Stabilization of the p-wave superfluid state in an optical lattice
Y.-J. Han, Y.-H. Chan, W. Yi, A. J. Daley, S. Diehl, P. Zoller, L.-M. Duan
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] A ug Stabilization of the p-wave superfluid state in an optical lattice
Y.-J. Han , Y.-H. Chan , W. Yi , A. J. Daley , S. Diehl , P. Zoller , and L.-M. Duan Department of Physics and MCTP, University of Michigan, Ann Arbor, Michigan 48109 Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, Austriaand Institute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck, Austria (Dated: November 15, 2018)It is hard to stabilize the p-wave superfluid state of cold atomic gas in free space due to inelas-tic collisional losses. We consider the p-wave Feshbach resonance in an optical lattice, and showthat it is possible to have a stable p-wave superfluid state where the multi-atom collisional loss issuppressed through the quantum Zeno effect. We derive the effective Hamiltonian for this system,and calculate its phase diagram in a one-dimensional optical lattice. The results show rich phasetransitions between the p-wave superfluid state and different types of insulator states induced eitherby interaction or by dissipation.
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The observation of the s-wave superfluid state in aFermionic atomic gas represents a remarkable break-through in the study of many-body physics with ultra-cold atoms [1]. The Feshbach resonance plays an impor-tant role in those experiments, enhancing the interatomicinteraction so that the superfluid phase can be enteredat a temperature that is experimentally achievable. Itis of great interest to realize the superfluid state withother pairing symmetries as well. The p-wave superfluidstate is the next candidate for observation, and has at-tracted the interest of many experimental and theoreticalgroups [2, 3]. The p-wave Feshbach resonance has beenrecently observed in experiments [2], and can push thesingle-component Fermi gas to the strongly interactingregion and open a door towards observation of the p-wave superfluid state in this system. However, comparedwith the s-wave Feshbach resonance, a key difficulty withthe p-wave resonance is that the inelastic collision loss inthis system is typically large [2, 4, 5], which forbids ther-malization of the gas within the system lifetime [5].In this paper, we dicuss how a dissipation-inducedblockade mechanism can stabilize the p-wave superfluidstate in an optical lattice in the strongly interacting re-gion. Such a dissipation-induced blockade has been re-ported in recent experiments to realize the Tonks gas[6] or for simulation of effective three-body interactions[7] with cold bosonic atoms. We apply this mechanismto stabilize the single-component Fermion system in anoptical lattice in the presence of the p-wave Feshbach res-onance. The p-wave Feshbach resonance has been con-sidered recently on single sites in a very deep optical lat-tice [4, 8]. Here, instead, we focus on the many-bodyphysics by deriving an effective Hamiltonian for this sys-tem, taking into account the atomic hopping in the re-duced Hilbert space caused by the dissipation-inducedblockade. This effective Hamiltonian provides a startingpoint to understand the quantum phases. We computethe phase diagram of the system explicitly with well-controlled numerical methods for an anisotropic lattice where the atom tunnelling is dominantly along one di-mension. The results show rich phase transitions betweenthe p-wave superfluid state, a dissipation-induced insula-tor state, the Mott insulator state, and different kinds ofmetallic states. Although these results are obtained fromone-dimensional calculations, we expect these phases tocorrespond also to similar phases in higher dimensions.We consider a single-component Fermi gas near a p -wave Feshbach resonance. If this strongly interacting gasis loaded into an optical lattice, many different Blochbands can be populated, in particular when the resonanceis broad (as it is the case for some recent experiments [2]).However, we can derive an effective single-band model forthis system that is independent of the interaction details.Following a strategy similar to the s-wave Feshbach res-onance case [9], we first analyze the local Hilbert spacestructure on a single lattice site. When we have zero orone atom on the site i , the states are simply denoted by | i i and | a i i = a † i | i i , respectively. For the case of twoatoms on a single site, the exact two-body physics hasrecently been calculated [8], and there are several two-body energy levels separated by an energy difference ofthe order of the band gap. If we assume that the systemtemperature is significantly below the band gap, only thelowest two-body state is relevant. We refer to this state asa dressed molecule level, and denote it by | b i i = b † i | i i .Note that the wave function of | b i i in general includescontributions from many of the original atomic orbitals[8]. It has a p-wave symmetry in space and is antisym-metric under exchange of the two atoms in this dressedmolecule.If more that two atoms coming to a single site, differentfrom the s-wave case, the state will not be stable due tobig three-particle inelastic collision loss [4]. At first sight,this seems to mean that the system will become unsta-ble. However, in a lattice, there is a subtle dissipation in-duced blockade mechanism [6] which forbids populationof the unstable three-particle state and thus stabilizesthe whole system. The basic idea is illustrated in Fig. 1. D t g t FIG. 1: (Color Online) Illustration of the dissipation-inducedblockade for multiple occupation (more than two) of a singlelattice site. Similar to a two level transition with a detuning∆ and a decay rate γ for the target level, the effective hop-ping in this case is suppressed by a factor t / ( γ + ∆ ), where t is the atomic tunneling rate. The three-particle state | i has a large bandwidth char-acterized by its inelastic collision rate γ and an energyshift characterized by the on-site atom-dressed-moleculeinteraction energy ∆ . If a single atom tunnel through abarrier with a hopping rate t to form this state | i , theprobability of getting | i is given by t / (cid:0) γ + ∆ (cid:1) (sim-ilar to a two-level transition with a detuning ∆ and abandwidth γ ). So the net collisional loss of the system isbounded by γ eff = γt / (cid:0) γ + ∆ (cid:1) ≤ t / (2∆ ) no mat-ter how large the inelastic collision rate γ is. Near theFeshbach resonance, the atom-dressed-molecule interac- tion energy ∆ is comparable with the lattice band gap(thus much larger than t ) [11], the net collisional loss γ eff is therefore small compared with the atomic hopping re-quired to thermalize the system [10]. The reduction ofpopulation in the noisy state | i is called the dissipation-induced blockade (or interpreted as the quantum Zenoeffect [6]; the blockade is actually induced by both dissi-pation and interaction when ∆ and γ are comparable).Due to this mechanism one can achieve many-body ther-mal equilibrium in a lattice even if there is big inelasticcollision loss.Due to the dissipation-induced blockade discussedabove, on each site we have only three relevant levels: | i i , | a i i , and | b i i as the low energy configurations. Theenergy difference between the sates | a i i and | b i i can betuned with the external magnetic field via the Feshbachresonance. We then take into account the atomic hop-ping, which changes the level configurations of the neigh-boring sites under the constraint that the atom numberbe conserved. Thus, there are only three possible pro-cesses, as illustrated in Fig. 2. Corresponding to theseconfiguration changes, the general Hamiltonian for thislattice system then takes the form H = X i h (∆ b † i b i − µ ( a † i a i + 2 b † i b i ) i − X h i,j i P (cid:2) t a + i a j + t ( b + i − b + j ) a i a j + t b + i b j a + j a i + H.c. (cid:3) P, (1)where µ is the chemical potential and ∆ is the energy de-tuning of the dressed molecule controlled with the mag-netic field. The value of ∆ characterizes the on-siteatomic interaction magnitude. The hopping rates for thethree processes illustrated in Fig. 2 are different, in gen-eral, due to the contributions from different bands (be-cause the dressed molecule is a composite particle withpopulation in multiple bands). The hopping takes placein the low energy Hilbert space specified by the projector P ≡ O i ( | i i h | + | a i i h a | + | b i i h b | ), and the summationin Eq. (1) is over all neighboring sites h i, j i .The Hamiltonian (1), although much simplified com-pared with multi-band models, is still complicated, and ingeneral it does not allow exact solutions. To understandsome basic physical properties of the system, we limitourselves in the following to the one-dimensional latticewhere the atomic hopping along the other two dimensionsare turned off with a high lattice barrier. In this case,we can solve this model with well controlled numericalsimulations, and the results show rich phase transitionsbetween the p-wave superfluid state and different kindsof insulator and metallic phases. We expect most of the a b t t t ab t t t Site i Site j
FIG. 2: (Color Online) Illustration of different atomic hop-ping processes over the two neighboring sites described by theHamiltonian (1), where each site has only three possible levelconfigurations. phases found in the one-dimensional case have counter-parts in higher dimensions. In particular, the p-wave su-perfluid state characterized by a quasi-long-range pairingorder with diverging pair susceptibility in one dimensioncan be easily stabilized to a true long-range order if weallow weak tunneling between the one-dimensional tubes[12].In the numerical simulation, we use the iTEBD algo-rithm, a recently developed method related to densitymatrix renormalization group techniques [13, 14], whichallows direct calculation of the physical properties in the
FIG. 3: (Color Online) The dressed molecule number n b (thedouble occupation probability) and the atom number n a (thesingle occupation probability) shown as the function of ∆ /t (with a fixed µ/t = − .
5) in Fig. (a) and µ/t (with a fixed∆ /t = − .
5) in Fig. (b). The non-analyticities of these curvessignal a number of quantum phase transitions from the dis-sipation induced insulator state (DII), to a p-wave superfluidstate (PS), to a Mott insulator state (MI), to a normal mixturestate (NM), and finally to normal Fermi gas phase (NFG). thermodynamic limit. The algorithm has been shown towork with high precision compared to known results forthe Hubbard model [15]. For simplicity, we take the hop-ping rates t = t = t = t , and use t as the energyunit. Then we have effectively only two parameters, ∆and µ (in units of t ) in the Hamiltonian (1). To figureout the complete phase diagram with respect to thesetwo parameters, we calculate ∂ h H i /∂ ∆ and ∂ h H i /∂µ as functions of ∆ or µ for the ground state of H , and usethe characteristics of these curves to identify the phasetransition points.In Fig. 3, we show n b = D b † i b i E and n a = D a † i a i E as functions of ∆ and µ . One can clearly see severalquantum phase transitions from this figure. First, witha fixed chemical potential µ = − . t , one has n b = 1and n a = 0 with a large negative detuning ∆ (corre-sponding to strong attractive atomic interaction). In thiscase, each site is doubly occupied with two atoms. Morethan two atoms cannot move to the same site becauseof the dissipation-induced blockade. So this is an insula-tor phase stabilized by the dissipation. As one increases∆ with ∆ > − . t , the number of atoms on each sitebegins to fluctuate. If one looks at the pair correlation D b † i b j E , it shows quasi-long-range behavior with slow al-gebraic decay. In Fig. 4(a), we show this correlation FIG. 4: (Color Online) The correlation functions shown forthe p-wave superfluid phase ((a) and (b)), the Mott insulatorphase ((c) and (d)), and the normal mixture phase ((e) and(f)). Figures (a), (c) and (e) show in k -space the Fourier trans-form P k of the pair correlation D b † i b j E . Figures (b), (d) and(f) show the charge density correlations of h n bi n bj i , h n bi n aj i and h n ai n aj i in real space. We take µ/t = − . in the k -space, defined as P k = (1 /N ) N X ρ =0 D b † i b i + ρ E e ikρ .The correlation P k is peaked sharply at k = 0. Thiscorresponds to the p -wave superfluid phase. The p -wavecharacter is inherited from the p -wave symmetry of thedressed molecule in space b i (or the atomic pair on thesame site). The p -wave nature of the pairing is also man-ifested in the atomic pair wavefunction at different sites h a i a j i , which is obviously antisymmetric under exchangeof the sites. In the one-dimensional case, the p -wave su-perfluid state is characterized by a quasi-condensate ofthe atomic pairs with a diverging pairing susceptibility.If one allows weak coupling between the one-dimensionaltubes in the optical lattice, the p -wave quasi-condensatecan easily be stabilized into a real condensate with a truelong range pairing order.If one further increases ∆ in Fig. 3(a), one entersa phase where the total particle number per site n =2 n b + n a is fixed at 1 (although the double occupationprobability D b † i b i E varies with ∆). This is a Mott insula-tor state with a finite gap to charge excitations. This canbe seen more clearly in Fig. 3(b), where we fix the detun-ing ∆, and show n b and n a as functions of the chemical FIG. 5: (Color Online) The complete phase diagram of thesystem versus two parameters µ/t and ∆ /t . The five differentphases are marked with the same notation as in Fig. 3. Theblack dashed lines a and b correspond to the parameters takenin Fig. 3(a) and Fig. 3(b). potential µ . For this phase, the number density doesnot vary with µ , so the system is incompressible as oneexpects for a Mott insulator phase. The correlation func-tions for this phase is shown in Fig. 4, where both thepair correlation and the charge density wave correlationsare of short range. As one further moves to the right sidein Fig. 3(a), there are two other phases: the normal mix-ture (NM) and the normal free gas (NFG). Both of thesetwo phases are of a metallic nature with a finite com-pressibility (see Fig. 3(b)). The difference is that in thenormal mixture phase, some sites are doubly occupied(with a finite D b † i b i E ). Several kinds of correlation func-tions for the normal mixture phase are shown in Fig. 4,and all of them decay rapidly with distance. In the NFGphase, the double occupation probability D b † i b i E reducesto zero, and one has a conventional single component freeFermion gas.We have calculated the phase transition points for all∆ and µ , and the result is summarized in Fig. 5 to givea complete two-parameter phase diagram. The five dif-ferent phases are marked there. Under real experimentalconditions, there is typically a weak global harmonic trapin addition to the optical lattice potential. As one movesfrom the trap center to the edge, the effective chemicalpotential µ gradually decreases under the local densityapproximation. So with a fixed interaction parameter ∆,a line in the phase diagram of Fig. 5 across different µ gives the phase separation pattern of the Fermi gas in aharmonic trap. With a large positive ∆, one has a Mottinsulator state in the middle, surrounded by a normalFermi gas. The phase transitions are most rich for small | ∆ | , where one can cross all the five different phases fromthe trap center to the edge. For large negative ∆, theregion of the p -wave superfluid state increases, but theMott insulator and the normal mixture states eventuallydisappear when ∆ < − t . As the p-wave superfluid state has a large stability region in the phase diagram, such aphase can be prepared experimentally by adiabaticallyramping the Hamiltonian parameters following certaintrajectories that suppress the three-particle occupation[7].In summary, we suggest that the p-wave superfluidstate near a Feshbach resonance can be stabilized inan optical lattice through a dissipation-induced block-ade mechanism. We have derived the effective Hamilto-nian to describe this strongly interacting system, takinginto account the restriction of the Hilbert space due tothis blockade mechanism. We solve the Hamiltonian inthe anisotropic one-dimensional lattice through exact nu-merical calculations, and the result suggests rich phasetransitions between the p-wave superfluid state and sev-eral kinds of insulator or metallic phases.We thank Ignacio Cirac, Jason Kestner, Wei Zhang,and Mikhail Baranov for helpful discussions. Thiswork was supported by the AFOSR through MURI, theDARPA, the IARPA, and the Austrian Science Founda-tion through SFB FOQUS. [1] C.A. Regal, M. Greiner and D.S. Jin, Phys. Rev. Lett. , 040403 (2004); M.W. Zwierlein et al. , Phys. Rev.Lett. , 120403 (2004); C. Chin et al. , Science ,1128 (2004); J. Kinast et al. , Science , 1296 (2005);M.W. Zwierlein et al. , Nature 435, 1047 (2005).[2] J. P. Gaebler, J. T. Stewart, J. L. Bohn, and D. S.Jin, Phys. Rev. Lett. 98, 200403 (2007); J. Zhang et al. ,Phys. Rev. A 70, 030702 (2004); J. Fuchs et al. (2008),arXiv:0802.3262; Y. Inada et al. Phys. Rev. Lett. 101,100401 (2008).[3] V. Gurarie, L. Radzihovsky, and A. V. 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