A Complete Study of the Ground State Phase
Diagrams of Spin-1 Bose-Einstein Condensates in a
Magnetic Field via Continuation Methods
Jen-Hao Chena, I-Liang Chernb,c,∗, Weichung Wangc
a
Department of Applied Mathematics, National Hsinchu University of Education, Hsinchu 300, Taiwan
b
Department of Mathematics, National Central University, Jhongli 32001, Taiwan
c
Institute of Applied Mathematical Sciences, National Taiwan University, Taipei 106, Taiwan
Abstract
We present a complete investigation of the ground state patterns and phase diagrams of the spin-1 Bose-Einstein condensates (BEC) confined in a harmonic or box potential under the influence of a homogeneous magnetic field. A pseudo-arclength continuation method with parameter switching technique is developed to study the BEC systems numerically. The continuation process is performed on the parameter space consisting of the independent interaction, spin-exchange interaction and the quadratic Zeeman (QZ) energy parameters. In the first stage of the parameter switching process, we fix the QZ energy term to be zero and vary the interaction parameters from zero to the desired physical values. Next, we fix the interaction parameters and vary the QZ energy parameter in both positive and negative regions. Two types of phase transitions are found, as we vary the QZ parameter. The first type is a transition from a two-component (2C) state to a three-component (3C) state. The second type is a symmetry breaking in the 3C state. Then, a phase separation of the spin components follows. In the semi-classical regime, we find that these two phase transition curves are gradually merged.
Keywords: Spin-1 Bose-Einstein condensate, continuation method, ground
state, quadratic Zeeman effect, symmetry breaking, phase transition, phase separation, phase diagram.
1. Introduction
Bose-Einstein condensates (BEC) with spin degree of freedom, called spinor BECs, have been achieved experimentally and attracted considerable interest
∗Corresponding author
Email addresses: jhchen@mail.nhcue.edu.tw(Jen-Hao Chen), chern@math.ntu.edu.tw (I-Liang Chern), wwang@ntu.edu.tw (Weichung Wang)
[1, 2, 3, 4, 5] recently. By trapping all Zeeman states of an atomic species in an optical trap, these spinor BEC possess a wealth of phenomena not found in single-component condensates, including spin vortex [6], spin textures [7, 8], do-main wall structure [9, 10], and spontaneous demagnetization dynamics [11, 12]. It is well known that the sign of atomic spin-exchange interaction determines the ground state property of spin-1 condensate, and leads to different quantum phases [3, 13, 14]. Moreover, under an applied external magnetic field, the spon-taneous symmetry breaking in a87Rb spinor BEC has recently been observed
[7]. The symmetry breaking phase transition in the ground state of the spin-1 antiferromagnetic BEC was also experimentally studied [15]. These experiments have opened up intriguing possibilities for exploring the ground state structure of spin-1 BEC under the effect between the spin-dependent interaction and the external magnetic field.
In the presence of an external magnetic field, the spin-1 BEC is subjected to the Zeeman effect [1, 16, 17, 18]. While the linear Zeeman (LZ) term has no in-fluence on the system due to conservation of the magnetization [17], the ground state and its phase diagram of the spin-1 BEC in a magnetic field are profoundly affected by the quadratic Zeeman (QZ) term which describes the Zeeman energy difference in a spin-flip collision [1]. The competition between the QZ effect and the spin-exchange interaction results in a rich variety of ground state patterns. Numerous studies on the spin-1 BEC under positive QZ effect have been re-ported, including spin-domain formation [19, 20], and phase separation [17, 18]. Recently, some experiments have been performed to observe the dynamics of the spin-1 condensates of sodium atoms by using an off-resonant microwave field to rapidly switch QZ effect from positive to negative [21, 22]. This motivates us to numerically investigate the ground state patterns and phase transitions in whole range (positive and negative) of QZ effect.
The ground state structure of the spin-1 condensate in the absence of an external magnetic field has been well examined both theoretically [23, 24, 25] and numerically [26, 14, 27]. The essential feature of these works is that, in ferromagnetic systems the ground state contains all three components which are constant multiples of a scalar wave function, while the ground state contains no zero-component in antiferromagnetic systems. Moreover, no bifurcation is found on the ground state solution curve as varying the spin-independent and spin-exchange interaction parameters.
For the spin-1 BEC in the presence of a homogeneous magnetic field, Lim et al. [28] proposed a numerical method based on the normalized gradient flow to compute the ground state solution of the system. They also numerically showed that for antiferromagnetic BEC in the presence of magnetic field, there exists a critical magnetization which is the transition between two- and three-component states. In [17], the authors theoretically predicted the existence of symmetry breaking phase transition of the antiferromagnetic system without the trapping potential. Matuszewski [18] also pointed out that in the case of harmonic trap-ping potentials an antiferromagnetic system can possess three distinct ground state patterns, while a ferromagnetic system possess two. However, the asym-metric solutions were drawn only schematically in their paper. In addition, the
phase diagram in the negative QZ effect regime has not been studied before. In this work, we aim to provide a complete investigation on the ground state patterns and the phase diagram of the spin-1 BECs confined in a harmonic or box trap with positive to negative QZ effect. Based on the pseudo-arclength continuation method (PACM) [29, 30, 31], we develop a numerical scheme to study these problems. First, together with the characteristics of the ground state of the spin-1 condensate without magnetic field [14], we trace the ground state solution curve varied by the spin-independent and spin-exchange coupling constants. We next treat the QZ effect as a continuation parameter, and take the target solution under zero magnetic field as the starting state to trace the ground state solution curve subject to nonzero magnetic field. We also detect the bifurcation point in the continuation process. The results reveal a rich variety phase transition phenomena in which symmetric or asymmetric ground states are observed. These ground states also show the miscibility or immiscibility in the three hyperfine components, and possess domain wall structures. Finally, we study the phase diagrams of antiferromagnetic systems in the semi-classical regime, in which the determination of the phase transitions is a demanding task due to the nearness of the bifurcation points. In the limiting case (zero kinetic energy term), the above two phase transition curves merge. To the best of our knowledge, some results have never been reported before.
The paper is organized as follows. In the next section, we introduce the model for the spin-1 BEC subject to an external magnetic field. In Section 3, we describe the numerical scheme based on the PACM to explore how the QZ effect and the spin-exchange interaction affect the ground state structures of the systems. In Section 4, we present the numerical results of the spin-1 BEC confined in the harmonic or box trap. The ferromagnetic or antiferromagnetic condensates with positive or negative QZ effects will be considered. Finally, conclusion of the paper is given in Section 5.
2. The Spin-1 BEC Models
At sufficiently low temperature, spin-1 atomic condensates in a nonzero ho-mogeneous magnetic field are described by the following dimensionless Hamil-tonian [16, 32, 18]: E[Ψ] = Z R3 " 1 X j=−1 ψj∗ −∇ 2 2 +Vext ψj+ 1 2gn|Ψ| 4+1 2gs(Ψ ∗SΨ)2 −pΨ∗SzΨ+qΨ∗Sz2Ψ # dr. (1) Here, Ψ(r) = (ψ1(r), ψ0(r), ψ−1(r))⊤ is the vectorial order parameter
corre-sponding to three hyperfine sublevels of the spin, mF = 1, 0, −1, at position
r = (x, y, z). The Ψ∗ denotes the conjugate transpose of Ψ. The term V ext
represents the external trapping potential. The nonlinear interaction among spin-1 condensate atoms is characterized by the spin-independent parameter gn
interaction is attractive. For gn >0 the spin-independent interaction is
repul-sive. The system is ferromagnetic if gs<0 and antiferromagnetic if gs>0. The
terms p and q are LZ and QZ energy parameters, respectively. These two pa-rameters are key factors in determining the ground state patterns of the spin-1 BECs in a magnetic field. The term S = (Sx, Sy, Sz) is the spin-1 Pauli operator
[3, 4] and Sx= 1 √ 2 0 1 0 1 0 1 0 1 0 , Sy = i √ 2 0 −1 0 1 0 −1 0 1 0 , Sz= 1 0 0 0 0 0 0 0 −1 . (2) Due to the elastic atomic collisions characterized by gn and gs, we have two
conserved quantities. One is the normalization of the wave functions
N = Z R3 |ψ1|2+ |ψ0|2+ |ψ−1|2 dr= 1, (3)
and the other is the total magnetization
M = Z R3 |ψ1|2− |ψ−1|2 dr, (4)
with −1 ≤ M ≤ 1. For convenience, we denote nj = |ψj|2as the density of each
spin component, n = n1+ n0+ n1 as the total density, and Nj =
R
R3njdr as
the corresponding particle number.
By minimizing the mean field energy functional (1) subject to (3) and (4), we arrive at a set of three coupled Gross-Pitaevskii equations (CGPE)
(µ + λ)ψ1 = L + p + q + gs(n1+ n0− n−1)ψ1+ gsψ¯−1ψ02, µψ0 = L + gs(n1+ n−1)ψ0+ 2gsψ−1ψ¯0ψ1, (µ − λ)ψ−1 = L − p + q + gs(n0+ n−1− n1)ψ−1+ gsψ¯1ψ02, (5)
where L = −∇22+Vext(r)+gnnis the spin-independent part of the Hamiltonian.
The Lagrange multipliers µ and λ corresponding to the constraints (3) and (4) are interpreted respectively as the chemical and magnetic potentials of the spin-1 BEC.
2.1. Settings for the numerical experiments
In this work, we consider the particular CGPE that the LZ effect is neglected and the wave functions have constant phases. For the Zeeman effect, it is well known that a spin-1 BEC in a magnetic field is subjected to the Zeeman effect. However, the LZ effect term p can be neglected due to conservation of the total magnetization in the system [7, 17, 15]. The p term can be absorbed into the magnetic potentials λ. We thus investigate the ground state structures of the spin-1 BECs under a variation of the QZ effect.
For the phases considerations, we write the wave functions ψj = ujeiθj,
where θj are constant absolute phases of the three hyperfine components. From
|∇ψj|2 = |∇uj|2+ u2j|∇θj|2, we see that a wave function with constant phase
has least kinetic energy. In this case, the spin-exchange Hamiltonian becomes 1 2gs(Ψ ∗SΨ)2= 1 2gs(n1− n−1) 2+ 2n 0(n1+ n−1+ 2u1u−1cos∆θ), (6)
where ∆θ = θ1+ θ−1− 2θ0 is called the relative phase. To achieve minimal
spin-exchange interaction, ∆θ must be taken as 0 or π. The former is called the phase-matched (PM) state, and the latter refers to the antipahse-matched (APM) state. The PM state is energetically favorable for the ferromagnetic condensate, while the APM state is for the antiferromagnetic case [16, 19, 17, 20].
Under the above considerations, the CGPE can be rewritten as (µ + λ)u1 = Lu1+ qu1+ gsn0(u1− σu−1) + u1(n1− n−1) µu0 = Lu0+ gsu0(u1− σu−1)2 (µ − λ)u−1 = Lu−1+ qu−1+ gsn0(u−1− σu1) + u−1(n−1− n1), (7) where σ = sign(gs). We note that uj’s are real-valued functions. It has been
shown that it is positive or identical to zero [25]. Since the term pM plays no role in the classification of ground states, we subtract them from the energy functional in our calculation. The corresponding reduced energy functional is
˜ E[Ψ] = Z R3 " 1 X j=−1 uj −∇ 2 2 + Vext uj +gn 2 n21+ n20+ n2−1+ 2n1n−1+ 2n0(n1+ n−1) +gs 2 (n1− n−1)2+ 2n0(u1− σu−1)2 + q(n1+ n−1) # dr. (8)
In short, we intend to explore the ground state patterns and phase diagrams of the spin-1 BECs in nonzero magnetic field by means of the CGPE (7) with two conservations (3) and (4). We also compute the corresponding energies via reduced energy functional (8) to confirm the validity of the ground states.
In particular, we investigate both ferromagnetic condensates87Rb (g
n >0
and gs <0) and antiferromagnetic condensates 23Na (gn >0 and gs >0) by
assuming the toal number of the cold atoms is 104. The spin component u j
is regarded as vanished when the normalized particle number Nj is less than
10−4. We assume a strong transverse trapping frequency which results in an
anisotropic cigar-shaped spin-1 condensate. That is, we assume Vext(x) = V0x2.
The ground state solutions for a quasi one-dimensional (1D) condensates are thus investigated by solving the 1D version of (7). Here, we take V0 = 0.5
for the case of the harmonic trapping potential, and V0 = 0 for box trapping
potential, i.e.,
Vext(x) =
(
0 when − L ≤ x ≤ L,
We then use the standard central finite difference method to approximate the model problem with the mesh size h = 0.05 in all computations. We terminate the Newton correction in the continuation process when the relative residual in the 2-norm is less than 10−12.
2.2. Remarks on ground states
First, for box trapping potential, the ground state is expected to be ho-mogeneous away from boundaries or domain walls. They can be computed by minimizing the energy functional without the kinetic energy term. Such approx-imation is called the Thomas-Fermi approxapprox-imation. The corresponding ground states are either a pure state or their combinations. The latter is a combination of non-overlapping pure states separated by domain walls [17, 18]. The pure states are listed below.
• A nematic state (NS) is a state in which all the atoms are in mF = 0
component, that is (0, u0,0).
• A magnetized state (MS) is a state in which all the atoms are in mF = +1
or −1 component, that is (u1,0, 0) or (0, 0, u−1).
• A two-component (2C) state is a state in which mF = ±1 components
are populated and miscible. This usually happens in antiferromagnetic systems. For ferromagnetic systems, however, mF = 1, 0 can be miscible.
Such state (u1, u0,0) is called the 2C{1,0}to distinguish from the previous
2C{1,−1}.
• A three-component (3C) state is a state in which all spin components are populated and miscible. Note that the 3C state is called the PM state if ∆θ = 0 and the APM state if ∆θ = π.
Second, we summarize some features of the ground states of the spin-1 with-out magnetic field as follows (see [14, 25] for more details).
• For the ferromagnetic system (gs < 0), the ground state is a constant
multiple of a single wave function. That is, Ψ = (γ1, γ0, γ−1)φ, where
γi ≥ 0 and φ is a scalar function. Such a solution is called single mode
approximation (SMA) in physics literatures. Further, the ground state preserves the population of the three components with
N3C= (1 − N0+ M 2 , 1 − M2 2 , 1 − N0− M 2 ), (10)
no matter how the strength of the coupling interactions change. (10) is obtained from the two conservation laws and minimizing the spin-exchange interaction energy. Recently, Lin and Chern [25] also theoretically proved the validity of the single-mode approximation for the spin-1 ferromagnetic BEC, and Bao et al. [36] applied this SMA ansatz to shorten the ground state computation.
• For the antiferromagnetic system (gs > 0), the ground state is a
two-component (2C) state, where ψ0 ≡ 0. The population of each spin
com-ponent always satisfies
N2C= (
1 + M 2 ,0,
1 − M
2 ). (11)
Finally, we have concluded that in the absence of the magnetic field, the ground state is a three-component (3C) solution (Nj 6= 0) if gs < 0 and is a
two-component (2C) solution (N0= 0) if gs>0.
3. The Parameter Switching Continuation Methods
The discoveries presented in Section 4 are attributed by the continuation methods equipped with the parameter switching technique. Without this pa-rameter switching technique, it would be non-trivial, if not impossible, to explore the new findings regarding the ground state phase diagrams of the spin-1 BECs in a magnetic field.
This particular continuation scheme is proposed on top of the pseudo-arclength continuation method [14, 29, 30, 31]. In the first stage, we employ the contin-uation algorithm proposed in [14] to trace the ground state solution curve by setting q = 0 and treating both gn and gs as continuation parameters. The
target point is the ground state of the spin-1 BEC in the absence of an external magnetic field. In the second stage, the solution curve starting from the target point in the previous stage is traced by treating q as continuation parameter, while gn and gs are fixed. In the study of the phase transitions, we detect the
bifurcation points in the second stage.
We elaborate the continuation scheme in the following sections. For conve-nience, the bold face letters or symbols are used to represent matrices or vectors. The approximations of wave functions ujand densities nj at grid points are
de-noted as uj and nj, respectively.
3.1. Pseudo-arclength continuation method (PACM)
Continuation methods are reliable and powerful tools for computing mul-tiform solutions of a system of nonlinear equations involving one or more pa-rameters. Various algorithms based on the continuation method have been successful in solving some challenging problems [31, 34, 14, 35, 27]. Recently, some numerical methods based on the gradient flow with discrete normalization (GFDN) have been proposed for computing ground states of spin-1 BEC systems [26, 28, 36]. The GFDN mainly computes the ground states with fixed physical parameters, while the continuation methods can be used to study not only the ground state patterns but also the bifurcation diagrams on the parameter space.
By letting u = (u⊤
1, u⊤0, u⊤−1, µ, λ)⊤ ∈ R3N +2, we write the discrete form of
the CGPE (7) with the constraints (3) and (4) as the following parameterized nonlinear equation system
where G : R3N +2× R → R3N +2. In particular, τ is the continuation parameter
depending on the problem under consideration and is incorporated into (12) by setting
gn= ¯gnτ, gs= ¯gsτ, or q = ¯qτ,
where ¯gn, ¯gs and ¯q are the desired physical constants and 0 ≤ τ ≤ 1. By
parametrizing the solution set (u, τ ) via arc-length in terms of s, we define the solution curve of (12) as
C = {x(s) = (u(s)⊤, τ(s))⊤ | G(x(s)) = 0, s ∈ R}. (13)
Note that the choices of the initial points of the solution curves are crucial. We perform the PACM by the predictor-corrector procedure based on Euler’s and Newton’s methods. Let xk = x(sk) = [u⊤(sk), τ (sk)]⊤be an approximating
point at the kth iteration. The Euler predictor is used to predict the next point
x(0)k+1= xk+ hk˙xk, (14)
where hk is a suitable step length and ˙xk = [( ˙uk)⊤,˙τk]⊤ is the unit tangent
vector to the solution curve at the current point. To obtain ˙xk, we solve the
linear system
DG(xk) ˙xk= 0,
where
DG(xk) = [Gu(xk), Gτ(xk)] ∈ R(3N +2)×(3N +3) (15)
is the corresponding Jacobian matrix with respect to s.
Next, to obtain the correction vector, we use Newton method to solve the nonlinear system ( G(u, τ ) = 0, ˙u⊤ku+ ˙τkτ = ˙u⊤ku (0) k+1+ ˙τk⊤τ (0) k+1, (16)
with the initial guess x(0)k+1 = [u (0)
k+1, τ
(0)
k+1]. We then solve the following
aug-mented system " Gu(x (i) k+1) Gτ(x(i)k+1) ˙u⊤i ˙τk # δ(i) = " G(x(i)k+1) (x(i)k+1− x(0)k+1) · ˙xk # (17) to obtain the Newton corrector
x(i+1)k+1 = x(i)k+1+ δ(i). (18)
When x(i+1)k+1 satisfies the convergence criterion, we let xk+1 = x(i+1)k+1 and
3.2. The parameter-switching technique
By using the PACM described in Section 3.1, we can compute the ground state solutions of the spin-1 BEC with desired coupling constants in zero or nonzero magnetic field. On top of the PACM, we further propose the parameter-switching technique to find the ground state solutions of the spin-1 BEC in the presence of the magnetic field.
In the first stage of the parameter-switching process, we compute the ground states of the spin-1 condensate in zero magnetic field under the given strength of coupling interactions (see [14] for more details). That is, we treat gn and gs
as the continuation parameters and track the solution curve, C(1)= {x = (u⊤, τ)⊤ | G(x) = 0 with g
n = ¯gnτ, gs= ¯gsτ and q = 0, for 0 ≤ τ ≤ 1},
(19) which starts from the ground state solution of linear Schr¨odinger equation (LSE). Note that the coupling constants gn and gs can be adjusted by
tun-ing the s-wave scattertun-ing length a0 and a2 via an appropriate setting of the
Feshbach resonances [13, 37]. This provides the justification for the choice of the continuation parameters gn and gs.
In spin-1 condensates, the ground state patterns depend on the strength of the applied magnetic field, or equivalently, the QZ parameter q. The parameter q can be tuned experimentally via laser or microwave dressing field [15, 21, 38, 39]. Moreover, q can vary from positive to negative [21, 22]. These experimental works interest us to explore the ground state patterns for both q > 0 and q < 0 regime numerically.
To investigate the ground state of a sp1 BEC numerically under the in-fluence of an external magnetic field, we further treat the QZ parameter q as the continuation parameter to track the solution curve
C(2)= {x = (u⊤, τ)⊤ | G(x) = 0 with gn = ¯gn, gs= ¯gs and q = ¯qτ, for 0 ≤ τ ≤ 1}.
(20) Note that C(2)starts from the termination points at C(1). The PACM used here
is the same as that in the previous stage. However, the Jacobian matrix now depends on q and bifurcations can occur along the solution curve. We detect the bifurcation points by monitoring the smallest eigenvalues of the augmented Jacobian matrix in continuation algorithm.
4. Ground States and Phase Transitions
We present the complete study of the ground states and their phase tran-sitions for the spin-1 BECs trapped in a harmonic or box potential under the influence of the magnetic field. All the findings are obtained by using the con-tinuation method proposed in Section 3. Table 1 summarizes all the scenarios considered in this study and the discoveries about the ground state patterns of the spin-1 BECs.
For a quick glance, the numerical results show that there are one or two bifurcation points along the ground state solution curve, depending on different
situations (shape of the trapping potential, the sign of gs and the sign of q).
Two types of phase transitions then occur: (i) a transition from 2C state to 3C state and vice versa (in the following, named qcr1), and (ii) a symmetry breaking
of the ground state (named qcr2). The component separation is also observed
for large |q| and domain walls are formed.
Detailed results are presented in the following sections. In Section 4.1, Case 1 and 2 concern the antiferromagnetic condensates (23Na) with box and harmonic
trapping potential. In Section 4.2, Case 3 and 4 concern the ferromagnetic condensates (87Rb) with box and harmonic trapping potential. In Section 4.3,
we assert the bifurcation diagrams in semi-classical regime numerically.
4.1. Antiferromagnetic condensates
For the antiferromagnetic system, we focus on the spin-1 23Na BECs with
the spin-independent interaction gn = 240.8 and the spin-exchange interaction
gs= 7.5 [28]. We compute the ground states with total magnetization M = 0.3.
By using the proposed continuation scheme, the 2C solution satisfying (11) is first chosen as the starting point to track the solution curve (19). Then, the termination point at this curve which is the ground state of system without magnetic field is used as the starting point for the computation of curve (20). Case 1: 23Na BEC with Vext(x) = 0. First, we study the case of the box
potential. By increasing q in the continuation algorithm, two bifurcation points (qcr1 = 0.01759 and qcr2 = 0.1028) are found with rich ground state patterns.
In Fig. 1, we plot the corresponding energy curves based on the reduced energy functional (8) and label these solution branches by 1, 1-1 and 1-1-1. The den-sities of the starting and termination states of these solution curves are plotted in Figs. 2 to 4. We highlight some observations from these figures.
• The solution curve 1 starts from the 2C ground state of the system in zero magnetic field. From Fig. 2, we can see that this 2C state is unchanged along the entire solution curve 1. This is due to the fact that the quadratic Zeeman energy functional R q(n1+ n−1) dx = qN is independent of the
2C state pattern. The populations N1 and N−1 can be determined from
N and M , see (11). This 2C state is the ground state when q < qcr1 and
becomes an excited state when q > qcr1.
• The branch 1-1 is bifurcated from the solution curve 1 at q = qcr1. This
bifurcation leads to the population transfer from n1 and n−1 to n0, and
n−1 is gradually depleted. From Fig. 3, we observe that the densities of
each spin component are symmetric, n0is immiscible with n1and n−1, and
two domain walls are formed. The ground states is this 3C (symmetric) state until q meets the next bifurcation point qcr2.
• The branch 1-1-1 is bifurcated from the branch 1-1 at q = qcr2. On
the branch 1-1-1, the corresponding solution is the ground state. It is asymmetric, and n0 is immiscible with n1 and n−1. Eventually, in high
states become immiscible MS+NS, as shown in Fig. 4. This symmetry breaking and component separation were pointed out in [17] but with only schematic structure. Here, we compute the bifurcation point qcr2precisely
and present details of this phase transition process.
In contrast to the case of positive q, the effect of negative QZ parameter reduces the component n0. Thus, the simulation for q < 0 becomes trivial,
since the ground state of the antiferromagnetic condensates in zero magnetic field is the 2C solution. The numerical results show this fact that the ground states from q = 0 to q = −1 are always the same 2C states.
We summarize the ground state patterns and their phase diagrams of the spin-123Na BEC in the box potential with various strengths of the QZ effect in
Table 2. The correspond energy and the particle number of u−1 are also listed
in the table.
Case 2: 23Na BEC with Vext(x) = 12x2. We now consider the spin-1 BEC
of 104 23Na atoms trapped in the harmonic potential. The solution curves for
the case of positive q are plotted in Fig. 5. The 2C solution curve is labelled by 1 and its solution branch by 1-1. We also depict the densities of the starting and termination states of these two curves in Figs. 6 and 7. In addition, for the case of negative q, the ground states are identical 2C states, and no bifurcation is found. We demonstrate how the ground state structure changes with various qin Table 3. We also make the following remarks for this simulation.
• From Fig. 5, we observe that there is only one bifurcation (qcr1= 0.02606).
The 3C APM states bifurcate from 2C states at q = qcr1. That is, this
phase transition leads to the increase in n0, but the depletion of n−1 and
n1.
• The solutions on curve 1 are identical 2C states with condition (11), as shown in Fig. 6. Note that they are the ground states when q < qcr1. For
q > qcr1, the n0state becomes positive.
• As we increase q, there are two situations depending on the strength V0of
the harmonic trap potential as well as the total magnetization M . Before we describe the two situations, let us first explain the mechanisms of sym-metry maintaining and symsym-metry breaking of this system. The harmonic trap potential introduces an external symmetric trapping force −∇V (x) toward the center. This symmetric force causes a symmetric profile of n, which reaches minimal potential energy. On the other hand, there is an internal repulsive force between n0 and n1 (n−1 as well). The separation
of n0and n1(n−1as well) decreases the spin-exchange interaction energy
and causes a symmetry breaking. The amount of this energy certainly depends on csas well as M .
Now, we explain the two situations of the ground states as we increase q. The first situation occurs when V0 is small. In this case, this symmetric
If we further increase q, then n−1 is almost negligible (but not zero) and
n1, n0 are separated. This is the MS+NS state in the Thomas-Fermi
regime.
The second situation occurs when V0 is large. The symmetric external
force is stronger than the internal repulsive force. In this case, the asym-metric profile (from the repulsive force between n1 and n0) costs higher
potential energy than the symmetry one (with no phase separation at the center). Thus, the ground state profile keeps symmetric. As we further increase q, as shown in Table 3, the solutions on the branch 1-1 eventually become the symmetric MS+NS state in which n−1 is almost negligible,
and n1 and n0 are separated. These results are in good agreement with
those in [17, 18, 28].
4.2. Ferromagnetic condensates
For the case of the ferromagnetic system, we consider the spin-1 condensate of 104 87Rb atoms, in which g
n= 885.4 and gs= −4.1 [28]. We first start the
continuation method from the 3C solution with (10) to get the ground state in zero magnetic field. The obtained solution is then used as the starting point to investigate the ground state pattern under the effect of magnetic field. In contrast to the antiferromagnetic case, the ground state patterns of the ferro-magnetic system are dramatically different for the negative QZ effect q. To our knowledge, there is no numerical study on the ferromagnetic spin-1 in the q < 0 regime.
Case 3: 87Rb BEC with Vext(x) = 0. First, by employing the proposed
continuation algorithm along the solution curve with parameters q = 0, gn =
885.4τ , gs = −4.1τ, 0 ≤ τ ≤ 1, we obtain the ground state of ferromagnetic
condensate in zero magnetic field, which is a 3C state. Then we take q as the continuation parameter from 0 to −1. Fig. 8 is the energy curves with q ranging from 0 to −1. Figs. 9 and 10 display the starting (q = 0) and target (q = −1) states on these solution curves. The observations from these computations are highlighted as follows:
• In Fig. 8, the solutions on curve 1 are symmetric. Along this curve (with q ranging from 0 to −1), the branch 1-1 bifurcates at qcr2 = −0.01389
which has lower energy. The solutions on the branch 1-1 are asymmetric. Thus, the ground states are symmetric for q > qcr2 and asymmetric for
q < qcr2, and qcr2is a symmetry-breaking bifurcation point.
• From Figs. 9 and 10, we observe that the decrease of q suppresses n0
for both symmetric and asymmetric solutions. As q < qcr2, n0 gradually
depletes to zero, n1 and n1 are gradually immiscible and separate.
Con-sequently, two domain walls are formed for symmetric solutions, whereas only one domain wall is form for the asymmetric ground state. This ground state is denoted by MS+MS.
For ferromagnetic condensates with positive q, we track the solution curve with q ranging from 0 to 5. The solutions start from a 3C state. As q increases, n0 increases, but n1 and n−1 decrease. Eventually, n−1 depletes to zero and
solution is a 2C miscible state consisting components 1 and 0. We denote it by 2C{1,0}. Along this solution curve, the solutions are all symmetric and no
bifurcation point is found. In Table 4, we collect the ground state patterns with various q for this simulation.
Case 4: 87Rb BEC with Vext(x) = 12x2. In this simulation, we study the
ground states of the spin-187Rb BEC confined in a harmonic trap. The
bifurca-tion diagram of the ground states with negative q is depicted in Fig. 11. Similar to the case of box potential, there is only one bifurcation (qcr2= −0.02125) on
this ground state solution curve. The qcr2causes the symmetry breaking phase
transition in the ground state. The properties of the ground state structure are similar to those in the box potential. We display the starting and target states of these solution curves in Figs. 12 and 13. In Table 5, we also show the ground state patterns with various q.
4.3. Bifurcation diagram in semi-classical regime
In this subsection, we further investigate the bifurcation diagrams in semi-classical regime by considering
(µ + λ)u1 = −ǫ2k ∇2 2 + gnn+ qu1+ gsn0(u1− σu−1) + u1(n1− n−1) µu0 = −ǫ2k∇ 2 2 + gnn]u0+ gsu0(u1− σu−1) 2 (µ − λ)u−1 = −ǫ2k ∇2 2 + gnn+ qu−1+ gsn0(u−1− σu1) + u−1(n−1− n1), (21) where the parameter 0 < ǫk≤ 1 is the strength of the kinetic-energy term. We
are interested in the variation of the bifurcation curves for various ǫk. In our
simulations, we choose ǫk= 0.1, 0.5 and 1. In the study below, the box potential
with V0= 0 is taken into account.
In Fig. 14, we present the ground state phase diagrams of the antiferromag-netic systems (23Na with g
n=240.8 and gs=7.5) with different ǫk by using the
proposed continuation algorithm. The two curves in these figures indicate two bifurcation points on the ground state solution curves versus magnetization M from 0.05 to 0.9. For small ǫk, the detection of the bifurcations becomes a
de-manding task due to the nearness of two phase transitions. We overcome this difficulty by tuning the step length used in Euler predictor (14). The proposed continuation algorithm thus enables us to determine precisely the bifurcations on the ground state solution curves. We highlight some observations found in these computations.
• In Fig. 14, we see that the two bifurcation curves qcr1(M ) and qcr2(M )
gradually merge as ǫkbecomes smaller and smaller. We expect they merge
• It is interesting to see that the bifurcation qcr2(M ) bends backward for
small M . Further, for large ǫk, the region between qcr1(M ) < q < qcr2(M )
becomes larger and larger. This means that, for small M , it is harder and harder to break the symmetry as we increase the strength of the applied magnetic field. This is mainly due to strong homogenization effect of the kinetic energy term.
5. Conclusion
Aiming at the ground state patterns and the phase diagrams of the spin-1 BECs, we have developed a numerical algorithm based on the PACM, in which the parameter-switching technique is used. By the proposed algorithm, we provide a complete investigation on the ground state patterns of spin-1 BECs over a broad range of physical parameters of interest.
For the spin-1 antiferromagnetic BEC with large enough positive QZ effect, the system undergoes two ground state phase transitions from a 2C state to a 3C symmetric state (2C→3C), then to an asymmetric phase separated state (3C→2C+NS) in the case of the box trap. While only one phase transition (2C→3C) occurs in the case of harmonic traps and results in the symmetric phase separated ground state. Next, we investigate the spin-1 BEC in negative qregime. Particularly, for ferromagnetic condensates, the effect of negative QZ shift leads to a symmetry breaking phase transition in both box and harmonic traps. We further study the phase transition diagrams on the (M, q) plane in the semi-classical regime. It is found that the two phase transition curves (2C→3C, 3C→2C+NS) merge as the the strength of the kinetic-energy term tends to 0. These results reveal that the proposed continuation algorithm is capable of accurately and efficiently finding all ground state patterns. Thus, a complete phase transition diagram of the spin-1 BECs in the presence of magnetic field is provided.
Acknowledgements
The authors are grateful to the anonymous referees for their useful comments and suggestions. This work is partially supported by the National Center for Theoretical Sciences and the National Science Council of the Republic of China under contract numbers: NSC 134-004 (Chen), NSC 102-2115-M-009-013 (Chern), and NSC 100-2628-M-002-011-MY4 (Wang).
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Table 1: Summaries of the ground state patterns of the spin-1 BECs.
BEC Potential QZ effect Ground state patterns
23Na Constant (Case 1) q >0* 2C → 3C → 2C+NS → MS+NS q= 0 2C q <0† 2C Harmonic (Case 2) q >0‡ 2C → 3C → 2C+NS → MS+NS q= 0 2C q <0† 2C 87Rb Constant (Case 3) q >0* 3C → 2C{1,0} q= 0 3C q <0† 3C → 2C+NS → MS+MS+NS → MS+MS Harmonic (Case 4) q >0‡ 3C → 2C {1,0} q= 0 3C q <0† 3C → 2C+NS → MS+MS+NS → MS+MS
*Only schematic drawings in literatures [17] and never been numerically studied. †Only been observed in experiments [21, 22] and never been numerically studied. ‡Agree with literatures [17, 18, 28].
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 6.3 6.4 6.5 6.6 6.7 6.8 6.9 7 7.1 7.2 q En e rg y 1 1-1 1-1-1 qcr1 qcr2
Figure 1: (Case 1) Energy curves of23
Na with M = 0.3 in the box potential. There are two bifurcation points qcr1= 0.01759 and qcr2= 0.1028.
−8 −6 −4 −2 0 2 4 6 8 0 0.005 0.01 0.015 0.02 0.025 0.03 Ω −8 −6 −4 −2 0 2 4 6 8 0 0.005 0.01 0.015 0.02 0.025 0.03 Ω
Figure 2: (Case 1) Densities of the starting state q = 0 (left) and the target state q = 1 (right) of curve 1 in Fig. 1. The mF = 1, 0, −1 components are depicted by blue dash-dotted,
green solid and red dashed lines, respectively.
−8 −6 −4 −2 0 2 4 6 8 0.005 0.01 0.015 0.02 0.025 0.03 Ω −8 −6 −4 −2 0 2 4 6 8 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 Ω
Figure 3: (Case 1) Densities of the starting state q = 0 (left) and the target state q = 1 (right) of curve 1-1 in Fig. 1. The mF= 1, 0, −1 components are depicted by blue dash-dotted, green
solid and red dashed lines, respectively.
−8 −6 −4 −2 0 2 4 6 8 0.005 0.01 0.015 0.02 0.025 Ω −8 −6 −4 −2 0 2 4 6 8 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 Ω
Figure 4: (Case 1) Densities of the starting state q = 0 (left) and the target state q = 1 (right) of curve 1-1-1 in Fig. 1. The mF = 1, 0, −1 components are depicted by blue dash-dotted,
Table 2: (Case 1) Ground state patterns of antiferromagnetic BEC (23
Na) with M = 0.3 in the box potential.
q= −1 q= 0 q= 0.02582 q= 0.1115 q= 1 q= 3 Energy ( ˜E) 5.2568 6.2568 6.2818 6.3309 6.6105 7.2114 N−1 0.35 0.35 0.2673 0.07373 0.0007 0.00008 State 2C 2C 3C 2C+NS 2C+NS MS+NS Profile 0 −8−6−4 −2 0 2 4 6 8 0.005 0.01 0.015 0.02 0.025 0.03 Ω −8−6 −4−2 0 2 4 6 8 0 0.005 0.01 0.015 0.02 0.025 0.03 Ω −8−6−4−2 0 2 4 6 8 0.005 0.01 0.015 0.02 0.025 Ω −8−6−4 −2 0 2 4 6 8 0.005 0.01 0.015 0.02 0.025 0.03 0.035 Ω −8−6−4 −2 0 2 4 6 8 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 Ω −8−6 −4−2 0 2 4 6 8 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 Ω 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 15.2 15.4 15.6 15.8 16 16.2 q En e rg y 1 1-1 qcr1
Figure 5: (Case 2) Energy curves of23
Na with M = 0.3 in the harmonic potential. There is one bifurcation point qcr1= 0.02606.
−8 −6 −4 −2 0 2 4 6 8 0 0.01 0.02 0.03 0.04 0.05 0.06 Ω −8 −6 −4 −2 0 2 4 6 8 0 0.01 0.02 0.03 0.04 0.05 0.06 Ω
Figure 6: (Case 2) Densities of the starting state q = 0 (left) and the target state q = 1 (right) of curve 1 in Fig. 5. The mF = 1, 0, −1 components are depicted by blue dash-dotted,
−8 −6 −4 −2 0 2 4 6 8 0.01 0.02 0.03 0.04 0.05 0.06 Ω −8 −6 −4 −2 0 2 4 6 8 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Ω
Figure 7: (Case 2) Densities of the starting state q = 0 (left) and the target state q = 1 (right) of curve 1-1 in Fig. 5. The mF= 1, 0, −1 components are depicted by blue dash-dotted, green
solid and red dashed lines, respectively.
Table 3: (Case 2) Ground state patterns of antiferromagnetic BEC (23
Na) with M = 0.3 in the harmonic potential.
q= −1 q= 0 q= 0.1537 q= 1 q= 8 Energy ( ˜E) 14.2659 15.2659 15.3686 15.6515 17.7593 N−1 0.35 0.35 0.0724 0.004 0.00008 State 2C 2C 2C+NS 2C+NS MS+NS Profile 0 −8 −6−4−2 0 2 4 6 8 0.01 0.02 0.03 0.04 0.05 0.06 Ω −10−8 −6−4−2 0 2 4 6 8 10 0 0.01 0.02 0.03 0.04 0.05 0.06 Ω −8−6−4 −2 0 2 4 6 8 0.01 0.02 0.03 0.04 0.05 0.06 0.07 Ω −100 −8−6−4 −2 0 2 4 6 810 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Ω −8 −6−4−2 0 2 4 6 8 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Ω −1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 21.4 21.6 21.8 22 22.2 22.4 22.6 q En e rg y 1 1-1 qcr2
Figure 8: (Case 3) Energy curves of87
Rb with M = 0.3 in the box potential. There is one bifurcation point qcr2= −0.01389.
−8 −6 −4 −2 0 2 4 6 8 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 0.022 Ω −8 −6 −4 −2 0 2 4 6 8 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 Ω
Figure 9: (Case 3) Densities of the starting state q = 0 (left) and the target state q = −1 (right) of curve 1 in Fig. 8. The mF = 1, 0, −1 components are depicted by blue dash-dotted,
green solid and red dashed lines, respectively.
−8 −6 −4 −2 0 2 4 6 8 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 0.022 Ω −8 −6 −4 −2 0 2 4 6 8 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 Ω
Figure 10: (Case 3) Densities of the starting state q = 0 (left) and the target state q = −1 (right) of curve 1-1 in Fig. 8. The mF = 1, 0, −1 components are depicted by blue dash-dotted,
green solid and red dashed lines, respectively.
Table 4: (Case 3) Ground state patterns of ferromagnetic BEC (87
Rb) with M = 0.3 in the box potential. q= −1 q= −0.2035 q= −0.01416 q= 0 q= 1 q= 5 Energy ( ˜E) 21.4417 22.2379 22.4109 22.4187 22.7659 23.9687 N−1 0.35 0.3438 0.1337 0.1225 0.0018 0.00007 State MS+MS MS+MS+NS 2C+NS 3C 3C 2C{1,0} Profile −8−6−4 −2 0 2 4 6 8 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 Ω −8−6−4 −2 0 2 4 6 8 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 Ω −8−6 −4−2 0 2 4 6 8 0.005 0.01 0.015 0.02 0.025 Ω −8−6−4−2 0 2 4 6 8 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 0.022 Ω −8−6−4 −2 0 2 4 6 8 0.005 0.01 0.015 0.02 0.025 0.03 0.035 Ω −8−6−4 −2 0 2 4 6 8 0.005 0.01 0.015 0.02 0.025 0.03 0.035 Ω
−1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 35 35.2 35.4 35.6 35.8 36 36.2 36.4 q En e rg y 1 1-1 qcr2
Figure 11: (Case 4) Energy curves of87
Rb with M = 0.3 in the harmonic potential. There is one bifurcation point qcr2= −0.02125.
−15 −10 −5 0 5 10 15 0.005 0.01 0.015 0.02 0.025 0.03 Ω −15 −10 −5 0 5 10 15 0 0.01 0.02 0.03 0.04 0.05 0.06 Ω
Figure 12: (Case 4) Densities of the starting state q = 0 (left) and the target state q = −1 (right) of curve 1 in Fig. 11. The mF = 1, 0, −1 components are depicted by blue dash-dotted,
−15 −10 −5 0 5 10 15 0.005 0.01 0.015 0.02 0.025 Ω −15 −10 −5 0 5 10 15 0 0.01 0.02 0.03 0.04 0.05 0.06 Ω
Figure 13: (Case 4) Densities of the starting state q = 0 (left) and the target state q = −1 (right) of curve 1-1 in Fig. 11. The mF = 1, 0, −1 components are depicted by blue
dash-dotted, green solid and red dashed lines, respectively.
Table 5: (Case 4) Ground state patterns of ferromagnetic BEC (87
Rb) with M = 0.3 in the harmonic potential. q= −1 q= −0.06099 q= −0.02195 q= 0 q= 1 q= 5 Energy ( ˜E) 35.1809 36.1352 36.10605 36.1474 36.4957 37.6991 N−1 0.35 0.2658 0.1399 0.1225 0.0021 0.00008 State MS+MS MS+MS+NS 2C+NS 3C 3C 2C{1,0} Profile 0 −15 −10 −5 0 5 10 15 0.01 0.02 0.03 0.04 0.05 0.06 Ω −15 −10 −5 0 5 10 15 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 Ω −15 −10 −5 0 5 10 15 0.005 0.01 0.015 0.02 0.025 0.03 Ω −15 −10 −5 0 5 10 15 0.005 0.01 0.015 0.02 0.025 0.03 Ω −15 −10 −5 0 5 10 15 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 Ω −15 −10 −5 0 5 10 15 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 Ω
0.05 0.1 0.15 0.2 0.25 0.3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 q M 2C 3C 2C+NS 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 q M 2C 3C 2C+NS 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 q M 2C 2C+NS 3C
Figure 14: (semi-classical regime) The ground state phase diagram of the spin-123
Na BEC with ǫk= 1, ǫk= 0.5, and ǫk= 0.1 (from top to bottom).
