Superfluid–Mott-insulator quantum phase transition in a cavity optomagnonic system 2022 Qian Cao.pdf

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QianCao,1LeiTan,1,2,∗andWu-MingLiu3
1LanzhouCenterforTheoreticalPhysics,KeyLaboratoryofTheoreticalPhysicsofGansuProvince,
LanZhouUniversity,LanZhou,Gansu,730000,China
2KeyLaboratoryforMagnetismandMagneticMaterialsofthe
MinistryofEducation,LanzhouUniversity,Lanzhou730000,China
3BeijingNationalLaboratoryforCondensedMatterPhysics,
InstituteofPhysics,ChineseAcademyofSciences,Beijing100190,China
Theemerginghybridcavityoptomagnonicsystemisaverypromisingquantuminformationpro-
cessingplatformforitsstrongorultrastrongphoton-magnoninteractiononthescaleofmicrometers
,thesuperfluid-Mottinsulatorquantumphasetransitioninatwo-

analyticalsolutionofthecriticalhoppingrateisobtainedbythemeanfieldapproach,secondper-

theincreasingcouplingstrengthandthepositivedetuningsofthephotonandthemagnonfavorthe
,the
,
-
sultsobtainedhereprovideanexperimentallyfeasibleschemeforcharacterizingthequantumphase
transitionsinacavityoptomagnonicarraysystem,whichwilloffervaluableinsightforquantum
simulations.
].
Hereafter,withthefabricationofoptomechanicalcav-
Quantumsimulationprovidesausefultoolforsolv-itysystemsatthedesiredfrequencyaccuracyandthe
ingmanyproblemssuchasquantumphasetran-couplingstrength,thecavityoptomechanicalarraysys-
sition,quantummagnetism,andhigh-temperaturetemprovidesanotherexperimentallyfeasibleavenueto
superconductivity[1].Quantumphasetransitionofansimulatethesuperfluid-Mottinsulatorquantumphase
interactionsystemcomposedofmultipleparticlesaretransition[31–33].Comparedwithcoupledcavityarrays
widelyinvestigated,suchasheavyfermionsinKondosystem,thephonon-mediatedcavityfieldandthetwo-
lattices[2],ultracoldatomsinopticallattices[3–5],andlevelsystemformpolaritonsbycoupling,whichprovide
theensembleoftwo-levelsystemsinteractingwithaaneffectiveon-siterepulsion,thenthesystemcanalso
bosonicfield(.,Dickemodel)[6].Especially,thesimulatethesuperfluid-Mottinsulatorquantumphase
superfluid-,theenhancedphonon-photon
bosons,whichformsoneoftheparadigmexamplesofcouplingfavorsthecoherenceofthesystem[34].
aquantumphasetransition,wasfirststudiedintheInanalogytothecavityoptomechanicalsystem,cav-
BoseHubbard(BH)modelduetothecompetitionofityoptomagnonicsystem,anewclassofhybridquan-
theon-siteinteractionandthehoppingtermtheoreti-tumsystemsbasedoncollectivespinexcitationsinfer-
callyandexperimentally[7–10].Giventheprecisecon-romagneticmaterials[35],hasreceivedincreasingatten-
trolofcouplingstrengths,thepropertiesofscalabilitytioninrecentyears,whichprovidesanewandpromising
andindividualaccessibilityofcoupledcavities[11–14],platformforstudyingmacroscopicquantumeffects[35–
theJaynes−CummingsHubbard(JCH)model,whichde-43].Thecollectivespinexcitationinferromagneticcrys-
scribesthedynamicsofthecoupled-cavityarrayswithtalsiscalledamagnon,whichcaninteractcoherently
arXiv:[quant-ph]20Jan2022eachembeddedwithinatwo-levelatomhasattractedwithmicrowavesandopticalphotonsaswellasphonons
tremendousattentions[15–17],magneto-optical,andmagnetostric-
theJCHmodelandextendedJCHmodel,thesuperfluid-tiveinteractions,respectively[44–47].Experimentally,
Mottinsulatorquantumphasetransitionoflightreminis-yttriumirongarnet(YIG)spheresarecharacterized
centoftheonesofatomsintheBHmodelareextensivelywithhighcollectivespinexcitationdensity,lowdissipa-
simulated[18–26].Moreimportantly,thequantumphasetion,greatfrequencytunabilityandthelongercoherence
transitionoflightdependscruciallyontheintrinsicatom-time,whicharewidelyusedinthestudyofmagnon-
photoninteractionintheJCHmodel,wheretheatom-photoncouplingduetoitsstrongandevenultrastrong
photoncouplingleadstotheformationofrepelledcol-couplings[48,49,35].Inaddition,thestrongcoupling
lectivepolaritonicexcitations,andthison-siterepulsivebetweencavityphotonandmagnonhasbeenobserved
potentialcompetewiththehoppingofphotonsbetweenatbothlowandhightemperatureexperimentally[50].
,thequantumBasedonthesefeaturesofcavityoptomagnonicsystem,
phasetransitionoflighthasgreatpotentialapplicationmanyintriguingphenomenahavebeenexplored,suchas
foranewsourceofquantum-correlatedphotons[1,23,27–magnondarkmodesandgradientmemory[51],coher-
2
ˆPˆ
entanddissipativemagnon-photoninteraction[52–59],andloweringoperators,=iNi=
thehigh-ordersidebandgeneration[60],theself-sustainedP†††
i(ˆaiaˆi+mˆimˆi+σˆiσˆi)isthetotalpolaritonnumber
pscillationsandchaos[61–63],non-Hermitianphysics[55,operator[77,78].Niisthetotalnumberofphotonic,
64,65],entanglement[66–70],magnon-inducednearlymagnonicandatomicexcitationsoftheithsiteinthe
perfectabsorption[71],magnonFockstate[72],-
squeezing[73],severalnovelpro-lationshipbetweenthetotalpolaritonnumberoperator
gressesassociatedwiththephotonblockadeincavityop-NˆiandtheHamiltonianHˆT,onecanfindthatNˆiisa
tomagnonicsystemarealsoinvestigated[74,75,76].
notethattheinterplayofthephotonblockadeandpho-inthegrand-canonicalensembleandthechemicalpoten-
-tialisµi,whichistheLagrangemultiplierinthegrand-
vantagesofmagnons,itisveryinterestingtofurtherex-canonicalensembleensuringtheconservationoftheto-
plorethesuperfluid-Mottinsulatorquantumphasetran-talexcitationnumberinthephasetransitionbetweenthe
sitionoflightinthehybridmacroscopicquantuminter-(ga)represents
faceofatoms,(atom),
Inthiswork,,ωcandωmarethefrequenciesofcavity
superfluid-Mottinsulatorquantumphasetransitioninaphoton,atomandmagnonrespectively,whereωm=γH,

JCHmodel,threenewdegreesoffreedomwereaddedfield,whichcanbegivenbytheHolstein-Primakoff(H-
bytheinclusionofYIGspheres,andtheeffectsoftheseP)transformation[35,79,80].Weintroducethedetun-
threenewdegreesonthequantumphasetransitionwereingbetweentheatomandthecavity,∆a=ωa−ωc,and
,theanalyticalsolutionsforlowex-thedetuningbetweenthemagnonmodeandthecavity
citationnumberareobtainedbasedonthemeanfieldap-modeis∆m=ωm−,thesecondtermof
proximation,thesecondorderperturbationtheoryandEq.(1)denotesthephotonhoppingbetweenthenearest-
,thephasedia-
gramsarediscussednumericallyusingthemean-fieldthe-thehoppingrateofphotonsκij=κbetweenadjacent
,µi=µisthesame
toshowthemechanismofthesuperfluid-Mottinsulatorforalloptomagnoniccavitiesforsimplicity.
transition.
Thispaperisorganizedasfollows:,we
describeacavityoptomagnonicarraysystemusedfor
studyingthesuperfluid-Mottinsulatorquantumphase

analyticalsolutionsofthissystem,andthenumericalso-

.

-dimensionalhybridcou-
a2Darrayofidenticalcoupledoptomagnoniccavities,
witheachcavitycontainingatwo-levelatom(TLA)in-cavitycontainsatwo-levelatomandaYIGsphereisplaced
teractingwiththephotonmode().Thetotalnearthemaximummagneticfieldofthecavitymodeandin
Hamiltonian(~=1)ofthesystemcanbewrittenasauniformbiasmagneticfield,whichestablishesthemagnon-
photoncoupling.
ˆXˆcpmX†Xˆ
HT=Hi−κijaˆiaˆj−µiNi(1)
ii,jiWeutilizethemeanfieldapproximationmethodto
studythesuperfluid-Mottinsulatorquantumphasetran-
cpm†††,wein-
Hˆ=ωcaˆaˆi+ωaσˆσˆi+ωmmˆmˆi
iiiitroducethesuperfluidorderparameterψ≡haiitostudy
††††
+gaσˆiaˆi+σˆiaˆi+Gmmˆiaˆi+mˆiaˆi(2),ψisacomplex
number,butitsphasefactorcanbegaugedawaywith-
wherei,jaretheindexesfortheindividualopto-,ψcanbetakento
=0,thesystem
sites,,thesystemis
Here,=0
††
(ˆai)andmˆi(mˆi)andψ6=0phasesdefinesaquantumphasetransitionin
arethephotonicandmagnoniccreation(annihilation).(3)oftheHamiltonianis
†
operators,(ˆσi)aretheatomicraisingobtainedbyusingthedecouplingapproximation,where
3
z=(2N+1)×(2(N+1)+1).Thematrixexpression

Xhcpm†2iaretheabbreviationoftheMeanfieldapproximationand
HT=Hi−zκψaˆi+aˆi−µNi+zκψ(3)
ithehoppingtermrespectively.
Therefore,wechoosetheeigenstatesasthebarestates
MF2
ofthecavityoptomagnonicsystem,whichiscomposedofH=zκψI+
MFhop
thedirectproductofthecavityphotonstates,magnonH(0)H(0)000
hopTMFhop
statesandatomstates,.,|photon(n),magnon(m),HHH00
(0)(1)(1)
atom(e,g)(2N+1)×0HhopTHMFHhop0
(1)(2)(2)
hop
(2N+1).So,wechooseacompletesetofbasisvectors00HTHMF0
(2)(3)
|N−m−1,m,ei,|0,N−1,ei,|N−m,m,gi,|1,N−1,gi,
....
|0,N,gitogivethematrixformoftheHamiltonianEq...

..
(4)afterthemeanfieldapproximationwithnrunning..
..
0,1,2,toN,whilemtakes0,1,2,...,N−,0000HMF
thematrixdimensionofHMFcanbedefined(2N+1)×(N)
(N)(4)
hop
(2N+1),andlikewise,thematrixdimensionofH(N)is
√

Nga√000
0N−1ga00

H2.
..
0000

√00ga0
HMF=(5)
(N)Nga√00

0N−1ga0

.
..H1

00ga
000(2N+1)×(2N+1)
ThespecificformsofH1andH2areintheAppendixA.
√

−N−1zκψ√000000
0−N−2zκψ00000

.
..
√
Hhop=(6)
(N)000−Nzκψ√000

0000−N−1zκψ00

.
..
00000−zκψ0(2N+1)×(2N+3)
Forexample,whenthetotalexcitationnumberisN=
0(N=1),thebasisvectorsareselectedas|0,0,gi(0q
12222
MFE1,−=∆−2µ+ωc−∆−2∆ωc+4ga+4Gm+ωc
|1,0,gi,|0,1,gi,|0,0,ei).AndthematrixofH(0)=(0)2
MF(10)
forN==1,thematrixdimensionofH(1)
shouldbesubstitutedby0q
12222
E1,−=∆−2µ+ωc+∆−2∆ωc+4ga+4Gm+ωc
ωa−µga02
MF(11)
H(1)=gaωc−µGm(7)
0Gmωm−µThesplittingbetweenstateswiththesameexcitation
Introducingthedetuningbetweenthephotonfre-numberofapolaritonisgivenby
quencyandthetwo-leveltransitionfrequency∆a
00q
(magnonfrequency∆m),theresultingHamiltonianis2222
δ=E−E=∆−2∆ωc+4g+4G+ω(12)
E1,−1,+amc
∆a−µga0
MF0Notethat,thesplittingδEdoesnotonlydependonthe
H(1)=gaωc−µGm(8)
0Gm∆m−µdetuningsofthephoton-atomandthephoton-magnon

theeigenvaluesaregiven(∆a=∆m=∆)
0say,inacavityoptomagnonicsystem,thestrongphoton-
E1,0=∆−µ(9)magnonandphoton-atomcouplingareallinvolvedin
4
(2)
thepolaritonmapping[77].Meantime,threenewdegreesE1,−isgivenbyEq.(B8).
offreedomareaddedinthenewmodelcomparedwithHereafter,accordingtoLandau’ssecond-orderphase
JCHmodel,whicharetheexcitationnumbermofthetransitiontheory,thephaseboundaryoftheMottinsu-
magnon,thecouplingstrengthGmandthedetuning∆mlatorphaseandthesuperfluidphasecanbedetermined
[82,83].
discussion,wecandeterminethattheexcitationnumberThen,thecriticalho