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Lattices of quantized vortices in polariton superfluids

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Lattices of quantized vortices in polariton superfluids. Comptes Rendus. Physique, Académie des

sciences (Paris), 2016, 17 (8), pp.893 - 907. �10.1016/j.crhy.2016.05.005�. �hal-01390913�

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Contents lists available atScienceDirect

Comptes Rendus Physique

www.sciencedirect.com

Polariton physics/Physique des polaritons

Lattices of quantized vortices in polariton superfluids

Réseaux de tourbillons quantifiés dans les superfluides de polaritons

Thomas Boulier, Emiliano Cancellieri, Nicolas D. Sangouard,Romain Hivet, Quentin Glorieux, Élisabeth Giacobino, Alberto Bramati

LaboratoireKastlerBrossel,UPMCSorbonneUniversités,CNRS,ENS-PSL,ResearchUniversity,CollègedeFrance,4,placeJussieu,case74, 75005Paris,France

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Availableonline13June2016

Keywords:

Polaritons

Quantumfluidsoflight Nonlinearoptics

Mots-clés : Polaritons

Fluidesquantiquesdelumière Optiquenonlinéaire

In thisreview,wewillfocus onthe descriptionofthe recent studies conductedinthe questfortheobservationoflatticesofquantizedvorticesinresonantlyinjectedpolariton superfluids. In particular, we will show how the implementation of optical traps for polaritons allowsfor therealizationof vortex–antivortexlatticesin confinedgeometries and how the development of a flexible method to inject a controlled orbital angular momentum (OAM) insuch systemsresults in the observation ofpatterns of same-sign vortices.

©2016Académiedessciences.PublishedbyElsevierMassonSAS.Thisisanopenaccess articleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

r é s um é

Danscetarticledesynthèse,nousnousconcentreronssurladescriptiond’étudesrécentes menéesdanslebutd’observerdesréseauxdetourbillonsquantifiésdansdessuperfluides de polaritons injectés de manière résonante. Enparticulier, nous montrerons comment l’implémentationdepiègesoptiquespourlespolaritonspermetlaréalisationderéseaux tourbillon–anti-tourbillon dans des géométries confinées etcomment le développement d’uneméthodeflexiblepourinjecterunmomentangulaireorbitalcontrôlé(OAM)dansde telssystèmespermetl’observationdemotifsdetourbillonsdemêmessignes.

©2016Académiedessciences.PublishedbyElsevierMassonSAS.Thisisanopenaccess articleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

In semiconductor microcavities [1,2], polaritons, the half-light, half-matter particles arising from strong coupling be- tween excitons and photons behave like weakly interacting composite bosons. Due to their excitonic part, they exhibit non-linearinteractions,whiletheir photonicpartallowscreatinganddetectingthemoptically. Inthissense,thepolariton systemispartofawiderfamilyofsystemswhereaneffectivephoton–photoninteractioncanbeengineered,resultingina hydrodynamical-likebehavior.Suchsystemsarelabeledasquantumfluidsoflight[3].

* Correspondingauthor.

E-mailaddress:alberto.bramati@lkb.upmc.fr(A. Bramati).

http://dx.doi.org/10.1016/j.crhy.2016.05.005

1631-0705/©2016Académiedessciences.PublishedbyElsevierMassonSAS.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

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workofKeelingandBerloff,theoreticallystudying[12]theformationofarotatingpolaritonBEC,pumpedoutofresonance trapped in a harmonicpotential. From the experimental point of view,the nucleation ofvortices has beenaddressed in non-resonant configurations,aswell asinresonant configurationsandinopticalparametric oscillator(OPO)schemes.For what concerns out-of-resonant schemes,vortexformation indisordered landscapeshas beenachieved[13] andhydrody- namicalnucleation ofvortexpairs havebeenobserved bycollidinga fluxofpolaritons onastructuraldefectof aplanar cavity [14].Regarding resonantly pumped systems, vortex formation hasbeen observed, similarly to theout-of-resonant case, by colliding amoving fluid ofpolaritons witha structuraldefectof aplanar cavity [9,15].In thesecases,however, a metallic mask has beenused to block thelaser pump inthe region behind the defect,inorder to allowthe phase of thescatteredpolaritonstoevolvefreely.Finally,vorticeshavealsobeenobservedintheopticalparametricoscillator(OPO) scheme[16,17].

Wewillshowinthefollowingthatthenewpossibilitiesofferedbyconfiningpolaritonsinopticallygeneratedpotential trapsallowcreatinginteraction-shapedvortex–antivortexlattices[18,19];moreover,wewillillustrateanewflexiblemethod basedontheuseofaspatiallightmodulator(SLM)forthedirectphaseimprintinapolaritonfluid;thisallowsthecreation ofa rotatingpolariton fluidinan annulargeometry,leading totheformationofachainofsame-sign vortices[20].These results constituteasignificant stepforwardinourunderstandingofthe quantumfluidsoflightandopen thewayto the studyofAbrikosov-likephysicsinthesesystems.

1.1. Sample

The planar microcavity used in the present work was built by using the molecular beam epitaxy technique at the

“École polytechniquefédérale deLausanne”(EPFL),Switzerland,by RomualdHoudré[21,22].Thesample ismadeofthree In0,04Ga0,96Asquantumwellsplacedattheantinodesofthecavityelectromagneticfield.

The cavityisGaAs-basedwithatypicallength2λ/nc forλ=835 nm (theexcitonicresonance),wherenc=3.54 isthe cavityGaAssubstrateopticalindex.TheBraggmirrorsformingthecavityaremadeof21(frontmirror)and24(backmirror) GaAs–AlAs pairs. Themeasured polariton linewidthis lessthan85μeV (limitedby thespectrometer resolution), andthe polariton lifetimeisestimatedatabout15psfromtimeresolvedmeasurements.Theback mirrorliesonaGaAssubstrate polishedtoallowworkingintransmissionmode.Duringepitaxialgrowth,avery smallangleoftheorderof104 degrees was implementedbetweentheBraggmirrorstocontinuouslychangethecavitythickness.Thisangleallowsforaveryfine controlofthephoton–excitondetuning.Thechangeinthecavitythicknessmodifieslinearlytheenergyofthecavityphoton with the position on the sample while the exciton energy remains constant. Exciton–photon detuning from +8 meV to

4 meVcanbeachieved, withanenergygradientofabout710μeV mm1.TheRabisplittingforoursample ismeasured tobe5.1meV.

1.2. Injectionmethods

There are two main methods to inject polaritons in a microcavity. One is the so-called“out-of-resonance” pumping and consists in injecting in the cavity high-energy photons with a laser. Depending on the efficiency of the relaxation phenomena, polaritonscanthermalizeandformaquasi-Bose–Einsteincondensate [8].Withthisinjectionmethod,alarge excitonpopulationiscreatedinadditiontothepolaritonpopulation.Thesecondtechniquetocreateafluidofpolaritonsis

“resonant injection”(or“quasi-resonant”),whichwewilluseinthefollowing.Itconsistsininjectinglaserphotonsdirectly atanenergyandin-planewavevectorresonantwiththelowerpolaritonbranch.Inthisway,theenergyandthewavevector oftheresultingpolaritonpopulationarefullycontrolled.Usingacontinuouslaser,wecancreatelargepolaritonpopulations whose properties(momentum,energy,density,phase,spin)are inheritedfromthe pump.Thiscontrolled injectionallows thestudyofpropagation-relatedphenomena[4,23,24],andquantumhydrodynamic experimentsareachievableunderthis pumpingscheme.Notethatthisisunliketheout-of-resonancesetup,wherethecoherenceandthein-planewavevectorof the pump arelost duringthe relaxationprocess.In thequasi-resonantscheme,one caninduce asmallenergy difference betweenthe laser andthepolariton energy.Withthis quasi-resonantpumping,the nonlinearitiesplay an importantrole andinduceabistablebehavioraccompaniedbymanyinterestingphenomena[3,4].

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1.3. Theoreticalmodel

AstandardwaytomodelthedynamicsofresonantlydrivenpolaritonsinaplanarmicrocavityistouseaGross–Pitaevskii equationforcoupledcavityandexcitonfields(C andX)generalizedtoincludetheeffectsoftheresonantpumpingand decay(¯h=1),

t

X

C

= 0

F

+

H0+

gX|X|2 0

0 VC

X

C

(1) withthelinearpolaritonHamiltonianH0 givenby:

H0=

ωXiγX R/2 R/2 ωC(k//)iγC

(2) wherethecavitydispersionisgivenby

ωC(k//)=ωC(0)+hk¯ 2//

2mC

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withthe cavityphotoneffective massmC=4×105m0,wherem0 isthebare electron mass.Thephoton effectivemass comesfromtheparabolicshapeofthecavitydispersioncurve.

¯

hωC= ¯hc

k2z+k2//≈ ¯hckz+h¯2k2//

2mC

withmC= ¯hkz c

wherekz isthemomentum componentperpendiculartothecavityplane andk// isthecomponentparalleltothe cavity plane.

Forthesimulationspresentedinthefollowing,wehaveassumedaflatexcitondispersionrelation ωX(k//)=ωX(0).The parameters RX,and γC arethe Rabisplitting andtheexcitonic andphotonicdecayrates,respectively,andhavebeen fixed to valuescloseto experimental ones:R=5.1 meV,γX=0.04 meV,and γC=0.06 meV. Inthismodel, polaritons areinjectedintothecavitybyacoherentandmonochromaticlaserfield,withapumpintensity fp andanarbitraryspatial profile F(r). Here, in contrast to other cases [25], we simulate the polariton system witha simplified two-field model, discardingtheroleoftheexcitonicreservoirsinceoursetupisbasedontheuseofacontinuous-wavelaserwithalinewidth orders of magnitudes smaller than the Rabi splitting, and therefore the injection of a reservoir of excitons is strongly suppressed. The exciton–excitoninteraction strength gX is setto 1 by rescaling both the cavity andexcitonic fields and thepump intensities. Alltheanalyzed quantitiesare takenwhenthesystemhasreacheda steady-statecondition aftera transientperiodof200ps.

2. Generationofvortex–antivortexlatticesinconfinedgeometries

Vortex dynamics have beeextensively studied in severalexperiments, which showedthe hydrodynamical nucleation ofvortex–antivortex (V–AV) pairs[9,10,12,26,27],single vortex stability in pulsedpumped regime andV–AVtrapping by natural disorder [16,28,29]. Some studies focused on the steady-state solution for polariton vortices held in controlled traps[30,31].

2.1. Setupdescription

Incontinuity withpreviousstudies oftrapped polariton vortices[26,27],ourgoalis heretostudythe geometricself- organization ofvortices ina steady-stateregime. Indeed,a steady-stateregime isobservable witha time-averagedsetup suchastheoneweuseinthisstudy.ToobservetheformationofastableV–AVlattice,wecreateatrappingpotentialby spatially modulatingtheGaussian pump beamwithan opaque maskandmaking useofpolariton–polaritoninteractions:

by cuttingthe centeroftheGaussian pump laser withthemask, thepolariton density(andthus thepolariton–polariton potentialenergy)ishigharoundthemaskedregionandlowinside.Theshapeofthetrapisfixedbythespatialgeometry ofthemaskedzone,createdwithametallicmaskwhoseshapewecanengineeratwill.

Inthisstudy,we usedeitherasquareoratriangularmaskwithsidesof45μm.Thelaserisfocusedonthemaskwith a waist larger than 45 μm. The mask image is projected onto the microcavityat normal incidence,as shown inFig. 1.

Therefore,laserintensityislocallysettozerointhemaskimage,placedatthebeam’scenter.

The differencein interactionenergy(or equivalently,polariton density) betweenthenon-masked(with highpolariton density)andmaskedregions (with low polariton density)acceleratespolaritonstowards thenon-pumped center.Indeed, polaritons“fall” into thetrap, their interaction energybeingconverted inkinetic energy,fromthe four(square) orthree (triangle)sides.ThisisillustratedinFig. 2.Theabsenceofpumpinsidethetrapensuresthefreeevolutionofthepolariton phase; onthecontrary,inthepumpedregion,thepolariton phaseislocked tothepump phase.Polaritonscominginside thetrapfromthetrapsidesmeetandinteract,givingrisetoV–AVpairsarrangedinalattice.Thelatticegeometrydepends

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Fig. 1.Pumpshaping.Schemeofthepumpingsetupforthepolaritontrap.AGaussianbeamisfocusedonasmallmetallicmaskinordertocutagiven shapeintothebeam’scenter.Inthisexample,themaskisatriangle,butitcanhaveanyshape.Themaskimageisthenfocusedonthesampleatnormal incidence,generatingboththepolaritonpopulationandthetrap.

Fig. 2.Trappingpotential.Lowerpolaritonbranch.(Left)Atlowpolaritondensity,thepumpenergyisblue-shiftedwithrespecttothebarelowerpolariton branch (LP)at k=0 μm1 and crossestheLPatawavevectorcorrespondingtotheRayleighring(greenline).Therefore, polaritonsdiffractedbythe mask borderscanbeinjectedintothe cavityat awavevectorkR correspondingtotheRayleighring(greenring). (Right)Athighlaserintensities,the lower polaritonbranchoutside ofthe maskedregionisrenormalizedduetopolariton–polaritoninteractions, andtherefore polaritonsareinjected at k=0 μm1(greendot).Insidethemaskedregion(centralregion),thepolaritondensitybeinglower,thepolaritonbranchisnotrenormalized.Therefore, whenpolaritonsenterthemaskedregion,theyacquireamomentum(greenring)inordertoconservetheenergy.

onthemask’sshape.Thetrapdepthdeterminesthepolariton speedwhenenteringthenon-pumpedregionaswellasthe totalpolaritondensityinthetrap.Thesystembeingcompletelysymmetric,itisboundtokeepatotalangularmomentum equaltozeroand, therefore,thenumberofgeneratedvorticesisalwaysequaltothenumberofantivortices.Inthissense, vorticesandantivorticesarealwaysgeneratedinpairs,althoughtheydonotnecessarilyformboundstates.

Inthefollowingsections,wewillpresentanddiscussthebehaviorofthesystemrespectivelyatlowandhighpolariton densities.

Forlowpolaritondensitiesinthetrap,weobserveaninterferencelatticecontainingtheV–AVpairs.Thepolaritonphase beingfreetoevolveandtheinteractionsbeingnegligibleinthisregime,thisisindeedwhatisexpected.Thelatticeiswell describedbyinterferencebetweenfour(three)polaritonpopulationscomingfromthefour(three)square(triangle)sidesto thecenter.

Thisisnotthecaseforhighpolariton densities,whereweobserveamodificationofthelatticegeometryandadrastic decreaseofthenumberofV–AVpairs.Byextrapolatingthisresult,wededucethatforahighenoughdensityinsidethetrap, all V–AVpairs mustdisappear, andtheinterferencepatternvanish.However, a drawbackinthe useofthisexperimental technique isthat themaskscutthe maximumintensityregion oftheGaussianbeam, leadingto highlossesofthepump powerandthustolimiteddensitiesinthetrap.Thisiswhyasecondschemeusingfourseparatepumpswasimplemented, sinceitallowsrealizingasimilarsystemminimizingthelosses.

3. Low-densityregime 3.1. Triangulartrap

Fig. 3showstheintensityandphaseoftheemissioninrealspacewitha triangularmask of20-μmside.The exciton–

photon detuning is δexc–photon=1.7 meV and the detuning between the laser and the lower polariton branch (LPB) is δlaser–LPB=0.12 meV.Weobserveahexagonformedby3vortices(inredintheinsetc)and3anti-vortices(inblue)inthe

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Fig. 3.Intensity(a)andphase(b)oftheemittedlightinrealspaceinthepresenceofatriangulartrap(yellow)of20-μmsidepumpedatnormalincidence bya100-μm-diameterGaussianbeam.Theexciton–photondetuningδexc–photon=1.7 meV andthelaser–LPBdetuningisδlaser–LPB=0.12 meV.Theinset in(b)showsthedetailofavortexwithaphasewindingof2πaroundthecore.Figure(c)showsthepositionsofthevortices(red)andtheanti-vortices (blue)inthetrap.Ahexagonformedby3vorticesand3anti-vorticesisobserved.(d)–(f):Numericalsimulationsinthesameconditionsasthoseofthe experiment.

Fig. 4.Intensity(a),phase(b)andvortex-distribution(c)intherealspaceoftheemittedlightwithatriangulartrap(yellow)of35-μmside.Theexciton–

photondetuningisδexc–photon=1.7 meV andthedetuningbetweenthelaserandtheLPBisδlaser–LPB=0.16 meV.Ahexagonalvortexlatticeisobserved.

(d)–(f):Numericalsimulationsinthesameconditionsasthoseoftheexperiment.

centerofthetriangulartrap.Thevortices canbeidentified onthephaseimagesbyarotation ofthephase of2π around theircore;conversely,theanti-vorticesexhibitanoppositephaserotationof2π.

The experiments are in good agreement withthe numerical simulations. Byincreasing the size of the trap, a larger numberofhexagonalcellsisobservedwithinthetriangleasseeninFig. 4,whereatriangulartrapwith35μmsideisused.

Theotherparameters aresimilartothosedisplayed inFig. 3(thedetuning laser–LPB,equalto0.16meV,isslightlyhigher inthislattercase;thishastheeffectofreducing thesizeofthelatticecells, duetotheincreaseofthewavevectorofthe injectedpolaritons).

3.2. Squaretraps

Fig. 5showsthephaseandtheintensityoftheemissionwitha45-μmsidesquare mask.Theexciton–photondetuning isδexc–photon=0.32 meV andthepump isbluedetunedfromthelow-polaritonbranchδlaser–LPB=0.6 meV.Theobserved intensity pattern(Figs. 5a, 5d) is formed ofsquare cells. The phase map (Figs. 5b, 5e) showsthat the lattice nodes are often,butnot necessarily,accompaniedby vortex–antivortexpairs.Notethat,whiletheinterferencepatternoffourplane wavescomingfromfourorthogonal directionswouldnotshow anyvorticity,herethe vortex–antivortexpairsare formed

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Fig. 5.Intensity(a),phase(b)anddistributionofvortices(c)intherealspaceemissionforatrapsquare(yellow)with45-μmside,pumpedbya100-μm- diameterGaussianbeamatnormalincidence.Thedetuningexciton–photonisδexc–photon=0.32 meV andthelaser–LPBdetuningisδlaser–LPB=0.6 meV.

(d)–(f):Numericalsimulationsunderthesameconditionsasthoseoftheexperiment.

Fig. 6.Theoreticalinterferencepatternresultingfromthesuperpositionofthree(a)andfour(b)planewaveswithwavevectork//.Thecharacteristicsize rislinkedtothewavevectorbytherelationk=2π/r.

due to the non-uniformdensity distribution ofpolaritons, originatingfrom thefinite polariton lifetime. This behavior is well reproduced by the numericalsimulations showninFig. 5e.It can beseen inFigs. 3, 4 and 5that the shape ofthe latticestronglydependsonthegeometryofthetrapandisinducedbytheinterferencepatternbetween3(foratriangular mask)or4beams(forasquaremask):inthelow-densityregime,theshapeandsizeofthelatticearedeterminedbylinear interferenceeffects.

Under a quasi-resonantpumping atlow intensity, theemission ofthe microcavityin themomentum spacetakes the formofascatteringring,whichistheRayleighring.Undertheseconditions,apumpatnormalincidencewithenergyh¯ωp generatespolaritonswiththesameenergyandaninplanemomentumk// suchas

¯ h2k2//

2m = ¯hδlaser–LPB (4)

Themodulusofthewavevectorissetbytherelationship(4),butnotits direction.Thepresenceofalow-amplitudedisor- deredpotentialinthecavityallowspolaritonscatteringinalldirectionsofspace.FollowingtheHuygens–Fresnelprinciple, insidethetrapthisisotropicscatteringinthepumpingregionresultsinapolaritonfloworthogonaltoeachsideofthetrap.

The dynamicsofpolaritons inthepresence ofatriangular trapis thensimilar tothat ofthree beamscomingfromeach sideofthetrap.Thesethreepolaritonflowsgeneratedfromasinglepumparemutuallycoherentandshowaninterference pattern inside the trap. A similar situationis observed forthe square trap, where the square lattice is the resultof the interferencebetweenfourpolaritonflows.

Toconfirmthishypothesis,westudiedthecharacteristicsizerofthelatticecellasafunctionofthemodulusofthe in-planepolaritonmomentumk// definedbyrelation(4).

Fig. 6,whichrepresentstheinterferencepatternsbetween3and4planewavesofthesamewavevectork//,illustrates theconceptofthecharacteristiccellsizeforsquareandhexagonallattices.ThecharacteristicsizedefinedinFig. 6andthe momentumk// arelinkedbytherelation:

k//=2π

r (5)

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