Weyl singularities in polaritonic multi-terminal Josephson junctions I. Septembre1J. S. Meyer2D. D. Solnyshkov1 3and G. Malpuech1 1Université Clermont Auvergne Clermont Auvergne INP

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Weyl singularities in polaritonic multi-terminal Josephson junctions

I. Septembre,1J. S. Meyer,2D. D. Solnyshkov,1, 3 and G. Malpuech1

1Université Clermont Auvergne, Clermont Auvergne INP,

CNRS, Institut Pascal, F-63000 Clermont-Ferrand, France

2Univ. Grenoble Alpes, CEA, Grenoble INP, IRIG, Pheliqs, 38000 Grenoble, France

3Institut Universitaire de France (IUF), 75231 Paris, France

We study theoretically analog multi-terminal Josephson junctions formed by gapped superﬂuids

created upon resonant pumping of cavity exciton-polaritons. We study the p-like bands of a 5-

terminal junction in the 4D parameter space created by the superﬂuid phases acting as quasi-

momenta. We ﬁnd 4/6 Weyl points in 3D subspaces with preserved/broken time-reversal symmetry.

We link the real space topology (vortices) to the parameter space one (Weyl points). We derive an

eﬀective Hamiltonian encoding the creation, motion, and annihilation of Weyl nodes in 4D. Our work

paves the way to the study of exotic topological phases in a platform allowing direct measurement

of eigenstates and band topology.

Introduction.– Topological singularities are singulari-

ties of both eigenvalues and eigenstates carrying a topo-

logical charge. Dirac points are 2D topological point

singularities. They transform into pairs of exceptional

points connected by a Fermi arc when adding non-

Hermiticity [1–3]. In 3D parameter spaces, Weyl points

(WPs) are Hermitian point degeneracies [4, 5] (diﬀerent

from 3D Dirac points [6, 7]). They are robust because any

Hermitian perturbation only moves the WPs in the pa-

rameter space, whereas it can destroy Dirac points. WPs

resemble exceptional points because they come in pairs

connected by Fermi arcs [5, 8]. The only way to annihi-

late both Weyl and exceptional points is to make points

of opposite charge meet [3, 9]. WPs can appear when

time-reversal (TR) symmetry and/or inversion symme-

try is broken. If TR symmetry is preserved, they come in

multiples of 4; they come in multiples of 2 otherwise [10].

Furthermore, additional symmetries make appear nodal

lines rather than points [11–16]. They are line singulari-

ties in 3D linked giving drumhead surface states [17–19].

Topological bands and states beneﬁt from outstanding

properties such as (in 2D) one-way edge propagation [20–

26], used in topological lasers [27–31].

Topological singularities described previously lie in a

parameter space which is usually the reciprocal space,

because it naturally comes as the matching of the real

space. However, some parameters that are easily tunable

experimentally can form additional dimensions of the pa-

rameter space. This enriches the exploration of topolog-

ical phase transitions. Such systems are called synthetic

topological matter. The fantastic freedom they oﬀer is

an enthralling playground for physicists [32–35]. It en-

ables to investigate physics beyond 3 dimensions [36–38]

as well as strongly correlated phenomena [39–41]. Topo-

logical photonics [42, 43], and especially polaritonics [44],

can take advantage of synthetic topological matter, no-

tably because the latter gives an experimental access to

the eigenstates and quantum geometry of topological sys-

tems [45, 46].

Andreev reﬂection occurs at the interface between a

superconductor and a non-superconducting material [47].

An incoming electron undergoes an anomalous reﬂection,

becoming a hole excitation with reversed wavevector,

charge, and spin. Usual Josephson junctions [48] contain

two interfaces between a non-superconducting material

(insulator, semiconductor, or metal) and a superconduc-

tor. Such junctions host Andreev bound states whose en-

ergy depends on the phase diﬀerence between the super-

conductors. This dependence can be described in terms

of synthetic bands, where the 1D parameter space is here

given by the phase diﬀerence [49, 50]. If the superconduc-

tors are topological, those synthetic bands can be topo-

logical when the energies cross at the Fermi energy, which

forms a topological singularity. In this case, the junction

is known to host Majorana fermions [51], which are very

promising for quantum computing [52]. Multi-terminal

Josephson junctions, where more than two superconduc-

tor wires are connected [53], is now a well-developed re-

search area [54–64]. The corresponding synthetic bands

demonstrate non-trivial topology (in arbitrary large di-

mensions) even with trivial superconductors.

Andreev reﬂection has been studied theoretically in

bosonic systems [65]. In cavity exciton-polaritons, an

analog superconductor can be created upon resonant

driving. The driving opens a gap in the energy spectrum

of the pumped modes, creating a “gapped superﬂuid” [66–

69]. It is possible to create analog Josephson junctions

by pumping two (or more) regions with diﬀerent pump

lasers (with well-deﬁned phases), the superﬂuids being

separated by a common non-superﬂuid region [70–72].

In [73], the existence of Andreev bound state analogs in

the normal region between two gapped superﬂuids has

been demonstrated, while the 1D bands parameterized

by the superﬂuid phase diﬀerence were found to be sep-

arated by a topological gap.

In this work, we theoretically study a multi-terminal

polaritonic Josephson junction, where a pentagonal nor-

mal region is connected to 5 gapped superﬂuids. Such

a junction hosts Andreev-like bound states which form

4D synthetic bands. We study the pair of bands corre-

arXiv:2210.11088v1 [cond-mat.mes-hall] 20 Oct 2022

sponding to the p-like conﬁned states of the pentagonal

trap. We study numerically a 3D subspace and observe

4 WPs. These WPs enable topological transitions in a

2D subspace between distinct topological phases with dif-

ferent Chern numbers. We model the 2 p-like bands by

an eﬀective 2×2Hamiltonian and explore the full 4D

parameter space, leading to additional features such as

3D subspaces with broken TR symmetry. This photonic

system has the advantage of allowing the access to the

eigenstates, making possible the direct measurement of

the Berry curvature and the detailed study of the WPs.

Our results may allow to ﬁnally observe Weyl singulari-

ties in multi-terminal Josephson junctions and pave the

way to synthetic topological matter in arbitrary large di-

mensions in polaritonics.

Model.– We consider a strongly-coupled planar mi-

crocavity hosting exciton-polaritons [68–70]. The exter-

nal drive P(r)is composed of Nregions, each pump-

ing a given area with an homogeneous amplitude Pi=

e−i(ωpt+φ0

i)(i∈[1,N]), where Ep=~ωpis the pump de-

tuning and φ0

ithe phase of the i-th region of the pump,

determining the phase of the corresponding superﬂuid

φi[71, 74]. This forms an analog N-terminal Josephson

junction as depicted in Fig. 1. The wave function ψof

the coherent pumped mode is described by the driven-

dissipative Gross-Pitaevskii equation:

i~∂ψ

∂t =−~2

2m∇2−iγ +α(r)|ψ|2+V(r)ψ+P(r),

(1)

where mis the exciton-polaritons eﬀective mass, γthe

decay (further taken equal to zero, it only adds a global

imaginary part), α > 0describes the repulsive interac-

tions in the pumped areas, and V(r)is a step-like po-

tential (in green in Fig. 1(c)) accounting for the etched

pattern. We assume that the stationary wave function ψs

is zero in all regions without pumping. In the Npumped

areas, the stationary wave function is given by the solu-

tion of the Gross-Pitaevskii equation for a spatially ho-

mogeneous system and reads: ψs,i =√neiφiwhere nis

the superﬂuid density and φiits phase. The spectrum of

the superﬂuid weak excitations in these areas contains a

gap ∆centered around the pump detuning Ep:

∆ = q(3αn −Ep) (αn −Ep)(2)

We consider weak excitations of the full system (the

normal and superﬂuid areas), looking for the solutions of

the following shape:

ψ(r, t) = e−iωptψs(r) + u(r)e−iωt +v∗(r)eiω∗t,(3)

where u(r), v(r)are the Bogoliubov coeﬃcients. We will

consider energies lying in the superﬂuid gap, so that these

coeﬃcients describe the proﬁle of propagative states in













(b)

(a)

(c)

DBR

Normal

Bound state

Andreev counterpart

, +E, majority component

, -E, minority component

e±2iLe±2iR

Figure 1: (a) 1D superﬂuid-normal-superﬂuid junction host-

ing bound states and their Andreev counterparts, linked by

the phases of the left/right superﬂuid φL,R. (b) Sketch of the

analog multi-terminal Josephson junction (here 5-terminal)

organized in a microcavity with a quantum well (QW) and

distributed Bragg reﬂectors (DBR) on a substrate. (c) Top

view of the junction. The superﬂuid density (brown) with 5

diﬀerent phases φiand the potential V(r)(green) proﬁles. In

the normal region, pumping and potential are absent.

the normal region and evanescent states in the superﬂu-

ids, which gives precisely a bound state. This wave func-

tion, inserted in Eq. (1), gives the Bogoliubov-de Gennes

equations:

Lαψ2

s(r)

−αψ∗2

s(r)−L∗u(r)

v(r)=~ωu(r)

v(r),(4)

where L= (k−Ep+ 2αn +V(r)) with k=~2k2/2m.

The case N= 2 is studied in [73]; N= 2 corresponds

to an analog Josephson junction with 2 terminals, con-

taining two normal/superﬂuid interfaces (phases φL,R),

as depicted in Fig. 1(a). A wave incident at each inter-

face undergoes both specular reﬂection (same energy, re-

versed wavevector) and Andreev reﬂection (opposite en-

ergy with respect to the pump Ep, reversed wavevector),

the latter being accompanied by a phase shift e±2iφL,R .

Considering the complete scattering process, Andreev

bound states can be found. These states are composed

of two parts of diﬀerent nature. First, the majority com-

ponent of proﬁle u(r)and energy E=~ωis qualitatively

the state coming from the quantum conﬁnement provided

by the interactions in the pumped regions. It exists with-

out Andreev process and corresponds to the state found

with a diagonal matrix in Eq. (4).The second part is the

minority component of proﬁle v(r)and energy −E, which

is not (in general) an eigenstate of the quantum well

formed by the junction. It appears only because of the

Andreev reﬂections at the interfaces and is therefore very

sensitive to the phase diﬀerence between the two super-

ﬂuids. We call it the minority component, or Andreev

counterpart of the bound state. The phase diﬀerence

between the two superﬂuids is a parameter that can be

tuned experimentally. Both the majority component and

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Weylsingularitiesinpolaritonicmulti-terminalJosephsonjunctionsI.Septembre,1J.S.Meyer,2D.D.Solnyshkov,1,3andG.Malpuech11UniversitéClermontAuvergne,ClermontAuvergneINP,CNRS,InstitutPascal,F-63000Clermont-Ferrand,France2Univ.GrenobleAlpes,CEA,GrenobleINP,IRIG,Pheliqs,38000Grenoble,France3InstitutUniver...

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Weyl singularities in polaritonic multi-terminal Josephson junctions I. Septembre1J. S. Meyer2D. D. Solnyshkov1 3and G. Malpuech1 1Université Clermont Auvergne Clermont Auvergne INP.pdf

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Weyl singularities in polaritonic multi-terminal Josephson junctions I. Septembre1J. S. Meyer2D. D. Solnyshkov1 3and G. Malpuech1 1Université Clermont Auvergne Clermont Auvergne INP

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