Weakly interacting spinor Bose–Einstein condensates with three-dimensional spin–orbit coupling
Song Shu-Wei1, 2, 3, †, , Sun Rui1, 2, 3, Zhao Hong1, 2, 3, Wang Xuan1, 2, 3, Han Bao-Zhong1, 2, 3
State Key Laboratory Breeding Base of Dielectrics Engineering, Harbin University of Science and Technology, Harbin 150080, China
Key Laboratory of Engineering Dielectrics and Its Application, Ministry of Education, Harbin University of Science and Technology, Harbin 150080, China
College of Electrical and Electronic Engineering, Harbin University of Science and Technology, Harbin 150080, China


† Corresponding author. E-mail: japical@live.cn

Project supported by the National Natural Science Foundation of China (Grant No. 11447178).


Starting from the Hamiltonian of the second quantization form, the weakly interacting Bose–Einstein condensate with spin–orbit coupling of Weyl type is investigated. It is found that the SU(2) nonsymmetric term, i.e., the spin-dependent interaction, can lift the degeneracy of the ground states with respect to the z component of the total angular momentum Jz, casting the ground condensate state into a configuration of zero Jz. This ground state density profile can also be affirmed by minimizing the full Gross–Pitaevskii energy functional. The spin texture of the zero Jz state indicates that it is a knot structure, whose fundamental group is π3(M) ≅ π3(S2) = Z.

1. Introduction

It is well known that the Bose–Einstein condensate (BEC) supplies us with nearly an ideal platform to investigate quantum phenomena with the help of lasers and magnetic fields. For example, the interactions between atoms can be manipulated by changing the strength of an applied magnetic or electric field for species that have a Feshbach resonance.

In 2008, Wu et al. theoretically explored the isotropic Rashba spin–orbit coupled Bose–Einstein condensation and spin–spiral-type condensate, and a half-quantum vortex was observed.[1] In 2009, artificial external gauge potentials coupled to neutral atoms were generated by controlling atom-light interaction.[2] Due to the presence of the spin–orbit coupling mechanism, the energy spectrum changes dramatically. The single particle energy minimum has finite momentum and the ground states are circularly degenerate, in which case the ground state of the condensate favors mainly the “stripe” or “plane wave” phase.[35] Therefore, spin–orbit coupled cold atom systems have attracted intensive attention over the last few years,[618] such as the studies on topics of Spin Hall effects,[19] Majorana fermions,[20,21] etc. Xu et al. theoretically investigated the interesting density patterns of the spin–orbit coupled spinor condensate in the presence of rotation.[2224] It was found that the condensate may become self-trapped as a result of the spin–orbit coupling and the nonlinearity, resembling the so-called chiral confinement.[25] Zhang et al. found that the condensate with spin–orbit coupling exhibited dynamical oscillations similar to the Zitterbewegung oscillation.[26] Deng et al. also investigated the spin–orbit coupled condensate in the presence of the dipole–dipole interactions.[27] For a weakly interacting bosons system, the spin–orbit coupled condensate favors a ground state with a zero or finite z component of the total angular momentum.[28,29]

For the spin–orbit coupled condensate of three dimensions, Anderson and Clark investigated the single-particle characteristics and found that the system showed a dimensional reduction from three to one and the single-particle spectrum could be well approximated by Landau levels in the radial coordinate and the splittings were inversely proportional to the spin–orbit coupling strength. Kawakami et al. found that the three-dimensional skyrmion appeared as the ground state of SU(2) symmetric BECs coupled with a non-Abelian gauge field.[30] Li et al. found that the topology of condensate wavefunctions manifested in the quaternionic representation.[31] The spatial distributions of the quaternionic phase show three-dimensional skyrmion configurations, and non-zero Hopf invariants of topological objects in S2 spin orientation was also observed. Recently, Liu and Yang from Beijing Normal University observed the novel topological object three-dimensional dimeron in trapped two-component BECs and explored both exact static and moving solitonic solutions in F = 1 BECs.[32,33] However, the SU(2) nonsymmetric BEC with three-dimensional spin–orbit coupling of Weyl type has not been definitely investigated as far as we know. Among the factors which would determine the nontrivial ground states, what role does the spin-dependent interaction play?

In the present paper, we investigate the weakly interacting two-component condensate with three-dimensional spin-orbit coupling of Weyl type. It is found that finite spin-dependent interaction drives the condensate into zero total angular momentum (Jz = 0) ground state. Numerically minimizing the full Gross–Pitaevskii energy functional, the state with only spin-dependent interaction favors a knot structure, whose topological Hopf charge is QH = 1. The paper is organized as follows. In Section 2, we show the single-particle energy spectrum and the eignstates as well. In Section 3, the ground state with weak interaction is investigated within the basis vectors obtained in Section 2. In addition, the imaginary time evolution method is also exploited to confirm the numerical results. Finally, a summary is presented in Section 4.

2. Single-particle spectrum

We begin with the single-particle Hamiltonian

where σ, , and λ are the Pauli matrices, the momentum operator, and spin–orbit coupling parameter, respectively. The atom mass and parabolic trapping frequency are m and ω, respectively. Scaled by the parabolic potential, the dimensionless single-particle hamiltonian can be obtained as

where the time, energy, and length are scaled by 1/ω, ħω, and respectively. The dimensionless spin–orbit coupling parameter is

Because of the spin–orbit coupling, the orbital angular momentum is not conserved any more. The total angular momentum J = L + S (L and S are the orbital angular momentum and spin angular momentum), however, is a conserved quantity since the total angular momentum operator Ĵ satisfies As a result, we can resort to the eigenstates of Ĵ2 and Ĵz to obtain the single-particle Hamiltonian spectrum in Eq. (1). The eigenstates of Ĵ2 and Ĵz read[34]


where s = 1 corresponds to the case that J and S are aligned (j = l + s) and s = −1 corresponds to the case that they are anti-aligned (j = ls). |j + s/2,m±1/2〉 denote the eigenstates of the orbital angular momentum operator with orbital angular momentum quantum number j + s/2 and magnetic quantum number m±1/2. The eigenvalues of Ĵ2 and Ĵz are j(j + 1)ħ2 and mħ, respectively. As proved in Ref. [34], the single-particle Hamiltonian can be reformed as

where 𝒜+ = σ· and 𝒜 = σ·. The operator vectors and = (x,y,z) are vectors of creation operators and annihilation operators, respectively.

Next, the basis vectors can be constructed by considering the quantum number n in the radial direction,

In the coordinate space, |n,l,m〉 reads

where and F are the spherical harmonic functions and the Kummer confluent hypergeometric function, respectively.

Since both Ĵ2 and Ĵz commute with the single-particle Hamiltonian, the Hamiltonian can be transformed into block-diagonalization form with respect to quantum numbers j and m. After some kind of tedious calculations (see also Ref. [34]), it can be found that

After diagonalizing the Hamiltonian matrix, one can obtain the eigenstates and the corresponding eigenvalues. Figure 1 shows the single-particle spectrum and the density profile of the ground state with quantum numbers j = 1/2,m = −1/2. For weak atomic interaction, it is only necessary to consider a finite number of eigenstates at the bottom of the energy spectrum.

Fig. 1. (a) Single-particle energy spectrum of the three-dimensionally spin-orbit coupled condensate with dimensionless spin–orbit coupling parameter κ = 2. (b) The isosurface of the density for the single-particle ground state with quantum number j = 1/2, m = −1/2 with |ψ|2 = 0.002 and |ψ|2 = 0.005. The color on the isosurface indicates the phase of the state order parameter.
3. Spin–orbit coupled condensate with weak interaction
3.1. Theoretical model

In the presence of the spin-dependent and spin-independent interactions, the model Hamiltonian in the second quantization form is given by


Ψ = (ψ,ψ)T (the superscript T stands for the transpose) denotes collectively the spinor field operators. c0 and c1 are spin-independent and spin-dependent interaction parameters, and in terms of the effective interaction between different species, they read c0 = (g1 + g12)/2 and c1 = (g1g12)/2. In int, the Einstein summation convention is used (a,a' = ↑,↓).

For a weakly interacting N-boson system, only several lowest energy levels are occupied. Thus, the field operator can be expanded as Ψ = fΨff, where Ψf denotes the single-particle state in Eq. (6), and f is the corresponding annihilation operator. The many-body Hamiltonian can be rewritten in the second quantization form as


and ɛf is the single-particle energy shown in Fig. 1(a). In Uijkl and Vijkl, the summation subscript parameters have values p,q,r,s =↑,↓. The many-body Hamiltonian can be solved numerically by using exact diagonalization in the Fock space. The numerical results from the exact diagonalization are also confirmed within the mean-field frame. In the mean-field frame, the operators f are replaced by complex numbers, and the mean-field energies are minimized under the constraint of total particle number N.[29]

3.2. Jz = 0 state with finite spin-dependent interaction

According to the single-particle spectrum in Fig. 1, it is sufficient to consider eigenstates with j = 1/2 and j =3/2 for weak interaction with c0(c1) < 0.001 and N < 100.[28]

In the absence of the spin-dependent interaction, the ground states with Jz = 0 and Jz ≠ 0 are degenerate. That means the states with Jz = 0 and Jz ≠ 0 have identical total energy. However, in the presence of the spin-dependent interaction, the condensate is cast into a zero Jz = 0 state. Figures 2(a) and 2(b) show the density profile and phase distribution of the minus Jz state. For the state with plus Jz, the three-dimensional vortex structure appears in another component of the condensate, and both the density and phase profiles are similar to that of the state with minus Jz. The density and phase profiles of zero Jz state in the absence of the spin-dependent interaction do not show any visible difference compared with the density and phase characteristics in Figs. 2(c) and 2(d). The color on the isosurface indicates the phase of the order parameter. In the presence of the spin-dependent interaction, the ground state favors Jz = 0. The corresponding density and phase profiles are shown in Figs. 2(c) and 2(d).

Fig. 2. The density isosurface of |ψ|2 = |ψ|2 =2×10−4. The color on the isosurface indicates the phase of the state order parameter. The state shown in panels (a) and (b) carries jz < 0 with Nc1 = 0, and the state shown in panels (c) and (d) carries jz = 0 with Nc1 = 0.001.

A similar conclusion can also be drawn by minimizing the full Gross–Pitaevskii energy functional realized by promoting the imaginary evolution of the following Gross–Pitaevskii equations[3437]

where g1 = c0 + c1 and g12 = c0c1. In the continuous gradient flow equations, the parameter μ physically corresponds to the chemical potential. These equations are the Gross–Pitaevskii equations for imaginary times and the order parameter ψi converges to the stationary state, claiming one of the local minima of the total energy functional. We solve the gradient flow equations from t to t + Δt with the two-order Runge–Kutta method.

Figures 3(a) and 3(b) show the sliced three-dimensional density profile with zero Jz. The density isosurface profile is the same as that in Fig. 2. By resorting to the spin texture, it is found that such a state is a knotted structure. Knots are topological structures characterized by the third homotopy group and the fundamental group of the order parameter manifold is π3(M) ≅ π3(S2) = Z, which can be characterized by a topological invariant called the Hopf charge:[32,38]

where ɛijk is the Levi–Civita symbol and

The pseudospin at the origin and the outmost periphery is (the boundary condition). Figure 3(c) shows the preimage of Sx = 0 (the torus). The color on the torus denotes the value of atan(Sy/Sz), indicating the vector orientation on the torus. In Fig. 3(c), we show the preimage for S = (0,1,0) (light green) and S = (0,0,1) (light blue). These two loops interlock once and only once, as illustrated in Fig. 3(d), indicating that QH = 1.

Fig. 3. The sliced three-dimensional density profile of (a) |ψ|2 and (b) |ψ|2 obtained by minimizing the full Gross–Pitaevskii energy functional. The spin–orbit coupling and spin-dependent interaction parameters are κ = 2.0 and c1 = 0.001, respectively.
4. Summary

In summary, we have investigated the weakly interacting two-component BEC with three-dimensional spin–orbit coupling of Weyl type. The role of the SU(2) nonsymmetric term, i.e., the spin-dependent interaction, is clarified. It is found that the finite spin-dependent interaction drives the condensate into Jz = 0 ground state. The mean-field numerical results show that the ground state shows a knot spin texture whose topological charge is QH = 1. This should help us to clarify the effect of the spin-dependent interaction in the process of the cold atom simulation. In addition, the results may shed light on the research of the topological nontrivial objects in cold atom systems.

1Wu C JIan M SZhou X F2008arXiv:0809.353210.1088/0256-307X/28/9/097102
2Lin Y Jcompton R LPerry A RPhillips W DPorto J VSpielman I B 2009 Phys. Rev. Lett. 102 130401
3Zhai H 2012 Int. J. Mod. Phys. 26 1230001
4Zhou XLi YCai ZWu C 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134001
5Song S WZhang Y CWen LWang H 2013 J. Phys. B: At. Mol. Opt. Phys. 46 145304
6Su S WGuo S CLiu I Ket al. 2015 New J. Phys. 17 033045
7Su S WLiu I KTsai Y CLiu W MGou S C 2012 Phys. Rev. 86 023601
8Aidelsburger MAtala MNascimbene STrotzky SChen Y ABloch I 2011 Phys. Rev. Lett. 107 255301
9Fu ZWang PChai SHuang LZhang J 2011 Phys. Rev. 84 043609
10Campbell D LJuzeliunas GSpielman I B 2011 Phys. Rev. 84 025602
11Cheuk L WSommer A THadzibabic ZYefsah TBakr W SZwierlein M W 2012 Phys. Rev. Lett. 109 095302
12Wang PYu Z QFu ZMiao JHuang LChai SZhai HZhang J 2012 Phys. Rev. Lett. 109 095301
13Zhang Y CSong S WLiu C FLiu W M 2013 Phys. Rev. 87 023612
14Liao RHuang Z GLin X MLiu W M 2013 Phys. Rev. 87 043605
15Zhu G BSun QZhang Y YChan K SLiu W MJi A C 2013 Phys. Rev. 88 023608
16Zhang S SYu X LYe JLiu W M 2013 Phys. Rev. 87 063623
17Chen G P 2015 Acta Phys. Sin. 64 030302 (in Chinese)
18Bao C GXie W FHe Y Z 2015 Chin. Phys. 24 060305
19Beeler M CWilliams R AJiménez-García KLeBlanc L JPerry A RSpielman I B 2013 Nature 498 201
20Liu X JJiang LPu HHu H 2012 Phys. Rev. 85 021603
21Liu X JHu H 2012 Phys. Rev. 85 033622
22Xu X QHan J H 2011 Phys. Rev. Lett. 107 200401
23Zhou X FZhou JWu C 2011 Phys. Rev. 84 063624
24Radić JSedrakyan TSpielman I BGalitski V 2011 Phys. Rev. 84 063604
25Merkl MJacob AZimmer F EOhberg PSantos L 2010 Phys. Rev. Lett. 104 073603
26Zhang Y PMao LZhang C W 2012 Phys. Rev. Lett. 108 035302
27Deng YCheng JJing HSun C PYi S 2012 Phys. Rev. Lett. 108 125301
28Hu HRamachandhran BPu HLiu X J 2012 Phys. Rev. Lett. 108 010402
29Song S WZhang Y CZhao HWang XLiu W M 2014 Phys. Rev. 89 063613
30Kawakami TMizushima TNitta MMachida K 2012 Phys. Rev. Lett. 109 015301
31Li YZhou X FWu C J2012arXiv:1205.2162
32Liu Y KYang S J 2015 Phys. Rev. 91 043616
33Liu Y KYang S J 2014 Europhys. Lett. 108 30004
34Brandon M ACharles W C 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134003
35Bao WChern I LZhang Y 2013 J. Comput. Phys. 253 189
36Bao WCai Y 2015 SIAM J. Appl. Math. 75 492
37Wang HXu Z 2014 Comput. Phys. Commun. 185 2803
38Kobayashi MNitta M2013arXiv:1304.6021v2