Phases instabilities and excitations in a two-component lattice model with photon-mediated interactions Leon Carl1Rodrigo Rosa-Medina1ySebastian D. Huber2

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Phases, instabilities and excitations in a two-component lattice model

with photon-mediated interactions

Leon Carl,1Rodrigo Rosa-Medina,1, †Sebastian D. Huber,2

Tilman Esslinger,1Nishant Dogra,3, ∗and Tena Dubcek2, ∗

1Institute for Quantum Electronics, ETH Z¨urich, 8093 Z¨urich, Switzerland

2Institute for Theoretical Physics, ETH Z¨urich, 8093 Z¨urich, Switzerland

3Cavendish Laboratory, University of Cambridge,

J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom.

(Dated: October 21, 2022)

Engineering long-range interacting spin systems with ultra cold atoms oﬀers the possibility to ex-

plore exotic magnetically ordered phases in strongly-correlated scenarios. Quantum gases in optical

cavities provide a versatile experimental platform to further engineer photon-mediated interactions

and access the underlying microscopic processes by probing the cavity ﬁeld. Here, we study a

two-component spin Bose-Hubbard system with cavity-mediated interactions. We provide a com-

prehensive overview of its phase diagram and transitions in experimentally relevant regimes. The

interplay of diﬀerent energy scales yields a rich phase diagram with superﬂuid and insulating phases

exhibiting density modulation or spin ordering. In particular, the combined eﬀect of contact and

global-range interactions gives rise to an antiferromagnetically ordered phase for arbitrarily small

spin-dependent light-matter coupling, while long-range and inter-spin contact interactions intro-

duce regions of instability and phase separation in the phase diagram. We further study the low

energy excitations above the antiferrogmagnetic phase. Besides particle-hole branches, it hosts spin-

exchange excitations with a tunable energy gap. The studied lattice model can be readily realized

in cold-atom experiments with optical cavities.

I. INTRODUCTION

Experiments with ultacold atoms in optical lattices

have substantially extended the scope of quantum simu-

lation of many-body systems [1,2]. Two key strengths

are the high-degree of tunability of diﬀerent energy

scales, and the possibility to involve the atomic spin de-

gree of freedom, facilitating the investigation of strongly-

correlated phenomena like superﬂuidity, quantum mag-

netism, high-temperature superconductivity and com-

plex out-of-equilibrium dynamics [3,4]. While con-

tact interactions naturally occur in ultracold atomic sys-

tems [2,5], long-range interactions have been more elu-

sive. Nonetheless, systems that are traditionally used to

study long-range interactions, such as dipolar quantum

gases, heteronuclear molecules and Rydberg atoms, suf-

fer from small long-range interaction strengths, low den-

sities and short lifetimes, respectively [6–8]. Quantum

gases coupled to optical cavities thus provide an alter-

native experimental platform to create photon-mediated

long-range interactions, whose strength and sign are con-

trolled by external laser ﬁelds [9,10]. This has facili-

tated theoretical [11–19] and experimental [20,21] inves-

tigations of lattice supersolid and charge density wave

phases in single-component spin systems. The atomic dy-

namics and many-body excitations can be accessed non-

destructively in real time by the light leaking from the

cavity [22]. Recently, the inclusion of an internal atomic

†rrodrigo@phys.ethz.ch

∗These authors contributed equally to this work; dubcekt@ethz.ch

spin degree of freedom has become feasible in such sys-

tems, leading to the observation of density and spin self-

organization [23–25]. Incorporating tunable long-range

spin interactions provides a natural path to further en-

rich the accessible phenomenology [26–36]. In combi-

nation with optical lattices, this approach will allow the

realization of strongly-correlated magnetic phases arising

due to the interplay of short- and long-range interactions.

Some magnetically ordered phases have been discussed

both in bosonic [37,38] and fermionic systems [39,40],

but a comprehensive theoretical study of the phase di-

agram and possible transitions in experimentally acces-

sible regimes is still missing, although it could notably

expedite their successful realization.

Here, we investigate an extended two-component Bose-

Hubbard (BH) model with cavity-mediated long-range

interactions—the lattice counterpart of the experiment

performed in Ref. [23] with a bulk Bose gas. The con-

sidered long-range interactions have a ‘density’ and a

‘spin’ contribution, which favor the two atomic compo-

nents to either occupy a common sublattice or two dif-

ferent ones, each breaking independently a lattice Z2-

symmetry. Their absolute and relative strengths can be

tuned via the intensity and the polarization of an exter-

nal laser ﬁeld, respectively. Additionally, the two atomic

components have diﬀerent intra- and inter-species con-

tact interactions. We extract the complete phase dia-

gram using a Gutzwiller approach in the case of ﬁxed

density at unity ﬁlling, and obtain density-modulated

and magnetically-ordered phases, both in the superﬂuid

and insulating regimes. Remarkably, the cooperation be-

tween short-range and long-range interactions results in

the formation of an antiferromagnetic Mott insulator for

arXiv:2210.11313v1 [cond-mat.quant-gas] 20 Oct 2022

arbitrarily small spin-dependent coupling strengths. In

some regimes, the competing contact-interaction energy

scales lead to the separation of the two spin compo-

nents [41–43]. In addition, long-range interactions can

introduce correlated phase-separated states in multicom-

ponent systems [44,45], and induce phase instabilities in

systems with only one component [17,46]. To further

elucidate the nature of the magnetically ordered phase,

we construct an eﬀective Hamiltonian for its low-energy

excitations via perturbation theory, and identify spin-

exchange branches with a tunable gap.

II. DESCRIPTION OF THE SYSTEM

We consider a balanced spin-mixture of two Bose-

Einstein condensates (BECs) coupled to a high-ﬁnesse

optical cavity. For concreteness, we consider 87Rb atoms

and a two-dimensional (2D) system extending in the

(x, z)-plane [Fig. 1(a)], closely resembling the experi-

ments in Refs. [21,23]. A λ-periodic quantized cavity

mode extends along the x-axis, is polarized in y-direction

and has a resonant frequency ωc. The two spin compo-

nents |↑i =|F= 1, mF= 1iand |↓i =|F= 1, mF=−1i

belong to the total angular momentum F= 1 manifold,

with the quantization axis deﬁned by a magnetic ﬁeld in

the z-direction. The mixture is loaded into a 2D λ/2-

periodic square optical lattice. The lattice arm along the

z-direction has a frequency ωpand linear polarization in

the (x, y)-plane, and fulﬁlls a dual role as a transverse

pump ﬁeld (TP). It is far red-detuned both from the

atomic and cavity resonance, ∆c=ωp−ωc<0, thus

acting dispersively on the atoms. The light scattered

from the TP into the cavity couples the atoms’ motional

and spin degrees of freedom to the cavity mode. The

single-particle Hamiltonian in the rotating frame of the

TP reads [47]

Hsp =ˆ

2m+ˆ

Vlat −∆c−U0cos22π

λˆxˆa†ˆa

+ cos 2π

λˆxcos 2π

λˆzηsˆ

X+ηvˆ

Pˆ

Fz,

(1)

with total momentum ˆ

p= (ˆpx+ ˆpz) and

~= 1 [23,47]. The ﬁrst two terms account

for the atom moving in the 2D lattice potential

Vlat =−Vcos22π

λˆx+ cos22π

λˆz.The operator ˆa†

denotes the creation operator associated to the intra-

cavity ﬁeld. The presence of the atom dispersively shifts

the cavity resonance frequency and leads to an eﬀective

detuning ˜

∆c= ∆c−U0cos2(2π

λˆx)<0, where U0<0 is

the maximal dispersive shift. The last term describes a

self-consistent interference potential that arises due to

light scattering between the TP and the cavity mode [47].

The scalar component of the atom-light interactions

couples the atomic motional degrees of freedom to the

real quadrature of the cavity ﬁeld, ˆ

X=ˆa+ ˆa†/√2,

giving rise to a λ-periodic spin-independent density

modulation. The vectorial coupling is mediated by the

imaginary quadrature, ˆ

P=iˆa†−ˆa/√2,and gives

rise to phase-shifted λ-periodic modulations for atoms in

the two spin states, since the z-component of the atomic

spin operator ˆ

Fyields ˆ

Fz|↑i = + |↑i and ˆ

Fz|↓i =−|↓i.

The associated coupling strengths are ηs=ηcos(φ) and

ηv=ηξ sin(φ),with ξ=αv

2αsgiven by the atom-cavity

coupling rate ηand the ratio of the scalar and vectorial

polarizabilities [48,49].

We adiabatically eliminate the intra-cavity ﬁeld in a

tight-binding approximation [14,21,47,50] and obtain

a many-body extended BH Hamiltonian,

H=ˆ

HBH +ˆ

HLong,(2)

with

HBH =−tX

m,<i,j>

(ˆ

b†

i,mˆ

bj,m + h.c.)

i,m

ˆni,m (ˆni,m −1) + U12 X

ˆni,↑ˆni,↓,

(3)

and

HLong =−Us

Kˆ

Θ2

D−Uv

Kˆ

Θ2

S.(4)

The ﬁrst term, Eq. (3), constitutes a two-component BH

model [1,2,41,42,51], comprising tunneling to the z

nearest neighbors, with rate t > 0 and repulsive inter-

and intra-spin contact interactions, U > 0 and U12 >0.

We assume identical intra-spin collisional interactions for

atoms occupying |↑i and |↓i, as is the case for 87Rb atoms

in the F= 1 hyperﬁne groundstate manifold [52]. The

operator ˆ

b†

i,m (ˆ

bi,m) denotes the bosonic creation (anni-

hilation) operator, while ˆni,m =ˆ

b†

i,mˆ

bi,m counts the total

number of atoms with spin m∈ {↑,↓} at site i= (ix, iz).

For suﬃciently large magnetic ﬁelds, both spin-changing

collisions and cavity-assisted Raman processes can be ne-

glected [52]. The second term, Eq. (4), consists of spin-

independent (‘scalar’) and spin-dependent (‘vectorial’)

global-range interactions that are mediated by the intra-

cavity ﬁeld. The scalar long-range interactions are as-

sociated with the operator ˆ

Θ2

D=Pi(−1)|i|ˆni2,where

|i|=ix+izand ˆni= ˆni,↑+ ˆni,↓. Its expectation value is

maximized for a spin-independent spatial density modu-

lation with all atoms occupying only even or odd sites.

The expectation value of the vectorial long-range oper-

ator, ˆ

Θ2

S=Pi(−1)|i|ˆ

Sz,i2with ˆ

Sz,i= ˆni,↑−ˆni,↓,is

maximized for a global antiferromagnetic ordering of the

atoms on the lattice, with all atoms in |↑i occupying

even sites and all atoms in |↓i occupying odd sites, or

vice-versa. The interaction strengths Us=ULcos2φand

Uv=ULξ2sin2φcan be tuned with respect to each other

via the angle φ. The overall interaction strength UL>0

depends on the lattice depth Vand the eﬀective detuning

∆c[47]. The total number of sites is denoted by K. We

emphasize that the energy scales of the tunneling, contact

and long-range interactions are all independently tunable

with respect to each other. The Hamiltonian, Eq. (2), is

invariant under a global spin-ﬂip ˆ

bi,↑→ˆ

bi,↓,and two

global rotations, ˆ

bi,m →eiφmfor each m∈ ↑,↓.Further-

more, the scalar and vectorial long-range interaction in-

troduce an additional Z2-symmetry associated to the two

sublattices deﬁned by even and odd sites. Henceforth,

the Hamiltonian has a U(1) ×U(1) ×Z2×Z2-symmetry.

(a)

(b)

AF-SS

CDW

AFM

Figure 1. (a) Schematic representation of a two-component

BEC (|↑i ,|↓i) conﬁned in a 2D optical lattice inside an opti-

cal cavity. The spin-mixture is illuminated by a transverse

pump ﬁeld (TP) with tunable polarization angle φin the

(x, y)-plane. (b) Mean-ﬁeld order parameters and associated

phases of the eﬀective Hamiltonian. The order parameters θD,

θSand ψcharacterize density modulation, spin-order and su-

perﬂuidity, respectively. For U12/U = 1, the possible ground

state conﬁgurations are a superﬂuid (SF), a charge density

wave (CDW), a lattice supersolid (SS), an antiferromagnetic

Mott-insulator (AFM) or an antiferromagnetic lattice super-

solid (AF-SS). Spin states are represented by red arrows, while

the black markers indicate spin-insensitive density conﬁgura-

tions.

III. GROUND STATE PHASE DIAGRAM

A. Method and Order Parameters

We explore the zero-temperature phase diagram at

unity ﬁlling, by using a Gutzwiller mean-ﬁeld approach

[53–55]. We assume a translationally invariant ground

state on each of the even (e) and odd (o) sublattices,

|ΨGi=

K/2

e=0

K/2

o=0 |φei|φoi.(5)

For each sublattice i∈ {e, o}, the wave function is given

|φii=

nmax

n=0

mmax

m=0

ai(n, m)|n, mii(6)

where |n, mii=(ˆ

b†

i,↑)n

√n!

(ˆ

b†

i,↓)m

√m!|0iis the local Fock state

with n≤nmax atoms in spin state |↑i and m≤mmax

atoms in state |↓i. The real ground-state coeﬃcients ae≡

(ae(n, m))n,m and ao≡(ao(n, m))n,m are optimized to

minimize the eﬀective mean-ﬁeld energy density

E(ae,ao) = hΨG|ˆ

H|ΨGi

K/2.(7)

The superﬂuid order parameter ψ:=1

4Pi,m ψi,m

with ψi,m =|hˆ

b†

i,mi| (m∈ {↑,↓}) signals the transition

from an insulating phase (ψ= 0) to a phase-coherent

superﬂuid phase exhibiting oﬀ-diagonal long-range or-

der (ψ > 0). The density θD=|hˆne−ˆnoi|,and spin

θS=|hˆ

Sz,e −ˆ

Sz,oi| order parameters indicate the degree

of global spatial density- and spin-ordering due to long-

range interactions [Fig. 1(b)].

B. Phases for a Uniform Mixture

We discuss the case of a balanced spin mixture at unity

ﬁlling,

ρm=Nm

(K/2) =hφe|ˆne,m|φei+hφo|ˆno,m|φoi= 1 (8)

for m=↑,↓. The choice to work at ﬁxed density is mo-

tivated by experiments with ultracold atoms, although

a qualitatively similar phase diagram arises in a grand

canonical ensemble [37,47]. In this section, we assume

U12 =U. The diﬀerent order parameters are shown in

Fig. 2. The competition of scalar and vectorial long-

range interactions gives rise to two qualitatively diﬀerent

scenarios.

For Us> Uv[Fig. 2(a,b)], we observe two distinct in-

sulating phases (ψ= 0) at low tunneling rates zt/U: For

large UL, a spin-degenerate charge density wave (CDW),

with θD>0 and θS= 0.For small UL, an antiferromag-

netic Mott insulator (AFM), with θS>0 and θD= 0.

Remarkably, the system favors an AFM for arbitrarily

small vectorial contributions Uv: the contact interaction

hinders the formation of a CDW and overcomes the ki-

netic energy cost to form a unity ﬁlling Mott insulator

(MI). There, the AFM conﬁguration is favored among all

possible MIs by the vectorial long-range interaction. The

discontinuity in the order parameters θDand θSat con-

stant tunneling as a function of UL/U signals a ﬁrst order

AFM– CDW phase transition. For t= 0, the boundary

between the phases is given by Us/U −Uv/U =1

2[47].

As tunneling increases, the system becomes superﬂuid

ψ > 0 and can either exhibit spin (θS>0) or density

ordering (θD>0). We denote these phases as antifer-

romagnetic lattice supersolid (AF-SS) and lattice super-

solid (SS), respectively. Meanwhile, a superﬂuid phase

(SF) with ψ > 0 and θD,S = 0 emerges at even larger

tunneling strengths.

In the regime Uv> Us[Fig. 2(c,d)], the system exhibits

solely spin ordered phases (θS>0,and θD= 0), as the

vectorial long-range and the contact interactions dom-

inate over the scalar long-range interaction. For small

UL, we identify a ﬁrst-order AFM– SF phase transition,

signaled by a discontinuous jump of ψand θS[Fig. 2(e)].

This is in contrast to the second-order MI– SF transition

in the absence of the long-range interactions (UL= 0).

For larger Uv/U the AFM phase extends towards higher

tunneling strengths. AFM– SF transitions in the context

of entanglement properties have recently been studied

in three-component BH models with long-range interac-

tions [38]. For larger UL, we observe second-order phase

transitions from AFM to AF-SS and from CDW to SS

phases, along lines of constant UL/U [Fig. 2(f,g)]. The

second-order phase transitions from AFM to AF-SS and

CDW to SS are supported by perturbative estimations,

cf. black lines in Fig. 2(a,c) [47,56].

0.2

0.6

1.0

1.4

1.8

UL/U

(a) (b)

0.0 0.2 0.4 0.6 0.8 1.0

zt/U

0.1

0.4

0.7

1.0

1.3

UL/U

(c)

0.0 0.2 0.4 0.6 0.8 1.0

zt/U

(d)

0.0 0.2 0.4 0.6

zt/U

0.0

0.5

1.0

θD/2, θS/2, ψ

AFM SF

(e)

0.0 0.2 0.4 0.6 0.8

zt/U

CDW SS SF

(f)

0.2 0.4 0.6 0.8

zt/U

AFM

AF-SS

(g)

0 0.4 0.8

012

θS

0 0.4 0.8

012

θS

012

θD

Figure 2. Mean ﬁeld phase diagrams for a balanced-spin mix-

ture at unity ﬁlling, ρ= 2.The calculations are performed

for Us/Uv≈4.64 (a,b) and at Us/Uv≈0.33 (c,d). (a,c) De-

pendence of superﬂuid order parameter ψ. The solid lines are

perturbative estimations for the transition. (b,d) Dependence

of density and antiferromagnetic order parameters θDand

θS.(e,f,g) Cuts along the phase diagrams for Us/Uv≈4.64

at constant UL/U = 0.63 and UL/U = 1.41 (e,f) and for

Us/Uv≈0.33 at UL/U = 1.3 (g), indicated by thin hori-

zontal lines in (a,c). The basis truncation nmax =mmax = 3

leads to a saturation of the superﬂuid order parameter at large

tunneling.

C. Phase Diagrams for diﬀerent U12/U

We now discuss the phase diagrams for diﬀerent ratios

of inter- and intra-spin interactions U12/U and scalar and

vectorial long-range interactions Us/Uv. Besides the ho-

mogeneous phases, we also calculate the regions of phase

separation between the two spin states (PS), which nat-

urally occur in two-component BH models with repulsive

inter-spin interactions [37,41–45,57,58]. Additional re-

gions of phase instability arise from the concurring long-

range interactions [17,46]. To calculate the energy of a

phase separated state, the system is divided into halves

(A, B): one with higher spin-up density (ρA

↑> ρA

↓) and

the other with higher spin-down density (ρB

↓> ρB

↑), while

imposing a density conservation constraint in each of the

halves, i.e., ρ≡ρA,B

↑+ρA,B

↓= 2 to ensure unity ﬁlling.

We further assume either hˆ

ΘDi= 0 or hˆ

ΘSi= 0. Phase

instability, on the other hand, is signaled by a negative

compressibility, ∂ρµ < 0 [46], where µ(ρ) = ∂ρE(ρ) is

the chemical potential as a function of the density ρ. We

calculate the derivative numerically by using the energy

densities E(ρ) extracted from the variational ansatz in

Eq. (S22) [47].

We ﬁrst discuss the results for Us/Uv>1, see Fig. 3(a-

c). For dominating intra-spin interactions [Fig. 3(a)],

U12 < U, the phases are identical as those obtained for

U12 =Uassuming a uniform mixture as discussed in

the context of Fig. 2. We additionally ﬁnd a region of

instability in the SS phase for U12 ≤U. Our observa-

tions of phase instability are qualitatively diﬀerent from

the results for spinless systems [17,46], which predict sta-

ble supersolid phases at integer ﬁlling in two-dimensional

systems. For U12 =U[Fig. 3(b)], the mixed CDW state

|φe, φoi=|↑↓,0i, the entangled state |φe, φoi=|↑↑,0i+

|↓↓,0iand the fully phase-separated (PS) conﬁguration

with ρA

↑=ρB

↓= 2 are degenerate. Increasing U12/U

from 0.9 to 1 shrinks the CDW region: CDW→AFM

transition boundary is shifted towards higher UL/U and

CDW→AF-SS transition boundary is shifted towards

lower zt/U. For U12 > U [Fig. 3(c)], the obtained CDW,

SS and SF are fully phase separated.

In the case of dominating vectorial long-rage interac-

tions Us/Uv<1 [Fig. 3(d-f)], the phase diagrams for

U12 < U and U12 =Uare qualitatively similar: Besides

AFM, AF-SS and SF phases, we ﬁnd extended regions

of instability in the AF-SS phase, which is similar to the

SS case. For U12 > U [Fig. 3(f)], the SF is replaced by

a fully PS SF, which is compatible with the results for

Us/Uv>1. We also ﬁnd that both the AF-SS and the

instability region shrink when U12/U is increased above

1. In contrast, boundaries between insulating regions

(PS CDW, AFM) and the phase separated non-insulating

state (PS SS, PS SF) do not change with U12/U, as the

energy of the PS phases and the AFM phase do not de-

pend on U12.

We note here two limitations of our simulations.

First, the identiﬁcation of diﬀerent phases in the phase

diagrams relies on numerical minimization [47] in a

high-dimensional landscape. This can lead to spuri-

ous solutions, such as the scattered instability points

in Fig. 3(a,b) and the irregular phase boundaries in

Fig. 2and 3. Second, there is a small region of fully PS

SS for U12 =Uand Us/Uv>1, cf. Fig. 3(b). This is due

to the relatively small Hilbert space (nmax =mmax = 3)

used for the simulations. We expect that a larger Hilbert

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Phases,instabilitiesandexcitationsinatwo-componentlatticemodelwithphoton-mediatedinteractionsLeonCarl,1RodrigoRosa-Medina,1,ySebastianD.Huber,2TilmanEsslinger,1NishantDogra,3,andTenaDubcek2,1InstituteforQuantumElectronics,ETHZurich,8093Zurich,Switzerland2InstituteforTheoreticalPhysics,ETHZurich...

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Phases instabilities and excitations in a two-component lattice model with photon-mediated interactions Leon Carl1Rodrigo Rosa-Medina1ySebastian D. Huber2

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