Anomalous supersolidity in a weakly interacting dipolar Bose mixture on a square lattice
AAnomalous supersolidity in a weakly interacting dipolar Bose mixture on a squarelattice
Ryan M. Wilson, Wilbur E. Shirley,
2, 3 and Stefan S. Natu Department of Physics, The United States Naval Academy, Annapolis, MD 21402, USA Condensed Matter Theory Center and Joint Quantum Institute, Department of Physics,University of Maryland, College Park, Maryland 20742-4111 USA California Institute of Technology, Pasadena, CA 91125 USA ∗ We calculate the mean-field phase diagram of a zero-temperature, binary Bose mixture on asquare optical lattice, where one species possesses a non-negligible dipole moment. Remarkably,this system exhibits supersolidity for anomalously weak dipolar interaction strengths, which arereadily accessible with current experimental capabilities. The supersolid phases are robust, in thatthey occupy large regions in the parameter space. Further, we identify a first-order quantum phasetransition between supersolid and superfluid phases. Our results demonstrate the rich features ofthe dipolar Bose mixture, and suggest that this system is well-suited for exploring supersolidity inthe experimental setting.
Introduction – The physics of emergent, competing or-ders is central to the rich phenomenology of many con-densed matter systems, such as high- T c superconductorsand frustrated magnets. Recently, exciting developmentsin the cooling and trapping of magnetic atoms [1–8] anddiatomic molecules [9–13] offer promise that the physicsof competing orders will be accessible in exceptionallyclean, controllable forms of synthetic quantum matter.One striking example is the predicted supersolid phaseof strongly dipolar bosons loaded in an optical lattice,where the system simultaneously exhibits crystalline or-der and superfluidity. Indeed, checkerboard and stripesupersolids are predicted to emerge in dipolar lattice sys-tems, in addition to a variety of structured insulatingphases [14–29]. The study of supersolidity predates ex-periments with ultracold atoms [30], and was first pro-posed as a potential manifestation of solidity in superfluid He [30, 31]. Despite significant, long-standing interestin this phase and controversy over its existence [32–36],a supersolid ground state has yet to be observed in anexperimental setting.In the context of ultracold atoms in optical lattices,it is often the case that the long-range dipolar inter-actions, which are responsible for discrete translationalsymmetry breaking and the formation of crystalline or-der [37–40], are typically very weak, being easily over-whelmed by atomic motion (hopping), repulsive local in-teractions, and finite temperature, which favor spatiallyuniform phases. Other proposals suggest that strongereffective dipolar interactions can be achieved by usinglarge densities, though local interactions can easily de-stroy supersolid order in this semiclassical regime [41].Here, we show that the challenge posed by dipolar in-teractions that are weak compared to the other energyscales in the system is readily overcome by working with a binary mixture of bosons, where a non-dipolar species iscospatial and interacting with the dipolar system [42, 43].We calculate the zero-temperature ( T = 0) phasediagram for this system in the experimentally rele- vant square lattice geometry, using a site-decoupledGutzwiller mean-field method [44]. A key parameter inour theory is the local interspecies interaction strength,which encourages translational symmetry breaking, andthus the formation of a supersolid. Our results demon-strate that the supersolid phase occupies an anomalouslylarge region in the phase diagram compared to scalardipolar Bose gases, and persists even for very weak dipo-lar interaction strengths. Thus, we propose that the bi-nary dipolar Bose mixture is a novel system for exploringthe interplay between superfluidity and crystalline order UU U t t ˆ a GS MS1 MS2 V nn FIG. 1: Schematic showing a mixture of equal-mass dipo-lar (blue arrows) and non-dipolar (red circles) bosons in asquare optical lattice. The local intraspecies interactions U are assumed to be species independent, U is the local inter-species interaction strength, and t is the hopping rate. Near-est neighbor (nn), next nearest neighbor, and next-next near-est neighbor dipolar interactions are shown schematically bythe black arrows in the bottom panel. This panel shows thespatial dependence of the superfluid order parameter (cid:104) ˆ a (cid:105) forthe dipolar species in the supersolid regime (see text). Theground state (GS) is a checkerboard supersolid. Two nearlydegenerate, metastable excited states are depicted as MS1and MS2. a r X i v : . [ c ond - m a t . qu a n t - g a s ] J a n in the cold atoms context, and is a promising candidatefor the experimental realization of supersolidity. Dipolar mixture – The tight-binding Hamiltonian for anequal-mass mixture of dipolar ( σ = 1) and non-dipolar( σ = 2) bosonic atoms reads (see Fig. 1)ˆ H = − t (cid:88) (cid:104) ij (cid:105) σ (ˆ a † σi ˆ a σj + h.c.) − (cid:88) i µ σ ˆ n σi + (cid:88) i,σσ (cid:48) U σσ (cid:48) n σi (ˆ n σ (cid:48) i − δ σσ (cid:48) ) + (cid:88) i 34 Hz [45],whereas U is typically many kHz. V nn should be abouttwice as large for Dy atoms under equivalent conditions.Thus, we can reasonably expect 0 . (cid:46) V nn /U (cid:46) . /ρ scaling. A key distinctionof our work is that we identify supersolid phases in thisrange of weak dipolar interaction strengths for a dipolarBose mixture. In contrast, dipolar interaction strengthsof V nn /U (cid:38) . T = 0 phase diagram of Eq. (1), and intro-duce a spatially varying superfluid order parameter (cid:104) ˆ a σi (cid:105) for each species σ [44]. Throughout, we find groundstates with either uniform or checkerboard spatial or-der. When dipolar interactions beyond nearest-neighborare considered, our method unveils a manifold of nearly-degenerate, metastable excited states with supersolid or-dering at multiple wave vectors [18]. Examples are de-picted schematically in Fig. 1 for a 4 × FIG. 2: (Left) Ground state phase diagram for U /U = 0 . V nn /U = 0 . 1, exhibiting a large supersolid region. Thelight blue lobes correspond to Mott insulators with spatiallyuniform total density, and checkerboard order in the individ-ual species. The dark blue lobes correspond to Mott insu-lators with checkerboard structure in the total density. Thered regions correspond to M n B / SS phases, the pink regionscorrespond to SS phases, and the white region correspondsto a spatially uniform superfluid (SF). The solid (dashed)black lines show second-order (first-order) phase transitions.(Right) Phase diagram at t = 0, varying U . All phases areMott insulators, with coloring equivalent to that in the leftpanel. (See text for details.) period boundary conditions (MS1 & MS2), and parame-ters t/U = 0 . µ /U = µ /U = 2 . U /U = 0 . 9, and V nn /U = 0 . 1. The metastable states are gapped fromthe ground state by an energy proportional to the dipo-lar interaction strength. Because uniform and checker-board orders possess an AB sublattice symmetry, weconsider only the nearest-neighbor part of the dipolarinteractions and specialize to a 2 × µ = µ ≡ µ . We vary µ as a free parame-ter in the theory, which controls the total atom number N = (cid:80) σi n σi , where n σi = (cid:104) ˆ n σi (cid:105) . Because the dipolarinteractions break the interspecies symmetry of the sys-tem, this choice produces a number imbalance that scaleswith V nn . The imbalance remains relatively small, how-ever, for the weak dipolar interactions we consider here.We focus on the regime 0 < U < U , which discouragesspatial demixing of the species. Results – In the left panel of Fig. 2, we present thephase diagram obtained for U /U = 0 . V nn /U =0 . t and µ . For larger t , correspond-ing to shallower lattice depths, the system is a spatiallyuniform superfluid (SF), characterized by non-zero val-ues of the k -space superfluid order parameters ˜ α σ ( k ) = (cid:80) i e i k · ρ i (cid:104) ˆ a iσ (cid:105) at k = ( k x , k y ) = (0 , t is de-creased, the superfluid order parameter(s) acquire weightat k = ( π, π ), signifying the transition to a checkerboardsupersolid phase. Supersolidity can manifest in two dis-tinct ways in this bosonic mixture: both species canexhibit supersolid order (the SS phase), or the dipolarspecies can transition directly from a SF to a checker-board Mott insulator while the non-dipolar species re-mains superfluid. In the latter case, the dipolar speciesforms an effective checkerboard potential for the non-dipolar species due to their mutually repulsive interac-tions, which results in superfluidity with density-waveorder, or supersolidity, for the non-dipolar species. Wedenote the Mott insulator phases by M n A n B , where n A ( B ) are the integer occupations of the A ( B ) sublattice sites.This phase diagram possesses a tri-critical point betweenthe ¯M , SS, and M / SS phases, though we do not studythis point in detail here.The SF-SS transition occurs at larger densities, for µ/U (cid:38) . 5, and is second-order, indicated by the solidblack line in Fig. 2. In contrast, the transition to theM /SS phase, where the dipolar species is in the M phase and the non-dipolar species is SS, occurs at smallerdensities and is strongly first-order, indicated by thedashed black line in this figure. We note that first-ordertransitions between purely insulating and SF phases werepredicted in previous theoretical studies of non-dipolarBose mixtures [49–51], and a first-order SF-SS transitionwas predicted for hard-core dipolar bosons on a triangu-lar lattice [28]. The presence of a first-order superfluid-supersolid phase transition for weak dipolar interactionsis a new feature of the system we consider here.We demonstrate the first-order nature of the M / SSto SF phase transition in Fig. 3, where the k -space su-perfluid order parameters at k = ( π, π ) are shown for µ/U = 1 . 25 in panel (a) and µ/U = 2 . 25 in panel (b),corresponding to horizontal cuts across the left panel ofFig. 2 (with U /U = 0 . V nn /U = 0 . / M insulator, through theM / SS and SS phases, to a SF phase. For µ/U = 1 . α ( π, π ) = 0 for all values of t/U , so only ˜ α ( π, π ) isshown in panel (a). Here, the transition from a M /M insulator to a M / SS supersolid is second-order, whilethe transition to a SF is clearly discontinuous, and first-order. By smoothly following our ground state solutionsfrom either side of the transition region, we find that˜ α ( π, π ) is multivalued for 0 . (cid:46) t/U (cid:46) . 06; this isindicative of hysteresis, which is a feature of first-orderphase transitions. Notice that a first-order transitionalso exists between the M /SS and SS phases, shownin Fig. 2, though the hysteresis area of this transition isnotably smaller.For sufficiently small t , corresponding to deeper lat-tices, both species enter Mott insulating phases, indi-cated by the blue lobes in the left panel of Fig. 2. The FIG. 3: (a) Supersolid order parameter as a function of t/U for V nn = 0 . U , U = 0 . U , and µ = 1 . U . (b) Sameas top, except for µ = 2 . U . The blue regions correspondto Mott insulators, the red regions correspond to M n B / SSphases, the pink corresponds to a SS, and the white regionscorrespond to uniform superfluids (SF). All transitions aresecond-order except the transition to a SF in (a), which isstrongly first-order. The double-valued order parameter ischaracteristic of hysteresis at a first-order transition. larger, light blue lobes correspond to insulating phaseswith a spatially uniform density of n atoms per site.When n is even, the individual species form checkerboardMott insulators, where n A + n A = n B + n B = n .When n is odd, degeneracies exist between insulatingphases with different combinations of n σA and n σB . Forexample, the first Mott lobe in Fig. 2 corresponds to n = 1, and has a degeneracy between the M /M andM /M phases. The third Mott lobe corresponds to n = 3, and has a degeneracy between the M /M andM /M phases. These phases are labeled ¯M and ¯M ,respectively, in Fig. 2. We note that this degeneracy isa consequence of our choice µ = µ , and is broken if weinstead enforce equal total atom number, N = N . In-terestingly, the Mott lobes with uniform n are separatedby smaller lobes, wherein the dipolar species forms acheckerboard insulator and the non-dipolar species formsa uniform insulator, resulting in a Mott insulator phasewith checkerboard ordering in the total density.To explore this further, we calculate the t = 0 phasediagram as a function of µ and U for V nn /U = 0 . U , and shrink linearly as U → U . For t > 0, theselobes melt into M n B /SS supersolid phases. At exactly U = U , these phases vanish, and the system only sup-ports insulting phases with uniform total density.The tendency for ground states to acquire checker-board density-wave order can be understood intuitivelyfor this system, as this minimizes the nearest neigh-bor contributions to the dipolar interaction energy ina square lattice geometry. This ordering is preferred FIG. 4: Ground state phase diagram for U /U = 0 . 99 and V nn /U = 0 . 01, exhibiting supersolidity for very weak dipolarinteractions. The light blue lobes correspond to Mott insu-lators with spatially uniform total density, the red regionscorrespond to M n B / SS phases, the pink regions correspondto SS phases, and the white region corresponds to a spatiallyuniform superfluid. The solid (dashed) black lines in the mainpanel show second-order (first-order) phase transitions. Thelower (upper) inset shows the supersolid order parameters for t/U = 0 . µ/U = 3 ( µ/U = 2). In the insets, thesolid line corresponds to the dipolar species ( σ = 1) and thedashed line corresponds to the non-dipolar species ( σ = 2).The first-order phase transition is apparent as a discontinuityin the order parameter near µ/U (cid:39) . 06 in the upper inset. by the interspecies interactions ( U > U /U > . t is sufficiently small and U iscomparable to U . This suggests that supersolidity maypersist for very small dipolar interaction strengths, pro-vided t and 1 − U /U are sufficiently small.In Fig. 4, we plot the phase diagram for U /U = 0 . V nn /U = 0 . 01, corresponding to very weak dipo-lar interactions, as a function of t and µ . We note thatvery small M / M and M / M Mott lobes exist near µ/U = 2 and µ/U = 3 and t ∼ 0, respectively, but areomitted from this diagram due to their vanishingly smallsize. Strikingly, supersolid regions exist at small t , be-tween adjacent checkboard Mott lobes, and still occupya significant region of the phase diagram. The insets inFig. 4 show the checkerboard supersolid order parameter˜ α i ( π, π ) for the dipolar species (solid black lines) and thenon-dipolar species (dashed black lines) for t/U = 0 . µ/U = 2,shown by the vertical black line to the left of the in-set. Here, the phase transitions from Mott insulator tosupersolid are all continuous, and second-order. The up-per inset corresponds to a cut near µ/U = 3. Here, the ¯ M -SS transition is second-order, while the transi-tion from M /SS to SS is discontinuous, and first-order;this is consistent with the phase diagram in Fig. 2 for V nn /U = 0 . 1. Additionally, the Mott insulator to SFtransitions are first-order for larger t , as indicated by thedashed black lines near the tips of the Mott lobes. Thisis consistent with the findings of Refs. [49–51], where afirst-order superfluid-insulator transition is predicted fora non-dipolar Bose mixture.Though our discussion has focused on interspecies in-teractions 0 < U < U , we note that supersolidity per-sists for U > U as well. We have performed anal-ogous calculations to those described above, but with µ > µ chosen to balance the total particle number. TheMott insulator states in this case are of the checkerboardform M n / M n for all non-vanishing (nearest-neighbor)dipolar interaction strengths. For V nn /U = 0 . 01 and U /U = 1 . 01, we find SS regions between the Mottlobes at small t , similar to those shown in Fig. 4. For V nn /U = 0 . U /U = 1 . 1, we find a large SS regionthat extends to larger t , well beyond the Mott lobes.While increasing U will eventually lead to phase sepa-ration at finite t , the Mott lobes will possess checkerboardorder for any U > 0, and we thus expect SS regions toexist between these Mott lobes for sufficiently small t . Discussion – In an experiment, the dipolar Bose mix-ture will inevitably have unequal masses, and thusspecies-dependent hopping. We note that the supersolidregions span a large range of t values, so supersolidityshould persist for moderate differences in the species-dependent hopping rates. Additionally, the presence oflow-lying metastable states at energies ∼ V nn above theground state suggests that very low temperatures will benecessary to realize pure checkerboard ground states. Forthe Er example discussed above, with V nn (cid:39) h × 34 Hz,temperatures on the order of a few nK are sufficient todiscourage population of these metastable states. Still,we note that these excited states are supersolid in nature,and should permit superfluid transport and show signa-tures of crystalline order in Bragg spectroscopy [52, 53] atsuper-critical temperatures. In previous theoretical stud-ies of single species dipolar systems, beyond mean-fieldeffects were found to enhance the Mott lobes, and onlyslightly diminish the supersolid regions [25, 54, 55]. Wetherefore expect the supersolid phases we find here to berobust against quantum effects at intermediate densities,even for V nn /U = 0 . 01 (c.f. Fig. 4).The subject of supersolidity in condensed matter sys-tems has a rich history, and remains an active area of re-search [56]. Despite evidence of supersolid phases in non-equilibrium systems [40], the observation of this phase asa ground state remains an open problem. Lattice ana-logues of supersolid phases in long-range interacting sys-tems are a promising avenue to explore this physics, asthe lattice naturally enhances correlations by suppress-ing the kinetic energy, while the long range interactionsintroduce a natural length scale for breaking discrete spa-tial symmetry. Here, we have shown that a dipolar Bosemixture on a square lattice is a promising candidate forrealizing supersolid ground states, even in the presenceof anomalously weak dipolar interaction strengths. Acknowledgements – We acknowledge B. M. Ander-son for helpful conversations in the early stages of thiswork. RW acknowledges partial support from the Officeof Naval Research under Grant No. N00014115WX01372,and from the National Science Foundation under GrantNo. PHY-1516421. WS acknowledges support from aJQI-PFC Seed Grant. SN thanks the LPS-CMTC, LPS-MPO-CMTC, JQI-NSF-PFC, and ARO-MURI for sup-port. ∗ Electronic address: [email protected][1] A. Griesmaier, J. Werner, S. Hensler, J. Stuhler, andT. Pfau, Phys. Rev. Lett. , 160401 (2005).[2] T. Lahaye, T. Koch, B. Fr¨ohlich, M. Fattori, J. Metz,A. Griesmaier, S. Giovanazzi, and T. Pfau, Nature ,672 (2007).[3] T. Lahaye, C. Menotti, L. Santos, M. Lewenstein, andT. Pfau, Rep. Prog. Phys. , 126401 (2009).[4] M. Lu, N. Q. Burdick, S. H. Youn, and B. L. Lev, Phys.Rev. Lett. , 190401 (2011).[5] M. Lu, N. Q. Burdick, and B. L. Lev, Phys. Rev. Lett. , 215301 (2012).[6] K. Aikawa, A. Frisch, M. Mark, S. Baier, A. Rietzler,R. Grimm, and F. Ferlaino, Phys. Rev. Lett. , 210401(2012).[7] G. Bismut, B. Laburthe-Tolra, E. Mar´echal, P. Pedri,O. Gorceix, and L. Vernac, Phys. Rev. Lett. , 155302(2012).[8] M. A. Baranov, M. Dalmonte, G. Pupillo, and P. Zoller,Chemical Reviews , 5012 (2012).[9] K.-K. Ni, S. Ospelkaus, M. H. G. de Miranda, A. Pe’er,B. Neyenhuis, J. J. Zirbel, S. Kotochigova, P. S. Julienne,D. S. Jin, and J. Ye, Science , 231 (2008).[10] J. Daiglmayr, A. Grochola, M. Repp, K. M¨ortlbauer,C. Gl¨uck, J. Lange, O. Dulieu, R. Wester, and M. Wei-dem¨uller, Phys. Rev. Lett. , 133004 (2008).[11] K. Aikawa, D. Akamatsu, J. Kobayashi, M. Ueda,T. Kishimoto, and S. Inouye, New J. Phys. , 055035(2009).[12] L. D. Carr, D. Demille, R. Krems, and J. Ye, New J.Phys. , 055049 (2009).[13] T. Takekoshi, L. Reichs¨ollner, A. Schindewolf, J. M. Hut-son, C. R. L. Sueur, O. Dulieu, F. Ferlaino, R. Grimm,and H.-C. N¨agerl, Phys. Rev. Lett. , 205301 (2014).[14] K. G´oral, L. Santos, and M. Lewenstein, Phys. Rev. Lett. , 170406 (2002).[15] D. L. Kovrizhin, G. V. Pai, and S. Sinha, Europhys. Lett. , 162 (2005).[16] V. W. Scarola and S. Das Sarma, Phys. Rev. Lett. ,033003 (2005).[17] S. Yi, T. Li, and C. P. Sun, Phys. Rev. Lett. , 260405(2007).[18] C. Menotti, C. Trefzger, and M. Lewenstein, Phys. Rev. Lett. , 235301 (2007).[19] I. Danshita and C. A. R. S´a de Melo, Phys. Rev. Lett. , 225301 (2009).[20] F. J. Burnell, M. M. Parish, N. R. Cooper, and S. L.Sondhi, Phys. Rev. B , 174519 (2009).[21] F. Cinti, P. Jain, M. Bononsegni, A. Micheli, P. Zoller,and G. Pupillo, Phys. Rev. Lett. , 135301 (2010).[22] C. Trefzger, M. Alloing, C. Menotti, F. Dubin, andM. Lewenstein, New J. Phys. , 093008 (2010).[23] L. Pollet, J. D. Picon, H. P. B¨uchler, and M. Troyer,Phys. Rev. Lett. , 125302 (2010).[24] M. Iskin, Phys. Rev. A , 051606(R) (2011).[25] C. Trefzger, C. Menotti, B. Capogrosso-Sansone, andM. Lewenstein, J. Phys. B , 193001 (2011).[26] J. M. Fellows and S. T. Carr, Phys. Rev. A , 051602(R)(2011).[27] T. Ohgoe, T. Suzuki, and N. Kawashima, Phys. Rev. B , 054520 (2012).[28] D. Yamamoto, I. Danshita, and C. A. R. S´a de Melo,Phys. Rev. A , 021601(R) (2012).[29] Z.-K. Lu, Y. Li, D. S. Petrov, and G. V. Shlyapnikov,Phys. Rev. Lett. , 075303 (2015).[30] G. V. Chester, Phys. Rev. A , 256 (1970).[31] A. J. Leggett, Phys. Rev. Lett. , 1543 (1970).[32] E. Kim and M. H. W. Chan, Nature , 225 (2004).[33] E. Kim and M. H. W. Chan, Phys. Rev. Lett. , 115302(2006).[34] I. A. Todoshchenko, H. Alles, H. J. Junes, A. Y. Parshin,and V. Tsepelin, JETP Lett. , 454 (2007).[35] H. Choi, D. Takahashi, K. Kono, and E. Kim, Science , 1512 (2010).[36] D. Y. Kim and M. H. W. Chan, Phys. Rev. Lett. ,155301 (2012).[37] A. van Otterlo and K.-H. Wagenblast, Phys. Rev. Lett. , 3598 (1994).[38] A. van Otterlo, K.-H. Wagenblast, R. Baltin, C. Bruder,R. Fazio, and G. Sch¨on, Phys. Rev. B , 16176 (1995).[39] G. G. Batrouni, R. T. Scalettar, G. T. Zimanyi, and A. P.Kampf, Phys. Rev. Lett. , 2527 (1995).[40] K. Baumann, C. Guerlin, F. Brennecke, and T. Esslinger,Nature , 1301 (2010).[41] A. B¨uhler and H. P. B¨uchler, Phys. Rev. A , 023607(2011).[42] R. M. Wilson, C. Ticknor, J. L. Bohn, and E. Timmer-mans, Phys. Rev. A , 033606 (2012).[43] W. E. Shirley, B. M. Anderson, C. W. Clark, and R. M.Wilson, Phys. Rev. Lett. , 165301 (2014).[44] D. S. Rokhsar and B. G. Kotliar, Phys. Rev. B , 10328(1991).[45] S. Baier, M. J. Mark, D. Petter, K. Aikawa, L. Chomaz,Z. Cai, M. Baranov, P. Zoller, and F. Ferlaino (2015),arxiv:1507.03500.[46] S. Ospelkaus, K.-K. Ni, D. Wang, M. H. G. de Miranda,B. Neyenhuis, G. Qu´em´ener, P. S. Julienne, J. L. Bohn,D. S. Jin, and J. Ye, Science , 853 (2010).[47] K.-K. Ni, S. Ospelkaus, D. Wang, G. Qu´em´ener,B. Neyenhuis, M. H. G. de Miranda, J. L. Bohn, J. Ye,and D. S. Jin, Nature , 1324 (2010).[48] M. Mayle, G. Qu´em´ener, B. P. Ruzic, and J. L. Bohn,Phys. Rev. A , 012709 (2013).[49] A. Kuklov, N. Prokof’ev, and B. Svistunov, Phys. Rev.Lett. , 050402 (2004).[50] A. Isacsson, M.-C. Cha, K. Sengupta, and S. M. Girvin,Phys. Rev. B , 184507 (2005). [51] D. Yamamoto, T. Ozaki, C. A. R. S´a de Melo, and I. Dan-shita, Phys. Rev. A , 033624 (2013).[52] J. Stenger, S. Inouye, A. P. Chikkatur, D. M. Stamper-Kurn, D. E. Prichard, and W. Ketterle, Phys. Rev. Lett. , 4569 (1999).[53] R. A. Hart, P. M. Duarte, T.-L. Yang, X. Liu, T. Paiva,E. Khatami, R. T. Scalettar, N. Trivedi, D. A. Huse, andR. A. Hulet, Nature , 211 (2015). [54] B. Capogrosso-Sansone, S. G. S¨oyler, N. Prokof’ev, andB. Svistunov, Phys. Rev. A , 015602 (2008).[55] B. Capogrosso-Sansone, C. Trefzger, M. Lewenstein,P. Zoller, and G. Pupillo, Phys. Rev. Lett.104