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Adiabatic melting of two-component Mott-insulator states M. Rodr´ıguez1,2, S. R. Clark1 and D. Jaksch1 1 Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, U.K. and 2 ICFO-Institut de Ciències Fotòniques, 08860 Castelldefels (Barcelona), Spain WeanalyzetheoutcomeofaMottinsulatortosuperfluidtransitionforatwo-componentBosegas withtwoatomspersiteinanopticallatticeinthelimitofslowrampingdownthelatticepotential. ThismanipulationoftheinitialMottinsulatingstatetransformslocalcorrelationsbetweenhyperfine states of atom pairs into multiparticle correlations extendingoverthewhole system. Weshow how 8 0 tocreatemacroscopictwinFockstatesinthiswayanthat,ingeneral,theobtainedsuperfluidstates 0 are highly depleted even for initial ground Mott insulator states. 2 PACSnumbers: 03.75.Lm,03.75.Mn,03.75.Dg n a J I. INTRODUCTION more intriguing due to the different phases that appear 1 in the T = 0 phase diagram depending on the ratios of 2 theinterparticleinteractionsandofthetunnelingratesof The experimental realization of degenerate atomic each component [12]. An analysis of the two-component ] gases [1] have opened up the possibility of engineering l symmetric case with all interparticle interactions equal l strongly correlated many-body quantum states. Such a canbefoundin[13]. In[9]westudiedtheadiabaticmelt- h states are fundamental for the development of quan- ing of an opticallattice in a two-componentsystem with - tum technologies such as those implementing quantum s different values for the inter-species and intra-species in- information protocols and quantum computation. A e teractions. We showed that one can create twin Fock m paramountexampleistherealizationofaMottinsulator stateswithhighfidelityviameltingofaninitialMIstate (MI) [2, 3] in the lowest Bloch band of an optical lattice . with two atoms per site in different internal states. We t whichcan serveasa quantumregister[4]. Bose-Einstein a inspected possible sources of experimental imperfections m condensates of alkali atoms have also been proposed as such as atom loss during melting or depletion caused systemstorealizemacroscopicsuperpositionstates[5]or - by final finite interactions. The possibility of creating d number correlated states (also called twin Fock states) macroscopic superposition states via the adiabatic melt- n which are useful for quantum metrology [6, 7, 8, 9]. ing of on-site entangled MI states in an infinitely con- o c In[9]weproposedamethodforengineeringtwinFock nected lattice was also briefly considered in [9]. An ex- [ statesviaatwo-componentMItosuperfluid(SF)transi- amination of the adiabatic melting in an infinitely con- tion. The proposed scheme manipulates hyperfine states nectedlattice, including the effect ofthe dynamicalrela- 1 ofatompairspinnedtosinglelatticesitesanddecoupled tive phase acquired by the state can be found in [14]. In v 2 from one another in the MI regime [4, 10]. The MI limit thispaperweextendtheresultsin[9,14]andanalyzethe 5 has already proven to be useful starting point for quan- melting of on-site superposition states in a lattice with 2 tum state engineering since the motional dynamics are only nearest neighbor hopping. We show that the final 3 frozen out and the system decouples to a set of single SFstatesobtainedfromsuchon-sitesuperpositionstates 1. site Hamiltonians to good approximation. In the case of arehighlydepletedeveniftheinitialstateisagroundMI 0 atwo-componentsystemwithdoublyoccupiedsitesspin- state. For the melted states we calculate the momentum 8 changingcollisions willoccur and canbe accuratelycon- distribution and the momentum space particle number 0 trolled with external magnetic or microwave fields [10]. correlations. These quantities are experimentally acces- v: As a result the spin-dynamics of atom pairs behaves like sibleeitherbytimeofflightexpansionofthecloudorvia i an effective two-level system undergoing Rabi-like oscil- shot noise measurements [15]. X lations. In contrastto the usual Rabi-modelfor the cou- r pling of a pair of single-particle states this represents a a This paper is organizedas follows. In sectionII we in- coherentcouplingbetweenapairoftwo-particlestatesin- troduce the two-mode Bose-Hubbard model (BHM) and ducedbycollisionalinteractions. Theselocaltwo-particle theinitialMottinsulatingstatesthatweconsider. Insec- correlationsarethentransformedinto multi-particle cor- tionIII wedescribetheadiabaticapproximationandthe relationsextendingoverthewholesystembymeltingthe no-crossing rule that we use to infer the final superfluid MI into a SF. This quantum melting can be experimen- states. In section IV we illustrate how one can create tally implemented by adiabatically ramping down the macroscopictwin Fock states if the inter-species interac- depth of the lattice potential. tion is greater than the intra-species interaction during Thebuildupoflong-rangecorrelationswhenaperiodic melting. In V we show using the symmetry properties potential is dynamically lowered is a highly non trivial of the initial states and a series of approximations, that process that occurs rather quickly in a one-component anon-site entangledMI groundstate melts into a highly system [2, 9, 11]. The quantum melting of a multi- depletedSF state inthe limitofintra-speciesinteraction component system is still an open problem that is even greater than the inter-species interaction. 2 II. TWO-COMPONENT BHM applicable to recently demonstrated ring-shaped optical lattices [17]. In this section we introduce the two-component BHM that describes the dynamics of ultra-cold atoms in an B. Symmetries optical lattice. We discuss its experimental feasibility and analyze the symmetries of the Hamiltonian. Finally we introduce the initial MI states under consideration. The most basic conserved quantity of the two- component BHM is the total number of atoms Nˆ = a nâ,Nˆ = nˆbintheeitheroftheinternalstatesaor i i b i i bP,respectivelyP. Withtheadditionalassumptionsoutlined A. Optical lattice setup a number of other symmetries arise which we exploit in this work. The homogeneityandperiodic boundarycon- Ourfocusisonastate-dependentopticallatticewhich ditions assumed results in translational invariance. To is sufficiently deep that two hyperfine states a and b of exploit this we introduce quasi-momentum operators for the atoms are be trapped in the lowest Bloch band of each component as the lattice. The resulting system is then accurately de- scribed [2] by the two-component BHM with the corre- 1 M sponding Hamiltonian (h¯ =1) cˆ†k = √M aˆ†ie−ikri, Xi=1 Hˆ = (J aˆ†aˆ +J ˆb†ˆb )+U nânˆb 1 M − a i j b i j i i dˆ† = ˆb†e−ikri. hXi,ji Xi k √M i Xi=1 V V + 2a nâi(nâi −1)+ 2b nˆbi(nˆbi −1), (1) where k is the quasi-momentum in the first Brillouin Xi Xi zone and r is the position of the site i. The quasi- i momentum states spanning the first Brillouin zone have where aî(ˆbi) is the bosonic destruction operator for an k = δk Km,..,0,..,K with δk = 2π/Md and d a(b)-atom localized in lattice site i, nâ = aˆ†aˆ and being th×e d{i−stance between}lattice sites. For M even i i i nnêbiig=hbˆbo†iˆrbsi,. wThhilee hpia,rjaimdeetneorteJsa(sbu)mismtahteiontuonvneerllinnegarmesat-- KKmm ==KM=/2(aMnd−K1)/=2. MTh/e2 t−ot1a,l mwhomileenftourmMofotdhde trix element for atoms in state a(b); V and U are system is then Pˆ = k(nâ + nˆb), where nâ = cˆ†cˆ a(b) k k k k k k the on-site intra- and inter-species interactionmatrix el- and nˆb = dˆ†dˆ , and iPs a conserved quantity. The total k k k ements, respectively. A crucialfeature of the optical lat- momentumgeneratescyclicshiftsofthelatticesites,e.g. tice setup is that the ratio between the tunnelling and i i+ 1 for the site indices of all operators, via the interaction matrix elements in Eq. (1) is experimentally un7→itary Sˆ = exp( 2iPˆπ/M). Its action on the bosonic controllable via the lattice laser intensities. Further in- creation operators−is dependent control of the interaction matrix elements U, V can be achieved with Feshbach resonances [16], or by Sâˆ†Sˆ† = aˆ† , i i+1 shifting the a and b atoms away from each other us- Sˆˆb†Sˆ† = ˆb† , ing state-dependent lattices [4]. All of these methods i i+1 of controlcan be exploited dynamically to permit differ- simultaneously for all sites i. The homogeneity of the ent regimesofthe Hamiltonianin Eq.(1) to be unitarily system also gives mirror reflection symmetry in which connected. Together with the long decoherence times of siteindicesforoperatorsinterchangeasi M i+1. In ultra-cold atoms this dynamical manipulation can often this casethe parity operatorRˆ is the con↔served−quantity be implemented on near-adiabatic timescales. and implements the transformation Basedonthe considerableflexibility ofthe opticallattice setup we make a number of simplifications in our Râˆ†Rˆ = aˆ† , i M−i+1 analysis. First, we restrict the couplings to the symmet- Rˆˆb†Rˆ = ˆb† , ric case where Ja = Jb = J and Va = Vb = V. Second, i M−i+1 we consider states of the system ψ in which the lat- | i simultaneously for all sites i. Finally, the homogeneity tice is commensurately filled with two atoms per site so and symmetry of the matrix elements result in color in- Nˆ ψ = N ψ has N = 2M, where Nˆ = (nâ +nˆb) | i | i i i i versionsymmetrya b. Likeparitythereisahermitian- is the total particle number and M is the tPotal number unitary operator Cˆ ↔which is the conserved quantity and of sites. Finally, we confine our attention to a 1D lattice implements the transformation withperiodicboundaryconditions(i.e. M+1 1)inad- ≡ dition to the spatialhomogeneity of the matrix elements Câˆ†Cˆ = ˆb†, i i in Eq. (1). This is not only a good approximation for the center of a large system, where trapping and bound- simultaneouslyforallsitesi. Thesethreesymmetriesare aryeffectshaveanegligibleinfluence,butisalsodirectly schematically represented in Fig. 1(a)-(c). 3 tal particle number N where N = N n and N = n a b − usingn=0,1,2,...,N. We denoteas thecompletely n S symmetric subspace of the direct sum , de- n N−n H ⊕H fined by Sˆ ψ = ψ , Rˆ ψ = ψ and Cˆ ψ = ψ for | i | i | i | i | i | i ψ where n = 0,...,N/2. As we shall n N−n | i ∈ H ⊕H now outline for the specific initial states considered here wefix N =2M throughoutandthe onlyrelevantsectors are those where n is restricted to be even or N/2. C. Initial states and MI ground states Our approach to exploit the two-component BHM to engineer quantum states is based on using two experi- mentallyaccessibleinitialstates. Bothofthesestatesare generated in the MI limit where interactions are domi- nant, i.e. U J and V J, and precisely with two ≫ ≫ atoms localized on every site. By controlling the spin- changing collisions a doubly occupied single-component MI formed in some additional hyperfine state can be transformed with near unit efficiency to the state [10] M Ψ = ab , (2) | abi | ii iY=1 with exactly the same number of a and b atoms in each FIG. 1: Schematic representations of (a) cyclic shifting, (b) lattice site. Note that Ψ is exclusively contained in ab mirror reflection and (c) color inversion symmetry for a 1D the symmetric subspace| .iBy applying a further π/2- M lattice with periodic boundary conditions. The colors repre- S Raman pulse to the state Ψ an on-site entangled MI ab sent atoms in two different hyperfinestates a and b. | i state M 1 Ψ = ( aa + bb ), (3) Symmetriesarebydefinitionrelatedtothepresenceof | aa+bbi √2 | ii | ii degeneracies[18]ormorepreciselytoadegeneracywhich iY=1 remains for all values of the couplings in the Hamilto- can be created. This state is an equal superposition of nian - a so-called permanent degeneracy. By using these all 2M possible combinations of the atom- pair states symmetries of Hˆ it follows that any of its eigenstates aa and bb over M sites. The states in the superposi- | i | i which are connected by a combination of these symme- tion are contained in the symmetric subspaces 2n with S try operators are necessarily degenerate. Moreover, any n = 0,1,..., M/2 , where denotes the integer part. ⌊ ⌋ ⌊·⌋ eigenstateswhichcannotbe mappedontoeachother by Specifically, in each sector 2n we form the state H symmetry operators will be non-degenerate if the sym- metriesofthe systemareexhaustive[19]. Othertypesof ψ2n =  aa bb  degeneracies can occur when the couplings attain a par- ps P | ii | ij (cid:12) (cid:11) iY≤n n<Yj≤M ticularvaluewhichindicatesthatatthispointthesystem (cid:12)   acquiresahighersymmetryascomparedtootherpoints. where ()standsforthenormalizedsuperpositionofall This enhancement of symmetry frequently occurs in the permutPat·ions of the lattice sites. Since the state ψ2n ps limits of zero or infinite coupling. An important exam- possesses full permutational symmetry it is mani(cid:12)festly(cid:11) ple in this work is when J =0 where Hˆ decouples into a cyclic-andmirror-symmetric. Ifwethensymmetri(cid:12)zethe setofsingle-siteHamiltoniansandacquiresfullpermuta- color as [33] tionalsymmetry. The breaking ofthe full permutational 1 symmetry for finite J splits the degenerate eigenstates Ψ2n ψ2n + ψN−2n , (4) ps ≡ √2 ps ps of the J = 0 Hamiltonian as we will see in proceeding (cid:12) (cid:11) (cid:0)(cid:12) (cid:11) (cid:12) (cid:11)(cid:1) (cid:12) (cid:12) (cid:12) sections. we obtain states which are each contained in 2n, re- S Particle number conservation for each component spectively. The state Ψaa+bb can then be seen to be a breaks up the system Hilbert space into sectors where binomial superposition| of theistates |Ψ2pnsi for all even-n Nˆ ψ = N ψ and Nˆ ψ = N ψ . The evolution symmetric subspaces as a a b b | i | i | i | i of each component of an initial state in each of these ⌊M/2⌋ 1/2 1 M sectors can therefore be treated independently. We shall Ψ = Ψ2n . (5) use Hn to denote the sectors which possess the same to- | aa+bbi √2M−1 nX=0 (cid:18)n(cid:19) (cid:12)(cid:12) ps(cid:11) 4 ThetwoinitialstatesinEqs.(2)and(3)areexactsym- metric ground states of the two-component BHM in the limit J =0 for different regimes of U/V. This is the MI regime where the particle-hole spectrum is gapped. In the case U < V where intra-species interaction is dom- inant, and in the subspace , the on-site state ab N/2 S | i is energetically favoredmaking Ψ the nondegenerate ab | i ground state. This is easily confirmed by comparing it to all other configurationsand moreoverestablishes that the ground state remains nondegenerate for non-zero J. For the opposite case U > V where the inter-species interactionisdominant“phase”-separatedconfigurations composed of on-site states aa = and bb = are FIG. 2: Schematic energy-level structure for the two- | i |◦i | i |•i energetically favored. Thus at J = 0 the full permuta- component BHM with only one varying parameter U/J and tional and color symmetry results in 2 N/2 degenerate fixed U/V. The no-crossing rule assures that the system n follows the instantaneous eigenstates and performs no level ground states [34] spanned by all confi(cid:0)gura(cid:1)tions of the crossings if the states have all the same symmetry, there is n states and N/2 n states distributed amongst |◦i − |•i only one varying parameter and the evolution is performed the M =N/2 sites. Some of these states canbe mapped sufficiently slow. bythecyclicshiftsymmetrySˆ,themirrorRˆ orthecolor Cˆ transformation. We denote as the sym- 2n 2n metrized(withrespecttoCˆ,SândRGˆ)gr⊂ounSdstateman- state. ifold of the BHM in Eq. (1) for J = 0 and U > V with A feasible way to analyze the properties of the final dimension denoted by g(n,M). melted states is to use the adiabatic approximation [20] Onthisbasisweseethateachofthestates Ψ2n com- | psi inwhichthesystemfollowstheinstantaneouseigenstates posing Ψ is one of manypossible groundstates in aa+bb ofthetwo-componentBHMinEq.(1)foreachvalueV/J | i the symmetric subspace . However, as will be de- 2n and U/J during the melting. The first requirement for G scribed in detail in Sec. VA, once J is non-zero this de- the analysis to be tractable is the lack of level crossings. generacy is partially split resulting in Ψ mapping aa+bb Thisconditionisgreatlyaidedbytheno-crossingrule[19] | i to a superposition of numerous low-lying excited states. which states that if the energy levels of a quantum sys- This is schematically shown in fig. 2. The reason for tem are plotted as a function of only one parameter, the this can be readily seen. For 0 < J/U 1 nearest- curvesforlevelsofthesamesymmetrydonotcross. The ≪ neighborhoppingwillreducetheenergyofconfigurations no-crossing rule can be followed by using the flexibility like whereaandbatomsformcontiguousre- of the optical lattice setup to ensure that the dynami- |•••◦◦◦i gions,comparedto configurationslike . Since calmeltingonlyvariestheratioV/J whilstkeepingU/V |•◦•◦•◦i Ψ contains contributions from all configurations aa+bb constant. Thustheanalysiscanproceedbyfollowingthe | i it necessarily maps to a superposition of numerous low- evolution of eigenstates in the symmetric subspace only. lyingexcitedstatesofthetwo-componentBHMinEq.(1) Also, the BHM is a non-integrable system that shows a for each subspace when J is non-zero. This feature 2n chaoticspectrumandthuspresentsenergylevelrepulsion S significantly complicates the analysis of this state even [23]. The work in [23] provides numerical evidence that under adiabatic conditions. forsmallsystemsthesymmetriesweconsiderareexhaus- tive forthe BHM (i.e. no crossingsareseen). We extend this assumption to arbitrary sized systems. The energy III. ADIABATIC QUANTUM MELTING levelsmayapproacheachotherformingso-calledavoided crossings, but for infinitely slow dynamics there will be Hereweexplorehowadiabaticquantummeltingwhich no transitions between eigenstates. For finite times the is an accessible coherent process that can be applied to Landau-Zener model gives the diabatic transition prob- the systemandperformsahighly non-trivialmany-body abilities [21] between two levels, and this generalizes to unitary transformation [35] can be exploited as a means Brundobler-Elser rule [22] for a multistate problem. of quantum state engineering. For both initial states In principle, adiabatic evolution requires that the dy- Ψab and Ψaa+bb wetakethequantummeltingasend- namical parameter V/J changes in an infinitely long | i | i ing in the non-interactingSF limit whereU =V =0. In time. A crude estimate of the time scale t needed r this limit the two-component BHM possesses a nonde- for adiabatic melting of an optical lattice is given by generate ground state t NV /MJ2 [11], where V /J stands for the r max max ≫ ΨNa,Nb = (cˆ0)Na(dˆ0)Nb vac , (6) ibnaittiicalcovanlduietioinn dthoeesMnIotresgcaimlee.witIhmtphoertsaynsttleymthseizeadbiuat- | sf i √N !N ! | i a b with the density N/M. Numerical computations in [9] for any sector in which all the a and b atoms Bose- indicate that the adiabatic time scale is on the order condense into tHheNibr correspondingzero quasi-momentum of t 3V /J2 for a finite realistically sized one- r max ∼ 5 dimensional system. These values are attainable in ex- perimentalsetupsandmaketheadiabaticapproximation a reasonable assumption. These estimates are valid for the case whenU =0 whenEq.(1)decouplesas twoone- componentBHM’s. Findinganalogoustimescalesforthe two-component system with U =0 is still an open prob- 6 lembutitisreasonabletoexpectthatthecorresponding time scales are also attainable in experimental setups. IV. MELTING THE PRODUCT MI : |Ψ i ab FIG. 3: (a) The spectrum of the two-component BHM Hˆ for M = 6 sites in the symmetric subspace S6, i.e. with We consider first the melting of the product MI state N = N = 6, as a function of V/J with V/U = 0.1 a b Ψ . Our analysis begins by examining the outcome throughout. To the far left (not plotted) where V/J ≫ 1 ab | i of quantum melting in a small system by using an exact the state |Ψabi is the non-degenerate ground state. To the numerical calculation. The results found then generalize rightwhereV/J =0thenon-interactingSFstate|Ψ6sfiisthe toanarbitrarilylargesystemandgiverisetoafinalstate non-degenerate ground state. As expected these two ground states are adiabatically connected (solid line). The first and whose usefulness for interferometry is analyzed. second excited states (dotted and dashed line, respectively) arealsoshownandbecomedegenerateintheV =0limit. (b) The overlap |hψgs|Ψ6sf,6i| between the instantaneous ground A. Exact numerics for a small system state |ψgsi and theSF state |Ψ6sf,6i as a function of V/J. Here we provide some numerical evidence for the claims made in the previous section by performing an exact numerical diagonalization of the two-component BHM Hamiltonian Hˆ for a system with M = 6 sites. For the state Ψ we need only consider the projection ab | i of Hˆ in the subspace . We then compute the ground 6 S state and low-lying excitations for a sequence of linearly decreasing interaction strengths V/J with a fixed ratio V/U = 0.1 throughout. In Fig. 3(a) the eigenenergies FIG. 4: (a) The melting of the product MI |Ψ i in to the are shown confirming that the ground state in re- ab S6 twin-Fock SF state |Ψtfi. (b) The two internal states a and mainsnondegenerateduringthemelting,andthatwithin b represent the arms of an interferometer with the rotations the symmetric subspace level repulsion ensures that no R1, R2 and the phase-shift φ induced by appropriate laser levelcrossingsoccurconsistentwiththeno-crossingrule. pulses. Finally a φ dependent observable is measured at M The final SF ground state is then Ψ6,6 in which the from which φ can be deduced. | sf i zeroquasi-momentummodeofeachcomponentisequally occupied. Thus the quantum melting transforms local initial correlations between pairs of atoms into correla- forarbitraryN. Quantummelting ofthe MI state Ψ ab | i tions between delocalized and macroscopically occupied thereforeprovidesadirectmeansofcreatingaN-particle modes. For this reason the final state Ψ6,6 is called a twin-Fock state Ψ , as depicted in Fig. 4(a). | sf i | tfi twin-Fockstate[6]. InFig.3(b)theoverlap ψ Ψ6,6 Twin-Fock states [6] have been proposed as useful in- |h gs| sf i| betweenthe instantaneousgroundstate ψ duringthe putstatesininterferometricexperimentswhichcouldpo- gs | i rampandthe SF twin-Fock Ψ6,6 is seento convergeto tentially allow measurement sensitivities which scale as unity as a V/J 0. | sf i 1/N. This is the Heisenberg limit for interferometry → ∝andimprovesonthestandardlimitobtainedfromanun- correlatedsourcewherethesensitivityscalesas 1/√N ∝ [25, 26, 27]. A typical experimental scheme is a Mach- B. Large systems and interferometry Zenhder interferometer (MZI) that consists of an initial beamsplitter andtwo interferometricpaths thatacquire We expand here the results that were investigated in a phase difference φ and are recombined in a final beam [9]. By using the fact that Ψab is a nondegenerate MI splitter, as shown in Fig. 4(b). The beam splitter op- | i groundstateandtheno-crossingruletheoutcomeofadi- erations could be implemented by rapid resonant π/2- abatic melting found for a small system can be readily Raman pulses. Finally one extracts the phase difference generalized to any size system as measuring a phase dependent quantity. Toanalyzethistwo-componentsystemwithinthecon- Ψ ΨN/2,N/2 = Ψ , (7) text of interferometry we will use the Schwinger boson | abi7→| sf i | tfi 6 formalism for constructing SU(2) operators [24, 25] For the initial (uncorrelated)MI state Ψ , which pos- ab sesses no off-diagonal correlations ρa |= ρbi = δ , this ij ij ij 1 Jˆ = (aˆ†ˆb +ˆb†aˆ ), yields a sensitivity at the standard limit ∆φ = 1/√2N. Jˆx = 21Xi (aî†ˆbi aî†ˆbi ), rFeolrattihonesfinρaiajl S=F ρ|Ψbijsf=i, w1h,icwhepsoesesetshseastlaosnygm-rpatnogteiccaollry- y 2i i i− i i the Heisenberg limit ∆φ = 1/ N2/2+N is recovered. Xi Thus the scaling ∆φ N−α pchanges from α = 1/2 to 1 ∝ Jˆ = (Nˆ Nˆ ), α=1duringthemelting. AninspectionofEq.(8)shows z a b 2 − that one needs non-zerovalues of the long-rangecorrela- and we denote ˆJ = (Jˆ,Jˆ,Jˆ). In many cases Jˆ is the tions ρij in order to havebetter scaling than N givenby x y z z the first term in Eq. (8). We showed in [9] that one can phase-dependent measurement made in the interferome- obtain sensitivities approaching the HL scaling even in ter. However,for the twin-Fock state Jˆ =0 so a differ- h i the presence of depletion if U =0. Recent results in [30] entquantityisrequired. Onepossibilityisthenextorder indicatethatourconclusionsabouttheeffectofdepletion angularmomentumJˆ2. Notethatinefficientdetectionof z are alsovalid for U >0. Moreover,we also showedin [9] such quantities can reduce the sensitivities back to the that one can obtain HL sensitivities (α 1) if particles standardlimit [28]and indirectmeasurementschemesof ∼ are lost during the melting process. Jˆ2 are needed [29]. z If we use Jˆ2 as the phase-dependent observable the z resulting sensitivity can be calculatedusing error propa- V. MELTING THE ENTANGLED MI : |Ψ i aa+bb gation theory. This gives ∆φ=∆Jˆ2/∂ Jˆ2 /∂φ where z | h ziφ | Jˆ2 and ∆Jˆ2 = ( Jˆ4 Jˆ2 )1/2 are the average In this section we analyze the melting of the entan- h ziφ z h ziφ − h ziφ and the spread respectively. To calculate one can gledMIstate Ψ . Inpreviouswork[14]wedemon- h·iφ | aa+bbi either rotate the state or the operators [25]. Since the stratedthatthis state maymeltintoa maximallyentan- action of the MZI is equivalent to the unitary operation gledstateifthelatticeisinfinitelyconnected. Inatypical exp( iφJˆ), applying it to the operators gives experimentalset-upwithonly nearestneighborhopping, y − the initial permutational symmetry for J = 0 is broken Jˆ2 = Jˆ2 cos2(φ)+ Jˆ2 sin2(φ) for finite J and the final SF state is highly excited. We h ziφ h zi h xi beginbyagainconsideringtheoutcomeofquantummelt- sin(φ)cos(φ) Jˆ,Jˆ , − h{ z x}i inginasmallsystem. Finallyafterapplyinganumberof approximations we compute the momentum correlations and of the final melted state for a 1D lattice. Jˆ4 = Jˆ4 cos4(φ)+ Jˆ4 sin4(φ) h ziφ h zi h xi + sin2(φ)cos2(φ) JˆxJˆz,JˆzJˆx A. Small system h{ } + Jˆ2,Jˆ2 + Jˆ ,JˆJˆ Jˆ { x z} { x z x z}i ForM =6theentangledMIstate Ψ isanequal − cos3(φ)sin(φ)h{Jˆx,Jˆz3}+{Jˆz,JˆzJˆxJˆz}i superpositionofall64possiblecombi|nataiao+nbsboiftheatom- − sin3(φ)cos(φ)h{Jˆz,Jˆx3}+{Jˆx,JˆxJˆzJˆx}i, pairstates|◦iand|•ioverM sites. FollowingEq.(5)it can be written as where Aˆ,Bˆ = AˆBˆ +BÂˆ is the anticommutator. For 1 sFtoactkesstw{atheisc,h}thariseyzieerldoseigenvectors of Jˆz, such as twin- |Ψaa+bbi = 8(cid:16)√2(cid:12)Ψ0ps(cid:11)+√12(cid:12)Ψ2ps(cid:11)+√30(cid:12)Ψ4ps(cid:11) + √20(cid:12) Ψ6 , (cid:12) (cid:12) ps ∆φ2 = sin2φ(hJˆx4i−hJˆx2i2)+cos2φhJˆxJˆz2Jˆxi. where the phase-separ(cid:12)(cid:12)ated(cid:11)s(cid:17)tates in each subspace 2n 4cos2φ Jˆ2 2 are S h xi The minimum valueofthis quantityoccursatφ=0 and Ψ0ps = S(|••••••i), ttthhheeeTeshoqenenuessa-iepltinaitvsyriittthiiycJvˆlxiertJeyˆddz2euJcˆncaxseniist=ybtoemhJ∆êax2xtφipr(iwr0cee)hssis=ceρhda12hheo=Jnˆlx2tdiisraˆ−ef†1loaˆy/r2sinauafctntheedrrsmtρuabsstieno=sgf. (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)ΨΨ2p4pss(cid:11)(cid:11)(cid:11) == S√1(3|0•h•√•6•S•(◦|i•)•, ◦••◦i)+√12S(|•••◦•◦i) hˆb†iˆbji using ij h i ji ij (cid:12) + √12S(|••••◦◦i)i, 1 Ψ6 = √2 ( )+√12 ( ) hJˆx2i= 14Xi (ρaii+ρbii)+Xi,j (ρaijρbji+h.c.). (8) (cid:12)(cid:12) ps(cid:11) +√2√06hS(|S••|••◦◦◦•◦◦i•)i◦.i S |••◦•◦◦(i9) 7 contrast subspaces and are three-fold degenerate 4 6 and the states Ψ4 Gand ΨG6 are both composed of 3 ps ps inequivalent M(cid:12)I con(cid:11)figurat(cid:12)ions.(cid:11) This situation becomes (cid:12) (cid:12) more typical with increasing system size. In the J = 0 limit these configurations are degenerate, however once J is non-zero they split in energy. In Fig. 5(a) the ener- gies for the ground state and three lowest lying excitations in are shown for a linear ramping of U/J with 4 S U/V = 0.1. This, together with a similar result for 6 S (not shown), confirm that no level crossings occur. For this reason we can map the MI configurations via their FIG. 5: (a) The low-energy spectrum of the two-component orderinginenergywiththeircorrespondinggroundstate BHM Hˆ for M = 6 sites in the symmetric subspace S4 as and excitations in the SF limit. The lowest energy con- a function of U/J with V/U = 0.1 throughout. The initial figurations adiabatically connected to the symmetric SF state |Ψaa+bbi projected in this subspace is ˛˛Ψ4ps¸. The adi- ground state are abatic evolution of this state involves the ground state and two lowest-lying excited states. In the SF limit with U = 0 1 the first-excited level is triply degenerate. Consistent with ( ) Ψ8,4 + Ψ4,8 , the no-crossing rule we see that energy levels within S4 re- S |••••◦◦i 7→ √2(cid:16)| sf i | sf i(cid:17) pel each other and only avoided crossings are seen. (b) The ( ) Ψ6,6 , momentum distribution hˆnai for one component (it does not S |•••◦◦◦i 7→ | sf i k matterwhichduetocolorsymmetry)ofthefinaladiabatically respectively,whilethetworemainingconfigurationsmap melted state |Ψ i derived from |Ψ i with M =6. f aa+bb to the lowest and second-lowest excitations in the SF limit. Quite generally there are many degenerate SF ex- citationsintheU =V =0limit. Forthesubspace the Here we have used () to denote the normalized equal S6 S · firsttwolowest-lyingexcitationsaredegenerate,whereas superposition of all equivalent configurations under the symmetry operations Sˆ,Rˆ and Cˆ. Eachphase separated the first three are for S4. To determine the appropriate statetowhicheachconfigurationmapstoweapplyfirst- state Ψ2n is a member of and is a superposition ps G2n orderperturbationtheory(seeAppendix A)inthe inter- of allt(cid:12)he g(cid:11)inequivalent symmetric states within it (note (cid:12) andintra-speciesinteractionwhichsplitsthesestatesand that g(0,6) = g(1,6) = 1, while g(2,6) = g(3,6) = 3). identifies the correct excitations. This then gives for TheweightsofthissuperpositionEq.(9)aregivenbythe S6 degeneraciesduetosymmetrizationoftheseinequivalent ( ) Ψ6 , states, which we denote by s(l) where l =1,.,g. S |••◦•◦◦i 7→ | ex,1i We nowproceedtomelteachofthese phase-separated ( ) Ψ6 , contributions separately. For the subspaces and S |•◦•◦•◦i 7→ | ex,2i 0 the corresponding states Ψ0 and Ψ2 aSre non- S2 ps ps and for 4 we have degenerate MI ground states.(cid:12)Ana(cid:11)logous(cid:12)to t(cid:11)he melting S (cid:12) (cid:12) otifo|nΨoafbithinetthweos-cuobmsppaocneeSnt6 eBxHaMctnwuimtherUica>ldViagcoonnafilrizmas- S(|•••◦•◦i) 7→ |Ψ4ex,1i, that there are no level crossings and so these two states S(|••◦••◦i) 7→ |Ψ4ex,2i, are adiabatically connected to their corresponding symmetric SF ground states where Ψn and Ψn denote the lowestand second- | ex,1i | ex,2i lowestSFexcitationsin (seeAppendix A). Notethat n ( ) 1 Ψ12,0 + Ψ0,12 , this calculation of the lSowest excitations works for ar- S |••••••i 7→ √2(cid:16)| sf i | sf i(cid:17) bitrary M, although it becomes cumbersome for larger 1 system size as we would need a very large number of SF ( ) Ψ10,2 + Ψ2,10 . S |•••••◦i 7→ √2(cid:16)| sf i | sf i(cid:17) excitations. Combining these results we can write the overall final In this notation we omit the dynamical phase eiα ac- SF state whichis adiabaticallyconnected to Ψ aa+bb 7→ | i7→ quired by the state during adiabatic melting [14]. In Ψ with M =6 as f | i 1 Ψ = eiα0 Ψ12,0 + Ψ0,12 +√6eiα2 Ψ10,2 + Ψ2,10 +√6eiα4 Ψ8,4 + Ψ4,8 +√12eiα4,1 Ψ4 | fi 8h (cid:16)| sf i | sf i(cid:17) (cid:16)| sf i | sf i(cid:17) (cid:16)| sf i | sf i(cid:17) | ex,1i ++√6eiα4,2 Ψ4 +√6eiα6 Ψ6,6 +√12eiα6,1 Ψ6 +√2eiα6,2 Ψ6 . | ex,2i | sf i | ex,1i | ex,2ii 8 themomentumdistributionsandmomentumcorrelations of the final states. In the MI regime for J = 0 and U > V the color- symmetricgroundstate manifoldis M degenerate. For n finiteJ,thedegeneracyisbrokenand(cid:0)g((cid:1)n,M)states(obtained after symmetrization in Sˆ,Rˆ) are not coupled by the evolution. As shown below for large M we can ap- proximate the initial MI state Ψ2n as an equal super- ps positionoftheg(n,M)states. I(cid:12)nthi(cid:11)sway,usingtheadi- (cid:12) abaticapproximationwecanmapthese g states into the lowestg statesinthe SFregime. Wecalculatethelowest FIG. 6: (a) Momentum space particle number correlations eigenstatesoftheBHMinthe SFregimeU =V =0and hˆnanâi of the final melted state |Ψ i for one component (it k q f theirdegeneraciesf(l)suchthat f(l)=g. Evenwith- does not matter which due to color symmetry) for M = 6. l out calculating the exact coefficiePnts of the final g states (b) The inter-species momentum space particle number correlations hˆnanˆbi for |Ψ i. in the SF eigenbasis, we can compute the momentum k q f distribution and the momentum space particle number correlationsofthe finalmelted state to goodapproxima- tion. where we have now written explicitly the dynamical phasesacquiredbyeachofthestatesinthesuperposition. Anobservableofexperimentalsignificanceisthemomen- 1. MI limit tum distribution for either component. In fact the momentum distribution nâ = nˆb is not dependent on h ±ki h ±ki IfU/V >1andJ =0thephase-separatedstate Ψ2n thepreciseformoftheexcitedstates Ψex sinceallstates ps in the same degenerate manifold hav|e theisame expecta- belongs to the ground state manifold 2n 2n.(cid:12) It i(cid:11)s G ⊂ S (cid:12) tion value ( nâ = 1 in this case). The resulting mo- constructedasanequalsuperpositionofthe N/2 atom- h ±3πdi 2 pair states and distributed over M =(cid:0) Nn/(cid:1)2 sites. mentum distributionis shownin Fig.5(b), andgivesthe |◦i |•i These atom-pair states belong to the ground state man- fraction of the population outside the zero-momentum ifold. However, only g(n,M) N/2 of them define the mode (i.e. depletion of the Bose-condensate) for either ≤ n component as D =1/12 8.3%. Both the intra-species, symmetrizedground-statemanif(cid:0)old (cid:1)2n andarenotcou- G nâ nâ , and the inter-≈species, nâ nˆb , momentum pled by evolution for finite J. We showedin section VA h ±k ±qi h ±k ±qi thate.g. g(3,6)=3andthatforM =6,thestate Ψ6 spaceparticlenumbercorrelationsdodependonthepre- ps cise form of the excited states Ψ . Neither the mo- isasuperpositionof3statesthatevolveinto3differ(cid:12)entfi(cid:11)- ex (cid:12) | i nalSFstates. Ingeneral,wecancalculatethedimension mentum distributions nor the momentum space particle of the subspace , g(n,M), by noticing that it is iso- number correlations depend on the relative dynamical 2n G morphictothenumberofcolorsymmetricconfigurations phases. In Fig. 6(a)(b) these correlations are plotted us- in a bracelet (c.f. Fig.1) with n beads of one color and ingtheexcitedstatesdeterminedfromfirst-orderpertur- M nbeadsoftheothercolor. Suchmathematicalstruc- bation theory with U/J =0.01 and V/U =0.1. − tures have been extensively studied [31, 32] and there areanalyticalformulasavailablefortheirproperties. For B. Large systems even M, the number of complementable bracelets with M beads and half of them of each color is given by [31] entTahneglfiendaMl SIFstsattaet|eΨoabat+abinbiedisvgiiaveandibaybatic melting the g(M/2,M)= 12[F(M2 )+2M2−2+(cid:0)MM24−−21(cid:1)] M2 odd g(M/2,M)= 1[F(M)+2M2−2+ M2 ] M even, Ψ = 1 ⌊M/2⌋ M 1/2 Ψ2n , (10) 2 2 (cid:0)M4(cid:1) 2 | fi √2M−1 (cid:18)n(cid:19) f where F(n)= 1 Φ(n/d) 2d−1 +Φ(2n/d)2d−1 and nX=0 (cid:12) (cid:11) 2n d|n d−1 (cid:12) Φ(x) is Euler’s toPtient function(cid:0). Fo(cid:1)r bracelets with odd where Ψ2n Ψ2n if we assume that U/V is fixed total number of beads M or even M and n = M/2, the ps 7→ f 6 during(cid:12)melti(cid:11)ng an(cid:12)d th(cid:11)at the symmetries considered are number of complementable bracelets is given by [32] (cid:12) (cid:12) exhaustive and thus no level crossings occur. The nu- g(n,M)= 1[Fold(G,0, d:dM dn )+H(n,M)]. merical calculation for M = 6 in section VA hints that 2 { | ∩ | } it is very complicated to obtain the final state for larger Here, Fold(G(, ),x, a ,...,a ) gives the last term of 1 k · · { } M whichareinaccessiblewithexactnumerics. Resolving the list x,G(x,a ),G(G(x,a ),a ),... generated by 1 1 2 { } the exact splitting of the states in each subspace 2n in the cumulative application of the function S the MI limit and in the SF limit is both complex and φ(y) M impracticalforanarbitrarilylargeM. Instead,weintro- G(x,y)=x+ y duceheresomeapproximationsthatallowustocalculate M (cid:18)n(cid:19) y 9 FIG.7: (a)NumberofMIgroundstates(d )thatbelongto MI eachoftheg(9,18)=765independentstatesforfiniteJ fora latticewithN/2=N =N =18andU >V. (b)Numberof a b independent states with a given degeneracy d . The total FIG.8: (a)ThemomentumdistributionofthefinalSFstate MI number of independent states is equal to g(9,18) = 765. (c) |ΨiobtainedviaadiabaticallymeltingtheMIstate|Ψ i f aa+bb Firstenergylevels(∗)intheSFregime(U =V =0)andtheir with N = N = M = 20 = N/2 and U/V > 1. (b) The a b degeneracyf (barplot)forM =18. Thepoints(◦)showthe depletion D = 1−hnai/N of the melted state |ΨN/2i for 0 a f quadratic approximation to the dispersion relation. We plot U/V > 1 as a function of the number of particles N. The only thefirst states such that P f(l)=g for g=765. solid line shows the best logarithmic fit −0.0748(lnN/2)2+ l 0.8201lnN/2−1.3633 to thecalculated values (shown as ◦). to the elements of the input list a ,...,a and 1 k { } assuming a quadratic approximation E (l) to the non- ap M−2 2 if M even, n odd interacting single-particle energy E(l). Due to the color H(n,M)= n−21 symmetry, the SF eigenstates e are degenerate. The  (cid:0)⌊n/2⌋(cid:1) otherwise. | li ⌊M/2⌋ degeneraciesof the states f(l) can be calculated numeri- (cid:0) (cid:1) cally. We show in Fig. 7 (c) the eigenenergiesof the first Note that we recover g(2,6) = g(3,6) = 3 obtained for SF states and their degeneracies for M =18. the states Ψ4ps and Ψ6ps given in Sec.VA. To obtain the momentum distribution of Ψ2n we do (cid:12) (cid:11) (cid:12) (cid:11) not need to compute the exact distribution|offthie final (cid:12) (cid:12) states Ψ in the e basis (see the Appendix A) ex,m l 2. Superposition states becaus|e all stiates in th{|e sia}me degenerate manifold have the same expectation value nâ = nˆb . We just need to h qi h qi We now need to write our initial state Ψ2n as a su- calculate the momentum distribution of the final eigen- ps perposition of the g(n,M) states that ev(cid:12)olve(cid:11)indepen- states el and their degeneracies f(l) and sum their (cid:12) {| i} dently for finite J. Using the no-crossing rule we can contributions up to the highest excited state such that mapthe initial superposition Ψ2n atJ =0 into a final f(l) g. ps l ≈ superposition Ψ2n . The co(cid:12)efficie(cid:11)nts of the superposi- POn the other hand, we saw in Sec. VA that f (cid:12) tion are given(cid:12)by th(cid:11)e the number of bracelet configura- we need the exact form of the final states Ψex,m tions s(l) that(cid:12)can be obtained by mirror inversion Rˆ or in order to obtain the momentum-number |correlai- cyclic permutation Sˆ of any of the independent states tions. Note, however, that fm=1hem|nˆknˆq|emi = l = 1,..,g(n,M) in the subspace . We show s(l) for f Ψ nˆ nˆ Ψ whPere we sum over a com- M =18 in Fig. 7 (a). Note that Gg2ns(l)= M . Numer- pPlemte=1dhegeenx,emra|tekmqa|nifeoxl,dm.iThus, if we make the further l n ical results in Fig. 7 (b) show thPat most (7(cid:0)7%(cid:1)) of the g approximation that the state Ψ2fn fully populates the inequivalent configurations have the same degeneracy s highestexcitedmanifold we ca(cid:12)ncalc(cid:11)ulateits momentum (cid:12) andthatthisbehaviorincreaseswithsizeM (forM =22 space particle number correlations without knowing the already 92% of the g bracelets present the same degen- exactstates Ψex,m . This is justifiedfor largeg because | i eracy) . We can thus assume that the number of states the exact form of the highest excited states is concealed isomorphictoeachindependentconfigurationisconstant by the lower manifolds that are fully populated. s= M /g. Withinthisapproximation, Ψ2n isthusan We show the momentum distribution of the melted n f equa(cid:0)l s(cid:1)uperposition of the first g eigens(cid:12)tates(cid:11)of the SF state Ψf in Fig. 8 (a). The final state is highly de- Hamiltonian. (cid:12) pleted|. Aiplot of the depletion D = 1 n /N of the 0 −h i term ΨN/2 as a function of N is shown in Fig. 8 (b). | f i Itindicatesthebehaviorofthe depletionofthefullstate 3. SF limit Ψ because this term has the highest binomial coeffi- f | i cientinthesum(10)andaverylargeg. Onecanseethat In Appendix B we show how to calculate the eigen- the depletion increases very rapidly with the number of states e of the Hamiltonian (1) in the SF regime particles. l |{ }i where U = V = 0. The lowest eigenstates in ascending The momentum-space particle number correlations of order in energy for large M can be calculated efficiently the melted state Ψ are shown in figure 9. We observe f | i 10 Acknowledgements M. R. thanks K. Surmacz, J.J. Garc´ıa-Ripoll and R. A. Molina for fruitful discussions and the Spanish MCyT under programme Juan de la Cierva for support. This research was supported by the EPSRC projects EP/E041612/1 and EP/C51933/1, and by the National Science Foundation under Grant No. NSF PHY05- 51164. M. R. and D. J acknowledge support from the EU through the STREP project OLAQUI. FIG. 9: (a) The momentum correlations hˆnaˆnai for one k q mode for the state |Ψ i obtained via the adiabatic melting f of|Ψaa+bbiwithN =40andU/V >1. (b)Theinter-species APPENDIX A: SPLITTING DEGENERATE SF momentum correlations hˆnakˆnbqi of |Ψfi with N = 40. Note EXCITATIONS that a twin-Fock state(7) would only show acentral peak in both distributions. In the non-interacting SF limit where U = V = 0 the excited states are made up by expelling pairs of atoms of opposite momentum from the ground state Eq. (6). that the populationoutside the zero momentummode is Of particular relevance to the exact calculation of the higherfor the inter-speciesmomentumcorrelations. Our melting of Ψ in a small system are excitations in aa+bb calculationsclearlyindicatethatmeltingtwocomponent thesubspac|es ,wihere N N =0,and ,where M a b M−2 Mott insulators leads to quantum states with a compli- N N =4SandM =N|/2.−Her|ewe shallSdemonstrate a b catedstructureandlongrangecorrelations. Anumberof |that−the|lowest-lying excitations in and are M M−2 approximationswere necessaryto arriveatthese results. split by the inter- and intra-species iSnteractioSn to first- Webelievethatanexacttreatmentisunlikelytoreveala order in V/J and U/J. simple structure which allows an intuitive interpretation For the subspace the two lowest-lying excitations M of the melting process. aredegenerate. ThesSeformanexcitationsubspacewhich isspannedbystateswhereasinglepairofatoms,ofeither component,isexpelledfromthecondensatetoequaland oppositemomentawiththelowestpossiblemagnitudeas VI. SUMMARY e = 1 c†d† +c† d† ΨM−1,M−1 , | 1i √2(cid:16) + − − +(cid:17)| sf i 1 We have analyzed the quantum melting of correlated e = c†c† ΨM−2,M +d† d† ΨM,M−2 (A,1) two-component MI states into the SF regime using the | 2i √2(cid:16) + −| sf i − +| sf i(cid:17) adiabaticapproximation. Basedontheexperimentalfact where is used to denote the smallest quasi-momenta that the one component MI-SF transition can be per- ± δk. The inter- and intra-speciesinteractionmatrix ele- formednear adiabatically,we haveapplied it to the two- ± ments in this excitation subspace are mode case. Using the no-crossing rule we have shown that quantum melting of a two-component MI in an op- V(M +1 2 )+U(M + 1 ) 2U 1 1 tical lattice provides a viable route for engineering twin  − M M q − M . Fock states useful in interferometric experiments. The  2Uq1− M1 V(M +1− M3 )+UM  twin Fock states present only macroscopic occupation of Upondiagonalizingthis2 2matrixtheresultinggapand the final SF ground state although in general this is not × non-degenerate excitations ΨM and ΨM (which the case. We haveshownthatdue to the breakingofthe ex,1 ex,2 translational symmetry when there is nearest neighbor will in general be some supe(cid:12)(cid:12)rposit(cid:11)ion of(cid:12)(cid:12)the sta(cid:11)tes |e1i and e )areobtainedinfirst-orderperturbationtheory. hopping the final SF connected to a ground initial state | 2i In a similar way for subspace the three lowest- can be highly depleted even in the limit of adiabatic be- SM−2 lying excitations are degenerate. This excitation sub- havior. space is again spanned by states where a single pair of Using a set of well justified approximations, we have atoms is expelled from the condensate to equal and op- calculated the momentum correlations and momentum posite momenta with the lowest possible magnitude as spaceparticlenumbercorrelationsforthefinalstatesob- 1 tainedviameltingon-siteentangledstatescreatedinthe e = c†c† ΨM−4,M+2 +d† d† ΨM+2,M−4 , MI regime. Our results clearly indicate that the melted | 1i √2(cid:16) + −| sf i − +| sf i(cid:17) SFstatespresentacomplicatedstructureandlongrange e = 1 c†d† +c†d† ΨM+1,M−3 + ΨM−3,M+1 , correlations. Further workis neededto ascertainifthese | 2i 2(cid:16) + − − +(cid:17)(cid:16)| sf i | sf i(cid:17) correlations can be exploited for interferometry or other 1 e = c†c† ΨM,M−2 +d† d† ΨM−2,M , tasks. | 3i √2(cid:16) + −| sf i − +| sf i(cid:17)