Spin textures in condensates with large dipole moments

Spin textures in condensates with large dipole moments

J. A. M. Huhtamäki, 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 School of Physics, Monash University, Victoria 3800, Australia

We have solved numerically the ground states of a Bose-Einstein condensate in the presence of dipolar 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 or 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 the nonlinear coupling and dipolar interactions. In an elongated harmonic potential, we observe a novel helical spin texture. The textures calculated according to the semiclassical model in the absence of external polarizing fields are predominantly analogous to previously reported results for a ferromagnetic spinor Bose-Einstein condensate, suggesting that the spin textures arising from the dipolar forces are largely independent of the value of the quantum number or the origin of the dipolar interactions.

Bose-Einstein condensate, dipole-dipole interaction, spin texture
PACS number(s): 03.75.Hh, 03.75.Mn, 75.10.Hk

I Introduction

Long-range interparticle forces in a quantum system with a large coherence length is an intriguing combination bound to exhibit a host of fascinating phenomena. Perhaps the most timely example of such a system is the gaseous atomic Bose-Einstein condensate (BEC) subject to magnetic dipole-dipole forces Lahaye2009 ().

The dipolar interaction potential, decreasing as in terms of the interparticle distance , dominates on length scales determined by the coherence length. Other two-body interactions present in the system, such as induced dipolar forces (van der Waals), weaken typically much faster () and become negligible already over distances of an average interparticle separation. A further interesting aspect of the dipole-dipole interaction is its anisotropy enriching the already diverse finite-size effects in trapped ultracold atomic gases. The magnetic dipolar interaction in condensates has been predicted to give rise to phenomena ranging from spin textures and spontaneous mass currents Takahashi2007 (); yi:020401 (); kawaguchi2006scg () to roton minimum in the excitation spectrum ODell2003 (); Santos2003 (), linking the field into the study of liquid He II.

The realization of condensates has provided means of probing dipolar effects experimentally due to the exceptionally large magnetic moments of the atoms PhysRevLett.94.160401 (). The ground states of a chromium condensate have been studied extensively Goral2000 (); santos:190404 (); diener:190405 (); Makela2007 (); He2009 (). Anisotropic deformation of an expanding chromium condensate due to dipolar forces has been observed PhysRevLett.95.150406 (), and dipole-induced spin relaxation in an initially polarized has been linked to the famous Einstein-de Haas effect in ferromagnets santos:190404 (); kawaguchi:080405 (). Also, collapse and subsequent -wave symmetric explosion of dipolar condensates have been recently studied in the case of both experimentally and theoretically Lahaye2008 (). Chromium condensates have been recently produced through optical methods Beaufils2008 ().

The strength of the magnetic dipolar interaction is determined by the atomic magnetic moment through the coupling constant , where is the permeability of vacuum. For example, for alkali condensates with total angular momentum quantum number , the magnetic moment is given by , where is the Bohr magneton and the Landé -factor. Such systems are subject to weak dipolar interactions, e.g., for , where is the mean-field density-density coupling constant. Here and are the -wave scattering lengths in the channels with total spin and , and is the atomic mass. Nevertheless, dipolar effects have been predicted to be observable in alkali BECs even in the presence of a magnetic field kawaguchi:110406 (), which was recently confirmed experimentally based on time-evolution study of a helical spin texture Vengalattore2008 (). It has also been proposed that spin echo in spinor BECs could be utilized in revealing dipole-dipole interactions Yasunaga2008 ().

The spontaneous occurrence of novel ground-state spin textures in the absence of external magnetic fields requires typically stronger dipolar interactions, . Hence, the condensates consisting of particles with magnetic moments of , as opposed to maximal magnetic moments of in alkali gases, seem more favorable for observing such effects. Moreover, the rare-earth-metal element , with a magnetic moment of , has been cooled down to temperatures Berglund2008 (). Also, recent developments in trapping and cooling of with the largest atomic magnetic moment of yields a promising candidate for observing the predicted spin textures Lu2010 (). Developments in the study of ultracold polar molecules provides means of investigating dipolar effects with large electric moments Bethlem1999 (); mancini:133203 (); ni2008hps ().

The study of alkali condensates based on a quantum mechanical mean-field treatment predict spin textures with the smallest possible value for the total angular momentum quantum number with internal degrees of freedom, namely . It is worthwhile to approach an analogous problem from the other extreme limit by treating the magnetic moments of the gas classically Takahashi2007 (); Takahashi2009 (). By comparing the results predicted by the two models, one may expect that if the predictions agree, they could be of universal character for all dipolar condensates and independent of the particular value of the quantum number . In general, the quantum mechanical order parameter has components and the short-range interaction term contains independent coupling constants. Hence, it would be very cumbersome to treat each value of separately with the complexity of the problem increasing along with .

Our semiclassical model is briefly described in Sec. II. The order parameter is written in an alternative form compared to previous studies Takahashi2007 (); Takahashi2009 () in order to simplify analysis and to increase numerical efficiency. The main results are explained in Sec. III: The ground states of the system in harmonic traps of various geometries are described and the collapse of the spin vortex state is analyzed briefly. The novel spin helix state is introduced before concluding remarks of Sec. IV.

Ii Model

In this Section, we construct a phenomenological mean-field model describing a trapped Bose-Einstein condensate with local as well as nonlocal interparticle interactions. The model is equivalent to the semiclassical approach previously studied in Takahashi2007 (); Takahashi2009 (), with the exception that now the order parameter field is written in a cartesian basis yielding a set of three Gross-Pitaevskii type of equations leading to more efficient numerics.

The local interaction is assumed to be of the standard -wave form, with coupling constant , where is the -wave scattering length. Henceforth, we will refer to its dimensionless form , expressed in natural trapping units: is the unit of energy with being the radial trapping frequency of the confining harmonic potential, and the radial harmonic oscillator length is the unit of distance. The number of confined atoms is denoted by . The nonlocal interaction is the anisotropic dipole-dipole interaction with the dimensionless coupling constant which is quantified in relation to throughout the article. Regardless of whether the origin of the dipolar interactions is considered to be magnetic or electric, we adopt notation and terminology as if it were of the former. For the trapped system to be stable, we find that for strong enough contact interaction, the value of should not exceed , a value very close to the number calculated in the case yi:020401 ().

The order parameter is taken to be a three-component real-valued vector . It is straightforward to show that all line defects in such order-parameter space are topologically unstable, because any closed curve on a sphere can be continuously transformed into a point which corresponds to a spin polarized state Mermin1979 (). Nevertheless, the energetically stable states can have nontrivial spin textures.

In the present model, the particle density is assumed to be related to the order parameter through and is normalized to unity, . Let us make the assumption that the system is ferromagnetic, and hence we may require that all spins are pointing into the same direction within a small enough region of space. With this simplification, the magnitude of the magnetization density is related to the particle density through


where is the magnetic moment of a single particle, and are the components of magnetization. By squaring, we obtain , which is satisfied if we define


relating the magnetization density to the order parameter. In the following, we omit writing the constant explicitly and assume it to be included in the coupling constant .

The energy functional can thus be written as


where is the single-particle Hamiltonian and is the external trapping potential expressed in natural trapping units. For now we will omit external rotation and mass currents in the system, and hence the kinetic energy is merely due to quantum pressure. The second term in Eq. (3) describes the local mean-field -wave interaction with the coupling constant , and the final term the nonlocal dipole-dipole interaction with


where is the relative coordinate and the unit vector along it.

Stationary states of the condensate are obtained by differentiating the energy functional with respect to the components of the order parameter with the particle number constraint taken into account through a Lagrange multiplier . Differentiation with respect to results in a set of three Gross-Pitaevskii equations


Here the functions are defined by


with being the th component of . These integrals may be further broken into convolutions. By applying the convolution theorem, we obtain


where stands for Fourier transform and is the Fourier transform of . The Fourier transforms are efficiently evaluated by using Fast Fourier Transform.

From the general form of the GP equations, Eq. (5), it is possible to conclude that the spin-polarized texture is not a stationary state in a confined three-dimensional system in the absence of external polarizing fields when . Namely, Eq. (5) is of the form , where , in general. In the spin-polarized state, we may choose, say, the -axis along the polarization, whence yielding from the general form above. When , this can be satisfied in regions of non-vanishing density only if . Hence, the bracketed expression in Eq. (7) must vanish identically. As and are non-vanishing in any finite volume , continuity of implies , which is a contradiction. Such conclusion can also be drawn from the quantum mechanical model by following similar arguments.

Apart from the quantum pressure term in Eq. (3), the spin model described above can be viewed as resulting from a classical energy functional. However, the present model can also be argued from the quantum mechanical spin- model constrained within the ferromagnetic manifold Ho1996 (). With maximally aligned spins, the order parameter at a fixed point is of the form , where are the hyperfine spin operators, and . The order parameter of the classical spin model is obtained if we neglect the phase factor and replace the quantum mechanical rotation operators by the classical equivalents and the eigenstate by the unit vector pointing along the -axis. Such substitution should be valid when quantum fluctuations of the spin operator become negligible. The relative fluctuations in the state are given by , which vanish in the limit of large . Possible mass currents arising from local spin-gauge symmetry are neglected when is set to unity, and the kinetic energy reduces merely to the quantum pressure term in Eq. (3).

Iii Results

We have solved the ground states of the system with various values of the coupling constants and , and the aspect ratio . Special emphasis is given to the pancake- and cigar-shaped systems, for which we choose and , respectively.

Figure 1: (Color online) Magnetization (arrows) and density (color) of the (a) flare and (b) spin vortex states for , in a trap with aspect ratio . Both quantities are shown in the plane, the density being nearly gaussian in the axial direction and small. For the chosen parameter values, the flare state in (a) is the energetically favored configuration. Panel (c) illustrates a spin vortex state with opposite spin winding compared to (b). Such state is not energetically favorable for the parameter values considered in this work. Each panel has dimensions .

iii.1 Ground states in the pancake-shaped limit

Let us first consider the case of a cylindrically symmetric harmonic trap with strong, , confinement in the axial () direction. In the presence of dipolar interactions, , the magnetic moments tend to lie predominantly in the plane perpendicular to the axial direction in order for the system to minimize dipolar interaction energy.

For small enough value of , the spin texture has typically the flare structure which has been studied previously using the semiclassical approach as well as the quantum mechanical mean-field model in the case Takahashi2007 (); yi:020401 (); kawaguchi2006scg (). Such state is illustrated in Fig. 1(a) for, and . The arrows denote the local direction of magnetization , whereas the color refers to the particle density . The repulsive interaction between parallel spins separated by a vector perpendicular to the spin vectors causes the magnetization to deviate from the polarized texture. The structure may also be thought of as resulting from the presence of two spin vortices located at the periphery of the cloud. The spin texture is flare-like also in the -plane () due to finite , which is in accordance with the picture that a single toroidal spin vortex encircles the cloud. In the flare state, the magnetization has even parity.

When the strength of dipolar interactions is increased, the ground state undergoes a second order phase transition into a state hosting a single spin vortex which is illustrated in Fig. 1(b) for and . The density is typically suppressed at the core of the vortex. The spin vortex state has also been studied previously within the semiclassical as well as the case Takahashi2007 (); yi:020401 (); kawaguchi2006scg (). Analogously to the flare state, the presence of the spin vortex results in a texture which favors dipolar interactions by reducing the repulsive interactions of parallel spins separated by a vector perpendicular to their magnetization. For example, close to the phase transition line in Fig. 2 with and , the differences in the kinetic, potential, contact interaction, and dipolar energies of the flare and the spin vortex states are , , , and , respectively, leading to a gain in total energy of in units of per particle.

Figure 1(c) illustrates a spin vortex state with opposite spin winding. For this texture, the angle between local magnetization and the –axis decreases as the vortex core 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 the present work. In a larger dipolar system, one can construct energetically low-lying spin vortex lattices by arranging the vortices presented in Figs. 1(b) and 1(c), and their negative counterparts () in an alternating square lattice. Both spin vortices presented in Fig. 1 have odd parity.

Figure 2: Ground-state phase diagram of a dipolar condensate in a harmonic trap with aspect ratio . The effective contact interaction coupling constant is represented in logarithmic scale on the vertical axis. The horizontal axis, measuring the strength of dipolar interactions through the ratio , has linear scale. The phase diagram is divided into three regions: flare (Fig. 1(a)), spin vortex (Fig. 1(b)), and the region where the spin vortex becomes unstable against collapse.

The ground-state phase diagram in the –parameter plane is shown in Fig. 2. The axes of the plane are chosen such that the abscissa is proportional to the particle number and the ordinate is independent of and proportional to the (bare) dipolar coupling constant . The spin vortex state is energetically favored for strong contact and dipolar interactions. The flare state dominates the phase diagram in the limit of weak contact interaction, , regardless of the strength of dipolar interactions. The phase transition point from flare to spin vortex state depends strongly on the value of . The spin vortex state becomes unstable towards collapse beyond the critical value of which depends only weakly on the value of for .

With strong enough contact interaction and weak dipolar interaction, the cores of the spin vortices are filled with particles whose magnetic moments are pointing in the axial direction. The condensate gains trapping as well as contact interaction energy by filling the vortex core. The dashed line in the phase diagram of Fig. 2 separates the states with filled cores from states with empty cores. On the line with , whereas the ratio is close to unity in the upper left corner. The finite axial magnetic moment due to the filled core breaks the inversion symmetry of the state. Instead, the components of have the following symmetry: , , where inverts the sign of the – and –coordinates keeping intact.

For large enough , the flare state develops continuously into a state with two spin vortices which have ferromagnetic cores as the strength of dipolar interactions is increased. The magnetic moments of the cores are pointing either into the same or opposite directions, the two states being nearly degenerate irrespective of the relative orientation. States hosting multiple spin vortices are found to be energetically unfavorable compared to single spin vortex states for the parameter values considered in this work.

The radial size of the spin vortex state diminishes significantly as the strength of dipolar interactions is increased. This suggests that the reason why the system becomes unstable at some critical value of could be due to inward collapse of the condensate. Local and global collapse of a dipolar condensate has been recently studied numerically Parker2009 ().

In order to understand why the spin vortex solution ceases to exist above the critical point, it is instructive to study scaling transformations of the form


where are the cylindrical coordinates, is the scaling parameter, determines the ratio between axial and radial scaling, and is chosen to ensure particle number conservation.

Close to the critical point of collapse, the spin vortex state in a pancake-shaped trap is the ground state of the system, and hence lies in a minimum of the energy functional. Under transformations of the form given in Eq. (8), the total energy becomes a function of the scaling parameter , . Deviation from the ground state always leads to increase in energy, and hence the second derivative of the total energy with respect to any one-parameter transformation must be positive, . The existence of a transformation for which this quantity vanishes indicates that the state becomes unstable against such variation.

Figure 3 shows the value of as a function of scaled by the value at . The solid curve is for , dashed for , and dash-dotted for . The curves are extrapolated (dotted lines) using the last few points to obtain an estimate for the critical value for which the minimum in the energy functional vanishes. The critical values are , respectively. The inset in Fig. 3 depicts the value of for which is minimized for each , the horizontal axis being the same as in the main graph. In the vicinity of the critical point for , , showing that the collapsing cloud shrinks both in radial and axial directions.

Figure 3: (Color online) Second derivative of the total energy with respect to a scaling transformation of the form given in Eq. (8) as a function of shown in units of the corresponding quantity at . The curves correspond to the parameter values (solid), (dashed), and (dash-dotted). The value of for which the second derivative vanishes indicates the critical strength beyond which the spin vortex state becomes unstable against collapse. The inset shows the ratio of axial and radial scaling for which the minimal value of the bracketed expression in the main figure is obtained.

Based on a spinor study, a critical value of has been previously reported for the existence of the spin vortex state yi:020401 (), where the parameters are chosen such that in the present study. As the current model is expected to be accurate, apart from possible mass currents, in the limit of large (classical) magnetic moments, this agreement suggests that the critical value for the collapse is universal and independent of .

iii.2 Ground states in the cigar-shaped limit

Let us now consider solutions to Eq. (5) in an elongated trapping geometry with . For definiteness, we will fix which corresponds to a number of atoms in a harmonic trap with radial frequency Hz. Regardless of the aspect ratio , the solutions are found to exist only within the interval , agreeing with the result previously reported in the study yi:020401 ().

The spins tend to lie predominantly along the axial direction for finite but sufficiently weak dipolar interactions. This ground state resembles the flare state in the pancake-shaped limit, and it has been discussed previously both in condensates as well as using the semiclassical model kawaguchi2006scg (); Takahashi2007 (). Figures 4(a)–(c) illustrate the spin textures in three radial cross-sections of the condensate. Here , , and the cross-sections are taken at , and , respectively. The color depicts the -component of magnetization, , and the color bar is scaled with respect to the maximum magnetization, , in the corresponding state. The arrows show the texture projected onto the -plane (henceforth referred to as the planar texture), with the length of the arrows scaled within each panel separately, except in Fig. 4(b), for which by symmetry. The stability range of the flare state depends strongly on the aspect ratio: For example, with , the flare state is the ground state for , with for , whereas with the flare state dominates the entire stability window.

Figure 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 and dipolar interaction strengths , and , respectively. The arrows illustrate the magnetization within a given radial cross-section projected onto the -plane, whereas the color refers to the axial magnetization normalized with respect to maximal magnetization within each state separately. Each panel has dimensions .

Typically, for stronger dipolar interactions, a spin vortex texture appears in the central region of the condensate, illustrated in Figs. 4(d)–(f) for and . As shown in Fig. 4(e), the planar texture near the center resembles the spin vortex texture in the pancake-shaped geometry discussed before, see Fig. 1(b). As one moves further away from the center, the planar texture deforms continuously towards the flare texture discussed in the previous paragraph, as depicted in Figs. 4(d) and 4(f). Although the magnetization in both the flare and the spin vortex states has the same symmetry, , , the phase transition is sharp, as illustrated below in Fig. 5(d).

In order to characterize the spin vortex state more precisely, we define the following quantities: The axial column density reads


This measures the number of atoms per unit length in the axial direction and is normalized to unity. The average twisting angle is given by


where and the averaging is taken over vectors whose length exceeds of the maximum of the planar magnetization . This quantity characterizes the twisting of the magnetization in plane, yielding zero (or ) for the flare-like textures, Figs. 4(a)–(c), and for the spin vortex texture, Fig. 4(e). The average twisting angle is essentially independent of the radial distribution of the density. Finally, we define the average tilting angle through


where the averaging is evaluated as above. The tilting angle is related to the pitch of the helical streamlines obtained by following the local direction of magnetization in the spin vortex state, c.f. Ref. yi:020401 ().

Figure 5: (Color online) The twisting angle (a), tilting angle (b), and the axial column density (c) for three values of the dipolar interaction strength shown with solid, dashed, and dash-dotted lines, respectively. The total magnetization , which is directed along the -axis by symmetry, is shown in (d) for the aspect ratios (dashed), (solid), and (dash-dotted). The dots in (d) refer to the values of used in (a), (b), and (c).
Figure 6: (Color online) (a) (solid) and (dashed) on the –axis as a function of in the spin helix state for and . The dash-dotted curve in the inset shows along the –axis in the range . The vertical axis spans the interval both in the main figure and the inset. (b) The column density 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 . (c) Wave vector of the spin helix state for shown with solid, dashed and dash-dotted curves, respectively. The peaks at the ends of the cloud are finite-size effects.

Figure 5(a) shows the twisting in the flare (solid) and spin vortex state (dashed and dash-dotted) for the dipolar interaction strengths , respectively. The flare state is chosen from the neighborhood of the transition point to a spin vortex state. However, remains nearly zero (or ) over the whole length of the cloud. Small deviation from zero shows that the flare state has even parity only approximately. As the dipolar interaction strength is increased, a spin vortex enters the system. Hence, the twisting angle decreases continuously from to along the length of the condensate. A plateau of forms in the central region of the system for strong dipolar interactions, . The width of the plateau decreases for increasing due to shrinking of the cloud.

The tilting angle is shown in Fig. 5(b) for the same parameter values as in Fig. 5(a). It remains relatively small in the flare state and experiences a sudden increase at the center of the system when the ground state hosts a spin vortex. For very strong dipolar interactions, in the central region, slightly even exceeding due to interaction with the axial magnetization of the core region. The small lobes in close to the top and bottom of the cloud are remainders of the flare state.

The strength of dipolar interactions affects the spatial density profile of the spin vortex state significantly. The axial column density is shown in Fig. 5(c) for the parameter values used in 5(a) and 5(b). Not only does the system shrink in the radial, but also the axial direction with increasing . Also, the column density appears to be slightly bimodal for strong enough dipolar interactions: the column density is enhanced in the central region of the condensate where the spin vortex lies in order for the system to gain dipolar energy. The width of the plateau in Fig. 5(a) due to the presence of the spin vortex matches the size of the central profile in the bimodal density distribution. The bimodality appears more vividly in elongated systems with . In the extreme limit of , the density in the spin vortex state is significantly reduced close to the center of the trap where .

Figure 5(d) depicts the total magnetization in the flare and spin vortex states as a function of . Due to symmetry, the total magnetization is along the axial direction. The dashed, solid and dash-dotted lines correspond to the aspect ratios , respectively. The sudden drops in total magnetization indicate the phase boundary between the flare and spin vortex states. Whereas the flare state dominates the entire stability window for very elongated trapping geometry (), the ground state in a prolate system () hosts a spin vortex already with . The phase transition points for different values of agree qualitatively with the analogous results in the study yi:020401 ().

It is reasonable to expect that there exist also stationary states with the opposite symmetry compared to the flare and spin vortex states, i.e., , . There indeed exist low-energy solutions to Eq. (5) with such symmetry, to which we refer to as spin helices. The spin helix state is found, e.g. with in the entire stability interval of the system. This state resembles closely the state studied in Ref. Vengalattore2008 (), where the helical spin texture is created by using a transient magnetic field gradient. According to our simulations, the stationary spin helix state exists also in a ferromagnetic system with dipolar interactions, which will be studied in more detail elsewhere. Dynamical instability of a similar structure in the absence of dipolar interactions has been studied recently Cherng2008 ().

The spin helix texture is illustrated for and in Fig. 4(g)–4(i), where the radial cross-sections are taken at , respectively. On the –axis, the magnetization lies in the -plane, as a consequence of the antisymmetry of . Further away from the –axis and perpendicular to the magnetization on the axis, becomes the dominant component. The whole planar texture rotates about the –axis as a function of the –coordinate, traversing typically through several cycles along the length of the condensate.

Energetically, the spin helix state appears to be favored by strong dipolar interactions and not too elongated geometries. For example, with , the helix becomes energetically favorable compared to the spin vortex state between . It is challenging to pinpoint the exact location of the phase transition point due to near degeneracy of the two states. Near the critical value of , the difference in the total energy between the spin vortex and helix states is per particle in favor of the helix, which is roughly of the total energy. For , the helix state appears to be the minimal energy texture only for , whereas for , the flare state lies lower in energy over the entire stability range of the system.

The number of cycles in the helix texture increases as the condensate is elongated, and thus for the sake of clarity we illustrate it as an excited state for and in Figs. 6(a)–(c). In Fig. 6(a), the solid and dashed curves show the components and along the –axis, respectively, and the component along the –axis is shown in the inset by the dash-dotted curve.

The spin helix can be thought of as two elongated stripes, polarized along the –axis in the opposite directions, intertwined around one another. The helical texture on the –axis arises due to continuous twisting of the magnetization via the -plane. In the case, quantized spin vortices of opposite winding penetrate through the axially polarized ferromagnetic stripes, forming an intertwined spin vortex pair. Intertwining of two mass vortices has been previously studied in relation to the splitting of a doubly quantized vortex in a scalar condensate PhysRevA.68.023611 (); Huhtamaki2006 ().

Figure 6(b) illustrates the column density for the same state as in Fig. 6(a), red denoting area of high and blue of vanishing particle density. With strong dipolar interactions density oscillation appear spontaneously due to the helix spin texture: for a fixed point in the -plane close to the surface region of the cloud, the axial magnetization is an oscillating function of . The particle density is suppressed in the vicinity of the nodes of and enhanced at the anti-nodes due to dipolar interactions.

As a measure of the pitch of the spin helix, we define the angle


The derivative yields the wave vector of the helix, which is plotted in Fig. 6(c) for and with the solid, dashed, and dash-dotted curves, respectively. The wave vector tends to increase for stronger dipolar interactions, which is reasonable because the dipolar coherence length decreases as . The peaks in at the top and bottom of the cloud are finite-size effects: The texture may adjust freely into an energetically favorable configuration at the edge as one of the boundary conditions due to continuity of the order parameter is liberated. Oscillations penetrate along the whole length of the condensate for . These oscillations enhance rapidly as the strength of dipolar interactions is increased even further.

The number of cycles in the spin helix state decreases as the aspect ratio is increased, until in spherical geometry, the direction of the spin on the –axis twists only through half a cycle along the length of the system, c.f. Fig. 6(a). Interestingly, the spin vortex state, for which the magnetization has the opposite symmetry with respect to inversion about the –axis, reduces to the spin helix state, rotated by about , where is the magnetization at the trap center in the helix state.

Iv Summary and Conclusions

We have studied spin textures arising from dipolar interactions in gaseous Bose-Einstein condensates of particles with large permanent dipole moments. The theory is based on a semiclassical model treating the dipole moments of the bosons classically.

The observed spin textures in clouds confined in harmonic trapping potentials agree qualitatively with previously reported results for an system yi:020401 (); kawaguchi2006scg (), such as . Moreover, the ground-state phase transition points with respect to the strength of dipolar interactions seem to agree roughly both with weak and tight axial trapping frequency. The qualitative agreement in the observations drawn from the two models suggests that similar textures and phase diagrams are to be expected also for ferromagnetic systems with and for condensates consisting of electric dipoles.

A major difference between our semiclassical model and the quantum mechanical model describing a magnetic system lies in the existence of mass currents: In the quantum mechanical case, phase gradients of the components of the order parameter emerge spontaneously possibly giving rise to mass currents which are absent in the semiclassical treatment, c.f. the last paragraph of Sec. II. The main effect from taking spontaneous mass currents into account regarding the present results would be that the phase transition line between the flare and the spin vortex states in Fig. 2 is shifted to the right due to increased kinetic energy of the spin vortex state. Spin dynamics of ferromagnetic condensates has been studied recently in the long-wavelength limit using a hydrodynamic model Lamacraft2008 ().

In addition to solving the ground states of the system for various parameter values, we investigate the spin vortex state in the extreme limit of the dipolar interaction strength . For larger dipolar interactions, the state becomes unstable against collapse of the cloud due to strong attractive forces overwhelming the quantum pressure term and repulsive interparticle interactions. The estimated point of instability agrees well with the value observed in the study yi:020401 ().

In the limit of tight axial confinement, two ground states are observed, namely, the flare and the spin vortex states. For prolate geometries, an additional spin helix texture appears as a low-energy stationary state. The helix is the ground state of the system only in slightly prolate condensates and for strong dipolar interactions. This state is most likely related to the state reported in yi:020401 (), and is especially interesting in relation to the experimental observation of dipolar effects in utilizing a similar spin texture Vengalattore2008 (). The magnetization pattern of the spin helix gives rise to spontaneous density oscillations in the stationary state for strong dipolar interactions. As in the case of the spin vortex and flare textures, the helix state ceases to exist for .

The authors would like to thank Japan Society for the Promotion of Science (JSPS) for financial support. Y. Kawaguchi is acknowledged for useful comments.


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