Metastable spin textures and Nambu-Goldstone modes of a ferromagnetic spin-1 Bose-Einstein condensate confined in a ring trap
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] D ec Metastable spin textures and Nambu-Goldstone modes of a ferromagnetic spin-1Bose-Einstein condensate confined in a ring trap
Masaya Kunimi
Department of Engineering Science, University of Electro-Communications, Tokyo 182-8585, Japan ∗ (Dated: September 28, 2018)We investigate the metastability of a ferromagnetic spin-1 Bose-Einstein condensate confined ina quasi-one-dimensional rotating ring trap by solving the spin-1 Gross-Pitaevskii equation. We findanalytical solutions that exhibit spin textures. By performing linear stability analysis, it is shownthat the solutions can become metastable states. We also find that the number of Nambu-Goldstonemodes changes at a certain rotation velocity without changing the continuous symmetry of the orderparameter. PACS numbers: 67.85.Fg, 03.75.Mn, 03.75.Lm, 14.80.Va
I. INTRODUCTION
Owing to recent developments in the experimentaltechniques associated with cold atomic gases, Bose-Einstein condensates (BECs) confined in multiply con-nected geometries have been realized [1–13]. Such sys-tems are suitable for investigating the fundamental prop-erties of superfluidity. In fact, many interesting featuresof superfluidity have already been observed, such as per-sistent current [1–4], phase slip and vortex nucleations[7, 8], hysteresis [10], and a current-phase relationship[12].Although the above experiments, except Ref. [4], wereperformed using scalar BECs, BECs with internal degreesof freedom (spinor BECs [14, 15]) confined in simply con-nected geometries have also been realized experimentally[16, 17]. In spinor BECs, there exist various topologi-cal defects [18–28], spin textures [29–33], and Nambu-Goldstone modes (NGMs) [34–39] due to their sponta-neous symmetry breaking.The previous theoretical works regarding multi-component systems (two-component Bose gases [40–44]and spinor BECs [45, 46]) in a ring trap concern the sta-bility of persistent currents. Other important properties,including metastability under external rotation, have alsobeen thoroughly investigated theoretically [47, 48] andexperimentally [7, 8, 10] for scalar BECs. However, acomplete understanding of the metastability of spinorBECs in multiply connected systems under external ro-tation has not yet been established.In this paper, we investigate the properties of spin-1BECs in a rotating ring trap within the framework ofthe mean-field approach. We present analytical solutionsof the spin-1 Gross-Pitaevskii equation (GPE) [34, 35]under a twisted periodic boundary condition. This so-lution exhibits spin textures. By performing linear sta-bility analysis, we show that these solutions can becomemetastable states. Furthermore, we determine the NGMsand find the change in the number of NGMs for a given ∗ E-mail: [email protected] rotational velocity without changing the continuous sym-metry of the order parameter. This change in the numberof NGMs is called a type-I − type-II transition [49]. II. MODEL
We consider N spin-1 bosons confined in a ro-tating ring trap. Within the mean-field approxi-mation, the system can be described by a three-component order parameter (a condensate wave func-tion) Ψ ( r , t ) ≡ [Ψ ( r , t ) , Ψ ( r , t ) , Ψ − ( r , t )] T , where Tdenotes the transpose. For simplicity, we treat the sys-tem as a quasi-one-dimensional torus. We assume thatthe spatial dependence of the order parameter is given by Ψ ( r , t ) ≡ Ψ ( x, t ) / √ S , where S is the cross section of thetorus and x represents the coordinate [50]. The energyfunctional of the system is given by E = Z L dx " ~ M X m (cid:12)(cid:12)(cid:12)(cid:12) ∂ Ψ m ( x, t ) ∂x (cid:12)(cid:12)(cid:12)(cid:12) + c n ( x, t ) + c F ( x, t ) , (1)where M is the mass of the boson; m denotes the mag-netic sublevels and can take the values of 1 ,
0, and − L is the length of the torus; n ( x, t ) ≡ P m | Ψ m ( x, t ) | is theparticle density; F ( x, t ) ≡ P m,n Ψ ∗ m ( x, t )( f ) mn Ψ n ( x, t )is the magnetization density vector; ( f ν ) mn ( ν = x, y, z )are the spin-1 matrices; c ≡ π ~ ( a + 2 a ) / M S and c ≡ π ~ ( a − a ) / M S are the spin-independent andspin-dependent interaction strengths, respectively; and a and a are the s -wave scattering lengths of the spin-0and -2 channels, respectively.The time-dependent GPE for spin-1 bosons [34, 35]is given by the functional derivatives i ~ ∂ Ψ m ( x, t ) /∂t = δE/δ Ψ ∗ m ( x, t ), i ~ ∂∂t Ψ ± ( x, t ) = L ± Ψ ± ( x, t ) + c √ F ∓ ( x, t )Ψ ( x, t ) , (2) i ~ ∂∂t Ψ ( x, t ) = L Ψ ( x, t )+ c √ F + ( x, t )Ψ ( x, t ) + F − ( x, t )Ψ − ( x, t )] , (3) L m ≡ − ~ M ∂ ∂x + c n ( x, t ) + mc F z ( x, t ) , (4)where F ± ( x, t ) ≡ F x ( x, t ) ± iF y ( x, t ). The effects of therotation are described by imposing the twisted periodicboundary condition [51, 52]Ψ m ( x + L, t ) = Ψ m ( x, t ) e iMvL/ ~ , (5)where − v is the rotational velocity of the ring. Thiscorresponds to the boundary condition in the rotatingframe [48]. We note that the twisted boundary conditionis invariant under the transformation v → v + lv , where v ≡ π ~ /M L and l ∈ Z .The Bogoliubov equation for spin-1 bosons [14] can de-rived by the linearizing the GPE around the stationarysolution Ψ m ( x, t ) = e − iµt/ ~ Ψ m ( x ), where µ is the chem-ical potential of the system, which is determined by thetotal number of particles: N = Z L dx X m | Ψ m ( x ) | . (6)SubstitutingΨ m ( x, t ) = e − iµt/ ~ h Ψ m ( x ) + u m ( x ) e − iǫt/ ~ − v ∗ m ( x ) e iǫ ∗ t/ ~ i , (7)into Eqs. (2) and (3) and neglecting the higher-orderterms of u m ( x ) and v m ( x ), we obtain the Bogoliubovequation (cid:20) H − H H ∗ − H ∗ (cid:21) (cid:20) u ( x ) v ( x ) (cid:21) = ǫ (cid:20) u ( x ) v ( x ) (cid:21) , (8) H ≡ H H H H ∗ H H H ∗ H ∗ H , H ≡ H H H H H H H H H , (9) u ( x ) = [ u ( x ) , u ( x ) , u − ( x )] T , (10) v ( x ) = [ v ( x ) , v ( x ) , v − ( x )] T , (11)where ǫ is the excitation energy. Here H ,ij and H ,ij are given by H = − ~ M d dx − µ + 2( c + c ) | Ψ ( x ) | + ( c + c ) | Ψ ( x ) | + ( c − c ) | Ψ − ( x ) | , (12) H = ( c + c )Ψ ∗ ( x )Ψ ( x ) + 2 c Ψ ∗− ( x )Ψ ( x ) , (13) H = ( c − c )Ψ ∗− ( x )Ψ ( x ) , (14) H = ( c + c )Ψ ( x ) , (15) H = ( c + c )Ψ ( x )Ψ ( x ) , (16) H = c Ψ ( x ) + ( c − c )Ψ ( x )Ψ − ( x ) , (17) H = − ~ M d dx − µ + ( c + c ) | Ψ ( x ) | + 2 c | Ψ ( x ) | + ( c + c ) | Ψ − ( x ) | , (18) H = ( c + c )Ψ ∗− ( x )Ψ ( x ) + 2 c Ψ ∗ ( x )Ψ ( x ) , (19) H = c Ψ ( x ) + 2 c Ψ ( x )Ψ − ( x ) , (20) H = ( c + c )Ψ ( x )Ψ − ( x ) , (21) H = − ~ M d dx − µ + ( c − c ) | Ψ ( x ) | + ( c + c ) | Ψ ( x ) | + 2( c + c ) | Ψ − ( x ) | , (22) H = ( c + c )Ψ − ( x ) . (23)We note that H † = H and H T2 = H hold, where † de-notes the hermitian conjugate. The boundary conditionsfor u m ( x ) and v m ( x ) are given by u m ( x + L ) = e + iMvL/ ~ u m ( x ) , (24) v m ( x + L ) = e − iMvL/ ~ v m ( x ) . (25)Throughout this paper, we use the parameters( L, c ) = (96 ξ , − . c ), where ξ ≡ ~ / √ M c n is thehealing length. These parameters correspond to the re-cent ring trap experiment [7] and spin-1 Rb [53]. Ourresults presented below are valid for other parameter re-gions as long as c < c ≫ | c | . III. RESULTS
First, we present trivial plane-wave solutions of theGPE. It can be shown that ferromagnetic plane-wave(FPW) and polar plane-wave (PPW) states Ψ F ( x ) = √ n e iM ( v + W F v ) x/ ~ [1 , , T , (26) Ψ P ( x ) = √ n e iM ( v + W P v ) x/ ~ [0 , , T , (27)are the stationary solutions of the GPE, where W F , W P ∈ Z are winding numbers and n ≡ N/L is the mean-particle density. The velocity dependences of the en-ergy for these states are shown in Fig. 1 [54]. Wefind that the ground state is the ferromagnetic state.The stability of these states can be determined by theBogoliubov excitation spectra. The low-lying excita-tion of the ferromagnetic state is a magnon mode aslong as c ≫ | c | and its expression is given by ǫ MF = ( E - E g ) / ( N | c | n ) v/v ( ± ± ± ( ± ± ± ± ± ± ± ± ± ± ± W F =0W F =1 W F =-1W F =2 W F =-2W P =2 W P =0 W P =1 W P =-1 W P =-2 FIG. 1. (Color online) Velocity dependence of the energyper particle in the rotating frame for L = 96 ξ and c = − . c , where E g /N = ( c + c ) n / W F , W P , and ( W, W ). − M v | v + W F v | + M v /
2. The low-lying magnon modebecomes negative, namely, Landau instability (LI) oc-curs for | v + W F v | > v /
2. The low-lying excitationsof the polar state are also magnon modes (doubly de-generated) and their expressions are given by ǫ MP = − M v | v + W P v | + ( M v / p c n /M v . Dynami-cal instability (DI) [Im( ǫ MP ) = 0] occurs in the polar statefor L/ξ > π p c / | c | .Next, we present a non-trivial solution of the GPE,which is expressed using the following ansatz:Ψ m ( x ) = √ n e iMvx/ ~ e iMW m v x/ ~ φ m , (28)where, W m ∈ Z is the winding number of the com-ponent m and φ m is a complex constant. We assumethat all components of φ m are non-zero. According tothe U(1) × SO(3) symmetry of the system and the re-quirement that the chemical potential be real valued, wecan choose φ and φ to be real and positive and φ − to be real without loss of generality. The { φ m } satisfy P m φ m = 1 due to the total particle number condition.When the winding numbers { W m } satisfy the relations W ≡ W − W = W − − W = 0, the condensate wavefunction (28) becomes a nontrivial solution of the GPE( W = 0 states correspond to trivial plane-wave states). The expressions of { φ m } are given explicitly by φ ± = (cid:12)(cid:12)(cid:12)(cid:12) W ± (cid:18) vv + W (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) s M z /N W ( v/v + W ) , (29) φ = vuut − M z /NW ( v/v + W ) " W (cid:18) vv + W (cid:19) , (30) M z N = 2 W (cid:18) vv + W (cid:19) (cid:20) − g ( v, W, W ) | c | n (cid:21) , (31)where M z ≡ R L dxF z ( x ) = LF z is the magnetizationof the z component ( F z does not depend on x ), φ − ispositive due to Eq. (33), and g ( v, W, W ) is a functiondefined for convenience as g ( v, W, W ) ≡ M (cid:20) W v − ( v + W v ) (cid:21) . (32)The parameter regions where the solution exists are givenby 0 ≤ φ ≤ (cid:12)(cid:12)(cid:12)(cid:12) vv + W (cid:12)(cid:12)(cid:12)(cid:12) ≤ | W | , (cid:18) vv + W (cid:19) ≥ W − | c | n M v . (33)The explicit expressions of the physical quantities suchas the chemical potential, total energy, total momentumof the rotating frame, and local magnetizations become µ = ( c + c ) n + 12 M ( v + W v ) + g ( v, W, W ) , (34) EN = µ −
12 ( c + c ) n − g ( v, W, W ) | c | n , (35) P ≡ − i ~ Z L dx X m (cid:20) Ψ ∗ m ( x ) ddx Ψ m ( x ) − c . c . (cid:21) = N M ( v + W v ) − M W v M z , (36) F x ( x ) = √ n φ ( φ + φ − ) cos (cid:18) πWL x (cid:19) , (37) F y ( x ) = √ n φ ( φ + φ − ) sin (cid:18) πWL x (cid:19) . (38)From the expressions for the magnetization density (37)and (38), this solution [we call it the three-componentplane-wave (TCPW) state] represents the spin texture.The spin rotates | W | times around the ring. We plotthe spin texture in Fig. 2. This texture is similar tothe polar core vortex (see Fig. 2 in Ref. [58]). We notethat similar solutions for infinite systems were reportedin Refs. [31, 59–61].The velocity dependences of the total energy for theTCPW states are shown by the solid red lines in Fig. 1.At v = − W v ± v /
2, the TCPW states with | W | = 1branches appear [see Eq. (33)]. These points are the sameas the points at which LI occurs in the FPW branches due F x F y F z F x F y F z (a) (b) (c) F x F y F z FIG. 2. (Color online) Spin textures for W = − v = − . v , (b) v = 0, and (c) v = 0 . v . to the magnon mode, indicating that the instability ofthe magnon mode in the FPW state triggers the TCPWstates, namely, the transition from the green line to thered line in Fig. 1. For | W | = 3 branches, the TCPWbranches continuously connect the PPW branches. Thesebifurcations occur when the right-hand side of the secondinequality in Eq. (33) becomes positive.To investigate the stability of the TCPW states, wesolve the Bogoliubov equation. From the boundary con-ditions (24) and (25), u ( x ) and v ( x ) can be expanded asa series of plane waves u ( x ) = e + iM ( v + W v − W v f z ) x/ ~ X n e ik n x √ L u ( k n ) , (39) v ( x ) = e − iM ( v + W v − W v f z ) x/ ~ X n e ik n x √ L v ( k n ) , (40)where k n ≡ πn/L is the wave number of the excitationand n ∈ Z . Substituting Eqs. (39) and (40) into Eq. (8)and using the wave function of the TCPW state (28) andthe translational symmetry of the system, we obtain theblock-diagonalized Bogoliubov equation for k space: (cid:20) H +1 ( k n ) − H ( k n ) H ( k n ) − H − ( k n ) (cid:21) (cid:20) u ( k n ) v ( k n ) (cid:21) = ǫ n (cid:20) u ( k n ) v ( k n ) (cid:21) , (41) H ± ( k n ) ≡ H ± ( k n ) H ( k n ) H ( k n ) H ( k n ) H ± ( k n ) H ( k n ) H ( k n ) H ( k n ) H ± ( k n ) , (42) H ( k n ) ≡ H ( k n ) H ( k n ) H ( k n ) H ( k n ) H ( k n ) H ( k n ) H ( k n ) H ( k n ) H ( k n ) , (43) where H ± ,ij ( k n ) and H ,ij ( k n ) are given by H ± ( k n ) = ǫ k n ± ~ k n ( v + W v − W v )+ M (cid:18) W v − W v − v (cid:19) + ( c + c ) n φ − c n φ − , (44) H ( k n ) = ( c + c ) n φ φ + 2 c n φ φ − , (45) H ( k n ) = ( c − c ) n φ φ − , (46) H ( k n ) = ( c + c ) n φ , (47) H ( k n ) = ( c + c ) n φ φ , (48) H ( k n ) = c n φ + ( c − c ) n φ φ − , (49) H ± ( k n ) = ǫ k n ± ~ k n ( v + W v ) + M ( v + W v ) − M ( W v ) + ( c − c ) n φ , (50) H ( k n ) = ( c + c ) n φ φ − + 2 c n φ φ , (51) H ( k n ) = c n φ + 2 c n φ φ − , (52) H ( k n ) = ( c + c ) n φ φ − , (53) H ± ( k n ) = ǫ k n ± ~ k n ( v + W v + W v )+ M (cid:18) W v + W v + v (cid:19) − c n φ + ( c + c ) n φ − , (54) H ( k n ) = ( c + c ) n φ − , (55) ǫ k n ≡ ~ k n M . (56)We note that H ± ( k n ) and H ( k n ) are the real and sym-metric matrices.The velocity dependence of the excited energy is shownin Figs. 3, 4, and 5. We find that the branch for | W | = 1 is a metastable state because the energy ishigher than the FPW branches and all excitation en-ergies ǫ j are real and positive, where ǫ j is the j th ex-cited energy and corresponding eigenvector is denoted by x j ≡ [ u j ( x ) T , v j ( x ) T ] T . We also find that LI or DI oc-curs in the branches for | W | > × SO(3) internalsymmetry [62]; however, the TCPW state breaks thissymmetry. Therefore, it can be expected that the sys-tem has a number of NGMs.First, we determine the number of the NGMs. Accord-ing to recent developments of the theory of the NGMs[63–65], the NGMs can be classified as two differenttypes: type I (unpaired) and type II (paired). The type-I NGM describes the mode related to one generator ofthe broken symmetry Q i . On the other hand, the type-II NGM describes one mode related to a canonical pair ε j / | c | n v/v Type-I − type-II transition FIG. 3. (Color online) Velocity dependence of the excitationenergy ǫ j for ( L, c , W, W ) = (96 ξ , − . c , − , − type-II transition point. of the two broken generators, which is analogous to thequantum mechanics; two variables ˆ x and ˆ p describe onedegrees of freedom because these are the canonical pair;[ˆ x, ˆ p ] = i ~ . The total number of the NGMs can be ob-tained by using the Watanabe-Brauner (WB) matrix [63–65] ρ ij ≡ − iL Z L dx Ψ ( x ) † [ Q i , Q j ] Ψ ( x ) . (57)In the present case, { Q i } is given by the spin matrices f x , f y , and f z and the 3 × I . The totalnumber of NGMs is given by n NGM ≡ n BG − rank( ρ ) / n BG (= 4 in the present case) is the number ofthe broken generators. Here rank( ρ ) / ρ reduces to ρ = 1 L M z − M z , (58)where we used M x = M y = 0 for the TCPW state.Therefore, we obtain the number of the NGM n NGM = 4 −
12 rank( ρ ) = ( M z = 04 for M z = 0 . (59)This result shows that the number of NGMs changesat M z = 0 ( v = − W v [66]). Although the num-ber of NGMs changes if the broken continuous symme-try changes, the broken continuous symmetry does notchange at M z = 0 in the present case. This is called atype-I − type-II transition [49]. We will see what happensat M z = 0 by calculating the wave functions below.Following Ref. [49], the zero-energy eigenstates of theBogoliubov equation (zero modes) originating from thespontaneous symmetry breaking are given by using thebroken generators and the order parameter: x B = (cid:20) Ψ ( x ) Ψ ∗ ( x ) (cid:21) = √ n (cid:20) e + iM ( v + W v − W v f z ) x/ ~ φ e − iM ( v + W v − W v f z ) x/ ~ φ (cid:21) , (60) x z = (cid:20) f z Ψ ( x ) f ∗ z Ψ ∗ ( x ) (cid:21) = √ n (cid:20) e + iM ( v + W v − W v f z ) x/ ~ f z φ e − iM ( v + W v − W v f z ) x/ ~ f z φ (cid:21) , (61) x x = (cid:20) f x Ψ ( x ) f ∗ x Ψ ∗ ( x ) (cid:21) = r n (cid:26) + e + iMW v x/ ~ (cid:20) e + iM ( v + W v − W v f z ) x/ ~ φ + e − iM ( v + W v − W v f z ) x/ ~ φ − (cid:21) + e − iMW v x/ ~ (cid:20) e + iM ( v + W v − W v f z ) x/ ~ φ − e − iM ( v + W v − W v f z ) x/ ~ φ + (cid:21)(cid:27) , (62) x y = (cid:20) f y Ψ ( x ) f ∗ y Ψ ∗ ( x ) (cid:21) = i r n (cid:26) − e + iMW v x/ ~ (cid:20) e + iM ( v + W v − W v f z ) x/ ~ φ + e − iM ( v + W v − W v f z ) x/ ~ φ − (cid:21) + e − iMW v x/ ~ (cid:20) e + iM ( v + W v − W v f z ) x/ ~ φ − e − iM ( v + W v − W v f z ) x/ ~ φ + (cid:21)(cid:27) , (63) φ ≡ [ φ , φ , φ − ] T , (64) φ + ≡ [ φ , φ − , T , (65) φ − ≡ [0 , φ , φ ] T , (66)which represent the global phase transformation and thespin rotation around the x , y , and z axes, respectively. We can show that x x and x y are not independent for M z = 0. To see this, we calculate the following inner -1.0-0.50.00.51.01.52.00.0 0.5 1.0 1.5 2.0 R e ( ε j ) / | c | n v/v | I m ( ε j ) | / | c | n v/v (a)(b) FIG. 4. (Color online) Velocity dependence of the (a) realpart and (b) imaginary part of the excitation spectra for(
L, c , W, W ) = (96 ξ , − . c , − , − product [14, 49, 67]:( x i , x j ) ≡ Z L dx x † i σ x j , (67) σ ≡ (cid:20) + I − I (cid:21) . (68)By direct calculations, we can show that ( x x , x y ) =( x y , x x ) ∗ = iM z and otherwise zero. These results meanthat x B and x z are always orthogonal to all other zero-modes and hence they give type-I NGMs. On the otherhand, x x and x y give one type-II NGM or two type-INGMs. For M z = 0, we can construct the wave functionwith positive norm by taking the linear combination for x x and x y : x ± ≡ ( x x ± i x y ) / √ x + = 1 √ (cid:20) ( f x + if y ) Ψ ( x )( f ∗ x + if ∗ y ) Ψ ∗ ( x ) (cid:21) = √ n e + iMW v x/ ~ (cid:20) e + iM ( v + W v − W v f z ) x/ ~ φ + e − iM ( v + W v − W v f z ) x/ ~ φ − (cid:21) , (69) x − = 1 √ (cid:20) ( f x − if y ) Ψ ( x )( f ∗ x − if ∗ y ) Ψ ∗ ( x ) (cid:21) = √ n e − iMW v x/ ~ (cid:20) e + iM ( v + W v − W v f z ) x/ ~ φ − e − iM ( v + W v − W v f z ) x/ ~ φ + (cid:21) . (70)The norm of x ± is given by ( x ± , x ± ) = ∓ M z . There-fore, for M z > < x − ( x + ) is the physicallymeaningful mode and it gives the type-II NGM. On theother hand, for M z = 0, the following relations hold:( x ± , x ∓ ) = ( x ± , x ± ) = 0. Therefore, x + and x − are thetype-I NGMs for M z = 0.We also determine the wave number of the zero modes.By comparing the expressions of the zero modes with theexpansion of the wave function for the excited states (39)and (40), we obtain the wave number of zero modes. Wefind that x B and x z have zero wave number and x + ( x − )has the wave number k + W ( k − W ). These results can beseen in Fig. 6. We summarize our results for the NGMsin Table I.Here we discuss the discrete symmetry of the system.At the type-I − type-II transition point, the order parame- TABLE I. Summary of the NGMs for the TCPW states.
Type-I ( k n = 0) Type-I ( k n = 0) Type-II ( k n = 0) M z > x B , x z x − M z = 0 x B , x z x + , x − M z < x B , x z x + ter is invariant under the π rotation around the z axis andtime-reversal operation: Ψ ( x ) = e − if z π T Ψ ( x ), where T is the time-reversal operator. This is because the twistedboundary condition (5) reduces to the periodic bound-ary condition at the transition point. According to thissymmetry, both x + and x − become physically meaning-ful modes because these modes have the opposite wavevector. -3.0-2.0-1.00.01.02.03.00.0 0.1 0.2 0.3 0.4 0.5 R e ( ε j ) / | c | n v/v | I m ( ε j ) | / | c | n v/v (a)(b) FIG. 5. (Color online) Velocity dependence of the (a) realpart and (b) imaginary part of the excitation spectra for(
L, c , W, W ) = (96 ξ , − . c , − , +1). Finally, we discuss the applicability of the above resultsto actual experiments. The TCPW states have differentwinding numbers, namely, different angular momenta foreach component. Such states can be prepared throughtwo-photon Raman transitions with circularly polarizedLaguerre-Gaussian and standard Gaussian beams [25].The type-I − type-II transition can be indirectly observedby utilizing the vanishment of the first excited energy(see Fig. 3) because the first excited state, which is amagnetic excitation, is converted to the zero mode at v = − W v , as described above. The magnon excitedenergy and its dispersion relation have been observed ina recent experiment using a magnon contrast interferom-etry technique [39]. By applying this technique to ringtrap experiments, the type-I − type-II transition can beobserved. IV. SUMMARY
We investigated the metastability of the spin texturesand excitations of ferromagnetic spin-1 BECs confined ina rotating ring trap using mean-field theory. We foundanalytical solutions of the GPE (TCPW solutions) thatexhibit spin textures analogous to polar-core vortices. ε n / | c | n k n ξ ε n / | c | n k n ξ (a)(b) FIG. 6. (Color online) Excitation energy as a function of k n for ( L, c , W, W ) = (96 ξ , − . c , − ,
0) at (a) v = 0 (thetype-I − type-II transition point) and (b) v = − . v . Theinsets show the magnified view around k n = 0. The pointsenclosed by blue boxes are zero-modes. By numerically solving the Bogoliubov equation, it wasshown how the TCPW states can become metastablestates. On the basis of these analytical solutions, wedetermined the number of NGMs by using the WB ma-trix [63–65]. We found that the number of the NGMschanges at v = − W v without changing the continuoussymmetry of the order parameter.In future work we hope to study the stability of theTCPW states in the presence of an external potentialthat breaks translational symmetry. An additional goalwould be to perform a many-body calculation for spinorBose gases in a ring trap. ACKNOWLEDGMENTS