Exciting the Goldstone modes of a supersolid spin-orbit-coupled Bose gas
Kevin T. Geier, Giovanni I. Martone, Philipp Hauke, Sandro Stringari
EExciting the Goldstone modes of a supersolid spin–orbit-coupled Bose gas
Kevin T. Geier,
1, 2, ∗ Giovanni I. Martone, Philipp Hauke, and Sandro Stringari INO-CNR BEC Center and Dipartimento di Fisica, Università di Trento, 38123 Povo, Italy Institute for Theoretical Physics, Ruprecht-Karls-Universität Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany Laboratoire Kastler Brossel, Sorbonne Université, CNRS,ENS-Université PSL, Collège de France; 4 Place Jussieu, 75005 Paris, France (Dated: February 5, 2021)Supersolidity is deeply connected with the emergence of Goldstone modes, reflecting the sponta-neous breaking of both phase and translational symmetry. Here, we propose accessible signaturesof these modes in harmonically trapped spin–orbit-coupled Bose–Einstein condensates, where su-persolidity appears in the form of stripes. By suddenly changing the trapping frequency, an axialbreathing oscillation is generated, whose behavior changes drastically at the critical Raman cou-pling. Above the transition, a single mode of hybridized density and spin nature is excited, whilebelow it, we predict a beating effect signaling the excitation of a Goldstone spin-dipole mode. Wefurther provide evidence for the Goldstone mode associated with the translational motion of stripes.Our results open up new perspectives for probing supersolid properties in experimentally relevantconfigurations with both symmetric as well as highly asymmetric intraspecies interactions.
Supersolidity is an exotic state of matter character-ized by the simultaneous spontaneous breaking of U (1) symmetry, yielding superfluidity, and of translational in-variance, yielding crystallization [1–3]. In the past, thisstate of matter has attracted considerable attention inthe context of solid Helium, where the experimental ef-forts to reveal the effects of superfluidity, however, havenot been conclusive [4, 5]. More recently, a renewed in-terest has emerged in the context of ultracold atomicgases, and experimental evidence of typical supersolidfeatures has been reported in Bose–Einstein condensatesinside optical resonators [6], spin–orbit-coupled configu-rations [7, 8], and in cold gases interacting with long-range dipole forces [9–11]. An important property ofsupersolidity with respect to ordinary superfluids is theappearance of additional gapless (Goldstone) modes inthe excitation spectrum, resulting from the broken trans-lational symmetry [2, 12–21]. While these modes havealready been the object of first experimental investiga-tion in harmonically trapped dipolar gases [22–25], so farno experimental observation has been reported in spin–orbit-coupled configurations.The purpose of this Letter is to characterize the exci-tation mechanisms of the Goldstone modes in the super-solid phase of a harmonically trapped spin–orbit-coupledBose–Einstein-condensed gas. We identify a characteris-tic beating effect that allows for the experimental ob-servation of a Goldstone excitation of spin nature, inanalogy to a similar procedure followed in dipolar su-persolids [22]. Moreover, we show that the locking ofthe polarization after a uniform spin perturbation pro-vides evidence for the Goldstone mode associated withthe rigid translation of the stripes [23].Spin–orbit-coupled Bose–Einstein condensates areknown to exhibit an intriguing variety of quantumphases, which are obtained by tuning the experimen-tally controllable Raman coupling, responsible for coher- ence between atoms occupying two different hyperfinestates [26–28]. Spin–orbit coupling gives rise to an ef-fective single-particle Hamiltonian of the form [29] H SP = 12 m ( p − (cid:126) k σ z ) + Ω2 σ x + δ σ z + V ( r ) , (1)where m is the atomic mass, and σ x and σ z are Paulimatrices. The spin–orbit term, fixed by the momentumtransfer (cid:126) k = (cid:126) k ˆ e x between two intersecting laser fieldsgenerating the Raman coupling with strength Ω and ef-fective detuning δ , is at the origin of non-trivial many-body effects which deeply differ from the ones causedby simple coherent coupling with negligible momentumtransfer, such as radio frequency or microwave coupling.We are in particular interested in the observability of theGoldstone modes in the presence of a harmonic trappingpotential V ( r ) = m ( ω x x + ω y y + ω z z ) / , with frequen-cies ω i and associated oscillator lengths a i = (cid:112) (cid:126) /mω i , i = x, y, z .In what follows, we assume that the two-body in-teraction between the atoms is described within theusual mean-field Gross–Pitaevskii theory of quantummixtures [30]. Then, writing the order parameter in thespinor form Ψ = (Ψ ↑ , Ψ ↓ ) T with the wave functions Ψ ↑ and Ψ ↓ describing the relevant hyperfine states, the en-ergy of the system is E = (cid:90) d r (cid:18) Ψ † H SP Ψ + g nn n g ss s z g ns ns z (cid:19) . (2)Here, we have identified three relevant interaction param-eters g nn = ( g ↑↑ + g ↓↓ +2 g ↑↓ ) / , g ss = ( g ↑↑ + g ↓↓ − g ↑↓ ) / ,and g ns = ( g ↑↑ − g ↓↓ ) / , fixed by proper combinations ofthe coupling constants g ij = 4 π (cid:126) a ij /m , where a ij arethe respective scattering lengths with i, j ∈ {↑ , ↓} . Theparticle density and the spin density entering Eq. (2) aredefined, respectively, as n ( r ) = | Ψ ↑ ( r ) | + | Ψ ↓ ( r ) | and s z ( r ) = | Ψ ↑ ( r ) | − | Ψ ↓ ( r ) | , the former being normalized a r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b to the total particle number N = (cid:82) d r n ( r ) . Averagesof an observable O are defined as (cid:104) O (cid:105) = (cid:82) d r Ψ † O Ψ /N ,or, with respect to an individual spin component i , as (cid:104) O (cid:105) i = (cid:82) d r Ψ ∗ i O Ψ i / (cid:82) d r | Ψ i | . Depending on the val-ues of the parameters entering the energy functional (2),the ground state of the system can be either in the single-minimum, in the plane-wave, or in the stripe phase, her-after also called supersolid phase (see, e.g., Ref. [31]).In order to explore the nature of the elementaryexcitations, we numerically solve the coupled time-dependent Gross–Pitaevskii equations, directly derivablefrom the variation of the action S = (cid:82) d t E [Ψ ↑ , Ψ ↓ ] − i (cid:126) (cid:82) d t d r (Ψ ∗↑ ∂ t Ψ ↑ +Ψ ∗↓ ∂ t Ψ ↓ ) with respect to Ψ ∗↑ and Ψ ∗↓ .More specifically, we compute the ground state of thesystem in presence of a static perturbation of density orspin nature by means of a non-linear conjugate gradientmethod [32]. At time t = 0 , the perturbation is suddenlyremoved, and the quenched system is evolved in time us-ing a time-splitting Fourier pseudospectral method [33].The frequencies of the induced collective oscillations areextracted from sinusoidal fits to the time traces of therelevant observables.We first investigate the case of symmetric intraspeciesinteractions, while the effects of strong asymmetries inthe couplings are explored towards the end of this Let-ter. The majority of previous works focuses on sym-metric configurations, as it is well realized, for instance,by Rb [29]. For our purposes, we choose a configura-tion close to Rb with a ↑↑ = a ↓↓ = 100 a , where a isthe Bohr radius. Naturally, Rb is characterized by avalue of g ↑↓ very close to g ↑↑ and g ↓↓ , yielding g ss ≈ ,and hence a small value of the critical Raman coupling Ω cr = 4 E r (cid:112) g ss / ( g nn + 2 g ss ) for the transition to thesupersolid phase [34, 35], where E r = (cid:126) k / m is therecoil energy. Increasing the value of Ω cr is desirable tolimit the effects of magnetic fluctuations and to observevisible consequences of the presence of stripes. To thisend, we consider configurations characterized by an ef-fective reduction of the coupling constant g ↑↓ . This maybe achieved by reducing the spatial overlap between thewave functions of the two spin components, for instance,with the help of a spin-dependent trapping potential sep-arating the two components [36, 37], or using pseudo-spinorbital states in a superlattice potential [7]. Here, we fol-low the former approach, using a quasi- harmonic trapwith frequencies ( ω x , ω y , ω z ) = 2 π × (50 , , ,where the two spin components are separated along the z -direction such that the effective interspecies cou-pling becomes ˜ g ↑↓ = 0 . g ↑↑ with ˜ g ↑↑ = g ↑↑ / √ πa z =˜ g ↓↓ [36]. The reported Raman coupling Ω correspondsto an effective coupling, accounting for the reduced spa-tial overlap [36]. Furthermore, we choose the param-eters k = √ π/λ Raman with λ Raman = 804 . [29], N = 10 , and δ = 0 .In Figs. 1a and 1b, we report the time dependenceof the root mean square radius in x -direction, x rms = /E r . . . . F r e q u e n c y ω / ω x stripe planewave singleminimum c ω B ( x rms ) ω B ( d x ) ω D ( x cm ) ω SD ( d x ) ω x / √ E r χ ω x t/ π . . . x r m s / a x Ω = 1 . E r − . . . a ω x t/ π Ω = 2 . E r − d x / a x b Figure 1. Collective modes for symmetric intraspecies inter-actions. (a, b) Oscillations of the observables x rms = (cid:112) (cid:104) x (cid:105) and d x = (cid:104) x (cid:105) ↑ − (cid:104) x (cid:105) ↓ after suddenly removing the per-turbation H pert = λmω x x with λ = 0 . . In the stripephase (a), a clear beating of two frequencies ω B ≈ . ω x and ω SD ≈ . ω x is visible in the observable d x , which isabsent in the plane-wave phase (b), where d x oscillates onlyat a single frequency ω B ≈ . ω x . (c) Dispersion ω (Ω) of the breathing mode (B), the spin-dipole mode (SD), andthe center-of-mass (dipole) mode (D), calculated for λ (cid:28) .The breathing and the spin-dipole modes are fully hybridizedabove the critical coupling Ω cr ≈ . E r , while below Ω cr ,a new Goldstone mode of spin nature appears. The dipolefrequencies ω D have been obtained from the center-of-massoscillation x cm = (cid:104) x (cid:105) after a sudden shift of the trap cen-ter. For Ω > Ω cr , they practically coincide with the bound ω x / (cid:112) k χ . The violation of this upper bound by ω D for Ω < Ω cr implies the emergence of a new low-energy mode. (cid:112) (cid:104) x (cid:105) , and of the relative displacement of the two spincomponents, d x = (cid:104) x (cid:105) ↑ − (cid:104) x (cid:105) ↓ , after the sudden removalof a static perturbation proportional to the operator x ,corresponding to a sudden decrease of the trapping fre-quency. For values of Ω larger than the critical Ramancoupling Ω cr ≈ . E r , both observables oscillate with thesame frequency (see Fig. 1b). The occurrence of a sin-gle collective mode of hybridized density and spin naturefor Ω > Ω cr can be understood as a consequence of thehydrodynamic behavior of the system and is caused bythe locking of the relative phase of the order parameterof the two spin components [38].When we enter the stripe phase, the scenario changesdrastically and we observe the appearance of a new os-cillation of spin nature, revealed by the beating in thesignal d x (see Fig. 1a). This oscillation is the finite-sizemanifestation of the gapless Goldstone spin branch ex-hibited by the supersolid phase in uniform matter [15].We have verified that the same modes can also be ex-cited by applying a perturbation proportional to xσ z ,corresponding to a relative displacement of the two spincomponents to one another.In Fig. 1c, we report the dispersion ω (Ω) of the result-ing breathing and spin-dipole excitations. The frequen-cies have been calculated for small perturbations in theregime of linear response, but the beating effect is clearlyvisible also for larger perturbation strengths (cf. Fig. 1a)that are closer to the onset of non-linearities. Similar dis-persion laws have been obtained in Ref. [39] by solvingthe Bogoliubov equations for a spin–orbit-coupled mix-ture in one dimension. With respect to Ref. [39], ourapproach explicitly exposes the beating effect betweenthe spin-dipole excitation and the compression mode inthe stripe phase, as well as the full hybridization of thetwo modes above Ω cr .In the limit Ω → , the spin-dipole frequency can becalculated analytically within the formalism of two-fluidhydrodynamics [30]. We find ω (Ω →
0) = ω x − ( g ns /g ss ) g nn /g ss − ( g ns /g ss ) , (3)yielding the value ω SD = 0 . ω x for the configuration con-sidered here, in agreement with Fig. 1c. The dispersionof the spin-dipole branch decreases as Ω approaches thetransition at the critical value Ω cr , and is expected tovanish at the spinodal point, corresponding to a value of Ω a little higher than Ω cr where the system develops adynamic instability associated with the divergent behav-ior of the magnetic polarizibility [40]. The decrease of ω SD as a function of Ω is a crucial consequence of spin–orbit coupling and of the presence of stripes. By con-trast, in the presence of radio frequency or microwavecoupling, the spin-dipole frequency increases with thecoupling strength, quickly approaching the value Ω ofthe spin gapped branch [41].In Fig. 1c, we also report the dispersion of the center-of-mass (dipole) mode, which is excited by suddenly re-moving a perturbation proportional to the operator x ,corresponding to a shift of the harmonic trap along the x -direction. Above Ω cr , the center-of-mass operator x and the spin operator σ z excite the same mode, similarlyto the case of the operators x and xσ z discussed above.Both the breathing and the dipole frequencies decreasewhen approaching the transition to the single-minimumphase, where the effective mass increases, inducing siz-able non-linear effects [40, 42].At the transition to the supersolid phase, both thebreathing and the dipole frequencies exhibit a smalljump, reflecting the first-order nature of the supersolid–superfluid transition. Entering the supersolid phase, oneexpects the emergence of the Goldstone mode that cor-responds, in uniform matter, to the rigid translation ofstripes. In a harmonic trap, the frequency of this motion ω x t/ π − . . . . . h σ z i − h σ z i . .
286 Ω = 2 E r Ω = 3 E r a − − x/a x n ( x ) a x t Ω = 2 E r ω x t/ π . . . b Figure 2. Evidence for the “zero-frequency” Goldstone modeassociated with the translation of the stripes. (a) Time evo-lution of the polarization (cid:104) σ z (cid:105) with respect to its equilibriumvalue (cid:104) σ z (cid:105) after removing the perturbation H pert = − λE r σ z with λ = 0 . in the stripe phase ( Ω = 2 E r ) and λ = 0 . in the plane-wave phase ( Ω = 3 E r ). In the latter case,the polarization oscillates around equilibrium at the dipolefrequency ω D ≈ . ω x . By contrast, in the stripe phase,the polarization remains locked for a time much longer than π/ω x , providing evidence for the “zero-frequency” Goldstonemode. The low-amplitude oscillations at the dipole fre-quency ω D ≈ . ω x shown in the inset indicate a weak exci-tation of the center-of-mass mode. (b) Time evolution of thedensity profile n ( x ) in the stripe phase for the same scenarioas in (a), explicitly revealing the excitation of the transla-tional motion of the stripes. In the linear regime, the velocityof the stripes is proportional to the perturbation strength λ according to v ( λ ) ≈ . λ (cid:126) k /m . is not exactly zero, but still much smaller than the oscilla-tor frequency ω x . The existence of this “zero-frequency”Goldstone mode can be inferred employing a sum-ruleargument, according to which a rigorous upper bound tothe lowest-energy mode excited by the operator x is givenby ω lowest ≤ ω x / √ E r χ , where χ is the magnetic po-larizibility [40]. This upper bound practically coincideswith the center-of-mass frequency ω D if Ω > Ω cr , whilebelow Ω cr the calculated value of ω D violates the bounddue to the large value of χ , revealing the existence of anew low-frequency mode.To shed light on the nature of this low-energy excita-tion, we apply a uniform spin perturbation proportionalto the operator σ z , causing a magnetic polarization of thesystem. After removing the perturbation [43], one wouldexpect the polarization to oscillate around its equilib-rium value, driven by the Raman coupling. In Fig. 2a,we show the time evolution of the polarization for typi-cal values of Ω in the stripe phase and in the plane-wavephase. Above Ω cr , after a short initial decrease reflect-ing the contribution of the high-frequency gapped spinbranch [38] to the static magnetic polarizability, the po-larization indeed oscillates around equilibrium with thecenter-of-mass frequency. In the stripe phase, we find /E r . . . . . F r e q u e n c y ω / ω x stripe planewave singleminimum c ω SD ( d x ) ω B ( x rms ) ω B ( d x ) ω D ( x cm ) ω x / √ E r χ ω x t/ π . . . x r m s / a x Ω = 0 . E r − . . . a ω x t/ π Ω = 1 . E r − d x / a x × − b Figure 3. Same as Fig. 1, but for strongly asymmetric in-traspecies interactions, as relevant for K. The asymmetryleads to a smooth crossover between the plane-wave and thesingle-minimum regime. Nevertheless, in the stripe phase, thebeating effects (a) and the additional frequencies (c) charac-teristic of the Goldstone mode are evident, while they areabsent in the plane-wave phase (b). instead that the polarization remains locked to its ini-tial value throughout the simulation time, with a resid-ual small-amplitude oscillation stemming from a weakexcitation of the center-of-mass mode by the spin oper-ator σ z . We have verified that the locking of the polar-ization survives longer than one thousand times the os-cillator time π/ω x , confirming the anticipated low fre-quency of the Goldstone mode. Remarkably, releasingthe spin perturbation has the effect of applying a boostto the stripes, causing their translation at a constant ve-locity proportional to the perturbation strength, as illus-trated in Fig. 2b. The translation of the stripes is practi-cally independent of the center-of-mass motion, therebyproviding direct evidence for the excitation of the “zero-frequency” Goldstone mode. By contrast, after suddenlyshifting the trap center, corresponding to a perturbationby the dipole operator x , the center of mass oscillatesaround equilibrium both in the superfluid and in the su-persolid phase (not shown). This shows that the “zero-frequency” Goldstone mode contributes only marginallyto the static dipole polarizibility, whereas its strong exci-tation by the operator σ z implies that it constitutes thepredominant contribution to the magnetic polarizibilityin the stripe phase.In the last part of this work, we focus on a configu-ration characterized by strongly asymmetric intraspeciesinteractions, g ↑↑ (cid:29) g ↓↓ ≈ g ↑↓ , yielding g nn ≈ g ss ≈ g ns .The main motivation is to explore the consequences of the high spin polarization on the stripe phase, relevant for thecase of K, which has recently become available for ex-periments on spin–orbit-coupled Bose–Einstein conden-sates [44]. We consider a set of scattering lengths givenby a ↑↑ = 252 . a , a ↓↓ = 1 . a , and a ↑↓ = − . a , realiz-able in K by using Feshbach resonances near a magneticfield of B ≈
389 G [44]. These values are consistent withthe stability condition g ↑↑ g ↓↓ > g ↑↓ . Furthermore, wechoose k = 2 π/λ Raman with λ Raman = 768 .
97 nm [44], ( ω x , ω y , ω z ) = 2 π × (50 , , , N = 10 , and δ = 0 .With these parameters, the transition to the supersolidphase occurs at Ω cr ≈ . E r , leading to fringes with highcontrast and hence to observable supersolid effects.Due to the strong asymmetry in the interspecies inter-actions ( g ↑↑ /g ↓↓ ≈ ), energetic minimization favors alarge value of the spin polarization. In particular, the po-larization is large also in the stripe phase, as opposed tothe symmetric case, where it instead vanishes. A simpleestimate of this effect can be obtained in uniform mat-ter, where, for Ω → , the variation of the energy (2)yields the expression s z /n = − g ns /g ss , which amountsto a polarization of (cid:104) σ z (cid:105) ≈ − . for our parameters.An important consequence of the strong asymmetry ofthe intraspecies interactions is that the central density,dominated by the majority component Ψ ↓ , is stronglyenhanced with respect to the usual values characteriz-ing symmetric configurations. This effect is accompaniedby a shrinking of the cloud radius and the occurrence offringes with high contrast in the minority component Ψ ↑ .Following the procedure employed in the symmetriccase, we consider the relative displacement d x of the twocomponents after a sudden quench of the trapping fre-quency (see Figs. 3a and 3b). Also in this case, we ob-serve a clear beating effect, revealing the occurrence of aGoldstone mode of spin nature in the stripe phase. Dueto the large polarization of the system, this mode mainlycorresponds to the motion of the minority component,while the majority component remains practically at rest.In Fig. 3c, we show the dispersion law of the resultingelementary excitations as a function of Ω , along with thedispersion of the center-of-mass mode excited by shiftingthe trap center. At Ω → , the spin-dipole frequency islarger than the center-of-mass frequency as a result of thenegative interspecies interaction g ↑↓ , in agreement withEq. (3). Due to the asymmetry of the intraspecies inter-actions, the transition between the plane-wave and thesingle-minimum phase is less sharp than in the symmetriccase and actually becomes a smooth crossover. Further-more, we find that the contrast of the stripes does notexhibit a jump at the supersolid–superfluid transition,but vanishes continuously, which may indicate that, dueto the smallness of g ↓↓ , the system is still rather far fromthe thermodynamic limit. Nonetheless, the qualitativefeatures of the excitation spectrum, including the occur-rence of the “zero-frequency” Goldstone mode, are similarto the symmetric case.In conclusion, we have provided accessible signatures ofthe Goldstone modes exhibited by the stripe phase of aharmonically trapped spin–orbit-coupled Bose–Einsteincondensate. The Goldstone modes are revealed by acharacteristic beating effect in the spin-dipole observable,following an experimentally straightforward density per-turbation, and by the dynamic excitation of the transla-tional motion of the stripes (“zero-frequency” Goldstonemode), following the release of a uniform spin pertur-bation. Both configurations with symmetric and highlyasymmetric intraspecies interactions provide promisingplatforms for the identification of these fundamental hall-marks of supersolidity.We thank Li Chen, Han Pu, Jean Dalibard, GabrieleFerrari, and the ICFO team led by Leticia Tarruell forfruitful discussions. This work is part of and fundedby the ERC Starting Grant “StrEnQTh” (project ID ), the Provincia Autonoma di Trento, and Q@TN— Quantum Science and Technology in Trento. Theauthors acknowledge support by the state of Baden–Württemberg through bwHPC. ∗ [email protected][1] D. Thouless, The flow of a dense superfluid, Ann. Phys. , 403 (1969).[2] A. Andreev and I. Lifshitz, Quantum theory of defects incrystals, Sov. Phys. JETP , 1107 (1969).[3] A. J. Leggett, Can a solid be "superfluid"?, Phys. Rev.Lett. , 1543 (1970).[4] S. Balibar, The enigma of supersolidity, Nature (London) , 176 (2010).[5] M. Boninsegni and N. V. Prokof’ev, Colloquium: Super-solids: What and where are they?, Rev. Mod. Phys. ,759 (2012), arXiv:1201.2227 [cond-mat.stat-mech].[6] J. Léonard, A. Morales, P. Zupancic, T. Esslinger, andT. Donner, Supersolid formation in a quantum gas break-ing a continuous translational symmetry, Nature (Lon-don) , 87 (2017), arXiv:1609.09053 [cond-mat.quant-gas].[7] J.-R. Li, J. Lee, W. 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