full measurement
$I_{J}(V_{J},B,I_{d})$ would allow reconstructing the spectra of Figure 1(e)
as well as verifying the predictions of Figure 2(b).

Josephson spectroscopy is well suited for detecting Andreev molecules, as
$E_{T}$ may be comparable to $\Delta/h$ (90 GHz for aluminum), well within the
spectrometer bandwidth. Conventional microwave spectroscopy 19, 25, with
greater sensitivity but narrower range, may also be used, as well as tunneling
spectroscopy 33, 34 instead of photonic spectroscopy.

In a second experiment, the existence of a non-local current-phase relation
can be directly determined by measurements of the switching current. This type
of experiment has been performed on superconducting atomic contacts 35,
graphene 36 and carbon nanotubes 37. As shown in Figure 4(b), a large
Josephson junction of critical current
$I_{0}\gg\textrm{{max}}\left(I_{L},I_{R}\right)$ (boxed cross) is wired in
parallel with the left weak link, $L$ (upper black cross), of the Andreev
molecule, forming an asymmetric SQUID. Similarly to the previous proposal,
$\delta_{L}$ and $\delta_{R}$ are controlled independently using the flux
$\Phi$ and control current $I_{d}$. Due to the asymmetry, the SQUID critical
current $I_{SQ}$ is essentially given by that of the large junction,
effectively at a phase difference of $\pi/2$, modulated by the supercurrent of
the $L$ junction, $I_{SQ}\approx I_{0}+I_{L}(\delta_{L},\delta_{R})$, where
flux quantization constrains $\delta_{L}=\Phi/\varphi_{0}-\pi/2$. By sending
current pulses or ramps and measuring the switching of the SQUID to a non-zero
voltage state, one can extract the current-phase relation
$I_{L}(\delta_{L},\delta_{R})$ and demonstrate that there is a non-local
component that depends on $\delta_{R}$.

We have shown that two superconducting weak links separated by a distance on
the order of the superconducting coherence length exhibit a non-local
Josephson effect, a phenomenon attributed to the hybridization of Andreev
Bound States between the two weak links and the formation of an Andreev
molecule. Our work provides a microscopic understanding of the “order-
parameter,” or Cooper-pair mediated, interaction between Josephson junctions.
The most prominent signatures of non-local effects are a modification of the
Andreev Bound State energy spectrum and the emergence of a $\varphi$-junction,
a weak link with non-zero supercurrent at zero phase difference. Such
signatures can be detected with well-established spectroscopy techniques and
switching-current measurements.

The Andreev molecule, with tunable energy gaps, may serve as a new type of
superconducting qubit, and its $\varphi$-junction behavior may find use in
magnetometry. In addition the large junction interaction holds promise for
applications employing arrays of coupled weak links, such as in quantum
information, metrology, or sensing. Arrays of junctions in the Andreev
molecule limit ($l\lesssim\xi_{0}$) may achieve new limits of qubit coupling,
quantum amplification, voltage stability, or magnetic field sensitivity. Such
arrays may also serve as quantum simulators of polymers or to implement model
Hamiltonians. By controlling the separation between weak links in the array as
well as the individual phase differences, the strength of the coupling between
Andreev states at adjacent weak links can be varied. For example, alternating
short and long separation would allow generating a Su-Schrieffer-Heeger (SSH)
type Hamiltonian modeling a polyacetylene molecule. It may also be possible to
introduce repulsive on-site interactions by using chains of weak links where
charging energy is important.

The predicted coupling between two weak links should also exist for exotic
Andreev Bound States where spin-orbit and Zeeman effects play a role, and
could be exploited for topological quantum computation with Majorana bound
states.

We acknowledge fruitful discussions with Yuli Nazarov, Régis Mélin, Benoît
Douçot, Yonatan Cohen, Yuval Ronen, Hugues Pothier, Fabien Lafont, Landry
Bretheau and Benjamín Alemán and support from Jeunes Equipes de l’Institut de
Physique du Collège de France. This research was supported by IDEX grant
ANR-10-IDEX-0001-02 PSL and a Paris “Programme Emergence(s)” Grant. This
project has received funding from the European Research Council (ERC) under
the European Union’s Horizon 2020 research and innovation programme (grant
agreement 636744).

Additional discussion on separation dependence of spectra; supplementary
spectra of Andreev molecules for different values of separation $l$; details
of the calculation of the Andreev molecule spectrum and supercurrent within
the Bogolubiov-de-Gennes formalism; discussion of Andreev molecules with
Josephson tunnel junctions.

## References

  * Wendin 2017 Wendin, G. Quantum information processing with superconducting circuits: a review. _Rep. Prog. Phys._ 2017, _80_ , 106001
  * Clarke and Braginski 2006 Clarke, J.; Braginski, A. I. _The SQUID Handbook Fundamentals and Technology of SQUIDs and SQUID Systems, Vol. 1_ ; Wiley-VCH: Weinheim, 2006
  * Benz and Hamilton 2004 Benz, S. P.; Hamilton, C. A. Application of the Josephson effect to voltage metrology. _Proc. IEEE_ 2004, _92_ , 1617–1629
  * Lamata et al. 2018 Lamata, L.; Parra-Rodriguez, A.; Sanz, M.; Solano, E. Digital-analog quantum simulations with superconducting circuits. _Adv. Phys.: X_ 2018, _3_ , 1457981
  * Likharev 1986 Likharev, _Dynamics of Josephson Junctions and Circuits_ ; CRC Press, 1986
  * Hansen and Lindelof 1984 Hansen, J. B.; Lindelof, P. E. Static and dynamic interactions between Josephson junctions. _Rev. Mod. Phys._ 1984, _56_ , 431–459
  * Deminova and Kovalenko 1979 Deminova, G.; Kovalenko, A. A calculation of interaction between two Josephson junctions. _IEEE Trans. Magn._ 1979, _15_ , 291–294
  * Brinkman and Golubov 2000 Brinkman, A.; Golubov, A. A. Coherence effects in double-barrier Josephson junctions. _Phys. Rev. B_ 2000, _61_ , 11297–11300
  * Golubov et al. 2004 Golubov, A. A.; Kupriyanov, M. Y.; Il’ichev, E. The current-phase relation in Josephson junctions. _Rev. Mod. Phys._ 2004, _76_ , 411–469
  * Freyn et al. 2011 Freyn, A.; Douçot, B.; Feinberg, D.; Mélin, R. Production of Nonlocal Quartets and Phase-Sensitive Entanglement in a Superconducting Beam Splitter. _Phys. Rev. Lett._ 2011, _106_ , 257005
  * Jonckheere et al. 2013 Jonckheere, T.; Rech, J.; Martin, T.; Douçot, B.; Feinberg, D.; Mélin, R. Multipair dc Josephson resonances in a biased all-superconducting bijunction. _Phys. Rev. B_ 2013, _87_ , 214501
  * Mélin et al. 2014 Mélin, R.; Feinberg, D.; Courtois, H.; Padurariu, C.; Pfeffer, A.; Duvauchelle, J. E.; Lefloch, F.; Jonckheere, T.; Rech, J.; Martin, T.; Doucot, B. D.C. Josephson transport by quartets and other Andreev resonances in superconducting bijunctions. _J. Phys.: Conf. Ser._ 2014, _568_ , 052006
  * Feinberg et al. 2015 Feinberg, D.; Jonckheere, T.; Rech, J.; Martin, T.; Douçot, B.; Mélin, R. Quartets and the current-phase structure of a double quantum dot superconducting bijunction at equilibrium. _Eur. Phys. J. B_ 2015, _88_
  * Mélin et al. 2016 Mélin, R.; Sotto, M.; Feinberg, D.; Caputo, J.-G.; Douçot, B. Gate-tunable zero-frequency current cross correlations of the quartet state in a voltage-biased three-terminal Josephson junction. _Phys. Rev. B_ 2016, _93_ , 115436
  * Mélin et al. 2017 Mélin, R.; Caputo, J.-G.; Yang, K.; Douçot, B. Simple Floquet-Wannier-Stark-Andreev viewpoint and emergence of low-energy scales in a voltage-biased three-terminal Josephson junction. _Phys. Rev. B_ 2017, _95_ , 085415
  * Pfeffer et al. 2014 Pfeffer, A. H.; Duvauchelle, J. E.; Courtois, H.; Mélin, R.; Feinberg, D.; Lefloch, F. Subgap structure in the conductance of a three-terminal Josephson junction. _Phys. Rev. B_ 2014, _90_ , 075401
  * Cohen et al. 2018 Cohen, Y.; Ronen, Y.; Kang, J.-H.; Heiblum, M.; Feinberg, D.; Mélin, R.; Shtrikman, H. Nonlocal supercurrent of quartets in a three-terminal Josephson junction. _PNAS_ 2018, _115_ , 6991–6994
  * Bretheau et al. 2013 Bretheau, L.; Girit, Ç. Ö.; Pothier, H.; Esteve, D.; Urbina, C. Exciting Andreev pairs in a superconducting atomic contact. _Nature_ 2013, _499_ , 312–315
  * Janvier et al. 2015 Janvier, C.; Tosi, L.; Bretheau, L.; Girit, Ç. Ö.; Stern, M.; Bertet, P.; Joyez, P.; Vion, D.; Esteve, D.; Goffman, M. F.; Pothier, H.; Urbina, C. Coherent manipulation of Andreev states in superconducting atomic contacts. _Science_ 2015, _349_ , 1199–1202
  * Della Rocca et al. 2007 Della Rocca, M. L.; Chauvin, M.; Huard, B.; Pothier, H.; Esteve, D.; Urbina, C. Measurement of the Current-Phase Relation of Superconducting Atomic Contacts. _Phys. Rev. Lett._ 2007, _99_
  * Krogstrup et al. 2015 Krogstrup, P.; Ziino, N. L. B.; Chang, W.; Albrecht, S. M.; Madsen, M. H.; Johnson, E.; Nygård, J.; Marcus, C. M.; Jespersen, T. S. Epitaxy of semiconductor–superconductor nanowires. _Nature Materials_ 2015, _14_ , 400–406
  * Albrecht et al. 2016 Albrecht, S. M.; Higginbotham, A. P.; Madsen, M.; Kuemmeth, F.; Jespersen, T. S.; Nygård, J.; Krogstrup, P.; Marcus, C. M. Exponential protection of zero modes in Majorana islands. _Nature_ 2016, _531_ , 206–209
  * Deng et al. 2016 Deng, M. T.; Vaitiekėnas, S.; Hansen, E. B.; Danon, J.; Leijnse, M.; Flensberg, K.; Nygård, J.; Krogstrup, P.; Marcus, C. M. Majorana bound state in a coupled quantum-dot hybrid-nanowire system. _Science_ 2016, _354_ , 1557–1562
  * Goffman et al. 2017 Goffman, M. F.; Urbina, C.; Pothier, H.; Nygård, J.; Marcus, C. M.; Krogstrup, P. Conduction channels of an InAs-Al nanowire Josephson weak link. _New J. Phys._ 2017, _19_ , 092002
  * Tosi et al. 2019 Tosi, L.; Metzger, C.; Goffman, M. F.; Urbina, C.; Pothier, H.; Park, S.; Yeyati, A. L.; Nygård, J.; Krogstrup, P. Spin-Orbit Splitting of Andreev States Revealed by Microwave Spectroscopy. _Phys. Rev. X_ 2019, _9_ , 011010
  * Bagwell 1992 Bagwell, P. F. Suppression of the Josephson current through a narrow, mesoscopic, semiconductor channel by a single impurity. _Phys. Rev. B_ 1992, _46_ , 12573–12586
  * Olivares et al. 2014 Olivares, D. G.; Yeyati, A. L.; Bretheau, L.; Girit, Ç. Ö.; Pothier, H.; Urbina, C. Dynamics of quasiparticle trapping in Andreev levels. _Phys. Rev. B_ 2014, _89_ , 104504
  * Hays et al. 2018 Hays, M.; de Lange, G.; Serniak, K.; van Woerkom, D. J.; Bouman, D.; Krogstrup, P.; Nygård, J.; Geresdi, A.; Devoret, M. H. Direct Microwave Measurement of Andreev-Bound-State Dynamics in a Semiconductor-Nanowire Josephson Junction. _Phys. Rev. Lett._ 2018, _121_ , 047001
  * Blonder et al. 1982 Blonder, G. E.; Tinkham, M.; Klapwijk, T. M. Transition from metallic to tunneling regimes in superconducting microconstrictions: Excess current, charge imbalance, and supercurrent conversion. _Phys. Rev. B_ 1982, _25_ , 4515–4532
  * Beenakker and van Houten 1991 Beenakker, C. W. J.; van Houten, H. Josephson current through a superconducting quantum point contact shorter than the coherence length. _Phys. Rev. Lett._ 1991, _66_ , 3056–3059
  * Bardeen et al. 1969 Bardeen, J.; Kümmel, R.; Jacobs, A. E.; Tewordt, L. Structure of Vortex Lines in Pure Superconductors. _Phys. Rev._ 1969, _187_ , 556–569
  * Buzdin and Koshelev 2003 Buzdin, A.; Koshelev, A. E. Periodic alternating 0- and $\pi$-junction structures as realization of $\varphi$-Josephson junctions. _Phys. Rev. B_ 2003, _67_ , 220504
  * Pillet et al. 2010 Pillet, J.-D.; Quay, C. H. L.; Morfin, P.; Bena, C.; Yeyati, A. L.; Joyez, P. Andreev bound states in supercurrent-carrying carbon nanotubes revealed. _Nature Physics_ 2010, _6_ , 965–969
  * Deng et al. 2018 Deng, M.-T.; Vaitiekėnas, S.; Prada, E.; San-Jose, P.; Nygård, J.; Krogstrup, P.; Aguado, R.; Marcus, C. M. Nonlocality of Majorana modes in hybrid nanowires. _Phys. Rev. B_ 2018, _98_ , 085125
  * Goffman et al. 2000 Goffman, M. F.; Cron, R.; Levy Yeyati, A.; Joyez, P.; Devoret, M. H.; Esteve, D.; Urbina, C. Supercurrent in Atomic Point Contacts and Andreev States. _Phys. Rev. Lett._ 2000, _85_ , 170–173
  * English et al. 2016 English, C. D.; Hamilton, D. R.; Chialvo, C.; Moraru, I. C.; Mason, N.; Van Harlingen, D. J. Observation of non sinusoidal current-phase relation in graphene Josephson junctions. _Phys. Rev. B_ 2016, _94_
  * Delagrange et al. 2016 Delagrange, R.; Weil, R.; Kasumov, A.; Ferrier, M.; Bouchiat, H.; Deblock, R. 0-$\pi$ quantum transition in a carbon nanotube Josephson junction: Universal phase dependence and orbital degeneracy. _Phys. Rev. B_ 2016, _93_ , 195437