Pseudo-Goldstone Excitations in a Striped Bose-Einstein Condensate
PPseudo-Goldstone Excitations in a Striped Bose-Einstein Condensate
Guan-Qiang Li,
1, 2
Xi-Wang Luo, ∗ Junpeng Hou, and Chuanwei Zhang † Department of Physics, The University of Texas at Dallas, Richardson, Texas 75080-3021, USA School of Arts and Sciences, Shaanxi University of Science and Technology, 710021 Xi’an, China
Significant experimental progress has been made recently for observing long-sought supersolid-likestates in Bose-Einstein condensates, where spatial translational symmetry is spontaneously brokenby anisotropic interactions to form a stripe order. Meanwhile, the superfluid stripe ground statewas also observed by applying a weak optical lattice that forces the symmetry breaking. Despiteof the similarity of the ground states, here we show that these two symmetry breaking mechanismscan be distinguished by their collective excitation spectra. In contrast to gapless Goldstone modesof the spontaneous stripe state, we propose that the excitation spectra of the forced stripe phasecan provide direct experimental evidence for the long-sought gapped pseudo-Goldstone modes. Wecharacterize the pseudo-Goldstone mode of such lattice-induced stripe phase through its excitationspectrum and static structure factor. Our work may pave the way for exploring spontaneous andforced/approximate symmetry breaking mechanisms in different physical systems.
Introduction.—
Spontaneous symmetry breaking playsa crucial role for the understanding of many impor-tant phenomena in different fields ranging from elemen-tary particles to condensed states of matter. For in-stance, crystalline and superfluidity orders are formedin the long-sought supersolids through spontaneouslybreaking spatial translational and U(1) gauge symme-tries [1]. While the early study of supersolidity focusedon solid He [2, 3] without conclusive experimental ev-idence [4, 5], ultracold atoms have emerged as a pow-erful platform in recent years for observing supersolid-like quantum phases [6–14]. Significant experimentalprogress has been made on the generation and measure-ment of supersolid-like superfluid stripe states in bothdipolar [10–13] and spin-orbit-coupled Bose-Einstein con-densates (BECs) [14], where spontaneous translationalsymmetry breaking is driven by dipolar or anisotropicspin interactions. In the latter case, the anisotropic spininteractions favor the occupation of both band minimaof the spin-orbit coupling induced double-well disper-sion, yielding superfluid stripe phase with periodic den-sity modulations [15–20].In the region where ground state symmetry cannotbe spontaneously broken by interactions, the symme-try breaking ground state may be achieved by apply-ing a weak symmetry breaking potential. Such forcedsymmetry breaking mechanism has been demonstratedrecently in a spin-orbit-coupled BEC, where the super-fluid stripe ground state is realized by applying a weakoptical lattice [21] that breaks translational symmetryexplicitly. Interestingly, the forced stripe ground stateshows similar (spin-)density patterns as the spontaneousone induced solely by the anisotropic atomic interactions.Therefore two questions naturally arise: Can we distin-guish the stripe ground states resulted from spontaneousand forced symmetry breaking mechanisms? If so, arethere interesting experimental observables? These ques-tions should also apply to general spontaneous and forcedsymmetry breaking ground states in other physical sys- tems.In this Letter, we address these two important ques-tions by investigating the collective excitations of theforced superfluid stripe ground state and showing thatthe emerging pseudo-Goldstone spectrum lies at theheart of understanding its forced symmetry breaking.The pseudo-Goldstone mode is an important concept infields ranging from standard model to solid-state ma-terials [22–24], with prominent examples including thepion (the lightest hardon) [25, 26] and longitudinal po-larization components of W and Z bosons in high-energy physics [27], phonon modes in superconductorsand/or superfluids [28–31] and magnons in magnets [32–34]. However, direct experimental observation of pseudo-Goldstone spectrum remains challenging. The capabilityof directly measuring excitation spectrum using Braggspectroscopy [35–40] in ultracold atomic gases thus pro-vides a powerful tool for probing pseudo-Goldstone spec-trum. Our main results are: i) In the strong anisotropic spin interaction region, thespontaneous superfluid stripe ground state hosts two gap-less Goldstone modes. A weak lattice breaks the transla-tional symmetry (i.e., the symmetry is approximate) andturns one gapless mode into a pseudo-Goldstone mode,which is characterized by the gap of the excitation spec-trum at the long wavelength limit (i.e., zero-momentumgap). The hybridization of the gapped pseudo-Goldstoneand the remaining gapless modes yields an avoided-crossing gap at a finite momentum. ii ) In the weak anisotropic spin interaction region,an increasing lattice potential forces a transition fromplane-wave to stripe ground states. The zero momen-tum pseudo-Goldstone gap first decreases to zero at thephase transition point and then reopens. In the forcedsuperfluid stripe region, the properties of gapped ex-citation spectrum (e.g., zero-momentum and avoided-crossing gaps) largely resemble those for spontaneous su-perfluid stripe phase subject to a weak lattice pertur-bation (i.e., approximate symmetry), demonstrating it a r X i v : . [ c ond - m a t . qu a n t - g a s ] J a n is a pseudo-Goldstone spectrum. The forced superfluidstripe ground state is experimentally more accessible androbust than the spontaneous one, opening the pathwayfor the direct observation of pseudo-Goldstone spectrumin experiments. iii) The structure factors of the two lowest energymodes show that gapless Goldstone and gapped pseudo-Goldstone branches correspond to density (phonon) andspin-density (magnon) modes, respectively. The staticstructure factor for the spin-density reveals some differ-ences between spontaneous and forced superfluid stripeground states due to their different stripe formationmechanisms. The excitation spectrum and structure fac-tor can be detected in experiments through Bragg scat-tering.
Model.—
We consider the experimental setup illus-trated in Fig. 1a. The BEC is confined in a cigar-shapedoptical dipole trap, with spin-orbit coupling along the x direction realized by two Raman laser beams, which cou-ple the two pseudospin states |↑(cid:105) and |↓(cid:105) (e.g., | , − (cid:105) and | , (cid:105) of Rb atoms within the F = 1 hyperfinemanifold) with momentum kick 2 k R ( k R is the recoil mo-mentum). In addition, we consider a weak optical lattice V L ( x ) = 2Ω L sin ( k L x ). The single-particle Hamiltonianin the spin basis (with momentum and energy units as (cid:126) k R and (cid:126) k R m ) reads H = ( i∂ x + σ z ) − δ σ z + Ω R σ x + V L ( x ) , (1)where Ω R is the strength of the Raman coupling and δ is the detuning of the two-photon Raman transition.Spin-orbit coupling induces a momentum-space double-well band dispersion, and the period of the optical latticeis set such that 2 k L equals to the separation between twoband minima.We first find the ground state ψ s ( x ) by imaginary-time evolution of the Gross-Pitaevskii (GP) equation i ∂ψ s ( x, t ) ∂t = H GP ( ψ s ) ψ s ( x, t ) , (2)where ψ s is the spinor wavefunction with s = ↑ , ↓ and H GP ( ψ s ) = H + gn + g | ψ ¯ s | with n the total density. Wehave assumed the intra-spin interaction as g ↑↑ = g ↓↓ = g and the anisotropic spin interaction as g = g ↑↓ − g ,with g ss (cid:48) the interaction strength between atoms in s and s (cid:48) states. The phase diagram in the g -Ω L plane isshown in Fig. 1b [typical (spin-)density distributions ofthe stripe state are shown in the inset]. The system favorsthe stripe phase (plane-wave phase) for large negative(positive) g (note that the stripe and plane-wave phasesbecome the unpolarized and polarized Bloch states in thepresence of a lattice). The phase boundary lies at theweak anisotropic interaction region around g = 0, andthe critical value of g increases with the lattice strength.Without the optical lattice, the stripe phase occupyingboth band minima can be formed in the system under the Stripe phase
Plane-wave phase (c) V L 在此处键入公式。 𝑥𝐸 𝐸 𝐸 Gap 在此处键入公式。 𝐸 𝐸 m* 在此处键入公式。 𝐸 Ω 𝐿 /𝐸 𝑅 在此处键入公式。 -1 (b) 𝑞 𝑥 (a) 𝑞 𝑥 BEC 𝑔 𝑛 / 𝐸 R 在此处键入公式。 Ω R 〈𝜎 𝑧 〉 𝑘 𝐿 𝑥 𝑛 -2 𝜋 𝑛 𝑠 -0.5 𝑧𝑦 FIG. 1: (a) Scheme to generate spin-orbit coupling and opticallattice for a trapped BEC. (b) Phase diagram in the g -Ω L plane, with Ω R = 2 . E R , δ = 0, gn = 1 . E R and n themean atom density. Two bold dashed lines correspond to theweak and strong spin interactions regimes for Figs. 2-4. Theinset shows the typical densities (normalized to n ) of spin-up(or spin-down) component (blue line) and the total density(green line). (c) Schematic illustration of the two (pseudo-)Goldstone modes without (left panel) and with (right panel)a weak and explicitly symmetry-breaking term. anti-ferromagnetic atomic interaction ( g <
0) [15–18].Typically, the stripe phase only exits for very weak Ω R and δ due to the weak anisotropy of interaction | g | in re-alistic experiments, making its observation difficult. Thismay be overcome by using atoms with strong anisotropicspin interactions, or alternatively, by adding a weak op-tical lattice that couples the two band minima directly.The latter approach has led to the recent observation oflong-lived superstripe state using Rb atoms [21]. Wewant to point out that, there is a tiny difference betweenthe ground-state stripe period at Ω L = 0 and the opticallattice period. The two periods would match as long asthe optical lattice strength is not extremely small.The forced stripe ground state induced by symmetry-breaking potential shows similar (spin-)density patternsas the spontaneous one induced solely by the anisotropicinteractions. To characterize and distinguish the stripestates formed under different symmetry breaking mecha-nisms, we consider the excitation spectrum. For sponta-neous stripe phase induced solely by interactions, boththe U (1) gauge and continuous translational symme-tries are broken spontaneously, leading to two gaplessGoldstone modes (as illustrated in the left panel ofFig. 1c) [41–43]. When the translational symmetry isweakly broken by a lattice perturbation (i.e., the symme-try is now approximate), we expect to observe the gapopening of one Goldstone mode (equivalent to an effectivemass m ∗ of the corresponding Goldstone boson). If thelower mode becomes a gapped pseudo-Goldstone mode,there should be an avoided crossing due to the hybridiza-tion of the two original Goldstone modes (as illustratedin the right panel of Fig. 1c).To obtain the spectrum of elementary excitations, wewrite the deviations of the wavefunctions with respect tothe ground state as ψ s ( x, t ) = e − iµt (cid:2) ψ s ( x ) + u s ( x ) e − iεt + v ∗ s ( x ) e iεt (cid:3) . (3) =0 =0.3 =0 =0.1 (c) (d) 𝐸 𝑅 𝑞 𝑥 /𝑘 𝐿 (a) (b) 𝜀 𝑗 / 𝐸 𝑅 𝜀 𝑗 / 𝐸 𝑅 Δ 𝜀 / 𝐸 𝑅 Δ 𝜀 / 𝐸 𝑅 𝑞 𝑥 /𝑘 𝐿 𝐸 𝑅 Ω 𝐿 /𝐸 𝑅 Ω 𝐿 /𝐸 𝑅 FIG. 2: (a,b) Low-energy spectra of the elementary ex-citations for the stripe phase. (c,d) Change of the zero-momentum gap ∆ ε with the lattice strength Ω L . (a,c) and(b,d) are for strong ( g n = − . E R ) and weak ( g n = − . E R ) anisotropic spin interactions, respectively. Thelattice strength Ω L = 0 . E R in (a), and Ω cL = 0 . E R in (b) is the phase transition point between the plane-wavephase and the stripe phase. The dashed lines in (c,d) repre-sent nonzero detunings. The dotted line in (d) correspondsto Ω cL . gn = 1 . E R and Ω R = 2 . E R . The amplitudes u s ( x ) and v s ( x ) satisfy normalizationcondition (cid:80) s (cid:82) d dx [ | u s ( x ) | − | v s ( x ) | ] = 1, with d thestripe period and µ the chemical potential. SubstitutingEq. (3) into Eq. (2), we obtain the Bogoliubov equationas ε [ u ↑ , u ↓ , v ↑ , v ↓ ] T = H [ u ↑ , u ↓ , v ↑ , v ↓ ] T . The expressionof the Bogoliubov Hamiltonian H is given in [44], andthe excitation spectra can be calculated numerically byexpanding u s ( x ) and v s ( x ) in the Bloch basis. Each ex-citation spectrum is periodic in momentum space withBrillouin zone determined by the stripe period. Pseudo-Goldstone spectrum.—
We focus mainly on theelementary excitations under the situation of the anti-ferromagnetic atomic interaction (i.e., g < R (cid:38) E R , the system prefers to form the stripe(plane-wave) phase under strong (weak) anisotropic spininteraction | g | in the absence of optical lattices. Wefirst consider the strong anisotropic spin interaction witha weak optical lattice (lower region in Fig. 1b). The op-tical lattice slightly breaks the space translational sym-metry of the system yet alters the excitation spectrumdramatically. The low-energy bands in the first Bril-louin zone with weak optical lattice are demonstratedin Fig. 2a. The double gapless spectrum disappears anda gap ∆ ε in the second band at zero Bloch momentum( q x = 0) is opened, which corresponds to the generationof the pseudo-Goldstone mode of the system at the long wavelength limit. The pseudo-Goldstone mode is gener-ated once the lattice is turned on. The change of thezero-momentum gap ∆ ε with the strength of the opti-cal lattice is given in Fig. 2c. The gap vanishes at zerolattice strength and increases with increasing optical lat-tice strength. The size of the gap almost changes linearlywith the optical lattice except near the zero momentum.A small detuning δ would hardly affect the spectrum,while a large δ may drive the system out of the stripephase and lead to a roton gap (∆ ε >
0) at Ω L = 0. Theeffect of δ is diminished at larger lattice strength.With decreasing anisotropic spin interaction | g | , thesystem is driven from the stripe phase into the plane-wave phase (i.e., the polarized Bloch state). The pseudo-Goldstone gap ∆ ε decreases to zero at the critical phaseboundary and then reopens as a nonzero roton gap. Fur-ther increasing lattice strength can drive the transitionfrom the polarized Bloch state to the stripe phase, whereroton gap ∆ ε decreases to zero at the critical phaseboundary and then reopens as the pseudo-Goldstonegap (see Fig. 2d). Notice that at the phase bound-ary, the excitation spectrum of the forced stripe stateis very similar to that for strong anisotropic interac-tion with two gapless Goldstone modes (see Fig. 2b).For weak anisotropic spin interactions ( g (cid:39) cL = 0 . E R . Beyond the critical lattice strengthΩ cL , the pseudo-Goldstone gap (see Fig. 2d) behaves sim-ilarly as that in the strong anisotropic interaction regime(see Fig. 2c), demonstrating the properties of the pseudo-Goldstone spectrum. Therefore, the forced superfluidstripe ground state, which is experimentally more accessi-ble and robust than the spontaneous one, opens the path-way for the direct observation of pseudo-Goldstone spec-trum in experiments utilizing techniques that have al-ready been used to study the spectrum of elementary ex-citations for spin-orbit-coupled BECs [37, 38], quantumgas with cavity-mediated long-range interactions [39] andBEC in a shaken optical lattice [40].The pseudo-Goldstone mode near the phase boundaryresults from the interplay between the interaction andoptical lattice, which tends to reduce the spatial modu-lation of the GP Hamiltonian H GP ( ψ s ) for the groundstate. In the vicinity of the phase boundary, H GP ( ψ s )preserves an approximate translational symmetry [44],which leads to a vanishing gap of the pseudo-Goldstonemode. On the other hand, the pseudo-Goldstone modefor strong | g | with a very weak lattice is induced bythe approximate translational symmetry of H , while H GP ( ψ s ) strongly breaks the translational symmetry. Inthe presence of nonzero detuning δ , the phase boundarybecomes a crossover boundary, and ∆ ε decreases to afinite value as the system goes from the stripe region tothe plane-wave region, where ∆ ε is almost a constant(see Fig. 2d). The effects of the Raman coupling and the 𝑞 a c / 𝑘 𝐿 (a) (b) Ω 𝐿 /𝐸 𝑅 Ω 𝐿 /𝐸 𝑅 Δ a c / 𝐸 𝑅 Δ a c / 𝐸 𝑅 𝑞 a c / 𝑘 𝐿 FIG. 3: Change of the size ∆ ac and position q ac of thenonzero-momentum gap with the lattice strength Ω L . Theparameters in (a) and (b) are the same as those in Figs. 2(c)and 2(d). The dashed lines represent nonzero detuning | δ | =0 . E R in (a) and | δ | = 0 . E R in (b), respectively. The graydotted line in (b) corresponds to the phase transition pointΩ cL = 0 . E R . atomic interactions on ∆ ε are given in [44]. Avoided spectrum crossing.—
In addition to the zero-momentum gap, there exists another avoided crossinggap ∆ ac (the minimum value of the gap between the firstand second bands), originating from the hybridization be-tween the pseudo-Goldstone and Goldstone modes. Thenonzero-momentum gap ∆ ac as a function of the latticestrength for strong anisotropic spin interaction is given inFig. 3a. The gap increases slowly with increasing latticestrength at the beginning, and then rapidly in the deeplattice region. In contrast, the avoided crossing point q ac first increases rapidly with the lattice strength, and re-mains saturated in the deep lattice region. Fig. 3b shows∆ ac as a function of lattice strength for weak anisotropicspin interaction. In the plane-wave phase, ∆ ac first in-creases with the lattice strength and then decreases tozero at the phase boundary, while q ac decreases directlyto zero. In the stripe phase, ∆ ac and q ac behave simi-larly as those for the strong anisotropic spin interaction(see Fig. 3a). A nonzero Raman detuning δ tends toincrease both the gap ∆ ac and its position q ac (see thedashed lines in Fig. 3a and 3b). The effect of the Ramancoupling on ∆ ac and q ac are given in [44]. Structure factors.—
As discussed above, the pseudo-Goldstone spectra for the weak and strong anisotropicspin interactions are very similar, although the groundstripe phases are achieved through different symmetrybreaking mechanisms. In experiments, the collectiveproperties of the excitation spectrum of the BECs canbe probed using Bragg spectroscopy, which measures thedynamical structure factors. For a scattering probe withmomentum (cid:126) q x and energy (cid:126) ω , the dynamical structurefactor takes the form [41, 47]: S ( q x , ω ) = (cid:88) j |(cid:104) j | ρ † q x | (cid:105)| ˜ δ ( (cid:126) ω − ε j ) (4)with | j (cid:105) the excited state, ε j the excitation energy, ρ q x = (cid:80) j e iq x x j / (cid:126) the density operator and ˜ δ ( · ) the Dirac deltafunction. The excitation strength Z j = |(cid:104) j | ρ † q x | (cid:105)| can 𝑆 ( 𝑞 𝑥 ) 𝑆 ( 𝑞 𝑥 ) 𝑆 𝜎 ( 𝑞 𝑥 ) 𝑆 𝜎 ( 𝑞 𝑥 ) 𝑞 𝑥 /𝑘 𝐿 𝑞 𝑥 /𝑘 𝐿 𝑞 𝑥 /𝑘 𝐿 𝑞 𝑥 /𝑘 𝐿 (a) (c) (b) (d) FIG. 4: Static structure factors and excitation strengths forthe density in (a,c) and spin-density in (b,d). (a,b) correspondto the excitation spectrum of the stripe phase in Fig. 2(a), and(c,d) correspond to the spectrum under the same parametercondition as Fig. 2(b) except Ω L = 0 . E R . The insets in(a) and (c) show the corresponding excitation strengths andspectra near q x = 0, respectively. be evaluated as: Z j = (cid:88) s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) d [ u ∗ js ( x ) + v ∗ js ( x )] e iq x x ψ ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (5)The integral of the dynamical structure factor gives thestatic density structure factor S ( q x ) = (cid:82) S ( q x , ω ) dω ,which is uniquely determined by the sum of the excitationstrengths for all energy bands. Similarly, we can definethe spin-density static structure factor S σ ( q x ) and the ex-citation strength Z σj by replacing ρ † q x with σ z ρ † q x . Suchspin structure factors may be probed by spin-dependentBragg spectroscopy using lasers with suitable polariza-tion and detuning [48–50]. S ( q x ) and S σ ( q x ) are relatedto density and spin-wave excitations, respectively. Bothof them include the contributions from all of consideredenergy bands.The static structure factors for strong anisotropic spininteraction with a weak lattice Ω L = 0 . E R are givenin Fig. 4a for the density and Fig. 4b for the spin-density, where the excitation strengths Z j and Z σj forthe first three excitation bands are also shown. S ( q x )and S σ ( q x ) increase with the quasi-momentum q x mono-tonically. S ( q x ) vanishes, but S σ ( q x ) has a non-zero min-imum value at q x = 0. The excitation strengths for thefirst and second bands exchange at the position of thenonzero-momentum gap q ac = 0 . k L , showing that thefirst and second lowest bands correspond to density andspin excitations, respectively. This feature could be usedto identify the pseudo-Goldstone modes in Bragg spec-troscopy experiments. S ( q x ) in the forced stripe phase for the weakanisotropic spin interaction (see Fig. 4c) has similar fea-tures as Fig. 4a, while S σ ( q x ) (see Fig. 4d) shows quitedifferent properties from Fig. 4b. In the forced stripephase, S σ ( q x ) decreases monotonically with q x with amaximum at q x = 0 (see Fig. 4d). At the phase transi-tion point, S σ ( q x ) diverges as q x →
0. The dependence of S ( q x ) and S σ ( q x ) on other parameters are shown in [44]. Conclusion.—
In summary, we show that the collectiveexcitation spectrum of a spin-orbit-coupled BEC can beused to distinguish the spontaneous and forced stripeground states induced by different symmetry breakingmechanisms. The lattice forced stripe phase, whichis experimentally more accessible and robust than thespontaneous one, can provide direct experimental evi-dence for the long-sought gapped pseudo-Goldstone spec-trum. While the present work focuses on spin-orbit-coupled BECs, similar ideas can be implemented to othersystems such as dipolar striped BECs [10–13], stripedBECs in optical superlattice [14] or in a cavity [6–9], tostudy pseudo-Goldstone modes in the presence of approx-imate/forced symmetry breaking.
Acknowledgements : We thank P. Engels, S. Moss-man, E. Crowell, and T. Bersano for helpful discussions.X.W.L., J.H. and C.Z. are supported by Air Force Of-fice of Scientific Research (FA9550-20-1-0220), NationalScience Foundation (PHY-1806227), and Army ResearchOffice (W911NF-17-1-0128). G.Q.L. acknowledges thesupports from NSF of China (Grant No. 11405100), theNatural Science Basic Research Plan in Shaanxi Provinceof China (Grant Nos. 2019JM-332 and 2020JM-507), theDoctoral Research Fund of Shaanxi University of Scienceand Technology in China (Grant No. 2018BJ-02), and theChina Scholarship Council (Program No. 201818610099). ∗ Corresponding author email: [email protected] † Corresponding author email: [email protected][1] M. Boninsegni and N. V. Prokofv, Colloquium: Super-solids: What and where are they?, Rev. Mod. Phys. ,759 (2012).[2] D. J. Thouless, The flow of a dense superfluid, Ann.Phys. , 403 (1969).[3] A. F. Andreev and I. M. 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Method.—
In order to calculate the excitation spectrum of the spontaneous and forced stripe phases, we first findthe ground states of the system. We adopt the following ansatz (cid:18) ψ ↑ ( x ) ψ ↓ ( x ) (cid:19) = (cid:88) K (cid:18) a K − b K (cid:19) e i ( K + k ) x , (S1)where K = (2 l − k L with the integer l = − L, ..., L + 1 represent the reciprocal lattice vectors, L is the cutoff of theplane-wave modes. The expansion coefficients a K and b K , together with k , are determined by minimizing the energyfunctional E = (cid:90) dxψ † ( x )[ H + 12 H int ( ψ ↑ , ψ ↓ )] ψ ( x ) , (S2)where ψ = ( ψ ↑ , ψ ↓ ) T is the two-component spinor wavefunction normalized by the atom number N = (cid:82) dxψ † ( x ) ψ ( x ),and H int ( ψ ↑ , ψ ↓ ) = diag[ gn + g | ψ ↓ | , g | ψ ↑ | + gn ] with density n = | ψ ↑ | + | ψ ↓ | . The results for the ground stateare calculated numerically by the imaginary-time evolution of the Gross-Pitaevskii (GP) equation. The correspondinginitial solution is given by the variational method considering the lowest-four modes in the wavefunction ansatz.To evaluate the spectrum of elementary excitations, the Bogoliubov equation is obtained by writing the deviationof the wavefunction with respect to the ground state as ψ s ( x, t ) = e − iµt (cid:2) ψ s ( x ) + u s ( x ) e − iεt + v ∗ s ( x ) e iεt (cid:3) . (S3)The perturbation amplitudes u s ( x ) and v s ( x ) with s = ↑ , ↓ are expanded in the Bloch form in terms of the reciprocallattice vectors: u s ( x ) = M +1 (cid:88) m = − M U s,m e i ( k + q x ) x + i (2 m − k L x , (S4) v s ( x ) = M +1 (cid:88) m = − M V s,m e i ( k + q x ) x − i (2 m − k L x , (S5)where q x is the Bloch wavevector of the excitations and M is the cutoff of the plane waves of the excited states. Ground state and phase diagram.—
Depending on the spin-orbit coupling and the atomic interactions, the spin-orbit-coupled BEC without the optical lattice potential has three different phases: stripe, plane-wave and zero-momentumphases [15–18, 41]. The parameter range for the existence of the stripe phase is very narrow, following with thesmall contrast and small wavelength of the fringes, which make the observation of the stripe state very difficult inexperiments. In contrast, the stripe phase in the spin-orbit-coupled BEC forced by the weak optical lattice has beenobserved recently [21]. The key feature is that the wavelength of the lattice beams and the Raman coupling strengthare chosen such that the lattice couples two minima of the lower spin-orbit band, where the static spin-independentlattice provides a 2 k L momentum kick while preserves the spin. Such forced stripe state has a long lifetime and ismore stable, and its existing parameter region is extended dramatically.With a large Raman coupling strength like Ω R = 2 . E R and without the optical lattice, the stripe phase and planewave phase appear at strong and weak anisotropic spin interaction regions, respectively. With the optical lattice, thereexists a magnetized feature related with the plane-wave phase (i.e., the polarized Bloch state) in the system, whichwas also revealed in previous studies [45, 46]. The stripe phase (i.e., unpolarized Bloch wave) for the two componentsexists at larger optical lattice strengths. In the formation of the stripe phase, the modulation depth of the densityincreases with the increasing lattice strength. The contrast of the total density C = ( n max − n min ) / ( n max + n min )reflects this change and shows the phase transition between the polarized Bloch state and the unpolarized Bloch state(the perfect stripe phase) for the weak anisotropic spin interaction [Fig. S1(a)].In Fig. 1(b) of the main text, the spin polarization is plotted with respect to the optical lattice strength and theanisotropic spin interaction between atoms. For the strong anisotropic spin interaction, the existence of the stripephase does not need the optical lattice. For the weak anisotropic spin interaction, there is a critical lattice strength Ω cL g n = ! : E R g n = ! : E R -10 -5 0 5 10012 V L ( x ) gn + g j A s ( x ) j V L + gn + g j A s ( x ) j 在此处键入公式。 Ω L / E R 在此处键入公式。 C E ff ec ti v e P o t e n ti a l k L x (a) (b) FIG. S1: (a) The modulation contrast of the ground state as a function of the strength of the optical lattice. The verticaldotted line represents the phase boundary between the plane-wave phase and the stripe phase for g n = − . E R . (b) Theeffective potential (i.e., gn + g | ψ ¯ s | + V L ) of H GP ( ψ s ) at the phase boundary. Other parameters are Ω R = 2 . E R , δ = 0 and gn = 1 . E R . (Ω cL = 0 . E R for gn = 1 . E R and g n = − . E R ) for the transition from the plane-wave phase (i.e., polarizedBloch state with (cid:104) σ z (cid:105) (cid:54) = 0) to the stripe phase (i.e., the unpolarized Bloch state with (cid:104) σ z (cid:105) = 0). We calculate thefirst- and second-order derivatives of the ground-state energy E with respect to the lattice strength Ω L . The jump inthe second order derivative with Ω L shows that the phase transition is the second-order. The phase transition pointcan also be identified from the excitation spectrum as discussed in the main text.At the phase transition point Ω cL , the spatial modulation due to the atom density in the GP Hamiltonian H GP ( ψ s )cancels with the spatial modulation of the lattice potential V L ( x ) [see Fig. S1(b)], therefore H GP ( ψ s ) is close toa constant at the ground state. H GP ( ψ s ) preserves the translational symmetry, leading to two gapless Goldstonemodes shown in the inset of Fig. 2(b) in the main text. Slightly above Ω cL , the translational symmetry becomesapproximate, leading to the pseudo-Goldstone gap. In this context, the region above Ω cL for the forced stripe phaseresembles the spontaneous stripe phase with a very weak lattice perturbation ( i.e. , approximate symmetry region). Bogoliubov equations.—
By substituting Eqs. (S3)-(S5) into Eq. (2) in the main text, the Bogoliubov equation canbe obtained as follows: H u ↑ u ↓ v ↑ v ↓ = ε u ↑ u ↓ v ↑ v ↓ , (S6)where the Bogoliubov Hamiltonian H = H ↑ Ω R + g ↑↓ ψ ↑ ψ ∗ ↓ gψ ↑ g ↑↓ ψ ↑ ψ ↓ Ω R + g ↑↓ ψ ∗ ↑ ψ ↓ H ↓ g ↑↓ ψ ↑ ψ ↓ gψ ↓ − gψ ∗ ↑ − g ↑↓ ψ ∗ ↑ ψ ∗ ↓ −H ∗↑ − Ω R − g ↑↓ ψ ∗ ↑ ψ ↓ − g ↑↓ ψ ∗ ↑ ψ ∗ ↓ − gψ ∗ ↓ − Ω R − g ↑↓ ψ ↑ ψ ∗ ↓ −H ∗↓ , (S7)with H ↑ = − ∂ /∂x + 2 i∂/∂x − δ/ V L ( x ) − µ + 2 g | ψ ↑ | + g ↑↓ | ψ ↓ | , (S8) H ↓ = − ∂ /∂x − i∂/∂x + δ/ V L ( x ) − µ + 2 g | ψ ↓ | + g ↑↓ | ψ ↑ | , (S9)and g ↑↓ = g + g . The time-independent GP equation becomes µψ = [ H + H int ( ψ ↑ , ψ ↓ )] ψ . (S10)The ground state ψ and the chemical potential µ are obtained by the imaginary-time evolution. The Bogoliubovexcitation energy ε with respect to q x is numerically obtained by diagonalizing the Bogoliubov Hamiltonian. + R = 1 : E R + R = 1 : E R + R = 2 : E R gn = 0 : E R gn = 1 : E R gn = 2 : E R + R = 2 : E R + R = 2 : E R + R = 2 : E R (a) (c) (d) (b) 在此处键入公式。 Ω L / E R 在此处键入公式。 Ω L / E R 在此处键入公式。 Ω L / E R 在此处键入公式。 Ω L / E R 入公式。 键入公式。 ∆ ac / E R 在此处键入公式。 𝑞 ac / k L Δ 𝜀 / 𝐸 𝑅 Δ 𝜀 / 𝐸 𝑅 FIG. S2: (a,b) The zero-momentum gap ∆ ε as a function of lattice strength Ω L for different Ω R (a) and gn (b). (c,d)∆ ac (c) and q ac (d) of the avoided crossing gap as functions of Ω L for Ω R = 2 . E R (dots), Ω R = 2 . E R (triangles) andΩ R = 2 . E R (squares), q ac reaches it minimum at Ω L = 0 . E R , 0 . E R and 0 . E R , respectively. The other parameters are δ = 0, gn = 1 . E R and g = − . E R . Besides the Raman detuning δ demonstrated in Figs. 2c, 2d and Fig. 3 in the main text, the effects of other tunableparameters (Raman coupling and atomic interactions) on the elementary excitations are shown in Fig. S2. The effectof the Raman coupling on zero-momentum gap ∆ ε is shown in Fig. S2(a), where the critical lattice strength for theminimum of the zero-momentum gap increases with the increasing Raman coupling strength. The effect of the atomicinteractions on ∆ ε is shown in Fig. S2(b), where the curves are shifted to larger optical lattice strength when theatomic interaction gn increases. The effects of the Raman coupling on the size and position of nonzero-momentumgap ∆ ac are shown in Figs. S2(c) and S2(d). The size of ∆ ac increases with the increasing Raman coupling strengthexcept at some crossing points. The minimum position of the nonzero-momentum gap shifts to the larger opticallattice strength for larger Raman coupling strength. Dynamical structure factor and Bragg spectroscopy measurement.—
The Bragg spectroscopy measures the dynamical Z Z Z S ( q x ) Z < Z < Z < S < ( q x ) (b) 𝑆 ( 𝑞 x ) 𝑆 𝜎 ( 𝑞 x ) 在此处键入公式。 q x / k L 在此处键入公式。 q x / k L (a) FIG. S3: Static structure factors and excitation strengths for density (a) and spin-density (b) at the critical lattice strengthΩ cL = 0 . E R . The insert in (a) corresponds to two lowest energy bands in the small momentum region. Other parametersare Ω R = 2 . E R , gn = 1 . E R and g n = − . E R . + L = 0 + L = 0 : E R + L = 0 : E R + L = 0 : E R j / j = 0 j / j = 0 : E R j / j = 0 : E R g n = ! : E R g n = ! : E R g n = ! : E R g n = 0 (b3) (b2) (b1) (a3) (a2) (a1) 𝑆 ( 𝑞 x ) 𝑆 𝜎 ( 𝑞 x ) 在此处键入公式。 q x / k L 在此处键入公式。 q x / k L 在此处键入公式。 q x / k L FIG. S4: Change of the static structure factors for density (a1-a3) and spin-density (b1-b3) with different parameters Ω L , δ and g n . (a1, b1) δ = 0, Ω R = 2 . E R , gn = 1 . E R and g n = − . E R ; (a2, b2) Ω R = 2 . E R , Ω L = 0 . E R , gn = 1 . E R and g n = − . E R ; (a3, b3) δ = 0, Ω R = 2 . E R , Ω L = 0 . E R and gn = 1 . E R . structure factor of the BEC, i.e. , the density response of the system to the external perturbation generated by thescattering probe of momentum (cid:126) q x , and energy (cid:126) ω [41, 47]. Denoting the linear perturbation V = V [ ρ † q x e − iωt + ρ − q x e iωt ], where ρ q x = (cid:80) j e iq x x j / (cid:126) is the Fourier transformation of one-body density operator with the probe momenta q x , the dynamical structure factor takes the form: S ( q x , ω ) = (cid:88) j |(cid:104) j | ρ † q x | (cid:105)| ˜ δ [ (cid:126) ω − ( E j − E )] . (S11)Here, | (cid:105) ( | j (cid:105) ) is the ground (excited) state with the energy E ( E j ). We can define the spin structure factor S σ in asimilar way, which can be measured using spin-dependent Bragg spectroscopy [48–50].The static structure factors and excitation strengths for the density and spin-density are given in Fig. S3(a) andS3(b) for the phase transition point Ω cL = 0 . E R . At Ω cL , the excitation spectrum contains two gapless Goldstonemodes although the anisotropic spin interaction is weak. S ( q x ) has the very similar feature as the spontaneous stripephase, while S σ ( q x ) shows a divergence at q x →
0. The effects of other external parameters on S ( q x ) and S σ ( q x ) aregiven in Fig. S4. As shown in Fig. S4(a1)-(a3), S ( q x = 0) = 0, independent of parameters. The dependence of S ( q x )on other parameters is generally very weak.In contrast, S σ ( q x ) shows strong dependence on other parameters. It increases with increasing optical latticestrength Ω L (Fig. S4(b1)), but decreases with increasing Raman detuning | δ | [Fig. S4(b2)]. S σ ( q x ) is larger forweaker anisotropic spin interaction [Fig. S4(b3)]. Interestingly, Fig. S4(b3) shows that S σ ( q x ) increases (decreases)monotonically with the momentum for strong (weak) anisotropic spin interaction, therefore has maximum (minimum)at q xx