Magnetic domains in spinor Bose–Einstein condensates

We discuss the structure of spin-1 Bose–Einstein condensates in the presence of a homogenous magnetic field. We demonstrate that the phase separation can occur in the ground state of antiferromagnetic (polar) condensates, while the spin components of the ferromagnetic condensates are always miscible...

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Datum:	2010
Hauptverfasser:	Matuszewski, M., Alexander, T. J., Kivshar, Y. S.
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Sprache:	English
Veröffentlicht:	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2010
Schriftenreihe:	Физика низких температур
Schlagworte:	К 80-летию со дня рождения В.Г. Барьяхтара
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spelling	irk-123456789-1174572017-05-24T03:02:50Z Magnetic domains in spinor Bose–Einstein condensates Matuszewski, M. Alexander, T. J. Kivshar, Y. S. К 80-летию со дня рождения В.Г. Барьяхтара We discuss the structure of spin-1 Bose–Einstein condensates in the presence of a homogenous magnetic field. We demonstrate that the phase separation can occur in the ground state of antiferromagnetic (polar) condensates, while the spin components of the ferromagnetic condensates are always miscible, and no phase separation occurs. Our analysis predicts that this phenomenon takes place when the energy of the lowest homogenous state is a concave function of the magnetization. We propose a method for generation of spin domains by adiabatic switching of the magnetic field. We also discuss the phenomena of dynamical instability and spin domain formation. 2010 Article Magnetic domains in spinor Bose–Einstein condensates / M. Matuszewski, T. J. Alexander, Y. S. Kivshar // Физика низких температур. — 2010. — Т. 36, № 8-9. — С. 883-890. — Бібліогр.: 36 назв. — англ. 0132-6414 PACS: 03.75.Lm, 05.45.Yv http://dspace.nbuv.gov.ua/handle/123456789/117457 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
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language	English
topic	К 80-летию со дня рождения В.Г. Барьяхтара К 80-летию со дня рождения В.Г. Барьяхтара
spellingShingle	К 80-летию со дня рождения В.Г. Барьяхтара К 80-летию со дня рождения В.Г. Барьяхтара Matuszewski, M. Alexander, T. J. Kivshar, Y. S. Magnetic domains in spinor Bose–Einstein condensates Физика низких температур
description	We discuss the structure of spin-1 Bose–Einstein condensates in the presence of a homogenous magnetic field. We demonstrate that the phase separation can occur in the ground state of antiferromagnetic (polar) condensates, while the spin components of the ferromagnetic condensates are always miscible, and no phase separation occurs. Our analysis predicts that this phenomenon takes place when the energy of the lowest homogenous state is a concave function of the magnetization. We propose a method for generation of spin domains by adiabatic switching of the magnetic field. We also discuss the phenomena of dynamical instability and spin domain formation.
format	Article
author	Matuszewski, M. Alexander, T. J. Kivshar, Y. S.
author_facet	Matuszewski, M. Alexander, T. J. Kivshar, Y. S.
author_sort	Matuszewski, M.
title	Magnetic domains in spinor Bose–Einstein condensates
title_short	Magnetic domains in spinor Bose–Einstein condensates
title_full	Magnetic domains in spinor Bose–Einstein condensates
title_fullStr	Magnetic domains in spinor Bose–Einstein condensates
title_full_unstemmed	Magnetic domains in spinor Bose–Einstein condensates
title_sort	magnetic domains in spinor bose–einstein condensates
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate	2010
topic_facet	К 80-летию со дня рождения В.Г. Барьяхтара
url	http://dspace.nbuv.gov.ua/handle/123456789/117457
citation_txt	Magnetic domains in spinor Bose–Einstein condensates / M. Matuszewski, T. J. Alexander, Y. S. Kivshar // Физика низких температур. — 2010. — Т. 36, № 8-9. — С. 883-890. — Бібліогр.: 36 назв. — англ.
series	Физика низких температур
work_keys_str_mv	AT matuszewskim magneticdomainsinspinorboseeinsteincondensates AT alexandertj magneticdomainsinspinorboseeinsteincondensates AT kivsharys magneticdomainsinspinorboseeinsteincondensates
first_indexed	2025-07-08T12:15:25Z
last_indexed	2025-07-08T12:15:25Z
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fulltext	© Michał Matuszewski, Tristram J. Alexander, and Yuri S. Kivshar, 2010 Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 8/9, p. 883–890 Magnetic domains in spinor Bose–Einstein condensates Michał Matuszewski, Tristram J. Alexander, and Yuri S. Kivshar Nonlinear Physics Center and ARC Center of Excellence for Quantum-Atom Optics, Research School of Physics and Engineering, Australian National University, Canberra ACT 0200, Australia E-mail: ysk@internode.on.net Received December 14, 2009 We discuss the structure of spin-1 Bose–Einstein condensates in the presence of a homogenous magnetic field. We demonstrate that the phase separation can occur in the ground state of antiferromagnetic (polar) con- densates, while the spin components of the ferromagnetic condensates are always miscible, and no phase separa- tion occurs. Our analysis predicts that this phenomenon takes place when the energy of the lowest homogenous state is a concave function of the magnetization. We propose a method for generation of spin domains by adia- batic switching of the magnetic field. We also discuss the phenomena of dynamical instability and spin domain formation. PACS: 03.75.Lm Tunneling, Josephson effect, Bose–Einstein condensates in periodic potentials, solitons, vor- tices, and topological excitations; 05.45.Yv Solitons. Keywords: Bose–Einstein condensates, homogeneous magnetic field, magnetic domains. 1. Introduction The analysis of the properties of spin domains and magnetic solitons is one of the major topics of the theory of crystalline magnetic structures [1]. However, the recent development in the physics of cold gases gave a birth to an exciting new field where the spinor dynamics and magnet- ic domain formation are the key ingredients of a new and seemingly different physics which, however, borrows many important results and techniques from the solid state physics. More specifically, the spin degree of freedom of spinor Bose–Einstein condensates (BECs) [2–4] leads to a wealth of new phenomena not possessed by single- component (spin-frozen) condensates. New spin-induced dynamics such as spin waves [3], spin-mixing [5] and spin textures [3,6] have all been predicted theoretically and ob- served in experiment. The observation of these spin- dependent phenomena became possible due to the devel- opment of optical traps [7] which trap all spin components, rather than just the low-magnetic-field seeking spin states of magnetic traps. However, the effect of an additional small non-zero magnetic field on the condensate in these optical traps was studied even in the seminal theoretical [4] and experimental [2] works. In fact the interplay of spin and magnetic field has been at the heart of some of the most impressive spinor BEC experiments, including the demonstration of spin domains [2], spin oscillations [8] and observation of spin textures and vortices [9]. A spin-1 BEC in a magnetic field is subjected to the well-known Zeeman effect. At low fields the effect is dom- inated by the linear Zeeman effect, which leads to a Lar- mor precession of the spin vector about the magnetic field at a constant rate, which is unaffected by spatial inhomo- geneities in the condensate [10]. At higher magnetic fields the quadratic Zeeman effect becomes important, and leads to much more dramatic effects in the condensate, such as coherent population exchange between spin components [8,11,12] and the breaking of the single-mode approxima- tion (SMA) [13,14], which assumes that all the spin com- ponents share the same spatial density and phase profile. The study of the behavior of a spin-1 condensate in the presence of a magnetic field began with the work of Sten- ger et al. [2], where the existence of magnetic (spin) do- mains were predicted and observed in the ground state of a polar 23Na condensate subject to a magnetic field gradient. At the same time, the ground states of both ferromagnetic and antiferromagnetic (polar) condensates in homogenous magnetic field were found to be free of spin domains in the local density approximation. It was later found that the SMA was broken in the ground state of a condensate con- fined in a harmonic trap even in a homogenous field [13]. Nevertheless, the SMA continued to be used in studies of spinor condensates for its simplicity and validity in a broad range of experimental situations [11,15], in particular when the condensate size is smaller than the spin healing length, which determines the minimum domain size. On the other Michał Matuszewski, Tristram J. Alexander, and Yuri S. Kivshar 884 Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 8/9 hand, the dynamical instability, leading to the spontaneous formation of dynamic spin domains, was found to occur in large ferromagnetic condensates prepared in excited initial states [9,16–18], while no such phenomenon was predicted to occur [16] or observed [19] in antiferromagnetic con- densates. Similar instabilities were found in the transport of both types of spin-1 condensates in optical lattices [20]. It seemed however that spin domains were only to be found in antiferromagnetic condensates in the presence of inho- mogeneous magnetic fields [2] or trapping potentials. In this work, first we overview the results of our recent works [21,22] and discuss the structure of spin-1 Bose– Einstein condensates in the presence of a homogenous magnetic field. We show that the translational symmetry of a homogenous BEC is spontaneously broken and phase separation occurs in magnetized polar condensates if the magnetic field is strong enough. An analogous phenome- non has been predicted and observed previously in binary condensates [23,24]. In contrast, the cases when domain formation is driven by inhomogeneous external potentials or magnetic field gradients may be referred to as potential separation according to the naming used in Ref. 24. Here, we show that for a range of experimental conditions, it is energetically favorable for the system to consist of two separate phases composed of different stationary states. Finally, we demonstrate numerically that this phenomenon can be observed in a polar condensate trapped in a harmon- ic optical potential. 2. Model We consider dilute spin-1 BEC in a homogenous mag- netic field pointing in the z -direction. The mean-field Hamiltonian of this system is given by the expression, 2 * 2 20 = ,0, = \| \| ( )\| \| , 2 2j j j j a j c H d n V H M− + ⎛ ⎞− ∇ψ ∇ψ + ψ + ψ +⎜ ⎟⎜ ⎟ ⎝ ⎠ ∑∫ r rh (1) where 0, ,− +ψ ψ ψ are the wavefunctions of atoms in mag- netic sub-levels = 1,0, 1m − + , M is the atomic mass, ( )V r is an external potential and 2= = \| \|j jn n ψ∑ ∑ is the total atom density. The asymmetric part of the Hamil- tonian is presented as: 22 = ,0, = \| \| , 2a j j j c H d E n − + ⎛ ⎞ ⎜ ⎟+ ⎜ ⎟ ⎝ ⎠ ∑∫ r F (2) where jE is the Zeeman energy shift for state jψ and the spin density is, † † †ˆ ˆ ˆ= ( , , ) = ( , , ) ,x y z x y zF F F F F FF ψ ψ ψ ψ ψ ψ (3) where , , ˆ x y zF are the spin matrices [25] and =ψ 0( , , )+ −= ψ ψ ψ . The nonlinear coefficients are given by 2 0 2 0= 4 (2 ) / 3c a a Mπ +h and 2 2 2 0= 4 ( ) / 3c a a Mπ −h , where Sa is the s-wave scattering length for colliding atoms with total spin S . The total number of atoms and the total magnetization = ,N n d∫ r (4) ( )= = ,zF d n n d+ −−∫ ∫r rM (5) are conserved quantities. The Zeeman energy shift for each of the components, jE can be calculated using the Breit– Rabi formula [26] 21= 1 4 1 , 8 HFS I BE E g B± ⎛ ⎞− + ±α +α μ⎜ ⎟ ⎝ ⎠ m 2 0 1= 1 4 1 , 8 HFSE E ⎛ ⎞− + +α⎜ ⎟ ⎝ ⎠ (6) where HFSE is the hyperfine energy splitting at zero mag- netic field, = ( ) /I J B HFSg g B Eα + μ , where Bμ is the Bohr magneton, Ig and Jg are the gyromagnetic ratios of electron and nucleus, and B is the magnetic field strength. The linear part of the Zeeman effect gives rise to an overall shift of the energy, and so we can remove it with the trans- formation ( ) / 2 ( ) / 2.H H N E N E+ −→ + + + −M M (7) This transformation is equivalent to the removal of the Larmor precession of the spin vector around the z-axis [21]. We thus consider only the effects of the quadratic Zeeman shift. For sufficiently weak magnetic field we can approximate it by 2 0= ( 2 ) / 2 /16HFSE E E E E+ −δ + − ≈ α , which is always positive. The asymmetric part of the Hamiltonian (2) can now be rewritten as 22 0= \| \| = ( ), 2a c H d En d ne⎛ ⎞−δ +⎜ ⎟ ⎝ ⎠∫ ∫r F r r (8) where the energy per atom ( )e r is given by [11] ( )2 2 22 2 0 0= \| \| = \| \| , 2 2 c n c n e E E m⊥−δ ρ + −δ ρ + +f f 2 2 2 0 0 0 0\| \| = 2 (1 ) 2 (1 ) cos .m⊥ ρ −ρ + ρ −ρ − θf (9) We express the wavefunctions as = exp ( )j j jn iψ ρ θ where the relative densities are = /j jn nρ . We also intro- duced the relative phase 0= 2+ −θ θ + θ − θ , spin per atom = / nf F , and magnetization per atom = =zm f + −ρ −ρ . The perpendicular spin component per atom is 2 2 2\| \| = x yf f⊥ +f . The Hamiltonian (1) generates the Gross–Pitaevskii eq- uations describing the mean-field dynamics of the system [ ] 2 * 2 0 2 0= ( ) ,i c n n n c t ± ± ± ∂ψ + + − ψ + ψ ψ ∂ m mh L (10) [ ] 0 2 0 2 0= ( ) 2 ,i E c n n c t + − + − ∂ψ −δ + + ψ + ψ ψ ψ ∂ h L where L is given by 2 2 0= / 2 ( )M c n V− ∇ + + rhL . Magnetic domains in spinor Bose–Einstein condensates Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 8/9 885 By comparing the kinetic energy with the interaction energy, we can define a characteristic healing length 0=2 / 2Mc nξ πh and spin healing length 2=2 / 2 .s Mc nξ πh These quantities give the length scales of spatial variations in the condensate profile induced by the spin-independent or spin-dependent interactions, respectively. Analogously, we define magnetic healing length as = 2 / 2B M Eξ π δh . In real spinor condensates, the 0a and 2a scattering lengths have similar magnitude. The spin-dependent inte- raction coefficient 2c is therefore much smaller than its spin-independent counterpart 0c . For example, this ratio is about 1:30 in a 23Na condensate and 1:220 in a 87Rb con- densate far from Feshbach resonances [27]. As a result, the excitations that change the total density require much more energy than those that keep ( )n r close to the ground state profile. In our considerations we will assume that the amount of energy present in the system is not sufficient to excite the high-energy modes, and we will treat the total atom density ( )n r as a constant. 3. Condensate without a trapping potential The ground states of spin-1 condensates in homogenous magnetic field have been studied in a number of previous works [2,13,28]. The most common procedure [2] involves minimization of the energy functional with constraints on the number of atoms N and the total magnetization M. The resulting Lagrange multipliers p and q serve as pa- rameters related to the quadratic Zeeman shift Eδ and the magnetization m . An alternative method, elaborated in [13], consists of minimization of the energy functional in the parameter space of physically relevant variables B and m . Most of the previous studies, however, were assuming that the condensate remains homogenous and well de- scribed by the single-mode approximation; in particular, the spatial structure observed in [2] resulted from the ap- plied magnetic field gradient, but the BEC was assumed to be well described by the homogenous model at each point in space (local density approximation). In Ref. 13, the breakdown of the single-mode approximation was shown numerically for a condensate confined in a harmonic po- tential. We correct the previous studies by showing that when the condensate size is larger than the spin healing length sξ , the translational symmetry is spontaneously broken and phase separation occurs in magnetized polar conden- sates if the magnetic field is strong enough. This pheno- menon takes place when the energy of the spin state with the lowest energy is a concave function of m for a given Eδ . On the contrary, the energy is always a convex func- tion of m for the ferromagnetic condensate, and no phase separation occurs. Note that phase separation has been pre- viously predicted in binary condensates [23,24] and in fer- romagnetic condensates at finite temperature [29]. We construct ground states of the condensate using homogeneous stationary solutions of the GP equations (10) ( )( , ) = e ,i t ij S j j jt n − μ +μ + θ ψ r (11) where 0= /S c nμ h is a constant and 0= 2+ −μ +μ μ due to a phase matching condition. Following [22], we distin- guish several types of stationary states. The states where only a single Zeeman component is populated ( = 1jn for a specified = ,0,j − or + ) are named −ρ , 0ρ , and +ρ , respectively. The state where 0 = 0n but , 0n n− + ≠ is the two-component (2C) state. The three-component states are classified according to the value of 0= 2+ −θ θ + θ − θ . The states with = 0θ are called phase-matched (PM) states, and the ones with =θ π are called anti-phase-matched (APM) states. For more details about this classification and the properties of the stationary states, refer to [22]. Two types of domain structures, depicted in Fig. 1, are composed of two different stationary states connected with a shaded region where all three components are nonzero. These two domain states have the advantage that the per- pendicular spin is nonzero only in the transitory region, hence their energy is relatively low in polar condensates. In fact, these are the only phase separated states that can be the ground states of a homogenous condensate. Their ener- gies per atom in the limit of infinite condensate size, which allows for neglecting of the relatively small intermediate region are 0 0= \| \| (1 \| \|) ,e m e m eρ +ρ ρ ρ± ± + − Fig. 1. Schematic structure of the phase separated states 0±ρ + ρ (a) and 02C + ρ (b). The shaded region, in which all three com- ponents are nonzero, has the approximate extent of one spin heal- ing length sξ or magnetic healing length Bξ , whichever is greater. The relative size of the domains is indicated with arrows. The corresponding wavefunction profiles obtained numerically with periodic boundary conditions in the case of 23Na for = 0.5m with 2/ ( ) = 0.8E c nδ (c) and 2/ ( ) = 0.23E c nδ (d). The n+ , 0n , and n− components are depicted by dash-dotted, da- shed, and dotted lines, respectively. The solid lines show the total density. aa bb cc dd 200200 200200 150150 150150 100100 100100 5050 5050 00 00 –200–200 –200–200 –100–100 –100–100 00 00 100100 100100 200200 200200 2C2C \|m\|\|m\|–– \|m\|\|m\|–– 1 – \|m\|1 – \|m\|–– \|m\|m \| – \|m\|\| – \|m\|2c2c –– x,x, mm�� x,x, mm�� 00 ��00 n , n n , n jj, a. u . , a. u . n , n n , n jj, a. u . , a. u . Michał Matuszewski, Tristram J. Alexander, and Yuri S. Kivshar 886 Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 8/9 2 2 =0 2 0 2 2 = \| 1 ,C C m m C C C m me e e m m+ρ ρ ⎛ ⎞ + −⎜ ⎟⎜ ⎟ ⎝ ⎠ (12) where = /m NM is the average magnetization and the magnetization of the 2C component 2Cm is a free parame- ter that has to be optimized to obtain the lowest energy state. Table 1. Ground states of spin-1 condensates in homogenous magnetic field. The states 02C + ρ and 0±ρ + ρ correspond to phase separation (see Fig. 1) Condensate Parameter range Ground state Ferromagnetic 2 2 \| \| E c n δ ≤ and = 0m 0ρ 2 < 2 \| \| E c n δ or 0m ≠ PM Polar = 0m 0ρ 2 2 2 E m c n δ ≤ 2C 2 2 1< < 2 2 m E c n δ and 0m ≠ 02C + ρ 2 1 2 E c n δ ≤ and 0m ≠ 0±ρ +ρ The ground states can be determined by comparing energies of the phase separated states with the energies of the homogenous solutions. The form of formulas (12) indi- cates that the phase separation will occur when the energy of the lowest homogenous state is a concave function of magnetization. The results for both polar and ferromagnet- ic condensates are collected in Table 1. In the cases when no phase separation occurs, our results are in agreement with those obtained in [13]. Note that we assumed that the condensate size is much larger than sξ and Bξ . For small condensates, the results of [13] are correct. In the case of high magnetic field strength, one of the Zeeman sub-levels is practically depleted [13] and the condensate becomes effectively two-component. The exis- tence of the 0±ρ + ρ phase in a polar condensate can then be understood within the binary condensate model [23,24]. We note that the experiment reported in Ref. 30, performed in this regime, can be viewed as the first confirmation of phase separation in spin-1 BEC in a homogenous magnetic field. However, the ground state was not achieved, and a multiple domain structure was observed. In Fig. 2 we present the phase diagram of polar conden- sates, obtained both numerically and using analytical for- mulas from Table 1. The ground state profiles for a quasi- 1D condensate were found numerically by solving the 1D version of Eqs. (10) [21] 2 2 0 2 0= ( ) ,i c n n n c t ± ± ± ∂ψ ⎡ ⎤+ + − ψ + ψ ψ⎣ ⎦∂ m m % % % % %% % % % %h L (13) *0 2 0 2 0= ( ) 2 ,i E c n n c t + − + − ∂ψ ⎡ ⎤− δ + + ψ + ψ ψ ψ⎣ ⎦∂ % % % % % %% % % %h L with 2 2 2 0= ( /2 ) /m x c− ∂ ∂ +% %hL , where 0 2 0=4 (2 )/3c a a⊥ω +% h , 2 2 0= 4 ( )/3c a a⊥ω −% h , \| \| =jdx Nψ∫ ∑ % , and ⊥ω is the transverse trapping frequency. We imposed periodic boun- dary conditions on ( )j xψ% and used the parameters corres- ponding to a 23Na BEC containing 4= 5.2·10N atoms con- fined in a transverse trap with frequency 3= 2 ·10⊥ω π . The Fermi radius of the transverse trapping potential is smaller than the spin healing length, and the nonlinear energy scale is much smaller than the transverse trap energy scale, which allows us to reduce the problem to one spatial di- mension [27,31]. The solutions were found numerically using the normalized gradient flow method [32,33], which is able to find a state which minimizes the total energy for given N and M, and fulfills the phase matching condi- tion. The stability of the resulting states was verified using numerical time evolution according to Eqs. (13). The slight discrepancy between numerical and analytical results can be accounted for by the finite size of the condensate (the box size was 10 sξ∼ ), and by the deviation from the as- sumption that the total density is constant (see the discus- sion at the end of Sec. 2). Due to the finite value of the ratio 2 0/c c there is a slight density modulation, as is evi- dent in Fig. 1,c,d. 4. Condensate trapped in a harmonic optical potential The results from the preceding subsection can be veri- fied experimentally in configurations involving toroidal or square-shaped optical traps [34]. However, in most expe- riments on BECs, harmonic potentials are used. The relev- ance of these results is not obvious in the case of harmonic trapping, since the coefficient 2/ ( )E c nδ , one of the main Fig. 2. Ground state phase diagram of the polar condensate. The symbols correspond to numerical data obtained for the parameters of 23Na, with solid triangles representing 2C, open circles 2C+ 0+ρ and open squares 0±ρ + ρ . The solid lines and shading are given by the analytical formulas from Table 1. 0.50.5 1.01.0 1.01.0 0.50.5 00 \|m\|\|m\|–– ��00��E/cE/c nn22 Magnetic domains in spinor Bose–Einstein condensates Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 8/9 887 parameters controlling the condensate properties, varies in space due to the varying total density n . The ground states in a highly elongated harmonic trap, where the parallel part of the potential has the form 2 2( ) = (1/2)V x M xω , are presented in Fig. 3 [33]. We can see that as the magnetic field strength is increased, phase separation occurs and the 0±ρ + ρ domain state is formed. However, in contrast to the previous case, the transition is not sharp, and in particular there is no distinct 02C + ρ phase for any value of the magnetic field. Note that the state in Fig. 3,a is also spatially separated due to different Thomas–Fermi radii of the −ψ and +ψ components; however, this is an example of potential separation, as op- posed to phase separation [24], since it is does not occur in the absence of the potential. On the other hand, Fig. 3,d shows that the components of ferromagnetic condensate are miscible even in the regime of strong magnetic field. In the regions where the wavefunctions overlap, the relative phase is equal to = 0θ for ferromagnetic and =θ π for polar ground states, since these configurations minimize the spin energy (9). The characteristic feature of phase separation in the po- lar BEC is that the = 0m domain tends to be localized in the center of the trap, as shown in Fig. 1,b and c. This can be explained by calculating the total asymmetric energy of the condensate (8), again assuming that the contribution from the intermediate region connecting the domains is negligible, ( ) 2 0 2 2 = ( \| \|) , 2 a c n H d n E d n c N E N n r r ρ ρ± ρ± ≈ −δ + = −δ − + 〈 〉 ∫ ∫ M (14) where n ρ±〈 〉 is the mean condensate density within the area of the ±ρ domain. We see that the energy will be the lowest if this domain is localized in the outer regions, where the condensate density is low. 5. Generation of spin domains We propose a method for generation of spin domains described in the previous section by adiabatic switching of the magnetic field. We start with a condensate with all the atoms in the = 1m sublevel, in the ground state of a har- monic potential. Subsequently, some of the atoms are transferred to the = 1m − component in the rapid adiabatic passage process. The magnetic field is then suddenly switched off. In this way we can obtain a condensate with Fig. 3. Ground state profiles in a harmonic trap potential. Phase separation occurs in the polar 23 Na condensate when the magnetic field strength is increased from = 0.1B G, 2 max/ ( ) = 0.09E c nδ (a) to = 0.12B G, 2 max/ ( ) = 0.13E c nδ (b) and = 0.25B G, 2 max/ ( ) = 0.56E c nδ (c). For comparison, the ground state of a 87Rb condensate is shown in for = 0.2B G, 2 max/ ( ) = 0.41E c nδ − (d). The n+ , 0n , and n− components are depicted by dash-dotted, dashed, and dotted lines, respectively. The solid lines shows the total density. Other parameters are 4= 2.1·10N , = 2 ·10ω π (23Na), = 2 ·7.6ω π (87Rb), 3= 2 ·10⊥ω π and = 0.5m . ––200200 ––100100 00 100100 200200 00 5050 100100 150150 200200 ––200200 ––100100 00 100100 200200 00 5050 100100 150150 200200 ––200200 ––100100 00 100100 200200 x,x, mmµµ x,x, mmµµ 00 5050 100100 150150 200200 ––200200 ––100100 00 100100 200200 x,x, mmµµ x,x, mmµµ 00 5050 100100 150150 200200 aa bb ddcc n , n n , n jj, a. u . , a. u . n , n n , n jj, a. u . , a. u . n , n n , n jj, a. u . , a. u . n , n n , n jj, a. u . , a. u . Michał Matuszewski, Tristram J. Alexander, and Yuri S. Kivshar 888 Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 8/9 arbitrary magnetization in the 2C state, which is the ground state in = 0B , see Fig. 2. Next, we gradually increase the magnetic field strength in an adiabatic process according to the formula final switch = tB B t (15) where = 0t at the beginning of the switching process, switcht is the switching time, and finalB is the desired final value of the magnetic field. The form of Eq. (15) assures that the quadratic Zeeman splitting grows linearly in time. We have confirmed that this condition improves the adia- baticity of the generation process. We present examples of the evolution of the condensate in Fig. 4. The left column shows the time dependence of the atom density in the in- itially unoccupied = 0m component, and the right column shows the final domain profiles. These should be compared with the ground state profiles in Fig. 3. The domains are generated for both the low and high magnetic field cases in times of the order of seconds, as shown in panels (a, d) and (b, e). However, when the switching time is significantly reduced, yielding the process no longer adiabatic, multiple metastable domains are formed as presented in panels (c, f). This picture is in qualitative agreement with the ex- periment [30], where metastable spin domains were formed in nonadiabatic process within 50–100 ms. Fig. 4. Generation of spin domains by switching the magnetic field on. The magnetic field is gradually increased from zero to final val- ue = 0.15B G during = 1t s (a, d), = 0.25B G during = 2t s (b, e), and = 0.25B G during = 0.5t s (c, f). The left column shows the time dependence of the atom density in the initially unoccupied = 0m component, and the right column shows the final domain pro- files. In the last case, corresponding to nonadiabatic switching, multiple domains are formed. Other parameters are the same as in Fig. 3. ––200200 ––200200 ––200200 ––100100 ––100100 ––100100 00 00 00 100100 100100 100100 200200 200200 200200 00 00 00 5050 5050 5050 100100 100100 100100 150150 150150 150150 200200 200200 200200 x, µmx, µm x , µ m x , µ m x , µ m x , µ m x , µ m x , µ m x, µmx, µm x, µmx, µm 100100 100100 100100 00 00 00 –100–100 –100–100 –100–100 n , n n , n , a. u . , a. u . jj n , n n , n , a. u . , a. u . jj n , n n , n , a. u . , a. u . jj 00 200200 400400 600600 800800 10001000 t, mst, ms t, mst, ms t, mst, ms 00 400400 800800 12001200 16001600 20002000 00 100100 200200 300300 400400 500500 aa dd bb ee cc ff Magnetic domains in spinor Bose–Einstein condensates Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 8/9 889 6. Spin domains and dynamical stability Our results presented above show that the domain struc- ture forming in polar condensates is absent in ferromagnet- ic BECs. This may seem to contradict the common under- standing of ferromagnetism and the results of the quenched BEC experiment Ref. 9. The conventional picture of a fer- romagnet involves many domains pointing in various di- rections separated by domain walls. Similar structure has been observed in Ref. 9. However, these cases correspond to the situations when there is an excess kinetic energy present in the system, due to finite temperature or excita- tion of the spatial modes. On the other hand, our study is limited to the ground states at = 0T . It is easy to see from Eq. (9) that in zero magnetic field the ground state of a ferromagnetic BEC will always consist of a single domain with maximum possible value of the spin vector \| \|= 1f , pointing in the same direction at all points in space. How- ever, when the temperature is finite, more domains can be formed each with a different direction of the spin vector. We emphasize that the domain structure of the ground state in polar condensates is very different from the do- mains formed when the kinetic energy is injected in the system as in Ref. 9. The latter constantly appear and disap- pear in a random sequence [9,16,17,21,35,36]. On the con- trary, the ground state domains are stationary and are posi- tioned in the center of the trap. They exist in the lowest- energy state, while the dynamical domains require an amount of kinetic energy to be formed. The ground state domains can be prepared in an adiabatic process, involving adiabatic rf sweep or a slow change of the magnetic field [30,35], while the kinetic domains require a sudden quench [9,35]. The dynamical instability of ferromagnetic condensates that leads to spontaneous formation of spin domains has been investigated theoretically [18,16,35] and observed in experiment [9]. An analogous phenomenon has been pre- dicted recently for polar condensates in presence of mag- netic field [21]. Here we correct the results of Ref. 21, by noting that the 0 = 1ρ state is stable in ferromagnetic con- densates for 2> 2 \| \|E c nδ , and the 2C ( 0 = 0ρ ) state is stable in polar BECs if 2< /2E mδ . Both states become the ground states for these values of parameters. By investigat- ing stability in various ranges of parameters, we are able to formulate a phenomenological law governing the dynami- cal stability of condensates: (i) The only stable state for both polar and ferromagnetic BECs in finite magnetic field is the ground state, as shown in Table 1; (ii) in zero mag- netic field, the same is true for ferromagnetic condensates; However, all stationary states of polar condensates are dy- namically stable in zero magnetic field [16,18,21]. The reason for the stability of polar condensates in vanishing magnetic field case is not yet clear. We note that the polar condensates in weak magnetic field may also be effectively stable on a finite time scale. As shown in Ref. 21, in this latter case the instability growth rate of unstable modes is proportional to the fourth power of the magnetic field strength. The time required for the development of instabil- ity may be much longer than the condensate lifetime [3]. 7. Conclusions We have studied the ground state of a spin-1 BEC in the presence of a homogenous magnetic field with and without an external trapping potential. We have found that without a trapping potential the translational symmetry can be spontaneously broken in polar BEC, with the formation of magnetic domains in the ground state. We have shown that these results may be used to understand the ground state structure in the presence of a trapping potential by map- ping the locally varying density in the trap to the homo- genous state. We have found that, depending on the mag- netic field, the antiferromagnetic BEC ground state in the trap displays pronounced spin domains for a range of poss- ible experimental conditions. 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Magnetic domains in spinor Bose–Einstein condensates

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