Cite this Article
Liu Yan, Zhang Su-Ying. The ground states and pseudospin textures of rotating two-component Bose–Einstein condensates trapped in harmonic plus quartic potential. Chinese Physics B, 2016, 25(9): 090304  
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The ground states and pseudospin textures of rotating two-component Bose–Einstein condensates trapped in harmonic plus quartic potential
Liu Yan, Zhang Su-Ying†,
Institute of Theoretical Physics, Shanxi University, Taiyuan 030006, China

 

† Corresponding author. E-mail: zhangsy@sxu.edu.cn

Project supported by the National Natural Science Foundation of China (Grant Nos. 91430109 and 11404198), the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20111401110004), and the Natural Science Foundation of Shanxi Province, China (Grant No. 2014011005-3).

Abstract
Abstract

The ground states of two-component miscible Bose–Einstein condensates (BECs) confined in a rotating annular trap are obtained by using the Thomas–Fermi (TF) approximation method. The ground state density distribution of the condensates experiences a transition from a disc shape to an annulus shape either when the angular frequency increases and the width and the center height of the trap are fixed, or when the width and the center height of the trap increase and the angular frequency is fixed. Meantime the numerical solutions of the ground states of the trapped two-component miscible BECs with the same condition are obtained by using imaginary-time propagation method. They are in good agreement with the solutions obtained by the TF approximation method. The ground states of the trapped two-component immiscible BECs are also given by using the imaginary-time propagation method. Furthermore, by introducing a normalized complex-valued spinor, three kinds of pseudospin textures of the BECs, i.e., giant skyrmion, coaxial double-annulus skyrmion, and coaxial three-annulus skyrmion, are found.

PACS: 03.75.Lm;03.75.Mn
Keyword:Bose–Einstein condensate;pseudospin texture;skyrmion
1. Introduction

With the implementation of Bose–Einstein condensates (BECs) in experiments,[1–3] the corresponding theoretical research has also made great progress. For the system of rotating BECs, the topological defects of single component BECs are always quantized vortices. For example, when the single BECs trapped in a harmonic plus quartic potential are driven to rotate, a series of singly quantized vortices will be created. Faster rotation generates more vortices, which finally condense into a giant vortex.[4,5] Different from the single BECs described by the scalar order parameter, the ground states and dynamic behavior of two-component BECs are always described by the vector order parameter.[6,7] With the change of the relative interaction strength between intra-component and inter-component, the two-component BECs can be divided into two kinds: miscible and immiscible. Different trap potential and angular velocity enable more intriguing topological structures.[8–11] In addition, no quantized vortex appears in two-component BECs at sufficiently slow rotations, but its effective velocity is nonzero. Thus in the rotating two-component BECs, there must be other new topological excitations such as skyrmion, giant skyrmion, and so on, which do not exist in single BECs. For two-component immiscible BECs, the spin textures are various for different external potential, s-wave scattering lengths or rotating angular frequency.[12,13] For example, in a harmonic trap, the interface of two components becomes curved for equal particle number but different intra-component scattering lengths, and the component with strong interaction curves towards the one with weak interaction. With the increase of angular frequency, the interface of two components becomes annular and the corresponding spin texture is a giant skyrmion.[14,15] The spin texture can also be coaxial double-annulus skyrmion if two-component immiscible BECs are confined in an annular trap made of harmonic plus linear potential.[16] More interesting phenomena have been presented in spin–orbit coupled BECs.[17–20]

In this paper, we discuss the ground states and the corresponding spin textures of rotating two-component BECs in an annular trap made of harmonic plus quartic potential. The quartic potential can increase the restriction on BECs, in order to study the BEC and its corresponding properties when the rotation speed exceeds the radial harmonic frequency. Firstly, we use the Thomas–Fermi (TF) approximation method to obtain the ground states of two-component miscible BECs for two cases: (i) the angular frequency increases while the width and the center height of the trap are fixed; (ii) the width and the center height of the trap increase while the angular frequency remains the same. Secondly, we make a comparison between the analytical solutions and the numerical solutions which are obtained by the imaginary-time propagation method. Thirdly, we study the ground states and the corresponding pesudospin textures of two-component immiscible BECs by introducing a normalized complex-valued spinor and using the imaginary-time propagation method.

2. Theoretical model

Consider two-component BECs confined in the following annular trap

where , V0 and r0 are dimensionless constants, is the radial coordinate. The structures of the annular trap for different r0 are shown in Fig. 1. From Fig. 1(b), we can see that the lowest point of the trap is and the highest center point of the trap is .

Fig. 1. (a) Schematic of the annular trapping potential with V0 = 0.5 and r0 = 5. (b) The cross sections of V(r) along x with different r0.

The trapped rotating two-component BECs can be characterized by the vector order parameter 𝜓 = (ψ1,ψ2)T, which is governed by the time-dependent coupled Gross–Pitaevskii (GP) equation with angular velocity Ω = Ωz

where Ujk = 4πħ2ajk/m (j,k = 1,2). When k = j, Ujk represents the intra-component interaction strength, and when k ≠ j, Ujk represents the inter-component interaction strength. Here ajk is the intra-component (k = j) or inter-component (k ≠ j) s-wave scattering length. Lz = −iħ(x∂y − y∂x) is the z-component of the angular momentum operator. The wave functions are normalized as

where N is the total number of particles in the condensates. After separating the wave function as ψj(r,t) = ϕj(x,y,t)φ(z) and re-scaling r̃ = r/a0, t̃ = ω⊥t, , Ω̃ = Ω/ω⊥, and L̃z = Lz/ħ, we obtain the quasi-two-dimensional dimensionless coupled GP equations by removing the wavy lines

where the interaction coefficients are defined as gjk = 4πNηajk, with the reduced parameter η = ∫ dz|φ(z)|4/∫dz|φ(z)|2. The wave functions are normalized as

Additionally, we make the following assumption:

where M ∈ [0,1) is an adjustable parameter. So

3. Two-component miscible BECs

If the interaction coefficients satisfy , the two-component BECs are miscible. We first use an equivalent Lagrangian formalism corresponding to Eqs. (4) and (5) as

where

is the time-dependent part of the Lagrangian function,

is the free-energy function, and μj is a Lagrange multiplier.

3.1. Thomas–Fermi approximation

Using the replacement , we have and ρj(r,t) = |ϕj (r,t)|2. The gradient of the phase ∇θj(r,t) = υ = Ω × r gives the dimensionless superfluid velocity. For the TF approximation, the curvature of the density is ignored. In this limit, the variation of the free energy ℱ with respect to |ϕj|2 yields

So we obtain the following TF density of the BECs:

3.2. Vortex lattice with central hole

Equation (15) or Eq. (16) allows a direct solution for the “classical” turning points where the TF density vanishes

and

where i = +, − correspond to the ± appearing on the right-hand side of Eqs. (17) and (18). The plus signs yield the outer squared radius and , and the minus signs denote the inner squared radius and . We discuss the density profile of ϕ1 and ϕ2 in two cases respectively:

Case 1 The width and the center height of the trap are fixed while the angular frequency increases.

The central hole in ϕ1 first appears when (so that and . According to the normalization condition (8), the critical rotation frequency Ω1h, at which the central density of ϕ1 first vanishes, can be expressed as

If and squared radius and satisfy the following conditions:

The normalization condition (8) yields

Combining Eq. (19) with Eq. (22), we have

and substituting it into Eq. (21), we obtain

It is very easy to get

Clearly, the area of the annular condensate ϕ1 remains constant for all Ω > Ω1h. The average radius R ≡ (R1+ + R1−)/2 increases and the width d = R1+ − R1− decreases with the increase of rotating angular frequency.

The central hole in ϕ2 first appears when (so that and . According to the normalization condition (9), the critical angular frequency Ω2h, at which the central density of ϕ2 vanishes, can be expressed as

If , the density profile of ϕ2 is an annulus and its inner and outer squared radius satisfy the following conditions:

The normalization condition (9) also yields

Combining Eqs. (26) and (29), we have

The relationship between the chemical potential and interaction strength is

According to Eqs. (27) and (30), we have

It is obvious that the area of the annular condensate ϕ2 remains constant for all Ω > Ω2h. The average radius R ≡ (R2+ + R2−)/2 increases and the width d = R2+ − R2− decreases with the increase of rotating angular frequency.

Case 2 The angular frequency is fixed while the width and the center height of the trap increase.

Similar to the case 1, we can obtain two critical values of r0, at which the central density of ϕ1 or ϕ2 first vanishes, by

The change of the annular condensates is the same as case 1 for all or .

According to the above analyses and additionally assuming Ω1h > Ω2h and , two-component miscible BECs can present three kinds of structures of the density distributions for different Ω or r0: (i) if Ω1h > Ω2h > Ω or , both components are disc shaped; (ii) if Ω1h > Ω > Ω2h or , ϕ1 is disc shaped and ϕ2 is annular shaped; (iii) if Ω > Ω1h > Ω2h or , both components are annular shaped. When g11 = g22 = g and Ω = 1, M = 0.4, V0 = 0.5, g12 = g21 = 200, we draw the phase diagram of the ground state density distribution of two-component miscible BECs with the varied r0 and g in Fig. 2, which shows that the solutions obtained by the above analytic method agree with the numerical solutions. In addition, we take Ω = 0.5, r0 = 11, V0 = 0.5, M = 0.4, g11 = g22 = 1000, and g12 = g21 = 800 to compare the analytic solution with the numerical solution, as shown in Fig. 3.

Fig. 2. r0–g phase diagram for parameters Ω = 1, M = 0.4, V0 = 0.5, and g12 = g21 = 200. The dashed line indicates the value of given by Eq. (33), and the solid line indicates the value of given by Eq. (34). The circles indicate that two components are annulus shaped; the plus signs indicate that the density distributions of the big component are disc shaped and the density distributions of the small component are annulus shaped; the points indicate that two components are both disc shaped.
Fig. 3. (a) The ground state density profile of ϕ1, (b) the ground state density profile of ϕ2, (c) the comparison of ϕ1 (solid line) with the analytic solution (dashed line) of Eq. (15), and (d) the comparison of ϕ2 (solid line) with the analytic solution (dashed line) of Eq. (16). Here we take M = 0.4, g11 = g22 = 1000, g12 = g21 = 800, Ω = 0.5, r0 = 11, and V0 = 0.5.
4. Two-component immiscible BECs

In this section, we illustrate the ground states and the pseudospin textures of two-component immiscible BECs, at which the interaction strength satisfies . Under this condition, the TF approximation is no longer applicable because it fails to describe the domain boundary region, where the quantum pressure term cannot be ignored. However, we can obtain the ground states by using the imaginary-time propagation method.[21]

We take g11 = 600, g22 = 1000, g12 = 800, M = 0.1, V0 = 0.5, and r0 = 4. Figure 4 shows that two-component immiscible BECs present three density distributions with the increase of the angular frequency: (i) ϕ1 is disc shaped and ϕ2 is annular shaped; (ii) ϕ1 is annular shaped and ϕ2 is a disc plus an annular; (iii) ϕ1 is annular shaped and ϕ2 is two annular shaped. The number of the quantum vortex increases with the increase of the angular frequency.

Fig. 4. The ground state density distributions and phases for different Ω: (a) Ω = 0.5, (b) Ω = 1, and (c) Ω = 2.5. The first and the third rows are the density distributions of ϕ1 and ϕ2. The second and the fourth rows are the phases corresponding to the density distributions of the first and third rows, respectively.

Besides, we take g11 = 600, g22 = 1000, g12 = 800, M = 0.1, V0 = 0.5, and Ω = 2. With the increase of the value of r0, two-component immiscible BECs present three other kinds of density distributions as shown in Fig. 5: (i) ϕ1 is a disc plus an annulus and ϕ2 is double annuli; (ii) ϕ1 is an annulus and ϕ2 is a disc plus an annulus; (iii) ϕ1 is an annulus and ϕ2 is double annuli. The bigger the value of r0 is, the greater the quantum vortices become.

Fig. 5. The ground state density distributions and phases for different r0: (a) r0 = 1, (b) r0 = 5, (c) r0 = 7. The first and the third rows are the density distributions of ϕ1 and ϕ2. The second and the fourth rows are the phases corresponding to the density distributions of the first and third rows, respectively.

In addition, we can introduce the pseudospin concept to study the corresponding pseudospin textures. Using a pseudospin-1/2 BECs to analysis this system, we introduce a normalized complex valued spinor χ(r) = [χ1(r),χ2(r)]T = [|χ1|eiθ1, |χ2|eiθ2]T, where |χ1|2 + |χ2|2 = 1, θj represents the phase of wave function ϕj, and . The pseudospin density is defined as , where σ is the Pauli matrix. The pseudospin density can be expressed as

Clearly, the modulus of the total spin is . The spin of ϕ1 points up and the spin of ϕ2 points down. For the parameters used in Fig. 4, the corresponding pseudospin density distributions Sx, Sy, and Sz are given in Fig. 6. Here, the red represents the spin pointing up and the blue represents the spin pointing down. At the interface of the two components, the spin turns from pointing up to pointing down or from pointing down to pointing up periodically. The period of the annular pseudospin texture is equivalent to that of the corresponding relative phase, because the pseudospin density is just a periodic function of their relative phase. With the increase of the angular velocity, the spin texture changes from giant skyrmions to coaxial double-annulus skyrmion.

In addition, for the parameters used in Fig. 5, the corresponding pseudospin density distributions Sx, Sy, and Sz are given in Fig. 7. With the increase of the value of r0, the corresponding pseudospin texture changes from coaxial three-annulus skyrmion, which has not been observed or studied in an annular potential and could not exist in harmonic trapping potentials, to coaxial double-annulus skyrmion. The period of the annular pseudospin texture is equal to that of the relative phase of two components.

Fig. 6. The pseudospin densities distributions and the relative phases for different Ω: (a) Ω = 0.5, (b) Ω = 1, and (c) Ω = 2.5. The first row is Sx, the second row is Sy, the third row is Sz, and the fourth row is the relative phases θ1 − θ2.
Fig. 7. The pseudospin densities distributions and the relative phases for different r0: (a) r0 = 1, (b) r0 = 5, and (c) r0 = 7. The first row is Sx, the second row is Sy, the third row is Sz, and the fourth row is the relative phases θ1 − θ2.
5. Conclusion

We have studied the ground states of two-component miscible BECs analytically and numerically. With the increase of the angular velocity, or with the increase of the central height and width of the external trap potential, two-component miscible BECs present three kinds of density distributions. In addition, we have also investigated the ground states and the spin textures of two-component immiscible BECs. By projecting the system into a pesudospin space, we found a new pesudospin texture, i.e., coaxial three-annulus skyrmion. The period of the annular pseudospin texture of the BECs is equal to that of the relative phase of the two different components.

Furthermore, the TF radius of the two-component miscible BECs confined in the dimensionless harmonic plus quartic potential as discussed in this paper, is of convergence under any parameter conditions. But the BEC confined in the dimensionless annular potential as [16] has TF radius of convergence only under the condition Ω2 < V0.

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