PHYSICAL REVIEW A 76, 063603 2007

Polarized states and domain walls in spinor Bose-Einstein condensates

H. E. Nistazakis,1 D. J. Frantzeskakis,1 P. G. Kevrekidis,2 B. A. Malomed,3 R. Carretero-Gonzlez,4 and A. R. Bishop5
Department of Physics, University of Athens, Panepistimiopolis, Zografos, Athens 15784, Greece Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003-4515, USA 3 Department of Interdisciplinary Studies, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel 4 Nonlinear Dynamical Systems Group, Department of Mathematics and Statistics and Computational Science Research Center, San Diego State University, San Diego, California 92182-7720, USA 5 Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA Received 21 June 2007; published 4 December 2007
2 1

We study spin-polarized states and their stability in the antiferromagnetic phase of spinor F = 1 quasi-onedimensional Bose-Einstein condensates. Using analytical approximations and numerical methods, we nd various types of polarized states, including patterns of the Thomas-Fermi type, structures featuring a pulse in one component inducing a hole in the other components, states with holes in all three components, and domain walls DWs . The stability analysis based on the Bogoliubovde Gennes equations reveals intervals of weak oscillatory instability in families of these states, except for the DWs, which are always stable. The development of the instabilities is examined by means of direct simulations. DOI: 10.1103/PhysRevA.76.063603 PACS number s : 03.75.Mn, 03.75.Hh, 03.75.Kk

I. INTRODUCTION

The development of the far-off-resonant optical techniques for trapping ultracold atomic gases has opened new directions in the studies of Bose-Einstein condensates BECs . By means of these techniques, atoms can be conned regardless of their hyperne spin state, thus avoiding freezing the atoms spin degree of freedom and paving the way for the study of the collective spin dynamics 1 . One of the major achievements in this direction was the experimental creation of spinor BECs 2,3 . The spinor condensate formed by atoms with spin F is described by a 2F + 1 -component macroscopic mean-eld wave function, which gives rise to various phenomena that are not present in single-component BECs, including formation of spin domains 4 , spin textures 5 , and multicomponent vectorial solitons of bright 68 , dark 9 , gap 10 , and bright-dark 11 types. Generally, the dynamics of the spinor F = 1 BEC is spinmixing 12 . However, there also exist spin-polarized states of the system, which are stationary solutions of the corresponding system of Gross-Pitaevskii GP equations 13,14 . The stability of such polarized states is different for the two distinct types of the F = 1 condensates, namely the ferromagnetic FM , such as in 87Rb, and polar alias antiferromagnetic AFM , such as 23Na, ones, where the spin-dependent interactions are, respectively, attractive and repulsive. Accordingly, as demonstrated in Refs. 15 and 7 , spinpolarized states are modulationally stable unstable in the AFM FM condensates. In this work, we focus on AFM spinor condensates, and study, in particular, spin-polarized states of the spinor BEC of 23Na atoms. Assuming that this spinor system is conned in a strongly anisotropic trap, we rst present the respective system of three coupled quasi-one-dimensional 1D GP equations. Then, employing the so-called single-mode approximation, which expresses the proles of all three components of the spinor condensate in terms of a single spatial
1050-2947/2007/76 6 /063603 8

mode 12,15 , we use analytical and numerical methods to nd the spin-polarized states of the system and study their stability via the Bogoliubovde Gennes BdG equations i.e., the linearization of the GP equations for small perturbations; this approach does not take into regard incoherent perturbations involving vapor uctuations above the condensate, but the addition of the vapor perturbations does not usually affect the stability of localized states in BECs 16 . The simplest possible form of these states is based on the Thomas-Fermi TF congurations in each hyperne component . We also present other spin-polarized states, including those in which one component is pulse-shaped, inducing a hole in the other two components, and structures with holes in all three components. Families of all these states feature regions of weak oscillatory instabilities, with values of the normalized instability growth rate 103. Development of the oscillatory instabilities is examined by means of direct simulations. It is found that the unstable structures with a hole in one component get weakly deformed, while the unstable states with holes in all the three components suffer stronger deformations. If three initially spatially separated components, kept in different harmonic traps, are placed in a single trap, spin domain-wall DW patterns are formed. A family of the DW solutions exists and is fully stable if, for a xed value of the traps strength, the chemical potential or the number of atoms exceeds a certain critical value. The paper is organized as follows. Section II presents the model. Section III deals with the TF states. In Secs. IV and V we examine three-component structures with one and multiple holes, respectively including their stability . Finally, conclusions are presented in Sec. VI.
II. MODEL AND SETUP

In the framework of the mean-eld approach, a spinor BEC with F = 1 is described by a vectorial order parameter, r , t = 1 r , t , 0 r , t , +1 r , t T, with the different
2007 The American Physical Society

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elds corresponding to three values of the vertical component of the spin, mF = 1 , 0 , + 1. Assuming that this condensate is loaded into a strongly anisotropic trap, with holding , we assume, as usual, that the wave frequencies x y , z , where the functions are separable, 0,1 = 0,1 x y , z represent the ground state of transverse components the respective harmonic oscillator. Then, following the standard approach 17 of averaging the coupled 3D GP equations in the transverse plane y , z , we arrive at the system of coupled 1D equations for the longitudinal components of the wave functions see Refs. 6,7,911 : i
t 1

= Hsi +

+ c21D
2 0 1, 2

c21D c21D

1 +
+1 2 0

We note that the above mean-eld description of the system, based on the three coupled GP Eqs. 4 and 5 , is accurate only for temperatures well below the critical transition temperature Tc. In such a case, the effects of quantum or thermal uctuations on the spinor BEC may be ignored. However, it is possible to study the dynamics of spin-1 atoms in nite temperatures adopting, e.g., a Hartree-Fock-Popov type of approximation 21 , or a BdG description of the thermal cloud 22,23 . In the present case, however, such a detailed study of the effect of the temperature is beyond our present scope. Spin-polarized states of the system, characterized by a constant population of each spin component, can be constructed in the form of
j

0 = Hsi

= n j x exp i j i jt ,

j = 1, 0, + 1,

2c21D

1 0 +1 ,

2 where the asterisk denotes complex conjugate and Hsi 1D 2 1 2 2 2 / 2m x + 2 m x x + c0 ntot is the spin-independent part of the Hamiltonian, with ntot = 1 2 + 0 2 + +1 2 being the total density m is the atomic mass . The nonlinearity coefcients have an effectively 1D form, namely c01D = c0 / 2 a2 and c21D = c2 / 2 a2 , where a = / m is the transverse harmonic-oscillator length which determines the size of the transverse ground state. The coupling constants c0 and c2 account for, respectively, the mean-eld spinindependent and spin-dependent binary interactions between identical spin-1 bosons, c0,c2 = 4
2

where n j and j are densities and phases of the components and j are their chemical potentials. Substituting this in Eqs. 4 and 5 , it is readily found that conditions for the existence of the spin-polarized states are 2 2
0 0

=
+1

+
1

+1 ,

= 0 or

/3m

Below, we assume that the chemical potentials for all components are equal: 1 = 0 = +1. To analyze the stability of a stationary spin-polarized state, s x = 1 x , 0 x , +1 x T, we perform linearization around the unperturbed state, assuming a perturbed solution,
j

a2 + a0 , a2 a0 ,

where a0 and a2 are the s-wave scattering lengths in combined symmetric collision channels corresponding to values of the total spin f = 0 and 2. The spinor condensate with c2 0 and c2 0 is of the FM and AFM types, such as, respectively, 87Rb and 23Na 13,18 . Measuring time, length, and density in units of / c01D n0, / mc01D n0, and peak density n0, we cast Eqs. 1 and 2 in the dimensionless form i
t 1

x,t = j x +

u j x e t +

xe

ei t ,

10

= Hsi

2 0

1,

4 i
t 0

= Hsi
1 2

1 2 2

+1

+2

1 0 +1 ,

5 traps

where Hsi strength is

2 1 x+ 2

x + ntot,

the
x

normalized

where u j and v j represent innitesimal perturbations with eigenvalues r + i i. Then, the solution of the ensuing BdG equations for and associated eigenfunctions u j, v j provides complete information about the stability of the underlying stationary state, s. Whenever it is unstable, we will also examine its evolution through direct simulations of the GP Eqs. 4 and 5 , using a nite-difference scheme in space and the fourth-order Runge-Kutta integrator in time. Typically, in the simulations the unstable state is initially perturbed by a uniformly distributed random perturbation of relative amplitude 5 104. At this point, and in this dimensionless 1D setting under consideration, it is also useful to introduce the distributions of the local spin average dened as see, e.g., Refs. 24 and 8 for the 1D case f x x;t = 2 Re f y x;t = 2 Im f z x;t =
+1 2 0 +1

3 2 a0 + 2a2 n0

+
2

/ntot , /ntot ,

11 12 13

and the FM or AFM character of the spinor condensate cor0 and 0, with responds, respectively, to c21D c01D a2 a0 = . a0 + 2a2 7 =

+1

/ntot ,

For spin-1 87Rb and 23Na atoms, this parameter is 4.66 103 19 and = 3.14 102 20 , respectively.

with the total normalized mean-eld spin given by f x ; t 2 = f 2 + f 2 + f z . For the spin-polarized states that we will x y present below, the y component of the mean-eld spin will always be f y = 0; additionally, in most cases apart from one

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pertaining to a certain domain-wall typesee Sec. V the z component will also be f z = 0. Thus, in most cases, the total mean-eld spin will be f = f x . Finally, to estimate relevant physical parameters, we assume a spinor condensate of 23Na atoms with 1D peak density n0 = 108 m1, conned in the harmonic trap with transverse and longitudinal frequencies =2 230 Hz and =2 13 Hz. In this case, the normalized trap strength is x = 0.1 this value is kept xed throughout this work , while the number of atoms, N, depends on chemical potential and the particular form of spin-polarized states. Typically, we 5, which corresponds to values of take in interval 1 N in the range of 3.5 103 N 3.5 104 atoms. For instance, at = 2 the number of atoms is N 104; in this case, the normalized time and space units correspond, respectively, to 1.2 ms and 1.83 m.
III. THOMAS-FERMI SPIN-POLARIZED STATES

0 1

0.5 0 0.5

0
0

(a)

10

20

30

1 30 20 10

(b)

10

20

30

1 0.5 0 30

(c)

20

10

10

20

30

The simplest spin-polarized states can be found in the framework of the single-mode approximation 12,15 . In anticipation of the fact that the three components n j x may be close to eigenmodes of a single effective potential, induced by a combination of the trap and nonlinearity, we introduce the ansatz n j x = q jn x . Here, the coefcients q j are the populations of each spin component in the steady state, subject to the normalization condition q1 + q0 + q+1 = 1. Then, Eqs. 4 and 5 lead to the following system: L+ 1+ L+ q 1 sq0 2 q1q+1 n q1 n = 0, 14

FIG. 1. Color online Top panels: Examples of two stable spinpolarized states for = 0 left panel and = right panel , both obtained for = 0.1 and = 2. Wave functions 1 are identical and are depicted by the solid line, while the wave function 0 is depicted by the dashed line. In the left panel, q1 = q+1 = 0.5 and q0 = 1, while in the right panel q1 = q+1 = 0.25 and q0 = 0.5. Bottom panel: The spatial distribution of the total mean-eld spin f; note that in this case f y = f z = 0 and f x = f for = 0 or , respectively.

nj =

qj p

1 2

2 2

x ,

19

1 q0 + 2s q1q+1 n n = 0,

15

1 1 where L 2 2 + 2 2x2 + n and s = 1 for = 0 or x = , respectively. In fact, Eqs. 14 and 15 are two equations for a single function, and it can be readily checked that they are tantamount to a single equation, viz.

1 2

2 x

1 2

1 , and n j = 0 elsewhere. Obviin the spatial region 2 2x2 ously, all three components of the TF solution have the same spatial width the TF radius , RTF = 2 / . In numerical simulations, we used a xed-point algorithm the Newton-Raphson method to nd exact spin-polarized solutions to Eqs. 4 and 5 , with proles close to those produced by the TF approximation of Eq. 19 . In particular, we used, as an initial guess, three identical proles of the form

2 2

x + pn

n = 0,

16
j

x = n x exp i

20

if the following relations are imposed upon populations q0 and q1, and the constant p in Eq. 16 is dened as follows: p 1+ , p 1, q0 = 2 q1q+1 q1 = q+1 for for = . = 0, 17 18

In uniform space = 0 , Eq. 16 possesses a solution with constant density, n = / p. As shown in Refs. 7,11,15 , such constant solutions to Eqs. 4 and 5 are modulation0. Below, we ally stable only in the AFM phase, with will only consider the case of the AFM condensate which, in particular, applies to the 23Na condensate, with = 3.14 102 . In the presence of a sufciently weak trap, Eq. 16 can be solved approximately by means of the TF approximation 25 . In this way, neglecting the kinetic-energy term 2 x n in Eq. 16 , we nd density proles of the three spin components:

1 = 0 or . Then, keeping the with n x = 2 2x2 and traps strength, , xed, we varied the chemical potential , and the numerical solution converged to stable spinpolarized states, which were indeed close to the approximate one given by Eq. 19 . Two typical examples are shown in = 0 left panel and the top panels of Fig. 1 for both cases, = right panel , with = 2. The numerically determined states are very close to their TF-predicted counterparts, with = 0, top left panel and q1 q1 = q+1 = 0.5 and q0 = 1 = q+1 = 0.25 and q0 = 0.5 = , top right panel . The distributions of the spin components are time-independent with f y = f z = 0, and f x = f for = 0 or , respectively. The total mean-eld spin f is shown in the bottom panel of Fig. 1. The stability of the TF states was examined too by means of both the linear-stability analysis and direct simulations . It has been concluded that these states are always stable.

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FIG. 2. Color online Spin-polarized state with a hole in the 1 components, and a pulselike shape of the 0 eld, for = 0.1 and = 0. The top left and right panels show unstable and stable states, with = 2 and = 3, respectively; solid and dashed lines depict components 1 and 0. The two panels in the second row display spectral planes r , i of the in stability eigenvalues for the same states. Note that the instability of the state pertaining to = 2 is of the oscillatory type, being accounted for by a quartet of eigenvalues with nonzero real parts. The solid and dashed lines in the third-row panel show the normalized number of atoms norm , N, in components 1 and 0, respectively, as a function of chemical potential . The bottom panel shows the maximal growth rate, max r , as a function of , which reveals the instability window for 1.81 2.15, with a maximum instability growth rate r max 1.3 103 at 2. The latter value corresponds to the unstable state shown in the top left panel. IV. SPIN-POLARIZED STATES WITH HOLES

FIG. 3. Color online Top panels: Contour density plots display the evolution of the weakly unstable solution shown in Fig. 2 the left and right panels represent, respectively, +1 1 and 0 components . Because of the extremely small growth rate of the instability, it manifests itself only at t 4000. Middle panels: The respective wave-function proles at t = 0 solid lines and t = 10 000 dashed lines ; as above, the left and right panels show, respectively, wave functions 1 and 0. Bottom panel: Snapshots of the distribution of the total mean-eld spin f = f x at t = 0 solid line and t = 10 000 dashed line .

Apart from the smooth spin-polarized states of the TF type, there exist other ones, which feature holes in some of the components, or in all of them. An example of such states is shown in Fig. 2. As seen in the two top panels of this gure, the 0 component is concentrated in the form of a pulse located at the traps center, while the 1 components feature a large hole at the same spot. This shape is explained by the fact that the interaction between components is repulsive, hence a peak hole in 0 1 induces a hole peak in 1 0 . The norm N of each component is shown, as a function of chemical potential , in the third-row panel of Fig. 2, and typical sets of the linear-stability eigenvalues for these states are displayed, for two different values of , in the second-row panels. The state with = 3 top right is

stable, as all the eigenvalues are imaginary, while the one with = 2 top left panel is unstable. Further analysis demonstrates that all such unstable states are destabilized by a Hamiltonian Hopf bifurcation, which gives rise to a quartet of eigenvalues with nonzero real parts. The instability inter2.15, with the largest instability growth val is 1.81 2 see bottom panel rate, max r 1.3 103, found at of Fig. 2 . A similar state with the eld proles in 1 and 0 exchanged, i.e., the 0 component featuring the hole, and 1 ones concentrated in narrow pulses, were also found. Moreover, such states were found too with either 0 or 1 having = . The results are not shown the opposite sign i.e., for here, as the stability properties of these states are qualitatively the same as in the above case. The evolution of unstable states is exemplied in Fig. 3, which displays results of direct simulations of Eqs. 4 and = 0 and = 2, that was 5 for the unstable state with presented in Fig. 2 top left panel . In the top panels of Fig. 3, contour plots of densities of the components of the solution are displayed as a function of time the densities of the +1 and 1 components are identical . It is clearly observed that the predicted oscillatory instability sets in at a very large time t 4000, which corresponds to t 5 s in physical

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1 1 0 30 30 0.2 0 1 0 1x10
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0 15 6000 10000

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1 15

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Max( )

2 1 0 1 2 3 4 5
(e)

20

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(b)

FIG. 4. Color online Same as Fig. 2 but for a state with one hole in each of the 1 components and two holes in the 0 component. In this case, the instability is driven by two quartets in the eigenvalue spectrum of small perturbations, which lead to instability in the interval 2.58 3.22. The maximum instability growth rate is max r 1.8 103 at 2.9. The unstable state shown in the top right panel corresponds to = 3.

FIG. 5. Color online Same as Fig. 3, for the unstable state shown in the top right panel of Fig. 4 it pertains to = 3 . The instability manifests itself at large times t 3500 and results in strong oscillatory deformation of the spinor condensate; this is clearly observed in the contour plots of the densities top panels , the snapshots of the wave function proles middle panels , and the snapshots of the mean-eld spin distribution bottom panel .

units ; this is a consequence of the extremely small growth rate of the instability. Eventually, the system settles down to a steady state, which is qualitatively similar to the initial one. In particular, as seen in the top panels of Fig. 3, after t 8000 the pulse in the 0 component broadens and its amplitude accordingly decreases, while the hole in the 1 components becomes shallower. This is clearly shown in the middle panels of Fig. 3, where snapshots of the spatial distributions of the wave functions at t = 0 solid lines and t = 10 000 dashed lines are shown. Respective snapshots of the mean-eld spin are also displayed in the bottom panel of Fig. 3; it is seen that the manifestation of the oscillatory instability does not affect f signicantly. Note that here, as in the previous case with the TF polarized states, f y = f z = 0 and f = f x. Apart from the states considered above, it is also possible to nd spin-polarized ones which feature, e.g., one hole in each of the 1 components, and two holes in 0. Examples of such a state are shown in the top panels of Fig. 4 the left one, for = 2, is stable, while the right one, for = 3, is unstable . As seen in this gure, one may consider components 1 and +1 as built of two overlapping pulses, which induce two holes in the 0 component due to the repulsive intercomponent interactions. Results of the stability analysis for these states are shown in Fig. 4. In this case, two quartets

of eigenvalues with nonzero real parts are found in the spectral plane see the right panel in the second row of Fig. 4 . These lead to instability in the interval 2.58 3.22 see
2 1.5 1 0.5 0 40

V V0 1

V+1

+1 20 0 20 40

FIG. 6. Color online Initialization of the system when three spatially separated traps, V1, V0 shown by dashed parabolas , and V+1 the solid parabola , with equal strengths = 0.1 and centers placed at x = 10, 0, and +10, hold the TF states of the 1, 0, and +1 components, respectively. Then, two traps V1 and V0 , centered at x = 10 and x = 0, are turned off, and a stationary solution, supported solely by the trap V+1 centered at x = 10, is looked for by means of the xed-point algorithm, using the conguration with the three mutually shifted components as an input.

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the bottom panel in Fig. 4 . The respective largest instability growth rate is r max 1.8 103 for 2.9, i.e., of the same order of magnitude as in the previous case. The development of the instability was studied, as above, in direct simulations, starting with initial conditions in the form of a perturbed solution pertaining to = 3. The result is shown in Fig. 5, in terms of the evolution of identical densities of the 1 components and the density of 0. Again, the instability manifests itself at large times t 3500, which corresponds to t 4.2 s in physical units , but in this case the nal result is a strong oscillatory deformation of the three components after t 5500 , contrary to the establishment of the new stationary pulse-hole state observed in Fig. 3. This deformation is clearly seen by the snapshots of the spatial distributions of the wave functions middle panels of Fig. 5 and the mean-eld spin bottom panel of Fig. 5 , at t = 0 solid lines and t = 10 000 dashed lines . Notice that the resulting states are asymmetric with respect to x x transformations, indicating the possibility of asymmetric states in the system such as domain walls see also below .
2

Similar states with one hole in 0 and two holes in each of the 1 components, as well as their counterparts corresponding to = , have also been found. They are not shown here; as in the previous case, the in stability of these additional states is similar to that reported in Fig. 4.

V. DOMAIN WALLS

In the above sections, we reported the spin-polarized states in which all three spin components were spatially overlapping, since they were conned by the same potential trap. However, it is also possible to use three different traps, each conning a different component, to initially separate them, and then allow the system to evolve in the presence of only one of these traps i.e., turning off the other two . In this section, we present spin-polarized states, including domainwall DW structures, obtained in this way. First, we describe the initialization of the system. We assume that the three TF-shaped components are initially

1
0

+1
20
1

0.5 0

1 0 15 30

0 30
(a)

15

(b)

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100 50 0
(c)

4
:f :f

1 0.5 0 30 15 0
(d)

1 0.5 0

: fz

15 30

30 15 0
(e)

15 30

FIG. 7. Color online Top left panel: Wave functions of the components +1 and 1 are identical in a stationary state found from the initial conguration prepared as shown in Fig. 6. The resulting spin-polarized state has the form of a domain-wall structure between the 0 and 1 components. The parameters are = 0.1 and = 2. The top right panel shows the wave functions of the domain-wall state found at = 1.43. Middle panel: The norm of each component in the domain-wall structure vs the chemical potential the dependences for = 0 and are identical . Bottom panels: The spatial distribution of the total mean-eld spin solid lines . For = 2 bottom left f = f x, while for 2 = 1.43 bottom right f = f 2 + f z f x; in the latter case, the f x and f z components are depicted by dashed and dashed-dotted lines, x respectively. 063603-6

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FIG. 8. Color online Evolution of the domain-wall structure shown in the top left panel of Fig. 7, to which a random perturbation was added. Shown in the left and right panels are spatiotemporal contour plots of the densities in components 1 identical to each other and 0, respectively.

loaded into three different traps, V j x , of the same strength, , centered at different positions: Vj x = 1 2
2

x j x 2,

j = 1, 0, + 1.

21

We choose x = 1 i.e., x = 10 for = 0.1 , which induces the initially separated TF congurations; see Fig. 6. After preparing this state, we turn off the traps V1 x and V0 x , keeping only the rightmost one, V+1 x , which now acts on all three components. The so dened initial conguration is fed, as an input, into the xed-point algorithm, to nd a spin-polarized state generated by it. Other possibilities, such as turning off potentials V1 and keeping V0, arranging the three components in a different way, etc., eventually lead to retrieving the spin-polarized states presented in the previous sections, while the approach outlined above keeping V+1 x and switching V1 x and V0 x off generates new DW patterns, which are displayed in Fig. 7, and could not be obtained otherwise in fact, the asymmetry of the procedure is instrumental in generating the new states . The most interesting spin-polarized DW states found following this procedure correspond to values of the chemical 1.43 or norm N 5400 , for = 0.1; for potential smaller , we typically found structures of the TF type. Two examples, one for = 2, and another exactly corresponding to = 1.43, are shown in Fig. 7. In the former case = 2 , the 0 component which has the larger norm is centered to the right of the midpoint of the remaining trap x+1 = 10 , while the identical 1 components are pushed to the left, due to the repulsion from 0, with a DW created between 1 and 0. Note that the total
30 15 x 0 15 30
(a)

mean-eld spin is f = f x and has a pulselike distribution shown in the bottom left panel of Fig. 7. In the state found at the above-mentioned special value, = 1.43, which is shown in the top right panel of Fig. 7, the shape of the 0 component is similar to that displayed in the left top panel for = 2, while the 1 components are not identical, in contrast to the previous example. In the present case, the 0 and 1 components overlap over a wider spatial region, and 1 changes its sign at x 13, featuring a structure resembling the waveform of a dark soliton embedded in a bright one 26 . Notice that in this case f z takes a small nonzero value and, thus, it has a small contribution to the total mean-eld spin; however, in fact, as seen in the bottomright panel of Fig. 7, the latter can be approximated as f 2 = f 2 + f z f x. x The stability of the DW states was also investigated in the framework of the BdG equations. It was concluded that there are no unstable eigenvalues, i.e., with a nonzero real part, in 5, or, equivalently, 5400 N 35 000 for interval 1.43 = 0.1 not shown here in detail . Thus the DWs are stable in this region. Verication of the stability, performed by direct simulations of Eqs. 4 and 5 , is illustrated in Figs. 8 and 9, for = 2 and = 1.43, respectively. It is obvious that these states indeed remain stable at very large times exceeding t = 10 000 i.e., 12 s in physical units .

VI. CONCLUSIONS

In this work, we have studied spin-polarized states in antiferromagnetic spinor F = 1 Bose-Einstein condensates. In particular, our analysis applies to a quasi-1D spinor condensate of sodium atoms. The considerations were based on analytical calculations and numerical computations of the coupled Gross-Pitaevskii equations for this setting. Assuming that all three hyperne spin components are conned in the same harmonic trap, we have found various types of spin-polarized states and examined their stability. The rst family consists of Thomas-Fermi congurations, considered analytically in the framework of the single-mode approximation which assumes the similarity of the spatial proles of all the components . Within their existence region, these states were found to be stable. Also identied were more complex patterns, which include states composed of one or more pulselike structures in one component, that induce holes in the other components, and states with holes in all three components. These states feature windows of weak
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FIG. 9. Color online Same as Fig. 8 but for the state shown in the top right panel of Fig. 7. The left, middle, and right panels show, respectively, the densities in the 1, 0, and +1 components. Noteworthy is a stationary dark-soliton-like structure, located at x 13 in the 1 component. 063603-7

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PHYSICAL REVIEW A 76, 063603 2007

instability. The development of the instability was investigated by means of direct numerical simulations, which demonstrate that it manifests itself at very long times, and results in a deformation of the states with a single hole in some of the components, which does not qualitatively change their form, and a stronger oscillatory perturbation and a potential eventual asymmetry of the states with multiple hoels. Fully stable families of spin-polarized states develop from congurations consisting of initially separated components that are held in three mutually shifted traps . These states form domain-wall structures between the components, at values of the chemical potential above a certain threshold. Just at the threshold we have found another spin-polarized state in which all the components partly overlap. It would denitely be interesting to investigate the existence and stability of higher-dimensional counterparts of the 1D spin-polarized states found in this work. In this connection, a relevant question for further analysis is whether spinor condensates support stable topological objects, such

as dark solitons or vortices. Moreover, the effect of temperature on the statics and dynamics of the spin-polarized states presented in this work is certainly another challenging issue deserving further investigation. Work in these directions is in progress. The work of H.E.N. and D.J.F. was partially supported from the Special Research Account of the University of Athens. H.E.N. acknowledges partial support from EC grants PYTHAGORAS I. P.G.K. acknowledges support from NSFCAREER, NSF DMS-0505663, and NSF DMS-0619492, as well as the warm hospitality of MSRI during the initial stages of this work. The work of B.A.M. was supported, in part, by the Israel Science Foundation through the Center-ofExcellence Grant No. 8006/03, and the German-Israel Foundation through Grant No. 149/2006. R.C.G. acknowledges support from NSF DMS-0505663. Work at Los Alamos National Laboratory is supported by the U.S. Department of Energy.

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