Probing the Flat Band of Optically-Trapped Spin-Orbital Coupled Bose Gases Using Bragg Spectroscopy
Wu Li, Lei Chen, Zhu Chen, Ying Hu, Zhidong Zhang, Zhaoxin Liang
PProbing the flat band of optically trapped spin-orbital-coupled Bose gases using Braggspectroscopy
Wu Li , Lei Chen , Zhu Chen , Ying Hu , Zhidong Zhang , and Zhaoxin Liang ∗ Shenyang National Laboratory for Materials Science,Institute of Metal Research, CAS, Wenhua Road, 72, Shenyang, China National Key Laboratory of Science and Technology on Computational Physics,Institute of Applied Physics and Computational Mathematics, Beijing 100088, China and Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, A-6020 Innsbruck, Austria
Motivated by the recent efforts in creating flat bands in ultracold atomic systems, we investigatehow to probe a flat band in an optically trapped spin-orbital-coupled Bose-Einstein condensate usingBragg spectroscopy. We find that the excitation spectrum and the dynamic structure factor of thecondensate are dramatically altered when the band structure exhibits various levels of flatness. Inparticular, when the band exhibits perfect flatness around the band minima corresponding to a near-infinite effective mass, a quadratic dispersion emerges in the low-energy excitation spectrum; in sharpcontrast, for the opposite case when an ordinary band is present, the familiar linear dispersion arises.Such linear-to-quadratic crossover in the low-energy spectrum presents a striking manifestation ofthe transition of an ordinary band into a flat band, thereby allowing a direct probe of the flat bandby using Bragg spectroscopy.
PACS numbers: 03.75.Kk, 67.85.-d, 64.70.Rh
I. INTRODUCTION
There have been intensive efforts in realizing flat bandsin various context of condensed-matter [1] and atomicphysics [2–6]. The motivation behind this search istwofold. First, a flat band, whose kinetic energy is highlyquenched compared to the scale of interactions, possessesmacroscopic level degeneracy, and as a result, interac-tions play a dominant role in affecting the system thathas given rise to many interesting quantum phases [3].Second, even more challenging is to create topologicalflat bands with nonzero Chern number [7], which canopen a new avenue for engineering a fractional topologicalquantum insulator [1, 2] without Landau levels promptedby the analogy to Landau levels [8] in condensed matterphysics. Motivated by the ongoing interests in creatingflat bands in ultracold atomic systems[2–7, 9], we addressbelow the problem of how to probe an arising isolated flatband in an optically trapped spin-orbital-coupled (SOC)Bose-Einstein condensate (BEC) [9] by using Bragg spec-troscopy.The key ingredient of our work consists in investigat-ing how the excitation spectrum and dynamic structurefactor of the system change when the band structurevaries its flatness. Our main results are as follows: (1) aquadratic dispersion (cid:15) ( k ) ∼ k emerges in the low-energyexcitation spectrum, if the band is perfectly flat in thevicinity of energy band minima (corresponding to an infi-nite effective mass); contrasting sharply, in the oppositecase when the BEC has an ordinary band, the famil-iar linear dispersion relation (cid:15) ( k ) ∼ k is found; (2) thestatic structure factor S ( k ) exhibits a crossover from lin-ear ( S ( k ) ∼ k ) to quadratic relation ( S ( k ) ∼ k ) in the ∗ E-mail: [email protected] momenta, when the band transforms from the ordinaryinto the flat band. Moreover, by relating the flatness ofthe band with the effective mass at the band minima andby using Feynman’s relation (cid:15) ( k ) = (cid:15) ( k ) /S ( k ) [10–12],we are able to directly connect the emerging quadraticdispersion in a perfect flat band case with the vanish-ingly small kinetic energy in the single-particle energy (cid:15) ( k ).The setting we consider to probe a flat band in a quasi-one-dimensional BEC with spin-orbit coupling(SOC)trapped in an optical lattice along x -direction is illus-trated in Fig. 1. In addition, we strongly confine theBEC in both y - and z - directions such that the dynamicsof the model system is effectively restricted to one dimen-sion. Experimentally, the setting in Fig. 1 can be realizedby combining Bragg spectroscopy [13, 14] and SOC [15–17] that are available in both BECs [18] and Fermi gases[19]. In particular, the one-dimensional(1D) SOC withequal Rashba and Dresselhaus contributions consideredin this work has been implemented in Ref. [18] by cou-pling two internal states of atoms Rb via Raman lasers.Very recently, Bragg spectroscopy has been employed toreveal the structure of the excitation spectrum in a BECwith SOC in the free space [20–22]. In particular, themeasurement of the static structure factor combined withFeynman’s relation [20–22] has allowed the experimentalverification of the emerging roton-maxon dispersion inthese systems. Building on this experimental progress inapplying Bragg spectroscopy in a free BEC with SOC,we propose that the Bragg spectroscopy in an opticallytrapped BEC with SOC in quasi-one-dimension can helpreveal the linear-to-quadratic transition in the low-energyspectrum predicted in this work and therefore provide adirect experimental probe of a flat band. Theoretically,the Gross-Pitaevskii equation (GPE) has been shown todescribe well, at the mean-field level, both the static and a r X i v : . [ c ond - m a t . qu a n t - g a s ] A p r K K Ω Ω θ R θ L y yZZ FIG. 1. (color online).On the left Bragg spectroscopy probesa single-particle flat band generated by an optically trappedspin-orbit-coupled Bose gas. Right panel: schematic setup forimplementing a flat band realized in 1D spin-orbit-coupledBose gas proposed by Ref. [9]. Here, Ω and Ω are the Rabifrequencies of the Raman lasers for generating SO coupling.The interference of other two counterpropagating laser beamslabeled by k and k generates an optical lattice. Bragg spec-troscopy can be described by the dynamic structure factor ofthe model system. the dynamic properties of a BEC with SOC [9, 23–26].The validity of the GPE can be tested a posteriori byevaluating the quantum depletion of the condensate. Fora more rigorous proof of validity of GPE, we refer to theSupplemental Material in Ref. [27].This paper is organized as follows. In Sec. II, we shallbegin with briefly describing the model system in which aflat band can arise following Ref. [9]. Then, in Sec. III weshow how Bragg spectroscopy can present as an efficienttool to quantitatively probe the presence of a flat band.Finally, Sec. IV is devoted to the discussion of observ-ing the described phenomena in a possible experimentalparameter regime and a summary of our work. II. EMERGING FLAT BANDS IN ASPIN-ORBIT-COUPLED BEC
The system considered in this work is illustrated inFig. 1, which consists of a BEC with 1D SOC that istrapped in a strongly anisotropic lattice potential. Thetransverse lattice confinement in the y - and z -directionsis sufficiently strong to freeze the atomic motion in thesedirections, allowing the atomic tunneling only in the x -direction [28]. This realizes an optically trapped quasi-1D BEC with 1D SOC in the x -direction, which can bewell described by the GPE [9, 23–26], i (cid:126) ∂ Ψ ∂t = ( H + H int ) Ψ , (1)with Ψ = ( ψ ↑ , ψ ↓ ) T being the two-component conden-sate wave functions. The Hamiltonian H describes non-interacting bosons in a 1D optical lattice with SOC, read-ing H = p x m + γp x σ z + Ω σ x + V × E R sin ( k L x ) , (2) where m is the bare atom mass, σ x and σ z are the x - and z -component of Pauli matrices, Ω is the Rabi frequencyfor generating SOC, γ = 2 π (cid:126) sin ( θ R / / ( λ R m ) with λ R being the wavelength of the two Raman lasers and θ R the angle between the lasers; and V labels the latticestrength in the unit of the recoil energy E R = (cid:126) k L / m ,with k L the wave vector of the lasers creating the opticallattice. The Hamiltonian H int describes the hard-core in-teraction between bosonic atoms, which can be generallywritten as H int = (cid:90) dx (cid:0) g n ↑ + g n ↓ + 2 g n ↑ n ↓ (cid:1) , (3)where n ↑ = | ψ ↑ | and n ↓ = | ψ ↓ | are the two-componentcondensate densities, and g ij = 4 π (cid:126) a ij /m ( i, j =1 or 2)is the coupling constant, with a ij the s -wave scatteringlength. In this work, we limit ourselves to the case when g = g = g = g = 4 π (cid:126) a/m >
0; in this regime, thestriped phase will not appear in the ground state. Forlater convenience, we rescale GP Eq. (1) into the dimen-sionless form by introducing x → k L x , t → (2 E R / (cid:126) ) t , γ → γ/ ( (cid:126) k L /m ), Ω → Ω / E R , and the dimensionlessinteraction coefficient c = √ ω y ω z k L aN/E R with N theatom number in one unit cell, and ω y and ω z the trappingfrequencies in the transverse directions.The physics of an optically trapped quasi-1D BEC withSOC is governed by the interplay among four parame-ters: the SOC parameters γ and Ω, lattice strength V and interaction c . Crucial to the emergence of flat bandin such systems, as pointed out in Ref. [9], is the in-terplay between the SOC parameters ( γ and Ω) and thelattice strength ( V ). The basic mechanism can be intu-itively described using the single-particle picture [9]: (i)Without the interaction ( c = 0) and the optical potential( V = 0), the single-particle Hamiltonian H can be castinto a dimensionless form H = (cid:32) k + γk ΩΩ k − γk (cid:33) , (4)which has two energy bands µ ± ( k ) = k / ± (cid:112) γ k + Ω separated by a band gap 2Ω at k = 0. (ii) When an opti-cal lattice ( V (cid:54) = 0) is added to Hamiltonian (4), a secondband gap will open at the edge of Brillouin Zone, withthe magnitude of the gap being dependent on V . (iii) Byengineering (via tuning γ , Ω and V ) the magnitude ofboth gap, a flat band can be realized. Strikingly, the ex-istence of flat bands stays robust against the mean-fieldinteraction in the BEC according to Ref. [9].In Figs. 2 (a1)-(d1), we have plotted the lowest Blochbands E g ( k ) for various choices of the SOC parameters ( γ and Ω) and lattice strength V by numerically solving Eq.(1) with fixed interaction parameter c (detailed numericalmethod can be found in Ref. [29–31]). The presence offlat band is manifest to the eye (see Fig.2 (b1), (c1) and(d1)), as compared to an ordinary band (see Fig.2 (a1)).Quantitatively, the global flatness of the bands can be E g / ( E R ) E g / ( E R ) E g / ( E R ) E g / ( E R ) q/kL ω j / ( E R ) ω j / ( E R ) ω j / ( E R ) (a2) (b2) ω j / ( E R ) (c2) −2 −1 1 2 3 −0.5−0.4−0.3 −1 −0.5 0 0.5 1−0.20.20.6 (a1) −0.738−0.736−0.734−0.732 −1 −0.5 0 0.5 1−0.7−0.5−0.3−0.1 (b1) (c1) −1 −0.5 0 0.5 10.40.60.81 −1 −0.5 0.5 1 −0.05−0.04−0.03−0.02 −1 −0.5 0 0.5 10.10.30.50.7 k/kL (d1) (d2) (b3) (c3) ω / ( E R ) ω / ( E R ) ω / ( E R ) ω / ( E R ) (a3) −0.5 0.5 1 q/kL (d3) ZZZS231ZZZ231 (a4) S (b4) −6 −4 −2 2 4 6 q/kL S ZZZ231 S ZZZ231 (c4) (d4)
ZZZ231
FIG. 2. (color online) ( a d
1) are the lowest Bloch band of an optically-trapped SO coupled BEC (in the insert, the lowesttwo bloch bands are plotted); ( a d
2) are the lowest three Bogoliubov bands ω j and the vertical lines represent the excitationstrengths Z j ( j = 1 , ,
3) toward the first three bands for q = 0 . k L , q = − . k L , q = 2 . k L respectively. ( a d
3) are the lowestexcitation spectrum while ( a d
4) are the excitation strength ( Z , , ) and static structure factor ( S ) via a given transferredmomentum q . Here, m ∗ labels the effective mass at the energy minima k min . The parameters are given as follows: c =0.0500 and( a , a , a , a γ = 0 . V = 0 . b , b , b , b γ = 1 . V = 1 . . c , c , c , c γ = 0 . V = 2 . . d , d , d , d γ = 0 . V = 1 . . measured by the ratio W between the bad gap and theband width [9].When the model BEC system is probed by the Braggspectroscopy, it is the excitation near the band minima k min that is addressed in the linear perturbation regime.Therefore, we expect the local flatness at k min to bedirectly probed in Bragg spectroscopy, rather than theglobal flatness measured by W .In order to characterize the local flatness near the bandminima, we have calculated the effective mass m ∗ ( k min )for various bands (in this work, whenever we use the no-tation m ∗ , we refer to the effective mass evaluated at k min ). Our calculation shows that an ordinary band has m ∗ ∼ /m ∗ = 0 .
65 in Fig. 2 (a3)), while incomparison, the flat band has much larger effective mass m ∗ >> m ∗ also varies sharply for various flat band, suchthat we can further discriminate between the sectionalflat band (see Figs. 2 (c1)-(d1)) and the global flat band(see Figs.2 (b1)), the former having much bigger effectivemass m ∗ than the latter. In other words, the sectionalband is locally much flatter near k min than the global flat band, even though its global flatness measured by W canbe actually smaller. Figure 2 (b1) presents a typical glob-ally flat band, which has 1 /m ∗ = 0 . /m ∗ = 0 . /m ∗ = 0 . m ∗ → ∞ for the sectional band in Fig. 2 (d1),we shall call it as a perfect flat band. As we shall see,the excitation behaviour of the model BEC can alter sig-nificantly when m ∗ and the flatness of band changes. III. PROBING FLAT BANDS USING BRAGGSPECTROSCOPY
We now discuss how the flatness of a band in aSOC BEC (see Fig. 1) can be revealed in Bragg spec-troscopy.Bragg spectroscopy consists here in generatinga density perturbation to the model system by using twoBragg laser beams that have momenta k , and a fre-quency difference ω ( ω is much smaller than their de-tuning from an atomic resonance [13, 14]). The lin-ear perturbation is described by the Hamiltonian V = V (cid:2) ρ † q e − iωt + ρ − q e + iωt (cid:3) , where ρ q = (cid:80) j e i q · r j / (cid:126) is theFourier transformed one-body density operator, and q = k − k is the probe momenta. Right after the pertur-bation, the dynamical structure factor [32, 33] is probed,which is written as S ( q , ω ) = (cid:88) e |(cid:104) e | ρ † q | (cid:105)| δ ( ω − ( E e − E g ) / (cid:126) ) , (5)with | (cid:105) ( | e (cid:105) ) being the ground (excited) state having theenergy E g ( E e ). From the dynamic structure factor, theexcitation spectrum can then be extracted [30, 31, 33].Let us calculate the excitation spectrum and the dy-namic structure factor S ( q, ω ) of the model system forvarious band structures, from an ordinary band to a per- fect flat band. For this purpose, we apply the Bogoliubovtheory [30, 34] to Eq. (1) and decompose the condensatewave function ( ψ ↑ , ψ ↓ ) T into the ground state wave func-tion ( φ ↑ , φ ↓ ) T and a small fluctuating term reading (cid:18) ψ ↑ ψ ↓ (cid:19) = e − iµt (cid:20)(cid:18) φ ↑ φ ↓ (cid:19) + (cid:18) u ↑ ( x ) u ↓ ( x ) (cid:19) e − iωt + (cid:18) v ∗↑ ( x ) v ∗↓ ( x ) (cid:19) e iωt (cid:21) . (6)By substituting Eq. (6) into Eq. (1) and ex-panding u ↑ , ↓ ( x ) and v ↑ , ↓ ( x ) in the Bloch form interms of u l and v l ( l labels the Bloch eigenstate),we obtain the Bogliubov-de Gennes (BdG) equations M ∆ φ = ω ∆ φ , with ∆ φ = ( u ↑ l , v ↑ l , u ↓ l , v ↓ l ) and (cid:82) dx (cid:16) | u ↑ l | − | v ↑ l | + | u ↓ l | − | v ↓ l | (cid:17) = 1, and the ma-trix M reads M = L ( ↑↓ ) ( k + q ) cφ ↑ Ω + cφ ↑ φ ∗↓ cφ ↑ φ ↓ − c (cid:16) φ ∗↑ (cid:17) − L ( ↑↓ ) ∗ ( k − q ) − cφ ∗↑ φ ∗↓ − Ω − cφ ∗↑ φ ↓ Ω + cφ ∗↑ φ ↓ cφ ↓ φ ↑ L ( ↓↑ ) ∗ ( − k − q ) cφ ↓ − cφ ∗↑ φ ∗↓ − Ω − cφ ∗↓ φ ↑ − c (cid:16) φ ∗↓ (cid:17) − L ( ↓↑ ) ( − k + q ) , (7)with L ( m m ) ( k ) = −
12 (2 in + ik ) + V sin ( x ) − iγ (2 in + ik ) − µ + 2 c | φ m | + c | φ m | , (8)and m , m = ↑ , ↓ . By solving the BdG equations nu-merically, the Bogoliubov excitation spectrum can be ex-tracted. Note that, for a BEC trapped in optical lattices,two different types of instabilities of the BEC, i.e. dy-namical instability and Landau instability [35], can breakthe superfluidity of the model system, both of which havebeen extensively studied in theory [35] and experiments[36]. In this work, in order to avoid dynamical insta-bility, which is relevant to our detailed calculations, wehave limited ourselves to the stable parameter regime[9]. Then the dynamic structure factor can be found via[30, 34] S ( q, ω ) = (cid:80) j Z j ( q ) δ ( ω − ω j ( q )), where Z j ( q ) and ω j ( q ) are the excitation strength and frequency from theground state to the j − th Bloch band, respectively. Inparticular, the static structure factor for the model sys-tem can be immediately read off as [30, 34] S ( q ) = (cid:88) j Z j ( q ) . (9)We present in Figs. (2) (a3)-(d3) the low-energy exci-tation spectrum of an optically trapped BEC with SOCcorresponding to the four bands in Figs. 2 (a1)-(d1), re-spectively. We find that, when the model BEC has anordinary band, the familiar linear relation (cid:15) ( q ) ∼ q arises(Fig. 2 (a3)); whereas, remarkably, when the model BEChas a perfect flat band, a quadratic dispersion (cid:15) ( q ) ∼ q emerges (Fig. 2 (d3)). Such distinct change in the ex-citation behaviour of the model system when the bandflatness varies is also clearly observed in the dynamicstructure, which is shown in Figs 2 (a4)- (d4). In partic-ular, the static structure factor (see black solid lines inFigs. 2 (a4)-(d4)) exhibits a crossover from a linear re-lation S ( q ) ∼ q to a quadratic relation S ( q ) ∼ q , whenthe band structure transforms from the ordinary into theperfect flat.The crossover from the linear dispersion (cid:15) ( q ) ∼ q to thequadratic dispersion (cid:15) ( q ) ∼ q in the excitation spectrumof the model BEC [37, 38], when the band structure tran-sits from an ordinary band to a locally perfect flat, canbe intuitively understood in connection with the effec-tive mass m ∗ near the band minima. As previously men-tioned, a perfect flat band is associated with an almostinfinite effective mass m ∗ → ∞ , therefore, the q term isexpected to varnish in the single-particle dispersion rela-tion (cid:15) ( q ) (corresponding to zero kinetic energy) and theleading term can only emerge as (cid:15) ( q ) ∼ q . Thus, byusing the above results S ( q ) ∼ q for a perfect flat bandand Feyman’s relation (cid:15) ( q ) = (cid:15) ( q ) /S ( q ), we immedi-ately have (cid:15) ( q ) ∼ q which explains the numerical resultsin Figs. 2 (c1) and (d1). In contrast, in the opposite caseof an ordinary band when m ∗ ∼
1, the single-particle ki-netic energy is finite such that (cid:15) ( q ) ∼ q . Hence from S ( q ) ∼ q and Feynman’s relation, we have the familiarlinear relation (cid:15) ( q ) ∼ q in the BEC. We point out that,while the existence of flat bands in an optically trappedquasi-1D BEC with SOC can be described with a single-particle picture, the emerging quadratic low-energy dis-persion when the band is perfectly flat is a many-bodyeffect, which results from the interplay between the in-teraction and the band’s flatness.Finally, we have also analyzed how the excitationstrength Z j is affected by the lattice strength V , in caseswhen the band is ordinary (Fig. 2(a1)) and when theband is flat (Figs. 2(b1)-(d1)), respectively. As is clearlyshown in Figs 2 (a2)-(d2), where the first three Bogoli-ubov bands are plotted, the dynamic structure factor issignificantly affected by the optical lattice compared tothe free-space case [25]. In particular, for a given valueof momentum transfer p , it is possible to excite severalstates corresponding to different bands. For example,Fig. 2 (d2) shows that when a density perturbation with q = 0 . k L is generated in the BEC, not only the first ex-citation strength Z = 0 .
37 obtained, but also the secondexcitation Z = 0 .
04. An important consequence is that,on the one hand, it is possible to excite the high-energystates with small values of p ; on the other hand, one canalso excite the low energy states in the lowest band withhigh momenta p outside the first Brillouin zone. Such ex-citation behaviour is shared by both the ordinary band(Fig. 2 (b3)) and the flat bands (Figs. 2 (b4)-(d4)) andtherefore the existence of flat bands cannot be revealedin the excitation strength Z j alone. IV. CONCLUSION
Overall, the crossover from linear to quadratic disper-sion in the low-energy excitation spectrum presents astriking manifestation of the transition of an ordinaryband into a perfect flat band, which permits the directprobe of flat band using the Bragg Spectroscopy. Theexperimental realization of our scenario amounts to con-trolling four parameters whose interplay underlies thephysics of this work: the lattice strength V , SOC param- eters γ and Ω, and the interatomic interaction c . All theseparameters are highly controllable in the state-of-the-arttechnologies: V can be changed from 0 E R to 32 E R ; both γ and Ω can be changed by varying the angle betweenthe two Raman lasers or through a fast modulation ofthe laser intensities [39]; in typical experiments to date,we can calculate the interaction coupling c = 0 .
05 withthe relevant parameters [18] of a = 100 a B with a B theBohr radius. Thus, we expect the phenomena discussedin this work be observable within the current experimen-tal capabilities.To conclude, we have found that the excitation spec-trum of an optically trapped quasi-1D BEC with SOCalters significantly when the band flatness varies. Inparticular, when the model BEC exhibits a perfect flatband (corresponding to m ∗ → ∞ at the band minima), aquadratic dispersion (cid:15) ( q ) ∼ q emerges in the low-energyexcitation spectrum; whereas, if the band is ordinary, thefamiliar linear dispersion (cid:15) ( q ) ∼ q arises. The variationin the flatness of band also alters the dynamic structuresignificantly.In particular, the static structure factor for aperfect band is quadratic in momenta S ( q ) ∼ q , in con-trast to the case of an ordinary band when S ( q ) ∼ q islinear. Based on these results, we propose to use Braggspectroscopy to probe the arising flat band in an opti-cally trapped quasi-1D BEC. The experimental verifica-tion of the new dynamic features predicted in this workis expected to provide a significant advance in our un-derstanding of systems exhibiting flat-band-related phe-nomena. V. ACKNOWLEDGMENTS
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