Spin-structures of the Bose-Einstein condensates with three kinds of spin-1 atoms

We have performed a quantum mechanic calculation (including solving the coupled Gross-Pitaevskii equations to obtain the spatial wave functions, and diagonalizing the spin-dependent Hamiltonian in the spin-space to obtain the total spin state) together with an analytical analysis based on a classical model. Then, according to the relative orientations of the spins SA, SB and SC of the three species, the spin-structures of the ground state can be classified into two types. In Type-I the three spins are either parallel or anti-parallel to each other, while in Type-II they point to different directions but remain to be coplanar. Moreover, according to the magnitudes of SA, SB and SC, the spin-structures can be further classified into four kinds, namely, p + p + p (all atoms of each species are in singlet-pairs), one species in f (fully polarized) and two species in q (a mixture of polarized atoms and singlet-pairs), two in f and one in q, and f + f + f. Other combinations are not allowed. The scopes of the parameters that supports a specific spin-structure have been specified. A number of spin-structure-transitions have been found. For Type-I, the critical values at which a transition takes place are given by simple analytical formulae, therefore these values can be predict.


Hamiltonian and the Ground State
Let the mixture of three kinds of spin-1 atoms be trapped by isotropic and harmonic species-dependent potentials ω m r J J 1 2 Note that, in the ground state (g.s.), every particles of a kind will condense to a spatial state (say, ϕ J ) which is most favorable for binding. Let Ξ denotes a normalized total spin-state. Then the g.s. can be in general written as ϑ is complete for all-symmetric spin-states. Let be a total spin-state of the mixture, in which S AB and S C are further coupled to S and M. It is recalled that S A , S B , S C , S and M are good quantum numbers, but S AB is not. Nonetheless, the states S S S (( ) ) The coupled gross-pitaevskii equations and the spatial wave functions. For the Hamiltonian given in Eq. (1), the associated CGP equations for ϕ A to ϕ C are 11,28 h . Under the TFA where the terms of kinetic energy have been neglected, in a domain where all the J ϕ are nonzero, the CGP can be written in a matrix form as Once the parameters are given, the six ′ Y n n ( ) (n n ′ ≠ ) are known, while the six ′ Z n n ( ) have not yet. This formal solution with ϕ = 0 n is denoted as Form II n , where the subscript specifies the vanishing wave function. When one and only one of the wave functions is nonzero in a domain (say, ϕ ≠ 0 J ), it must have the unique form as Obviously, J ϕ in this form must descend with r. This form is denoted as Form I J , where the subscript specifies the survived wave function.
If a wave function (say, ϕ J ) is nonzero in a domain but becomes zero when r r o ≥ , then a downward form-transition (say, from Form III to II j ) will occur at r o . Whereas if ϕ J is zero in a domain but emerges from zero when ≥ r r o , then an upward form-transition (say, from Form II J to III) will occur at r o . r o appears as the boundary separating the two connected domains, each supports a specific form. In this way, the formal solutions serve as the building blocks, and they will link up continuously to form an entire solution of the CGP. They must be continuous at the boundary because the two sets of wave functions by the two sides of the boundary satisfy exactly the same set of nonlinear equations at the boundary.
Recall that there are three unknowns A ε , ε B and ε C contained in the formal solutions. Taking the three addi- into account, the three unknowns can be obtained. Then, under the TFA, the CGP is completely solved. The details are shown below.
The spatial wave functions. The spin-structures in multi-species BEC is caused by the inter-species interactions. Obviously, they would act more effectively when the three species are distributed closer to each other. Therefore, in the following examples, we take the miscible states into account, in which all the three species have nonzero distribution at the center (r = 0). An example is given in Fig. 1, where the wave functions in zone I to IV are in Form III, Form II 1 , Form I 3 , and empty, respectively. www.nature.com/scientificreports www.nature.com/scientificreports/ For this example, we know that the boundary r a (at which 0 (14)). They give the outmost boundary of ϕ A , ϕ B and C ϕ , respectively. Taking the normalization into account, we obtain Since Z A , Z B , and Z C have been obtained as given above, A ε and ε B can be further obtained via Eq. (10). Then, the entire solution of the CGP together with the chemical potentials are completely known.
Nonetheless, the realization of the miscible state is based on a number of assumptions. First, it is assumed that all the wave functions are nonzero at the center, thus Z 0 A > , Z 0 B > , and Z 0 C > are required. Second, ϕ A is assumed to descend with r in zone I and ϕ B is assumed to descend with r in zone II, thus Y 0 A > and Y 0 > ). Thus, the type as shown in Fig. 1 can be realized only if the parameters are given inside a specific scope. A comprehensive discussion on the scope of parameters for each spatial type of solution is the base for obtaining the phase-diagrams, but this is beyond the scope of this paper.
The total spin-state. Making use of the spin basis-state, we define a set of basis-states for the g.s. as 0 01 CC and γ = . 0 08 where the subscript S denotes a specific set of the good quantum numbers S S S S ( ) A B C . When a magnetic field is not applied, the label M can be neglected. Accordingly, a candidate of the g.s. can be expanded as When the values of the good quantum numbers in S are presumed, the coefficients d S AB can be obtained via a diagonalization of H spin in the space expanded by ψ S S , AB . The matrix elements are where the summation of J covers A, B and C, , and so on. Carrying out the diagonalization of S ′ H S S , AB AB , the lowest eigenstate is Ψ S and the corresponding energy is denoted as E S . Let the presumed values in S be varied. If S E arrives at its minimum when = o S S, then the g.s.
To extract information on spin-structure from Ψ gs , we calculate the averaged angle between the two spins S A and S B as J is the operators for the total spin of the J-species. Similarly, we have Examples are given below.

Classical model (Type-I).
Neglecting all spin-independent terms, the spin-dependent energy of the g.s. can be written asˆ∑ Based on Eq. (27), we propose a classical model to facilitate qualitative analysis. In this model, the total spin of the J-species is considered as a vector S J → with norm S J ranging from 0 to N J , JJ θ + is the angle between → S J and → + S J . The magnitudes and orientations of the three → S J together describe an intuitive picture of the spin-structure. The classical analog of E spin is defined as 10:2727 | https://doi.org/10.1038/s41598-020-59540-z www.nature.com/scientificreports www.nature.com/scientificreports/ The effect of the inter-species force is embodied by will be lying along the same direction. Whereas when Q 0 JJ > + (repulsive), along opposite directions. Note that two of the spins will define a plane and will pull the third lying on the same plane. Therefore, the spin-structures of the 3-species condensates are assumed to be coplanar (this assumption will be checked later). Thus, in what follows, 2 A B C BC CA θ θ . When these variables lead to the minimum of E spin M , they specify a coplanar spin-structure of the g.s. In order to find out the minimum, we calculate the partial derivatives of E spin M . They are given in the appendix. . These two cases belong to the Type-I.
For Type-I When S J of a species is given at 0, N J and in between, let the corresponding phase of the J species be denoted by p, f and q, respectively. Let p be a point with the coordinates S S S ( , , ) A B C bound by a cuboid as shown in Fig. 3. Let p g.s. be the point where E spin M arrives at its minimum. There are the following possibilities.
The case p g.s. is located inside the cuboid (i.e., not on the surfaces of the rectangle). In this case < < are necessary to hold. This leads to a set of homogeneous However, the matrix of this set is in general not singular. Therefore, there is no nonzero solution. Even, for a specific choice of the parameters, the matrix is singular, the nonzero solution can be multiplied by a variable common number ς. One can see that E spin M varies with ς monotonically. In order to minimize E spin M , ς should be given either in its upper or lower limit but not inside. Thus, p g.s. cannot locate inside the cuboid. It implies that the three species cannot all be in the q-phase.
Let a rectangle on the surface of the cuboid be denoted as p p p p 1 4 8 5 , etc. (refer to Fig. 3). There are six rectangles classified into two kinds. The three containing the common vertex P 1 belong to the first kind, the other three containing p 7 belong to the second kind. www.nature.com/scientificreports www.nature.com/scientificreports/ The case p g.s. is located on a rectangle of the first kind. If this case is realistic, the g.s. would have at least one species in p-phase. For instance, if p g.s. were located on p p p p 1 4 8 5 With similar arguments, p g.s. cannot be located on p p p p 1 5 6 2 and p p p p 1 2 3 4 as well, but it can be located at the point P 1 . It implies that the case with one or two species in P-phase is prohibited, while all species in P is possible. This fact coincides with the finding found in 2-species condensates, in which the P-phase is extremely fragile when it is accompanied by an f or a q. Therefore, the P + f or P + q structures do not exist, but the P + P structure is allowed [20][21][22][23] .
With the prohibition of the above two cases, p g.s. can only access P 1 and the three rectangles of the second kind, but those edges each being a common edge of two rectangles belonging to two kinds should be excluded.  Table 2. In these tables, we have defined

Spin-structure Constraint
When all species are ferromagnetic in nature (i.e., all Q are required. This leads to the constraint listed in the second row of Table 1. This structure can be realized only if Q 0 C > (i.e., the C-species is polar in nature), whereas Q A and Q B can be negative or weakly positive. If they are positive and large, the inter-species interaction should be even stronger to ensure that the inequalities hold. The equality for S C implies that the intra-force and the inter-force imposed on the C-atoms arrive at a balance. The energy E ffq M is given in Table 2.

The structures f//q//f (B-species in q) and q//f//f (A-species in q)
can be similarly discussed. These three together are called the double-f-structure (double-f-str).
The case p g.s. is located in the interior of the rectangles of the second kind. When p g.s. is in the interior of p p p p  Table 1. This structure can be realized only if both the Band C-species are polar in nature, whereas Q A can be negative or weakly positive. Besides, the condition > | | Q Q Q B C BC 2 is necessary. One can prove that the constraint listed in the third row leads to β < 0 M is a product of a positive value and β ABC . Thus, β < 0 ABC is a necessary condition for the f//q//q structure. The structures q//f//q and q//q//f can be similarly discussed. The three together are called the single-f-str.
The case p g.s. is located at p 1 . When all the three species are polar in nature ( is positive. If the inter-species forces are zero or weak, this positive term would be dominant. This leads to 0 ABC β ≥ . In this case all the species are in p and the structure is therefore denoted as increases, ABC β will decrease. Once β ABC becomes zero, the energy of the single-f-str will be lower than E ppp M (refer to Table 2), and the transition p + p + p → single-f-str will occur. With these in mind, the possible spin-strs of the g.s. are p + p + p, single-f-str, double-f-str, and f//f//f depending on the parameters. Spin-structure transition. We aim at the effect caused by the variation of the inter-species forces. Note that , the first transition is from p + p + p to a single-f-str as mentioned above. Recall that the single-f-str must have 0 ABC β ≤ while the p + p + p has β > 0 ABC , therefore 0 ABC β = is the critical point of transition. One can prove that the two sets of constraint for two different single-f-strs (say, f//q//q and q//f//q) cannot both be satisfied Otherwise, two contradicting inequalities 0 ABC β > and 0 ABC β ≤ would both hold. This fact implies that, for a given set of parameters, only one of the three single-f-strs can survive. Therefore, p + p + p can only transit to a specific single-f-str depending on the parameters. Besides, the transitions among the three single-f-strs (say, f//q//q q → //f//q) are prohibited.
When | | increase further, a q-phase can be changed to a f-phase. Therefore, the single-f-str → double-f-str transition will occur (as shown below). One can prove that the three sets of constraints for the three double-f-strs do not compromise with each other as before. Thus, a single-f-str can only transit to a specific double-f-str

Model Energy
In Fig. 4 = . q 0 405 2 as listed in Table 3. The dimensionless parameters are given as N 120  Table 3 (except q 1 , but still acceptable). Thus, the related analytical formulae Eqs. (33, 34, 35) are useful for qualitative evaluation. For other chains of transition, the analytical formulae for the critical points can be similarly obtained.

Classical model (Type-II).
When all + Q JJ are positive (Fig. 2c) or only one of them is positive (Fig. 2d,  } JJ are considered as constants. Thus, the situation is the same as for Type-I. With the same arguments as those for Type-I, we deduce that p g.s. can only access p 1 and the three rectangles of the second kind, but those edges each being a common edge of two rectangles belonging to two kinds should be excluded.  are required to be zero. These lead to (refer to Eqs. (48) and (49) The two angles obtained in this way should ensure that the two second order derivatives given in Eqs. (46)  and the subspace of parameters that supports this structure are also known.
When p g s . . is located in the interior of p p 7 3 as an example, = S N A A , S N B B = , and the structure is denoted as f + f + q. The constraints appear as (refer to the second row of Table 1): The angles are subjected to the two coupled equations (refer to Eqs. (48) and (49)) depends on the angles. Solving these equations (say, numerically), we can obtain BC θ and θ CA . Then, the energy + + E f f q and the subspace of parameters that supports this structure can be known as before. The cases of f + q + f and q + f + f can be similarly discussed.
When . . p g s is located in the interiors of the rectangles of the second kind, say, p p p p 7 6 2 3 , then S N A A = and the structure is denoted as f + q + q. The constraint imposed on this structure is listed in the third row of Table 1 Table 4. For the structure f + f + f of the Type-II., the angles (in degrees) between the spins against the increase of Q CA . The data for θ + JJ are from the model (refer to Eqs. (37) and (38)), those for JJ θ + are from the diagonalization of H spin (refer to Eqs. (24), (25) and (26)). The parameters are given as  Table 4 demonstrates that the results given by Eqs. (24), (25) and (26) are quite accurate. In particular, the sum of the three θ + { } JJ given in the last column is very close to 2π. This supports the assumption of coplanar structure.
Final remarks. Features of the spin-structures of 3-species condensates with spin-1 atoms have been extracted from a model and have been checked via a QM calculation. Note that the effect of the spatial wave function is embodied in the factors ∫ ϕ r d J 4 and ∫ ϕ ϕ included in Q J and + Q JJ , respectively. Since we do not aim at specific kinds of atoms, they are just considered as parameters to avoid the solving of the CGP (of course, this step is necessary when specific species are aimed). The results from the model are found to be consistent with those from the QM calculation. In summary: • The structures can be described by the norms of the three spins S { } J and the average angles θ + { } JJ between them. When the three species are polar in nature (i.e., all > c 0

J2
) and the inter-forces are weak, the mixture is in the p + p + p phase.
• The spin-structures not in p + p + p are all coplanar. They can be first classified according to the relative orientations of S { } J as intuitively shown in Fig. 2. The case that all inter-forces are attractive (i.e., all < ′ c 0 JJ 2 ) is shown in Fig. 2a, only one is attractive (say, c 0 AB2 < ,) in Fig. 2b, all are repulsive (all c 0 JJ 2 > ′ ) in Fig. 2c, and only one is repulsive (say, > c 0 AB2 ) in Fig. 2d. • The spin-structures can be further classified according to the norms of the spin. In addition to p + p + p, there are other three structures, namely, the singlef -str (where one species is in f ), the doublef -str (two species in f ), and the f + f + f (all in f ). Note that the single-p-str, the double-p-str, and the q + q + q do not exist.
Thus, p and f (or p and q) cannot coexist, just as found in 2-species BEC. If not in p + p + p, at least a species must be fully polarized, also similar to 2-species cases. • Starting from the p + p + p, when | | ′ c JJ 2 increases, more species will tend to be in f -phase. Therefore, a chain of phase-transitions p+p + p f → + q + → q f + f + → q f+ f + f will occur. In the parameter space, there are a number of critical surfaces. When the point (representing a set of parameters) vary and pass through one of the surfaces, a transition will occur. For Type-I (Fig. 2a,b) the equations describing the surfaces have been quite accurately obtained (refer to Eqs. (33), (34) and (35)). Thus, the critical points at which the transitions occur can be predicted. Moreover, the analytical formulae demonstrate the competition among contradicting physical factors, thereby the inherent physics could be understood better. For Type-II (Fig. 2c,d), analytical analysis based on the model becomes complicated. Nonetheless, the results from the model have been checked to be also valid.
• The spin-structures found above might also appear in K-species BEC ( >