Oscillations of a quasi-one-dimensional dipolar supersolid
B. Kh. Turmanov, B. B. Baizakov, F. Kh. Abdullaev, M. Salerno
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b Oscillations of a quasi-one-dimensional dipolar supersolid
B. Kh. Turmanov , , B. B. Baizakov , F. Kh. Abdullaev , , M. Salerno Physical-Technical Institute, Uzbek Academy of Sciences, 100084, Tashkent, Uzbekistan Department of Theoretical Physics, National University of Uzbekistan, Tashkent 100174, Uzbekistan Dipartimento di Fisica E.R. Caianiello, and INFN Gruppo Collegato di Salerno,Universita di Salerno, Via Giovanni Paolo II, 84084 Fisciano, Salerno, Italy (Dated: March 1, 2021)The properties of a supersolid state (SS) in quasi-one-dimensional dipolar Bose-Einstein conden-sate is studied, considering two possible mechanisms of realization: (i) due to repulsive three-bodyatomic interactions, (ii) and quantum fluctuations in the framework of the Lee-Huang-Yang (LHY)theory. The proposed theoretical model, based on the minimization of the energy functional, allowsevaluating the amplitude of the SS for an arbitrary set of parameters in the governing Gross-Pitaevskii equation (GPE). To explore the dynamics of the supersolid first we numerically constructits ground state in a box-like or parabolic trap then impose a perturbation. In oscillations of theperturbed supersolid we observe the key manifestation of SS, namely the free flow of the superfluidfraction through the crystalline fraction of the system. Two distinct oscillation frequencies of thesupersolid associated with the amplitude (Higgs mode) and phase (Goldstone mode) of the wavefunction are identified from numerical simulations of the GPE.
I. INTRODUCTION
The existence of an unusual state of matter, called supersolid , was theoretically predicted more than half a centuryago (see review articles [1, 2]). In this state, matter exhibits both the crystalline order and superfluidity at the sametime. More definitely, some fraction of the total mass is rigid as a crystal, while the rest flows through the systemfreely, without dissipation. The last property is considered to be the key signature of the supersolid state (SS) andits experimental observation has been the main objective pursued over the years of research. To date, there aresome experimental pieces of evidence for this phenomenon in solid He, but the theoretical interpretation of observedproperties remains inconclusive, as summarized in [3]. Recent studies have discovered, that quantum gases with strongdipole-dipole atomic interactions can be another medium, where the SS clearly shows up [4–7]. The advantage hereis that the quantum gases appear to be a highly controllable coherent system. The physical mechanism behind theemerging SS in this system is the delicate balance between the long-range dipolar and short-range contact interactionsamong the atoms of the gas. The stabilizing effect of quantum fluctuations is essential for the existence of quantumdroplets and dipolar supersolids (see review articles [8–10]. The short-range interactions originate from two-bodyand three-body atomic collisions, characterized by the s -wave scattering length a s , while the quantum fluctuationeffects are described in the local density approximation by the Lee-Huang-Yang (LHY) theory [11, 12]. The quantumfluctuations and three-body atomic interactions were shown to play similar roles in the process of stabilization ofself-bound quantum droplets [13]. It is natural to suggest, that they play similar roles in stabilizing the supersolidsas well. Formation of an array of quantum droplets in dipolar Bose-Einstein condensates (BEC), stabilized solelyby the three-body atomic interactions, was theoretically demonstrated [14, 15]. A new kind of dipolar quantumdroplets, which can be made of strongly interacting bosons, was reported in [16]. In the strongly interacting Tonks-Girardeau regime, the governing equation of Ref. [16] reduces to the nonlocal Gross-Pitaevskii equation (GPE) withquintic nonlinearity, previously studied in the context of flat-top localized states in quasi-1D dipolar condensates[17, 18], while in the weakly interacting limit it reduces to usual nonlocal GPE with cubic nonlinearity [19, 20]. Ina recent study [21], the authors have derived the extended GPE, taking into account the next order (beyond LHY)quantum fluctuation effects. The additional term in the energy functional has a cubic dependence on the gas density,i.e. formally similar to the effect of attractive three-body atomic interactions. It follows from the above argumentsthat the nonlocal GPE with cubic-quintic nonlinearity and usual LHY term can be a more adequate framework fordescribing the SS in dipolar quantum gases, therefore it is adopted in the present study.The basic properties of condensed matter systems can be revealed from the analysis of their excitation spectra.The spectrum of elementary excitations in dipolar quantum gases of Dy and
Er, confined to a harmonic trap,was studied in [7, 22]. It was experimentally demonstrated, that in the regime of BEC the system exhibits ordi-nary quadrupole oscillations with a single frequency, while in the supersolid regime two distinct frequencies arise,corresponding to oscillations of the superfluid and crystalline phases. An important result of these works is thatthe two-frequency response of a supersolid, originally predicted for infinite-size systems, was confirmed for finite-sizetrapped quantum gases. Supersolid behavior of a dipolar BEC was addressed through numerical simulations of theBogolyubov - de Gennes equations in a tubular periodic confinement [23]. In this work, the coherence of the systemacross the phase transitions BEC - supersolid - isolated quantum droplets was analyzed. Description of the groundstate and collective excitations of a supersolid in dipolar quantum gases, based on the variational approach, waspresented in [24, 25]. A comprehensive introduction to the nonlocal GPE with the contribution of quantum fluctua-tion effects in a quasi-one-dimensional setting can be found in [26]. Despite impressive experimental and theoreticalprogress in exploring the SS in dipolar quantum gases, some static and dynamic aspects of trapped supersolids remainless studied.Our objective in this work consists in exploring the conditions, which can support a robust SS in a quasi-one-dimensional dipolar Bose-Einstein condensate with competing long-range dipole-dipole and short-range contact atomicinteractions, in the presence of quantum fluctuation effects. A theoretical model will be developed to estimate theamplitude of the emerging SS. It should be noted that the amplitude of the SS is directly linked to the experimentallymeasurable quantity, called density contrast [27, 28]. Therefore the proposed model can be useful for the analysis andinterpretation of experimental results. Besides, we numerically investigate the dynamics of a supersolid, subject to aweak perturbation, to demonstrate its key property, namely the free flow of the superfluid fraction of the quantum gasthrough its crystalline fraction. When perturbed, the crystalline and superfluid components of the system performundamped oscillations with different frequencies [7, 22, 29], which is the hallmark property of supersolids. In ournumerical simulations by recording the time-dependence of the width of the oscillating supersolid, and Fourier-analyzing the obtained data, we identify these two distinct frequencies associated with the Higgs mode, correspondingto an oscillation of the amplitude of the density modulation, and the Goldstone mode, which corresponds to a shiftof the position of the density modulation [30].The paper is structured as follows. In the next Sec. II we present the main equations, the spectrum of elementaryexcitations, and the variational approach for the amplitude of the supersolid. Sec. III is devoted to SS dynamicssubject to perturbation. In Sec. IV we summarize our findings.
II. THE MODEL AND MAIN EQUATIONS
The model is based on a normalized quasi-1D nonlocal GPE, derived from its full three-dimensional version (fordetails of 3D to 1D reduction see [26, 31]) iψ t + 12 ψ xx + V ( x ) ψ + q | ψ | ψ + γ | ψ | ψ + p | ψ | ψ + g ψ ( x, t ) Z + ∞−∞ R ( | x − x ′ | ) | ψ ( x ′ , t ) | dx ′ = 0 , (1)where ψ ( x, t ) is the mean-field wave function of the condensate, V ( x ) is the trap potential, q, p, g are the coefficients oftwo-body, three-body and dipole-dipole atomic interactions, respectively. The parameter γ quantifies the contributionof quantum fluctuations, R ( x ) = √ π exp (cid:0) x (cid:1) (cid:0) x (cid:1) erfc ( | x | ) − | x | is the response function which characterizesthe degree of nonlocality of the medium [32]. The Eq. (1) is obtained for the condensate, which is tightly confinedin the radial direction, with the transverse and axial trap frequencies satisfying the condition ω ⊥ ≫ ω x . Under suchconfinement, the condensate acquires a highly elongated form in the axial direction commonly termed as cigar-shapedcondensate. Below we assume the trap potential V ( x ) to have tight tubular geometry so that its presence is specifiedthrough periodic boundary conditions for the wave function, as accepted in theoretical work [23] and experimentalsetup [6, 36]. To impose perturbation upon the stationary supersolid we employ a weak parabolic potential V ( x ) = αx with α ≪ t → ω ⊥ t , x → x/a ⊥ , ψ → p | a bg | ψ , with a bg being the background value of the s -wave scattering length, a ⊥ = p ¯ h/mω ⊥ is the radial harmonic oscillator length, m is the atomic mass. The dimensionless coefficients ofEq. (1) are expressed through the actual parameters of the system as follows q = − a s / | a bg | , p = − g m ω ⊥ / π ¯ h a s , g = − µ µ / (4 π ¯ h | a bg | /m ), where g characterizes the strength of three-body atomic interactions, µ , µ are thepermeability of vacuum and atomic magnetic moment, respectively. In these notations q > g < p < γ = 128 a s (1 + 1 . ε dd ) / √ πa ⊥ , where ε dd = mµ µ / π ¯ h | a s | is theratio between the dipolar and s -wave scattering lengths. The LHY energy correction in dipolar condensates leads to arepulsive term ∼ γ n / D in the GPE (1) provided that the one-dimensional density of the gas n D is sufficiently large( n D a s > .
6) [33]. It is relevant to mention that the coefficient p in Eq. (1) may also be associated with the beyondLHY description of quantum fluctuation effects [21]. A. Dispersion relation for elementary excitations
The spatial period of emerging density modulations in a dipolar condensate, which can transform in suitableconditions into SS, can be defined from the dispersion curve of elementary excitations. To derive the dispersionrelation we consider the ground state wave function of the form ψ = √ n e iθt , (2)where n is the constant background density, θ is the uniform phase θ = n (cid:18) q + pn + γn / + g Z ∞−∞ R ( | x | ) dx (cid:19) . (3)Next, we introduce weak perturbation to the ground state ψ ( x, t ) = ψ + δψ ( x, t ), assuming δψ ( x, t ) = η ( x, t ) e iθt and | η ( x, t ) | ≪ n . Substituting ψ ( x, t ) into Eq. (1) and linearizing around ψ ( x, t ) one obtains iη t + 12 η xx + n (cid:18) q + 2 pn + 32 γn / (cid:19) ( η ( x, t ) + η ∗ ( x, t ))+ gn Z ∞−∞ R ( x − x ′ )( η ( x ′ , t ) + η ∗ ( x ′ , t )) dx ′ = 0 . (4)Splitting the perturbation field into real and imaginary parts η ( x, t ) = u ( x, t ) + iv ( x, t ), and taking the Fouriertransform of the resulting equation, yields the following coupled system for the transformed components˙˜ u ( k, t ) = 12 k ˜ v ( k, t ) , (5)˙˜ v ( k, t ) = − (cid:20) k − n (cid:18) q + 2 pn + 32 γn / + g ˜ R ( k ) (cid:19)(cid:21) ˜ u ( k, t ) , where the overdot denotes the derivative with respect to time, while the tilde stands for the Fourier transform˜ f ( k ) = R ∞−∞ f ( x ) e ikx dx , and ˜ R ( k ) is given by˜ R ( k ) = 2 (cid:20) k e k / Ei (cid:18) − k (cid:19)(cid:21) , (6)with Ei( z ) being the exponential integral function [34]. Notice that the above equations are equivalent to a singleharmonic oscillator equation ¨˜ u + Ω ( k )˜ u = 0 with the frequencyΩ( k ) = k (cid:18) k − n ( q + 2 pn + 32 γn / + g ˜ R ( k )) (cid:19) / . (7)For small values of the wave vector k the above frequency defines the energy spectrum E ( k ) = v s k of the elementaryexcitations obtained from the Bogoliubov-de Gennes analysis with a sound velocity v s = s − n (cid:18) q + 2 pn + 32 γn / + g ˜ R ( k ) (cid:19) . (8)For larger values of k and suitable parameters the dispersion relation (7) features a local minimum at k = k rot , corre-sponding to quasi-particles known as rotons , whose existence in dipolar condensates was theoretically predicted [35]and experimentally confirmed [36, 37]. Although initially the roton was linked to local vorticity in superfluid He [38],nowadays it is viewed as the precursor of a crystallization instability [39].Figure 1 illustrates the dispersion relation (7) for different sets of parameter values. The real and positive values ofthe frequency Ω( k ) > k ∼ k rot implies that the roton quasi-particles aremore effectively generated in the condensate compared to other types of excitations [31]. The supersolid emerges asthe frequency (7) becomes imaginary (red and black solid lines in Fig. 1) and roton instability sets in [40]. The steadygrowth of the amplitude of density waves, caused by the roton instability, is counterbalanced by the repulsive LHYterm and/or three-body atomic scattering effects. It should be stressed that the balance solely between the attractivetwo-body ( q >
0) and repulsive dipolar ( g <
0) interactions is unstable (corresponding to the curve (a) in Fig. 1).Therefore, the presence of repulsive LHY and/or three-body interaction terms is essential for the existence of stableSS in dipolar condensates. The spatial period of the supersolid represents its most important characteristic which isdefined by the wave vector of rotons λ = 2 π/k rot . Another important parameter is the amplitude of the SS, expressedthrough its field contrast as a = max( | ψ ( x ) | ) − min( | ψ ( x ) | ), where max (min) denotes the maximum (minimum) overthe space. Below we study the existence of SS in dipolar quantum gases employing the variational approach, exact Ω ( k ) abcd FIG. 1: The excitation spectrum according to Eq. (7) for A = 2 √ q = 1 and different values of the dipolar ( g ), quintic ( p )and LHY ( γ ) coefficients: (a) g = − p = 0, γ = 0; (b) g = − . p = 0, γ = 0; (c) g = − . p = − . γ = − . g = − . p = − . γ = − . k rot = 4 . k rot = 4 . k ) becomes imaginary. diagonalization of the Hamiltonian and direct numerical integrations of the GPE (1). B. Variational approach for supersolids in dipolar quantum gases
It was pointed out earlier that supersolids in dipolar condensates can exist due to the stabilizing effect of repulsivequantum fluctuations and/or three-body atomic interactions. The existence conditions and some characteristics of thesupersolid can be evaluated using the variational approach. A variational theory based on minimization of the GPEenergy functional was developed for the former [24, 25] and latter [41] cases, using the cosine modulated trial function ψ ( x ) = √ n (cid:0) cos θ + √ θ cos( kx ) (cid:1) , with n being the average linear density of the condensate. The variationalparameters θ and k characterize the amplitude and period of the density modulations, respectively. This ansatz isvalid for weakly modulated density waves in the absence of trapping potential.Below we develop a variational approach for supersolids in dipolar quantum gases using an alternative trial function ψ = A + a cos( kx ) , (9)where the variational parameters ( A, a, k ) are introduced, meaning the amplitude of the background, amplitude andwave vector of density modulations, respectively. The normalization to the average linear density of the condensate n = lim L →∞ L − Z L/ − L/ | ψ ( x, t ) | dx ! = a A (10)is adopted. Similar (sine-modulated) trial function was previously employed in Ref. [28]. These trial functions aresuitable when the axial confinement is absent ( α = 0). By substitution of the ansatz (9) into the energy density,corresponding to Eq. (1) without the harmonic trap E = 12 | ψ x | − q | ψ | − γ | ψ | − p | ψ | − g | ψ | ∞ Z −∞ R ( | x − x ′ | ) | ψ ( x ′ , t ) | dx ′ , (11)and integration over the space variable E = lim L →∞ (cid:16) L − R L/ − L/ E dx (cid:17) , one obtains the following GP energy functional E = a k − q (cid:0) n + 16 a n − a (cid:1) − γ r n − a (cid:0) n + 32 a n − a (cid:1) − p (cid:0) n + 48 a n − a n − a (cid:1) − g (cid:20) n √ π + a (cid:18) n − a (cid:19) ¯ R ( k ) + a
16 ¯ R (2 k ) (cid:21) . (12)The response function in the Fourier space ˜ R ( k ), defined by Eq. (6), has the following properties˜ R (0) = 2 , ˜ R ′ ( k ) = k + 42 k ˜ R ( k ) − k , (13)where the prime stands for the derivative with respect to k . Using the relations Eq. (13), the expression for theamplitude of density modulations a can be derived from minimization of the energy ( dE/dk = 0) a = 2 k + 8 gn − gn ( k + 4) ˜ R ( k ) g (( k + 1) ˜ R (2 k ) − k + 4) ˜ R ( k ) + 14) ! / (14)and the vanishing of the roton gap (Ω( k ) = 0) at some k >
0. The supersolid amplitude Eq. (14) implicitly dependson the system parameters through k = k rot = f ( q, p, γ, g, n ). Calculations according to Eq. (7) and Eq. (14) forparameter values specified for the black line (d) in Fig. 1 gives for the roton minimum k rot = 4 .
1, and SS amplitude a = 0 . n = 12 .
12, which is in good agreement with the results of GPE ( a = 0 . L = 9 λ with periodic boundary conditions. a) b) - - - x | ψ | FIG. 2: a) Stable supersolid state (red solid line) produced by self-consistent procedure applied to GPE (1), starting form theinitial condition ψ ( x ) = A + a cos( kx ) with A = 2 √ a = 0 . k = 4 . | ψ ( x, t ) | showingthe time evolution of the SS according to GPE (1). Parameter values: q = 1, g = − . p = − . γ = − . λ = 2 π/k ≃ . . The validity of the variational Eq. (14) is restricted to weak (sinusoidally modulated) supersolids. When it iscomposed of strongly localized spikes of density, the trial function (9) is no longer appropriate.
III. DYNAMICS OF THE PERTURBED SUPERSOLID
The basic properties of a supersolid can be studied by observing its collective dynamics. Oscillations of the supersolidcan be induced in different ways, such as variations of the trapping potential or changing the atomic interactions.Two modes of supersolid oscillations, associated with the amplitude (Higgs) and phase (Goldstone) of the condensatewave function are of special interest [43] because they can help to clarify the correspondence between models in highenergy physics and condensed matter systems. Excitation of the Goldstone mode in spin-orbit-coupled Bose gas bysuddenly changing the trapping harmonic potential has been explored in a recent paper [44].Below we present the results of numerical simulations where the oscillations of the supersolid, confined to a box-like [45] or harmonic trap, have been recorded and Fourier analyzed. The starting point is the stationary (ground) stateof a trapped SS, which is numerically obtained by self-consistent diagonalization of the Hamiltonian associated withthe GPE (1) for parameter values corresponding to the roton minimum of the dispersion curve (7). The oscillationsof the supersolid can be induced either by slight modification of the trap potential, or variation of the coefficient ofatomic interactions. Figure 3 illustrates the excitation of the amplitude mode due to the slight axial squeezing of thebox-like trap. a) b) - - - | ψ | FIG. 3: Excitation of the amplitude mode due to reduction of the box-trap length L = 9 λ by 10 %. a) The superfluid linksconnecting the spikes of the crystalline component become periodically stronger (blue dashed line, t = 5) and weaker (red line, t = 9) as time progresses. b) The density plot showing the periodic dynamics of the amplitude mode according to GPE (1).Parameter values: q = 1, g = − . p = − . γ = − . A = 2 √ k = 4 . λ = 2 π/k = 1 . The excitation of the lattice (phase) mode becomes more distinguished if the perturbation of the supersolid isachieved by imposing the weak parabolic potential. In Fig. 4 the time-dependent effective width of the supersolid W ( t ) = N Z L/ − L/ x | ψ ( x, t ) | dx ! / (15)is depicted, where L is the domain length, N = R L/ − L/ | ψ ( x ) | dx is the conserved quantity associated with the norm ofthe wave function. Under the action of the harmonic potential, the supersolid performs collective oscillations involvingboth the amplitude and phase modes. The amplitude mode is illustrated in Fig. 3. The phase mode shows up as aperiodic variation of the inter-peak distance of the crystalline component (see Fig. 4).To identify the frequencies of particular modes we perform Fourier analysis of the time-dependent width (see Fig. 5a),calculated according to Eq.(15), with ψ ( x, t ) being the numerical solution of the GPE (1). The Fourier transfromshown in Fig. 5b clearly reveals the two main frequencies of the supersolid oscillations, associated with the amplitudeand phase of the wave function. A. Estimation of parameter values
Now we estimate the parameters of our model using the quantities relevant to experiments [7, 22, 27, 29]. Consider
Dy atoms, whose magnetic dipole moment, background s -wave scattering length and atomic mass are µ Dy = 10 µ B , a s = − a , m Dy = 2 . × − kg, respectively, with µ B , a being the Bohr magneton and Bohr radius. Thecondensate of N = 4 × atoms is held in a tight quasi-1D trap with radial confinement frequency ω ⊥ = 2 π ×
110 Hz.The corresponding radial harmonic oscillator length, which is the adopted length scale in this work, is a ⊥ = 0 . µ m.The unit of time is ω − ⊥ = 1 . × − s. The ratio between the strengths of dipolar and contact interactions,computed with above defined parameters, is found to be ǫ d = µ µ Dy m Dy / π ¯ h a s ≃ .
34, therefore in the groundstate we have the dipolar interaction dominated regime. The coefficients of two-body contact interactions and long-range dipolar interactions in physical units are q = 4 π ¯ h a s /m Dy = − . × − kg · m / s and C d = µ µ Dy =1 . × − kg · m / s , yielding the dimensionless parameter g = C d /q ≃ −
4. The strength of three-body atomicinteractions K = K r + i K i in dysprosium condensate was reported to be in the range K r = (3 . ÷ . × − ¯ h · m / sfor real conservative part and K i = 7 . × − ¯ h · m / s for imaginary part, characterizing the three-body recombination a) b) - - - È Ψ È FIG. 4: Oscillations of the supersolid induced by changing the strength of the harmonic trap V ( x ) = αx from α = − . α = − .
1. Both the amplitude and phase modes are excited. Significant exchange of matter betweenthe superfluid and crystalline components is evident. a) The lattice (Goldstone) mode shows up as periodically stretching andsqueezing of the crystalline component. The wave profile of the supersolid at particular time instances t = 20 (blue dashed line)and t = 22 (red solid line). b) The density plot showing the collective dynamics of the perturbed supersolid. The parametervalues are the same as in Fig. 3. a) b) ωℱ FIG. 5: a) Breathing oscillation of the effective width of the supersolid, shown in Fig. 3a, subject to a weak harmonic confinement V ( x ) = 0 . x . The superposition of the two frequencies is evident. b) The Fourier transform of the effective width F (W)reveals two main frequencies, corresponding to the amplitude ( ω H ≃ . ω G ≃ .
4) modes, respectively. Theparameter values are the same as in Fig. 3. rate [14]. Since the imaginary part of this parameter is smaller than its real part by three orders of magnitude, wehave omitted it in the GPE (1), leaving only the conservative part g = K r . The coefficient of quintic nonlinearity,computed using this value and above defined parameters is found to be in the range p = − (0 . ÷ . γ ∼ − . L = 9 λ a ⊥ ≃ µ m. The amplitudeof the background wave in dimensionless units is defined as A = p | a s | N/ L ≃
6, where N stands for the number ofatoms in the condensate. The linear density of the condensate is n l = N/ L ∼ × µm − .The above estimates for the dimensionless parameters derived from experimentally feasible quantities are in therange of values, used in our numerical simulations, except the contribution of quantum fluctuations. Some deviationscan be adjusted by changing the tunable parameters of the system, such as the s -wave scattering length, frequencyof radial confinement, and the strength of dipolar interactions. As for the roles of the repulsive three-body atomiccollisions and quantum fluctuations, these are two possible factors responsible for the existence of stable supersolids[7, 41], and for a complete description of the problem, both terms should be considered. In the particular caseaddressed in this work, we find the contribution of the three-body atomic collisions to be prevailing. IV. CONCLUSIONS
We have explored the static and dynamic properties of a supersolid state in quasi-one-dimensional dipolar quantumgases. The model is based on the nonlocal GPE involving the repulsive three-body atomic interactions and quantumfluctuation terms, which are responsible for the stability of the supersolid. The analytic approach, based on mini-mization of the GP energy functional is developed to estimate the amplitude of the supersolid, which is linked to theexperimentally relevant quantity called the density contrast. The supersolid states with weak and strong modulationof the condensate density are produced by numerical methods. The amplitude of the weakly modulated supersolidwell agrees with the theoretical prediction. The oscillations of the static supersolid are induced by changing the pa-rameters of the trap potential. In numerical simulations we observed significant dynamic exchange of matter betweenthe superfluid and crystalline components of the condensate. 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