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QUANTUMPHASETRANSITIONSWITHPHOTONSANDPOLARITONS Fernando G.S.L. Brandaõ, Michael J.Hartmann, andMartinB. Plenio InstituteforMathematicalSciences,Prince’sGate,53ExhibitionRoad,Imperial CollegeLondon,LondonSW7 2PG, UnitedKingdom QuantumOpticsandLaserSciencegroup,BlackettLaboratory,ImperialCollege London,LondonSW7 2BW We show that a system of polaritons - combined atom and photon excitations - in an 7 array of coupled cavities, under an experimental set-up usually considered in electromag- 0 0 netically induced transparency, is described by the Bose-Hubbard model. This opens up 2 the possibility of using this system as a quantum simulator, allowing for the observation n of quantum phase transitions and for the measurement of local properties, such as single a site observables. All the basic building blocks of the proposed setting have already been J 1 achievedexperimentally, showingthefeasibility ofitsrealization inthenearfuture. 3 1 1. INTRODUCTION v 3 The exponential growth of the Hilbert space dimension with the number of particles of the 0 0 systemisahurdlefortheclassicalsimulationofquantummany-bodysystems,inparticularnear 2 oratcriticality. Apossiblesolutiontothisconundrum,pioneeredbyFeynman20yearsago[1], 0 7 istheuseofquantumsimulators,i.e. artificialquantumsystemswhichgeneratetheevolutionof 0 rather complex Hamiltonians in a controlled and systematic manner. An emblematic example / h of a quantum simulator is a quantum computer, capable of simulating any quantum evolution. p - Yet,theoverwhelminglyhighdegreeofaccuracyrequiredtobuildausefulquantumcomputer- t n operatingbelowthefault-tolerancethreshold-rendersunlikelytheavailabilityofsuchadevice a inthenearfuture. Anaturalapproachisthentoconsiderlessgeneralquantumsimulators,engi- u q neered to simulate specific many-body models, but requiring much less stringent experimental : v conditions. i X Cold atoms trapped in an optical lattice constitutes an important example of this second r kind of simulator [2,3,4,5]. By adjusting the experimental parameters properly, the Bose- a Hubbard (BH) Hamiltonian - extensively used in modeling strongly correlated many-particle systems-canbegenerated. Thisidea,inturn,culminatedinseveralimportantexperimentsem- ulating condensed-matter phenomena, such as the superfluid-to-Mott-insulatorquantum phase transition [3] and the generation of a Tonks-Girardeau gas for the first time [4]. Moreover, the realization of the BH model opened up the possibility of studying several interesting aspects of many-body quantum systems, such as disorder, Anderson localization, and quantum mag- netism[5]. Despitetheirimpressivesuccess, opticallattices are in alltheirapplicationslimited by an intrinsic draw back: due to the separation between neighboring lattice sites of only half oftheemployedopticalwavelength,itis extremelychallengingto access themindividually. Recently, we have proposed an alternativeway to create an effective Bose-Hubbard model, which,incontrastto opticallattices,allowsforthemanipulationand measurementoftheprop- erties of individualconstituent particles [6] (see Refs. [7] for related proposals). This proposal canberealizedincoupledarrays ofopticalcavitieseachinteractingwithanensembleofatoms Figure1: (taken from Ref. [6]) Aschematicviewofthemodelconsidered. leading to the dynamical evolution of polaritons, combined atom-photon excitations, in this setting. An exciting feature of this proposal is the possibility of observing strong-correlation phenomena in a system of photons, which emerges as one of the limits of the polariton excitations. Ourmodelconsistsofanarrayofcavitiesinanarbitrarygeometry,whereeach cavityinter- acts with an ensemble of atoms, which are driven by an external laser. Photon hopping occurs between neighboring cavities due to the overlapp of their photonic wavefunctions (see fig. 1) while the repulsive force between two polaritons occupying the same cavity is generated by a largeKerrnonlinearitythatoccursifatomswithaspecificlevelstructure,usuallyconsideredin thecontextofElectromagneticallyInducedTransparency[8,9,10,11,12],interactwithlight. By varyingtheintensityofthedrivinglaserandhencethestrengthofthegeneratedKerrnonlinear- ity, the system can be driven through the superfluid-to-Mott-insulator transition. In particular, the drivinglaser may be adjusted for each cavity individuallyallowingfor a much wider range oftuningpossibilitiesthan inan opticallattice, includingattractiveinteractions[6]. Our system can be described in terms of three species of polaritons, where under the con- ditionsspecified below,thespecieswhichis leastvulnerableto decay processesisgovernedby theeffectiveBose-Hubbard(BH) Hamiltonian, 2 Heff = κ p†R~ (pR~)2 + J p†R~ pR~′ + h.c. . (1) X (cid:16) (cid:17) X (cid:16) (cid:17) R~ R~,R~′ h i p† creates a polariton in the cavity at site R~ and the parameters κ and J describe on site repul- R~ sionand intercavity hoppingrespectively. 2. DERIVATIONOFTHE MODEL We start by considering a periodic array of cavities, which we describe here by a real, periodic dielectric constant. The electromagnetic field can be expanded in Wannier functions, Figure2: (taken from Ref. [6])Diagramofthelevelstructureconsidered. w~ , each localized at one single cavity at location R~. In terms of the creation and annihilation R~ operators oftheWanniermodes,a†R~ and aR~,theHamiltonianofthefield can bewritten, 1 H = ωC (cid:18)a†R~aR~ + 2(cid:19)+2ωCα a†R~aR~′ +h.c. . (2) X X (cid:16) (cid:17) R~ R~,R~′ h i Here is the sum of all pairs of cavities which are nearest neighbors of each other. α is R~,R~′ givenPbyhanoiverlapintegralofneighboringWannierfunctions[13]andcanbeobtainednumer- ically for specific models. Since α 1, we neglected rotating terms which contain products ≪ of two creation or two annihilation operators of Wannier modes in deriving (2). The model (2) provides an excellent approximation to many relevant implementations such as coupled photoniccrystal micro-cavities or fiber coupled toroidal micro-cavities. Furthermore, it allows for the observation of state transfer and entanglement dynamics and propagation for Gaussian states[14,15]. The on site interaction potential for polaritons is generated in each cavity by atoms of a particular level structure that are driven with an external laser in the same manner as in Elec- tromagnetically Induced Transparency, see figure 2: the transitions between levels 2-4 and 1-3 coupleviadipolemomentstothecavityresonancemode,whereasthetransitionsbetweenlevels 2 and3 are coupledtothelaserfield. InarotatingframewithrespecttoH = ω a a+ 1 + N ω σj +ω σj +2ω σj , 0 C † 2 j=1 C 22 C 33 C 44 the Hamiltonian of the atoms in one cavity read(cid:0)s, H =(cid:1) PN εσ(cid:0)j +δσj +(∆+ε)σj +(cid:1) I j=1 22 33 44 N Ω σj + g σj a + g σj a + h.c. . Here σPj = (cid:0)k l projects level l of a(cid:1)tom j=1 L 23 13 13 † 24 24 † kl | jih j| jPto le(cid:0)vel k of the same atom, a creates one ph(cid:1)oton in the cavity, ω is the frequency of the † C cavity mode, Ω the Rabi frequency of the driving by the laser, g and g the parameters of L 13 24 the dipole coupling of the cavity mode to the respective atomic transitions and ∆, δ and ε are detunings. Alltheseparameters are assumedtobereal. Neglecting the coupling to level 4, g = 0, and two photon detuning, ε = 0, the Hamil- 24 tonianH canbewrittenasamodelofthreespeciesofpolaritons[10,16]withthecreation(and I annihilation)operatorsp†0 = B1 (cid:16)gS1†2 −ΩLa†(cid:17)andp†± = qA(A2±δ) (cid:16)ΩLS1†2 +ga† ± A±2δS1†3(cid:17), where g = √Ng13, B = g2 +Ω2L, A = √4B2 +δ2, S1†2 = √1N Nj=1σ2j1 and S1†3 = m√1NutaPtioNj=n1rσel3ja1t.ioInnsthaendlimfoirt ogf2p4la=rge0aatnodmεn=um0bethrse,HNam≫ilt1o,npi†0a,npH†+Ianthdups†−dPessactirsibfyesboinsdoenpicencdoemn-t non-interactingbosonicparticles, [HI]g24=0,ε=0 = µ0p†0p0 +µ+p†+p+ +µ−p†−p−, (3) where the polariton frequencies are given by µ = 0, µ = (δ A)/2 and µ = (δ + A)/2. 0 + − − We will now focus on the dark statepolaritons, p†, which will be shownto be described by the 0 Hamiltonian(1). For g , ε , ∆ µ µ , µ µ , thecouplingg and thetwophotondetuning 24 + 0 0 24 | | | | | | ≪ | − | | − − | ε do not induce interactions between different polariton species [6]. Assuming further that np(np −1)g24 ≪ |∆|,wherenp isthenumberofpolaritonsp†0,thecouplingtolevel4canbe ptreated perturbatively. Theresultis an energy shiftofn (n 1)κwith p p − g2 Ng2 Ω2 κ = 24 13 L . (4) − ∆ (Ng2 + Ω2)2 13 L Notethatκ > 0for∆ < 0andviceversa. Thetwophotondetuningε,inturn,leadsanalogously to an energy shift of εg2B−2 for the polariton p†0, which plays the role of a chemical potential intheeffectiveHamiltonian. The interaction between cavities, derived above in terms of the photonic operators, can be reexpressed in terms of the polaritons scpecies. Due to the large separation in the polaritons frequencies, 2ω α µ µ , µ µ , the hopping does not mix different polariton C + 0 0 species. The|Hamilt|on≪ian|for t−hepo|la|rit−on−sp†R~|(thedark state polaritonp†0 at siteR~) thus takes on theform (1), with 2ω Ω2 J = C L α, (5) Ng2 +Ω2 13 L wherewehaveassumedanegligibletwophotondetuning,ε 0. ≈ The number of polaritons in one individual cavity can be measured via resonance fluores- cence. Tothatend,thepolaritonsaremadepurelyatomicexcitationsbyadiabaticallyswitching off the driving laser [16]. The process is adiabatic if gB 2 dΩ µ , µ [12], which − dt L ≪ | +| | −| means it can be done fast enough to prevent polaritons from hopping between cavities during the switching. Hence, in each cavity, the final number of atomic excitations in level 2 is equal to theinitialnumberofdark statepolaritons. By measuringthenumberofpolaritonsinseveral individualcavities in possiblyrepeated resonance flourescence experiments one can obtain the localnumberfluctuationswhichallowtodistinguishbetweenthedifferentphasesofthesystem. 3. FEASIBILITY OF THEOBSERVATIONOFPHASETRANSITIONS Promisingcandidatesforanexperimentalrealisationarephotonicbandgapcavities[17,18] andtoroidalorsphericalmicro-cavities,whicharecoupledviataperedopticalfibres[19]. These cavities can be produced and positioned with high precision and in large numbers. They have avery largeQ-factor (> 108)forlightthatis trapped as whisperinggallery modesand efficient coupling to optical fibres [20] as well as coupling to Cs-atoms in close proximity to the cavity via the evanescent field [21,22] have been demonstrated experimentally. Photonic crystals represent anappealingalternativeas theyofferthepossibilityforthefabricationoflargearrays of cavities in lattices or networks [13,23]. These cavities have been realised with Q-factors of 106 and higherQ-factors of2 107 havebeen predicted [18]. × As levels1 and 2 are metastable,spontaneousemissioncan only occurfrom levels3 and 4. However, due to pratically negligible population of these two levels, the main loss channel is cavity decay. For successfullyobservingthe dynamicsand phases of theeffectiveHamiltonian (1), the repulsion term κ needs to be much larger than the damping rate for the dark state polaritons, Γ [6]. Choosing Ω √Ng , one can currently achieve a ratio of κ/Γ 5.2 for L 13 ≪ ∼ photonic band gap cavities (for predicted Q-factors this can go up to κ/Γ 170) [18], while ∼ for toroidal micro-cavities the impressive large value of κ/Γ 1.1 103 is possible [24,21], ∼ × makingbothidealcandidates foran experimentalimplementation. Asanilustration,anumericalsimulationoftheMott-insulator-to-superfluidphasetransition is shown in figure 3. We consider three coupled cavities with periodic boundary conditions, which are prepared such that there is initially one polariton p† in each. We take parameters R~ for toroidal microcavities from [24], g = g = 2.5 109s 1, Γ = Γ = 1.6 107s 1 24 13 − 3 4 − × × (spontaneous emittion rate from levels 3 and 4), Γ = 0.4 105s 1 (cavity decay rate), N = C − × 1000, ∆ = 2.0 1010s 1, and 2ω α = 1.1 107s 1. To drive the system through the − c − − × × transition, The Rabi frequency of the driving laser is imcreased from Ω = 7.8 1010s 1 to L − × Ω = 1.1 1012s 1 throughout the experiment, while all other parameters remain constant. L − × Fig. 3 shows in the left part, Ω (inset), the on-site interaction strength κ and the hopping rate L J duringthetransition. TherightpartshowsthatnumberfluctuationsF1 = h(p†1p1)2i−hp†1p1i2 are small in the Mott phase and increase when driving into the superfluid phase. The average numberofpolaritonsin one cavity n1 = hp†1p1i stays closeto unityacross thewholesimulated timerangeconfirmingthat polaritonlossis indeednegligibleon thistimescale. 4. CONCLUSION In summary, we have demonstrated, that polaritons, combined atom - photon excitations, can form an effective quantum many particle system, which is described by a Bose-Hubbard Hamiltonian. We have demonstrated numerically that the observation of the Mott insulator to superfluid phase transition is indeed feasible for experimentally realizable parameters. In contrasttoearlierrealizations,ourscenario hastheadvantagethatsinglesitesoftheBH model canbeaddressedindividuallyandoffersthepossibilityofcreatingaBHtypeHamiltonianwith an attractiveonsitepotential. REFERENCES [1] R.P. Feynman, Found.Phys.16, 507(1986) [2] D.Jaksch et al, Phys.Rev. Lett.81,3108(1998) Figure3: (takenfromRef.[6])left: LogplotofofκandJ andlinearplotofthetime-dependent Ω (inset). right: expectation value (n ) and fluctuations (F ) for the number of polaritons in L 1 1 onesiteconsideringdamping(see maintextfordetails). [3] M.Greiner, O. Mandel,T.Esslinger,T.W.Ha¨nsch, and I. Bloch, Nature415, 39(2002) [4] B. 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