Spin textures in condensates with large dipole moments
J. A. M. Huhtamäki, M. Takahashi, T. P. Simula, T. Mizushima, K. Machida
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] J u l Spin textures in condensates with large dipole moments
J. A. M. Huhtam¨aki, , M. Takahashi, T. P. Simula, , T. Mizushima, and K. Machida Department of Physics, Okayama University, Okayama 700-8530, Japan Department of Applied Physics/COMP, Aalto University School of Science and Technology,P.O. Box 15100, FI-00076 AALTO, Finland and School of Physics, Monash University, Victoria 3800, Australia
We have solved numerically the ground states of a Bose-Einstein condensate in the presence ofdipolar interparticle forces using a semiclassical approach. Our motivation is to model, in particular,the spontaneous spin textures emerging in quantum gases with large dipole moments, such as Cror Dy condensates, or ultracold gases consisting of polar molecules. For a pancake-shaped harmonic(optical) potential, we present the ground state phase diagram spanned by the strength of thenonlinear coupling and dipolar interactions. In an elongated harmonic potential, we observe anovel helical spin texture. The textures calculated according to the semiclassical model in theabsence of external polarizing fields are predominantly analogous to previously reported results fora ferromagnetic F = 1 spinor Bose-Einstein condensate, suggesting that the spin textures arisingfrom the dipolar forces are largely independent of the value of the quantum number F or the originof the dipolar interactions. I. INTRODUCTION
Long-range interparticle forces in a quantum systemwith a large coherence length is an intriguing combina-tion bound to exhibit a host of fascinating phenomena.Perhaps the most timely example of such a system is thegaseous atomic Bose-Einstein condensate (BEC) subjectto magnetic dipole-dipole forces [1].The dipolar interaction potential, decreasing as r − in terms of the interparticle distance r , dominates onlength scales determined by the coherence length. Othertwo-body interactions present in the system, such as in-duced dipolar forces (van der Waals), weaken typicallymuch faster ( r − ) and become negligible already overdistances of an average interparticle separation. A fur-ther interesting aspect of the dipole-dipole interaction isits anisotropy enriching the already diverse finite-size ef-fects in trapped ultracold atomic gases. The magneticdipolar interaction in condensates has been predicted togive rise to phenomena ranging from spin textures andspontaneous mass currents [2–4] to roton minimum in theexcitation spectrum [5, 6], linking the field into the studyof liquid He II.The realization of Cr condensates has providedmeans of probing dipolar effects experimentally dueto the exceptionally large magnetic moments of theatoms [7]. The ground states of a chromium condensatehave been studied extensively [8–12]. Anisotropic de-formation of an expanding chromium condensate due todipolar forces has been observed [13], and dipole-inducedspin relaxation in an initially polarized Cr has beenlinked to the famous Einstein-de Haas effect in ferro-magnets [9, 14]. Also, collapse and subsequent d -wavesymmetric explosion of dipolar condensates have beenrecently studied in the case of Cr both experimentallyand theoretically [15]. Chromium condensates have beenrecently produced through optical methods [16].The strength of the magnetic dipolar interaction is de-termined by the atomic magnetic moment µ M through the coupling constant g ′ d = µ µ M / π , where µ is thepermeability of vacuum. For example, for alkali con-densates with total angular momentum quantum number F = 1, the magnetic moment is given by µ M = µ B g F ,where µ B is the Bohr magneton and g F = 1 / g -factor. Such systems are subject to weak dipo-lar interactions, e.g., g ′ d /g ′ ∼ − for Rb, where g ′ = 4 π ~ ( a + 2 a ) / m is the mean-field density-density coupling constant. Here a and a are the s -wave scattering lengths in the channels with total spin 0and 2, and m is the atomic mass. Nevertheless, dipolareffects have been predicted to be observable in F = 1alkali BECs even in the presence of a magnetic field [17],which was recently confirmed experimentally based ontime-evolution study of a helical spin texture [18]. It hasalso been proposed that spin echo in spinor BECs couldbe utilized in revealing dipole-dipole interactions [19].The spontaneous occurrence of novel ground-state spintextures in the absence of external magnetic fields re-quires typically stronger dipolar interactions, g ′ d /g ′ ∼ − –10 − . Hence, the Cr condensates consisting ofparticles with magnetic moments of 6 µ B , as opposed tomaximal magnetic moments of 1 µ B in alkali gases, seemmore favorable for observing such effects. Moreover, therare-earth-metal element Er, with a magnetic momentof 7 µ B , has been cooled down to µ K temperatures [20].Also, recent developments in trapping and cooling of Dywith the largest atomic magnetic moment of 10 µ B yieldsa promising candidate for observing the predicted spintextures [21]. Developments in the study of ultracoldpolar molecules provides means of investigating dipolareffects with large electric moments [22–24].The study of alkali condensates based on a quantummechanical mean-field treatment predict spin textureswith the smallest possible value for the total angular mo-mentum quantum number with internal degrees of free-dom, namely F = 1. It is worthwhile to approach ananalogous problem from the other extreme limit by treat-ing the magnetic moments of the gas classically [2, 25].By comparing the results predicted by the two models,one may expect that if the predictions agree, they couldbe of universal character for all dipolar condensates andindependent of the particular value of the quantum num-ber F . In general, the quantum mechanical order param-eter has 2 F + 1 components and the short-range interac-tion term contains F + 1 independent coupling constants.Hence, it would be very cumbersome to treat each valueof F separately with the complexity of the problem in-creasing along with F .Our semiclassical model is briefly described in Sec. II.The order parameter is written in an alternative formcompared to previous studies [2, 25] in order to simplifyanalysis and to increase numerical efficiency. The mainresults are explained in Sec. III: The ground states ofthe system in harmonic traps of various geometries aredescribed and the collapse of the spin vortex state is an-alyzed briefly. The novel spin helix state is introducedbefore concluding remarks of Sec. IV. II. MODEL
In this Section, we construct a phenomenologicalmean-field model describing a trapped Bose-Einstein con-densate with local as well as nonlocal interparticle inter-actions. The model is equivalent to the semiclassical ap-proach previously studied in [2, 25], with the exceptionthat now the order parameter field is written in a carte-sian basis yielding a set of three Gross-Pitaevskii type ofequations leading to more efficient numerics.The local interaction is assumed to be of the stan-dard s -wave form, with coupling constant g ′ = 4 π ~ a/m ,where a is the s -wave scattering length. Henceforth, wewill refer to its dimensionless form g = 4 πN a/a r , ex-pressed in natural trapping units: ~ ω r is the unit of en-ergy with ω r being the radial trapping frequency of theconfining harmonic potential, and the radial harmonicoscillator length a r = p ~ /mω r is the unit of distance.The number of confined atoms is denoted by N . Thenonlocal interaction is the anisotropic dipole-dipole inter-action with the dimensionless coupling constant g d whichis quantified in relation to g throughout the article. Re-gardless of whether the origin of the dipolar interactionsis considered to be magnetic or electric, we adopt no-tation and terminology as if it were of the former. Forthe trapped system to be stable, we find that for strongenough contact interaction, the value of g d should not ex-ceed ∼ g/
4, a value very close to the number calculatedin the F = 1 case [3].The order parameter is taken to be a three-componentreal-valued vector ψ = ( ψ x , ψ y , ψ z ). It is straightfor-ward to show that all line defects in such order-parameterspace are topologically unstable, because any closedcurve on a sphere can be continuously transformed intoa point which corresponds to a spin polarized state [26].Nevertheless, the energetically stable states can havenontrivial spin textures. In the present model, the particle density is assumedto be related to the order parameter through n ( r ) = P k ψ k ( r ) and is normalized to unity, R d r n ( r ) = 1. Letus make the assumption that the system is ferromagnetic,and hence we may require that all spins are pointing intothe same direction within a small enough region of space.With this simplification, the magnitude of the magneti-zation density is related to the particle density through | M ( r ) | = µ M n ( r ) = µ M X k ψ k ( r ) = sX k M k ( r ) , (1)where µ M is the magnetic moment of a single particle,and M k are the components of magnetization. By squar-ing, we obtain P k M k ( r ) = µ M P k ψ k ( r ) n ( r ), which issatisfied if we define M k ( r ) = µ M ψ k ( r ) p n ( r ) , (2)relating the magnetization density to the order parame-ter. In the following, we omit writing the constant µ M explicitly and assume it to be included in the couplingconstant g d .The energy functional E tot [ ψ x , ψ y , ψ z ] can thus bewritten as E tot = Z X k ψ k ˆ hψ k d r + g Z n d r + g d Z Z D ( r , r ′ ) d r d r ′ , (3) where ˆ h = − ∇ + V trap ( r ) is the single-particle Hamilto-nian and V trap = (cid:0) x + y + λ z (cid:1) is the external trap-ping potential expressed in natural trapping units. Fornow we will omit external rotation and mass currents inthe system, and hence the kinetic energy is merely due toquantum pressure. The second term in Eq. (3) describesthe local mean-field s -wave interaction with the couplingconstant g , and the final term the nonlocal dipole-dipoleinteraction with D ( r , r ′ ) = (cid:2) M ( r ) · M ( r ′ ) − M ( r ) · e R ) (cid:0) M ( r ′ ) · e R (cid:1)(cid:3) /R , (4) where R = r − r ′ is the relative coordinate and e R theunit vector along it.Stationary states of the condensate are obtained bydifferentiating the energy functional with respect to thecomponents of the order parameter with the particlenumber constraint taken into account through a La-grange multiplier µ . Differentiation with respect to ψ j results in a set of three Gross-Pitaevskii equations ˆ hψ j + gnψ j + g d (cid:20) P k M k I k n ψ j + √ nI j (cid:21) = µψ j . (5) Here the functions I j are defined by I j ( r ) = Z " M j ( r ′ ) − e jR X l M l ( r ′ ) e lR /R d r ′ , (6)with e lR being the l th component of e R . These integralsmay be further broken into convolutions. By applyingthe convolution theorem, we obtain I j ( r ) = F − "X l F [ M l ( r )] ˆ f lj ( k ) , (7)where F stands for Fourier transform andˆ f lj ( k ) = − π (cid:0) δ lj − k l k j /k (cid:1) is the Fourier trans-form of f lj ( r ) = (cid:0) δ lj − r l r j /r (cid:1) /r . The Fouriertransforms are efficiently evaluated by using Fast FourierTransform.From the general form of the GP equations, Eq. (5),it is possible to conclude that the spin-polarized textureis not a stationary state in a confined three-dimensionalsystem in the absence of external polarizing fields when g d = 0. Namely, Eq. (5) is of the form A jk ψ k = b j , where b j = g d √ nI j = 0, in general. In the spin-polarized state,we may choose, say, the z -axis along the polarization,whence ψ x = ψ y = 0 yielding b x = b y = 0 from thegeneral form above. When g d = 0, this can be satisfiedin regions of non-vanishing density only if I x = I y = 0.Hence, the bracketed expression in Eq. (7) must vanishidentically. As ˆ f zx and ˆ f zy are non-vanishing in any finitevolume d k , continuity of ψ z implies M z ( r ) = 0, whichis a contradiction. Such conclusion can also be drawnfrom the quantum mechanical model by following similararguments.Apart from the quantum pressure term in Eq. (3),the spin model described above can be viewed as re-sulting from a classical energy functional. However,the present model can also be argued from the quan-tum mechanical spin- F model constrained within theferromagnetic manifold [27]. With maximally alignedspins, the order parameter at a fixed point r is ofthe form ψ = √ ne iθ e − i ˆ F z α/ ~ e − i ˆ F y β/ ~ e − i ˆ F z γ/ ~ | z i = √ ne i ( θ − F γ ) e − i ˆ F z α/ ~ e − i ˆ F y β/ ~ | z i , where ˆ F α are the hyper-fine spin operators, and ˆ F z | z i = ~ F | z i . The order param-eter of the classical spin model is obtained if we neglectthe phase factor e i ( θ − F γ ) and replace the quantum me-chanical rotation operators by the classical equivalentsand the eigenstate | z i by the unit vector pointing alongthe z -axis. Such substitution should be valid when quan-tum fluctuations of the spin operator ˆ F become negligi-ble. The relative fluctuations in the state | z i are givenby h ( ˆ F − ~ F ) i / h ˆ F i = 1 / ( F + 1), which vanish in thelimit of large F . Possible mass currents arising from lo-cal spin-gauge symmetry are neglected when e i ( θ − F γ ) isset to unity, and the kinetic energy reduces merely to thequantum pressure term in Eq. (3). III. RESULTS
We have solved the ground states of the system withvarious values of the coupling constants g and g d , and theaspect ratio λ . Special emphasis is given to the pancake-and cigar-shaped systems, for which we choose λ = 10and λ = 0 . .
50, respectively.
FIG. 1: (Color online) Magnetization M ( r ) (arrows) and den-sity n ( r ) (color) of the (a) flare and (b) spin vortex states for g = 100, g d /g = 0 .
15 in a trap with aspect ratio λ = 10. Bothquantities are shown in the z = 0 plane, the density beingnearly gaussian in the axial direction and M z small. For thechosen parameter values, the flare state in (a) is the energeti-cally favored configuration. Panel (c) illustrates a spin vortexstate with opposite spin winding compared to (b). Such stateis not energetically favorable for the parameter values consid-ered in this work. Each panel has dimensions 8 a r × a r . A. Ground states in the pancake-shaped limit
Let us first consider the case of a cylindrically sym-metric harmonic trap with strong, λ = 10, confinementin the axial ( z ) direction. In the presence of dipolar in-teractions, g d >
0, the magnetic moments tend to liepredominantly in the plane perpendicular to the axialdirection in order for the system to minimize dipolar in-teraction energy.For small enough value of g d /g , the spin texture hastypically the flare structure which has been studied pre-viously using the semiclassical approach as well as thequantum mechanical mean-field model in the F = 1case [2–4]. Such state is illustrated in Fig. 1(a) for, g = 100 and g d /g = 0 .
15. The arrows denote the lo-cal direction of magnetization M ( r ), whereas the colorrefers to the particle density n ( r ). The repulsive interac-tion between parallel spins separated by a vector perpen-dicular to the spin vectors causes the magnetization todeviate from the polarized texture. The structure mayalso be thought of as resulting from the presence of twospin vortices located at the periphery of the cloud. Thespin texture is flare-like also in the x – z -plane ( y = 0)due to finite M z , which is in accordance with the picturethat a single toroidal spin vortex encircles the cloud. Inthe flare state, the magnetization has even parity.When the strength of dipolar interactions is increased,the ground state undergoes a second order phase tran-sition into a state hosting a single spin vortex which isillustrated in Fig. 1(b) for g = 100 and g d /g = 0 .
15. Thedensity is typically suppressed at the core of the vortex.The spin vortex state has also been studied previouslywithin the semiclassical as well as the F = 1 case [2–4].Analogously to the flare state, the presence of the spinvortex results in a texture which favors dipolar interac-tions by reducing the repulsive interactions of parallelspins separated by a vector perpendicular to their mag-netization. For example, close to the phase transitionline in Fig. 2 with g = 1000 and g d /g = 0 .
05, the differ-ences in the kinetic, potential, contact interaction, anddipolar energies of the flare and the spin vortex statesare ∆ E kin = − .
16, ∆ E pot = 0 . E nl = − .
11, and∆ E dip = 0 .
28, respectively, leading to a gain in totalenergy of ∆ E tot = 0 .
077 in units of ~ ω r per particle.Figure 1(c) illustrates a spin vortex state with oppo-site spin winding. For this texture, the angle between lo-cal magnetization and the x –axis decreases as the vortexcore is circled around in the counterclockwise direction,whereas for the state in Fig. 1(b), the angle increases.Such state is found only as an excited solution in thepresent work. In a larger dipolar system, one can con-struct energetically low-lying spin vortex lattices by ar-ranging the vortices presented in Figs. 1(b) and 1(c), andtheir negative counterparts ( M −→ − M ) in an alternat-ing square lattice. Both spin vortices presented in Fig. 1have odd parity. Spin Vortex
Filled Core Empty Core U n s t a b l e Flare c o up li n g c o n s t a n t g .
05 0 .
10 0 .
15 0 .
20 0 .
25 0 . ratio g d /g FIG. 2: Ground-state phase diagram of a dipolar condensatein a harmonic trap with aspect ratio λ = 10. The effectivecontact interaction coupling constant g is represented in log-arithmic scale on the vertical axis. The horizontal axis, mea-suring the strength of dipolar interactions through the ratio g d /g , has linear scale. The phase diagram is divided intothree regions: flare (Fig. 1(a)), spin vortex (Fig. 1(b)), andthe region where the spin vortex becomes unstable againstcollapse. The ground-state phase diagram in the ( g, g d /g )–parameter plane is shown in Fig. 2. The axes of the planeare chosen such that the abscissa is proportional to theparticle number N and the ordinate is independent of N and proportional to the (bare) dipolar coupling con-stant g ′ d . The spin vortex state is energetically favoredfor strong contact and dipolar interactions. The flarestate dominates the phase diagram in the limit of weakcontact interaction, g .
50, regardless of the strengthof dipolar interactions. The phase transition point fromflare to spin vortex state depends strongly on the valueof g . The spin vortex state becomes unstable towardscollapse beyond the critical value of g d /g ≈ .
25 which depends only weakly on the value of g for g & g & R n ( r = 0 , z ) dz/ max R n ( r, z ) dz = 0 . r = p x + y , whereas the ratio is close to unity inthe upper left corner. The finite axial magnetic momentdue to the filled core breaks the inversion symmetry ofthe state. Instead, the components of M have the follow-ing symmetry: ˆ P z M x,y = − M x,y , ˆ P z M z = M z , where ˆ P z inverts the sign of the x – and y –coordinates keeping z intact.For large enough g , the flare state develops continu-ously into a state with two spin vortices which have fer-romagnetic cores as the strength of dipolar interactions isincreased. The magnetic moments of the cores are point-ing either into the same or opposite directions, the twostates being nearly degenerate irrespective of the rela-tive orientation. States hosting multiple spin vortices arefound to be energetically unfavorable compared to singlespin vortex states for the parameter values considered inthis work.The radial size of the spin vortex state diminishes sig-nificantly as the strength of dipolar interactions is in-creased. This suggests that the reason why the systembecomes unstable at some critical value of g d /g couldbe due to inward collapse of the condensate. Local andglobal collapse of a dipolar condensate has been recentlystudied numerically [28].In order to understand why the spin vortex solutionceases to exist above the critical point, it is instructiveto study scaling transformations of the formˆ T σ ( τ ) ψ k ( r, z ) = c σ ( τ ) ψ k ([1 + τ ] r, [1 + στ ] z ) , (8)where ( r, z ) are the cylindrical coordinates, τ is the scal-ing parameter, σ determines the ratio between axial andradial scaling, and c σ ( τ ) is chosen to ensure particle num-ber conservation.Close to the critical point of collapse, the spin vor-tex state in a pancake-shaped trap is the ground state ofthe system, and hence lies in a minimum of the energyfunctional. Under transformations of the form given inEq. (8), the total energy becomes a function of the scal-ing parameter τ , E σ tot ( τ ) = E tot [ ˆ T σ ( τ ) ψ k ( r, z )]. Devia-tion from the ground state always leads to increase inenergy, and hence the second derivative of the total en-ergy with respect to any one-parameter transformationmust be positive, ∂ τ E σ tot ( τ ) (cid:12)(cid:12) τ =0 >
0. The existence of atransformation for which this quantity vanishes indicatesthat the state becomes unstable against such variation.Figure 3 shows the value of min σ { ∂ τ E σ tot ( τ ) (cid:12)(cid:12) τ =0 } asa function of g d /g ∈ [0 . , .
30] scaled by the value at g d /g = 0 .
10. The solid curve is for g = 10 , dashed for g = 10 , and dash-dotted for g = 10 . The curves areextrapolated (dotted lines) using the last few points toobtain an estimate for the critical value for which theminimum in the energy functional vanishes. The criti-cal values are g d /g = (0 . , . , . σ for which ∂ τ E σ tot ( τ ) (cid:12)(cid:12) τ =0 is minimized for each g d /g , the horizon-tal axis being the same as in the main graph. In thevicinity of the critical point for g . , σ >
0, showingthat the collapsing cloud shrinks both in radial and axialdirections. . . m i n σ { ∂ τ E σ t o t ( τ ) (cid:12)(cid:12) τ = } .
10 0 .
15 0 .
20 0 .
25 0 . g d /g − . . σ min vs. g d /g FIG. 3: (Color online) Second derivative of the total energywith respect to a scaling transformation of the form givenin Eq. (8) as a function of g d /g shown in units of the cor-responding quantity at g d /g = 0 .
10. The curves correspondto the parameter values g = 10 (solid), g = 10 (dashed),and g = 10 (dash-dotted). The value of g d /g for which thesecond derivative vanishes indicates the critical strength be-yond which the spin vortex state becomes unstable againstcollapse. The inset shows the ratio of axial and radial scalingfor which the minimal value of the bracketed expression inthe main figure is obtained. Based on a spinor F = 1 study, a critical value of g d /g ≈ .
24 has been previously reported for the exis-tence of the spin vortex state [3], where the parametersare chosen such that g ≈ F . B. Ground states in the cigar-shaped limit
Let us now consider solutions to Eq. (5) in an elon-gated trapping geometry with λ ∈ [0 . , . g = 10 which corresponds to anumber of N ≈ . × Rb atoms in a harmonic trap with radial frequency ω r = 2 π ×
100 Hz. Regardless ofthe aspect ratio λ , the solutions are found to exist onlywithin the interval g d /g ∈ [0 , . F = 1 study [3].The spins tend to lie predominantly along the axialdirection for finite but sufficiently weak dipolar interac-tions. This ground state resembles the flare state in thepancake-shaped limit, and it has been discussed previ-ously both in F = 1 condensates as well as using thesemiclassical model [2, 4]. Figures 4(a)–(c) illustrate thespin textures in three radial cross-sections of the conden-sate. Here λ = 0 . g d /g = 0 . z = − a r ,
0, and 12 a r , respectively. Thecolor depicts the z -component of magnetization, M z ( r ),and the color bar is scaled with respect to the maximummagnetization, max | M ( r ) | , in the corresponding state.The arrows show the texture projected onto the x – y -plane (henceforth referred to as the planar texture), withthe length of the arrows scaled within each panel sepa-rately, except in Fig. 4(b), for which M x = M y = 0 bysymmetry. The stability range of the flare state dependsstrongly on the aspect ratio: For example, with λ = 0 . < g d /g . . λ = 0 .
20 for 0 < g d /g . .
08, whereas with λ = 0 . λ = 0 .
20 and g d /g = 0 . P z M x,y = − M x,y , ˆ P z M z = M z , the phasetransition is sharp, as illustrated below in Fig. 5(d).In order to characterize the spin vortex state more pre-cisely, we define the following quantities: The axial col-umn density reads n z ( z ) = Z n ( r ) dxdy. (9)This measures the number of atoms per unit length in theaxial direction and is normalized to unity. The average twisting angle is given by α ( z ) = (cid:28) arccos (cid:20) ˆ r xy · M xy ( r ) M xy ( r ) (cid:21)(cid:29) , (10)where ˆ r xy = ( x e x + y e y ) / p x + y and the averagingis taken over vectors M xy whose length exceeds 1 %of the maximum of the planar magnetization M xy = q M x + M y . This quantity characterizes the twistingof the magnetization in plane, yielding zero (or π ) forthe flare-like textures, Figs. 4(a)–(c), and π/ FIG. 4: (Color online) Spin textures in the flare, (a)–(c),spin vortex, (d)–(f), and spin helix, (g)–(i), states in a cigar-shaped trapping geometry with aspect ratio λ = 0 .
20 anddipolar interaction strengths g d /g = 0 . , . . x – y -plane,whereas the color refers to the axial magnetization M z nor-malized with respect to maximal magnetization within eachstate separately. Each panel has dimensions 12 a r × a r . spin vortex texture, Fig. 4(e). The average twisting an-gle is essentially independent of the radial distribution ofthe density. Finally, we define the average tilting angle through β ( z ) = (cid:28) arctan (cid:20) M xy ( r ) M z ( r ) (cid:21)(cid:29) , (11)where the averaging is evaluated as above. The tiltingangle is related to the pitch of the helical streamlinesobtained by following the local direction of magnetizationin the spin vortex state, c.f. Ref. [3].Figure 5(a) shows the twisting α ( z ) in the flare (solid)and spin vortex state (dashed and dash-dotted) for thedipolar interaction strengths g d /g = (0 . , . , . α ( z ) remains nearly zero (or π ) over the wholelength of the cloud. Small deviation from zero showsthat the flare state has even parity only approximately.As the dipolar interaction strength is increased, a spinvortex enters the system. Hence, the twisting angle de-creases continuously from π to 0 along the length of thecondensate. A plateau of α ( z ) ≈ π/ .
100 1 . π / β [ r a d ] π α [ r a d ] t t t −
25 0 25 −
25 0 25 −
25 0 25 0 0 .
10 0 . (a) (b)(c) (d) n z [ N / a r ] M t o t [ N µ M ] z [ a r ] g d /g FIG. 5: (Color online) The twisting angle α ( z ) (a), tilt-ing angle β ( z ) (b), and the axial column density n z ( z ) (c)for three values of the dipolar interaction strength g d /g =(0 . , . , . M tot , which is di-rected along the z -axis by symmetry, is shown in (d) for theaspect ratios λ = 0 .
10 (dashed), 0 .
20 (solid), and 0 .
50 (dash-dotted). The dots in (d) refer to the values of g d /g used in(a), (b), and (c). g d /g & .
15. The width of the plateau decreases for in-creasing g d /g due to shrinking of the cloud.The tilting angle β ( z ) is shown in Fig. 5(b) for the sameparameter values as in Fig. 5(a). It remains relativelysmall in the flare state and experiences a sudden increaseat the center of the system when the ground state hosts aspin vortex. For very strong dipolar interactions, β ( z ) ≈ π/ π/ β ( z ) close to the top andbottom of the cloud are remainders of the flare state.The strength of dipolar interactions affects the spatialdensity profile of the spin vortex state significantly. Theaxial column density n z ( z ) is shown in Fig. 5(c) for theparameter values used in 5(a) and 5(b). Not only doesthe system shrink in the radial, but also the axial di-rection with increasing g d /g . Also, the column densityappears to be slightly bimodal for strong enough dipo-lar interactions: the column density is enhanced in thecentral region of the condensate where the spin vortexlies in order for the system to gain dipolar energy. Thewidth of the plateau in Fig. 5(a) due to the presence ofthe spin vortex matches the size of the central profilein the bimodal density distribution. The bimodality ap-pears more vividly in elongated systems with λ < . M x , y ( , , z ) (cid:2) N µ M / a r (cid:3) −
30 30 z [ a r ] x z ✻ ✲ (a) (b) (c) ∂ z θ [ r a d / a r ] . −
40 40 z [ a r ] FIG. 6: (Color online) (a) M x (solid) and M y (dashed) on the z –axis as a function of z in the spin helix state for λ = 0 . g d /g = 0 .
20. The dash-dotted curve in the inset shows M z along the y –axis in the range [ − , a r . The vertical axisspans the interval [ − . , . Nµ M /a r both in the main figure and the inset. (b) The column density n ( x, z ) = R n ( r ) dy illustrating density oscillations characteristic of the spin helix state for strong dipolar interactions. The parameters are as in(a), and the field of view is 8 a r × a r . (c) Wave vector of the spin helix state for g d /g = (0 . , . , .
23) shown with solid,dashed and dash-dotted curves, respectively. The peaks at the ends of the cloud are finite-size effects.
In the extreme limit of g d /g = 0 . n ( r ) inthe spin vortex state is significantly reduced close to thecenter of the trap where r ≈ M tot = R M ( r ) d r in the flare and spin vortex states as a func-tion of g d /g . Due to symmetry, the total magnetiza-tion is along the axial direction. The dashed, solidand dash-dotted lines correspond to the aspect ratios λ = (0 . , . , . λ = 0 . λ = 0 .
50) hosts a spin vortex alreadywith g d /g = 0 .
01. The phase transition points for dif-ferent values of λ agree qualitatively with the analogousresults in the F = 1 study [3].It is reasonable to expect that there exist also station-ary states with the opposite symmetry compared to theflare and spin vortex states, i.e., ˆ P z M x,y ( r ) = M x,y ( r ),ˆ P z M z ( r ) = − M z ( r ). There indeed exist low-energy so-lutions to Eq. (5) with such symmetry, to which we referto as spin helices. The spin helix state is found, e.g.with λ = (0 . , . , .
50) in the entire stability interval0 < g d /g ≤ .
235 of the system. This state resemblesclosely the state studied in Ref. [18], where the helicalspin texture is created by using a transient magnetic fieldgradient. According to our simulations, the stationaryspin helix state exists also in a ferromagnetic F = 1 sys-tem with dipolar interactions, which will be studied inmore detail elsewhere. Dynamical instability of a similarstructure in the absence of dipolar interactions has beenstudied recently [29].The spin helix texture is illustrated for λ = 0 .
20 and g d /g = 0 .
030 in Fig. 4(g)–4(i), where the radial cross-sections are taken at z = ( − . a r , , . a r ), respectively.On the z –axis, the magnetization lies in the x – y -plane, as a consequence of the antisymmetry of M z . Furtheraway from the z –axis and perpendicular to the magneti-zation on the axis, M z becomes the dominant component.The whole planar texture rotates about the z –axis as afunction of the z –coordinate, traversing typically throughseveral cycles along the length of the condensate.Energetically, the spin helix state appears to be fa-vored by strong dipolar interactions and not too elon-gated geometries. For example, with λ = 0 .
50, the he-lix becomes energetically favorable compared to the spinvortex state between 0 . < g d /g < .
10. It is challeng-ing to pinpoint the exact location of the phase transitionpoint due to near degeneracy of the two states. Nearthe critical value of g d /g = 0 . E tot ≈ . ~ ω r per particle in favor of the helix, whichis roughly 2% of the total energy. For λ = 0 .
20, the helixstate appears to be the minimal energy texture only for g d /g & .
20, whereas for λ = 0 .
10, the flare state lies2%–7% lower in energy over the entire stability range ofthe system.The number of cycles in the helix texture increasesas the condensate is elongated, and thus for the sake ofclarity we illustrate it as an excited state for λ = 0 .
10 and g d /g = 0 .
20 in Figs. 6(a)–(c). In Fig. 6(a), the solid anddashed curves show the components M x and M y alongthe z –axis, respectively, and the M z component along the y –axis is shown in the inset by the dash-dotted curve.The spin helix can be thought of as two elongatedstripes, polarized along the z –axis in the opposite di-rections, intertwined around one another. The helicaltexture on the z –axis arises due to continuous twistingof the magnetization via the x – y -plane. In the F = 1case, quantized spin vortices of opposite winding pene-trate through the axially polarized ferromagnetic stripes,forming an intertwined spin vortex pair. Intertwining oftwo mass vortices has been previously studied in relationto the splitting of a doubly quantized vortex in a scalarcondensate [30, 31].Figure 6(b) illustrates the column density n ( x, z ) = R n ( r ) dy for the same state as in Fig. 6(a), red denot-ing area of high and blue of vanishing particle density.With strong dipolar interactions density oscillation ap-pear spontaneously due to the helix spin texture: for afixed point in the x – y -plane close to the surface regionof the cloud, the axial magnetization M z is an oscillatingfunction of z . The particle density is suppressed in thevicinity of the nodes of M z and enhanced at the anti-nodes due to dipolar interactions.As a measure of the pitch of the spin helix, we definethe angle θ ( z ) = arctan (cid:20) M y (0 , , z ) M x (0 , , z ) (cid:21) . (12)The derivative ∂θ/∂z yields the wave vector of the he-lix, which is plotted in Fig. 6(c) for λ = 0 .
10 and g d /g = (0 . , . , .
23) with the solid, dashed, anddash-dotted curves, respectively. The wave vector tendsto increase for stronger dipolar interactions, which is rea-sonable because the dipolar coherence length decreases as ξ d ∝ g − / d . The peaks in ∂θ/∂z at the top and bottomof the cloud are finite-size effects: The texture may ad-just freely into an energetically favorable configuration atthe edge as one of the boundary conditions due to con-tinuity of the order parameter is liberated. Oscillationspenetrate along the whole length of the condensate for g d /g = 0 .
23. These oscillations enhance rapidly as thestrength of dipolar interactions is increased even further.The number of cycles in the spin helix state decreasesas the aspect ratio λ = ω z /ω r is increased, until in spher-ical geometry, the direction of the spin on the z –axistwists only through half a cycle along the length of thesystem, c.f. Fig. 6(a). Interestingly, the spin vortex state,for which the magnetization has the opposite symmetrywith respect to inversion about the z –axis, reduces to thespin helix state, rotated by ( ± ) π/ e z × M h , where M h is the magnetization at the trap center in the helixstate. IV. SUMMARY AND CONCLUSIONS
We have studied spin textures arising from dipolar in-teractions in gaseous Bose-Einstein condensates of parti-cles with large permanent dipole moments. The theoryis based on a semiclassical model treating the dipole mo-ments of the bosons classically.The observed spin textures in clouds confined in har-monic trapping potentials agree qualitatively with previ-ously reported results for an F = 1 system [3, 4], such as Rb. Moreover, the ground-state phase transition points with respect to the strength of dipolar interactions seemto agree roughly both with weak and tight axial trappingfrequency. The qualitative agreement in the observationsdrawn from the two models suggests that similar texturesand phase diagrams are to be expected also for ferromag-netic systems with
F > g d /g ≈ /
4. For larger dipolar interac-tions, the state becomes unstable against collapse of thecloud due to strong attractive forces overwhelming thequantum pressure term and repulsive interparticle inter-actions. The estimated point of instability agrees wellwith the value observed in the F = 1 study [3].In the limit of tight axial confinement, two groundstates are observed, namely, the flare and the spin vortexstates. For prolate geometries, an additional spin helixtexture appears as a low-energy stationary state. Thehelix is the ground state of the system only in slightlyprolate condensates and for strong dipolar interactions.This state is most likely related to the S state reportedin [3], and is especially interesting in relation to the ex-perimental observation of dipolar effects in Rb utilizinga similar spin texture [18]. The magnetization patternof the spin helix gives rise to spontaneous density oscil-lations in the stationary state for strong dipolar interac-tions. As in the case of the spin vortex and flare textures,the helix state ceases to exist for g d /g & / Acknowledgments
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