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Magnetized vector solitons in a spin-orbit coupled spin-1 Bose-Einstein condensate with Zeeman coupling
Physics Letters A 383 (2019) 2883–2890
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Physics Letters A www.elsevier.com/locate/pla
Magnetized vector solitons in a spin-orbit coupled spin-1 Bose-Einstein condensate with Zeeman coupling Ping Peng a,b , Guan-Qiang Li b , Li-Chen Zhao a , Wen-Li Yang a , Zhan-Ying Yang a a b
School of Physics and Shaanxi Key Laboratory for Theoretical Physics Frontiers, Northwest University, Xi’an 710069, China School of Arts and Sciences, Shaanxi University of Science and Technology, Xi’an 710021, China
a r t i c l e
i n f o
Article history: Received 24 October 2018 Received in revised form 3 June 2019 Accepted 6 June 2019 Available online 6 June 2019 Communicated by R. Wu Keywords: Spin-1 BEC Vector solitons Spin-orbit coupling Zeeman coupling Spin polarization Multiscale perturbation method
a b s t r a c t The property of matter-wave vector solitons in a spin-1 Bose-Einstein condensate with spin-orbit and Zeeman couplings is investigated by multiscale perturbation method. The excitation spectrum and the corresponding state vectors of the system are obtained analytically, and they can be adjusted by the parameters of the system. The bright and dark vector solitons are formulated by reducing the threecomponent coupled Gross-Pitaeviskii equations to a standard nonlinear Schrödinger equation, which has the solutions of the bright and dark solitons with positive or negative mass depending on the product of the effective dispersive and nonlinear coefficients. The moving vector solitons are demonstrated by adjusting specific momentum near the energy minimum. Finally, the magnetized features of the vector solitons are discussed by the spin polarization of the system. © 2019 Elsevier B.V. All rights reserved.
1. Introduction As a self-reinforcing solitary wave, soliton can traverse at a constant velocity without changing its shape due to a balance between the dispersion and nonlinear interaction [1]. The solitary waves have been observed and investigated in a variety of contexts, ranging from water waves [2] and plasmas [3] to nonlinear optics [4] and atomic Bose-Einstein condensates (BECs) [5]. The research progress of the ultracold atomic gases in recent twenty years demonstrated BEC is an ideal platform for studying the solitary waves and their related phenomena [6]. Many achievements have been obtained for matter-wave solitons in BECs, including bright [7–12] and dark [13–15] solitons in single-component system and so-called vector solitons in two-component one [16–18]. The realization of the spin-orbit coupling (SOC) in two-component spin-1/2 BECs makes it possible to generate even more exotic states, such as striped solitons [24–26] and semi-vortex and mixed-mode solitons [27]. Not only that, the critical and supercritical collapse of the self-attractive BECs in higher dimensions can be suppressed by the SOC and the stable higher dimensional solitons in free space without ground state may be realized in spin-orbit coupled BECs [28–30]. Recently, the developments of the three-component spin-1 BECs have further enriched the investigation of solitons in matter waves [19–23]. The spinor-1 BECs introduce more tunable parameters, for example, the interaction between the three species, not only generating new effects, but also bringing in novel nonlinear structures, which have no counterparts in the scalar and two-component BECs [31,32]. The interatomic interactions in spinor BECs lead to inherent nonlinearity under the coupled Gross-Pitaevskii mean-field description, and the interaction strength could also be tuned by magnetic or optical Feshbach resonances [33], which make the investigation of the system complex but necessary. The addition of the SOC in the spin-1 BECs make the study even more interesting, many new results were revealed, such as ground state [34–37], phase transition [38–41], synthetic dimension [42], spin squeezing [43], and so on. The theoretical work on solitary waves in SO-coupled spin-1 BECs has increased significantly [44–47], and many other nonlinear aspects of these systems such as vortex have been also explored [48,49]. In this letter, we aim to investigate the vector solitons in a spin-orbit coupled spin-1 BEC with Zeeman coupling. The paper is organized as follows. Section 2 describes our theoretical models of a spin-1 BEC with spin-orbit and Zeeman couplings. In Sec. 3, we give our
E-mail address: [email protected] (Z.-Y. Yang). https://doi.org/10.1016/j.physleta.2019.06.006 0375-9601/© 2019 Elsevier B.V. All rights reserved.
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analysis for the system by a multiscale perturbation method. The excitation spectrum and the corresponding states of the system without considering the atomic interactions are obtained analytically. Meanwhile, the three-component coupled Gross-Pitaeviskii equations (cGPEs) for the system are reduced to a standard nonlinear Schrödinger equation (NLSE) and the effective dispersive and nonlinear coefficients for the latter equation are also given as two analytical expressions. The information of the physical parameters of the system are included in the coefficients, and their effects on the characteristics of the solution are discussed. In Sec. 4, the bright and dark vector solitons with positive or negative mass depending on the product of the dispersive and effective nonlinear coefficients are constructed. The magnetized features of the vector solitons are also discussed by the spin polarization of the system in this section. These features are suitable not only for bright soliton but also for dark soliton. Finally, a brief discussion and a short conclusion are given in Sec. 5. 2. Theoretical model The spinor BEC with SOC can be realized experimentally by counter-propagating Raman lasers along x to couple the three hyperfine states with momentum transfer of the Raman process 2k R [36,41]. The order parameter of the spin-1 BEC in the mean-field framework is (r, t ) = (1 , 0 , −1 )T , containing all necessary information of the system. The ground-state properties of the system can be obtained by minimizing the energy functional E († , ) =
dr † H 0,3D +
g 03D 2
(† )2 +
g 23D 2
(† F)2
with H 0,3D = H 0 (x) + [¯h2 (k2y + k2z )/2m +
V (r)] 1, where 1 is the 3 × 3 unit matrix. The density of the atomic gas n(r) = † (r)(r) meets the normalized constraint drn(r) = N. The atomic interactions g 03D = 4π h¯ 2 (a0 + 2a2 )/3m and g 23D = 4π h¯ 2 (a2 − a0 )/3m are related to the s-wave scattering lengths in the spin-0 2 (a0 ) and the spin-2 (a2 ) channels. The external harmonic trapping potential is represented by V (r) = 12 m[ωx2 x2 + ω⊥ ( y 2 + z2 )]. When the trapping frequency along the y and z axes is much larger than along the x axis (ω⊥ ωx ), the original mean-field model for the spin-1 BEC can be safely analyzed in a quasi-one-dimensional (quasi-1D) trapping geometry. The dynamics of the system is described by
i
∂ (t , x) = H 0 (x)(t , x) + g 01D |(t , x)|2 (t , x) + g 21D [(t , x)† F(t , x)] · F(t , x), ∂t
(1)
where F = ( F x , F y , F z ) is the spin-1 operator [47,50]. The single-particle Hamiltonian is written as
⎛ ⎜
H 0 (x) = h¯ ⎝
h¯ (kx +2k R )2 2m
R 2
0
−δ
R h¯ k2x 2m
2
− ε R 2
⎞
0
⎟ ⎠,
R 2 h¯ (kx −2k R )2 2m
+δ
(2)
3D 2 the interactions between the atoms are g 01D ¯ /(mω⊥ ). The Raman coupling, represented by R , induces tran,2 = g 0,2 /(2π l⊥ ), and l⊥ = h sitions between the three spin components of the spin-1 BEC. The spinor nature of the system allows to introduce effective linear and quadratic Zeeman terms in external magnetic field, with √strength δ and ε , respectively. 3/ 2 1 j ( ψ Using t = ω⊥ t, x = l− x, and t , x ) = l ( t , x )/ N ( j = −1, 0, 1) as dimensionless scales, the mean-field equation for the spin-1 j ⊥ ⊥ BEC with SOC and Zeeman coupling can be written as follows
∂ψ1 ∂ψ1 ∗ = H (x)ψ1 − i γ − δψ1 + R ψ0 + g 2 |ψ1 |2 + |ψ0 |2 − |ψ−1 |2 ψ1 + g 2 ψ02 ψ− 1, ∂t ∂x ∂ψ0 = H (x)ψ0 − ε ψ0 + R (ψ1 + ψ−1 ) + g 2 |ψ1 |2 + |ψ−1 |2 ψ0 + 2g 2 ψ1 ψ−1 ψ0∗ , i ∂t ∂ψ−1 ∂ψ−1 = H (x)ψ−1 + i γ + δψ−1 + R ψ0 + g 2 |ψ−1 |2 + |ψ0 |2 − |ψ1 |2 ψ−1 + g 2 ψ02 ψ1∗ . i ∂t ∂x i
(3) (4) (5) 2
Here, the tildes overhead the corresponding variables have been omitted for convenience. The Hamiltonian H (x) = − 12 ∂∂x2 + V (x) + 1 2 g0 j =−1 |ψ j | . Eqs. (3)-(5) are called as the coupled Gross-Pitaevskii equations (cGPEs), including complex linear and nonlinear inter−3 actions. The nonlinear interactions are described by g 0,2 = g 01D hω⊥ ), which can be controlled by the external fields [31–33]. The ,2 Nl⊥ /(¯ strength of SOC is defined as γ ≡ 2l⊥ k R , and the other parameters are normalized by the transformations δ/ω⊥ → δ , R /2ω⊥ → R , ε /ω⊥ → ε . These external field parameters altogether determine the properties of the solutions for the system. The trapping potential will be ignored in the next sections for analyzing the motions of the solitons since it doesn’t affect the physics in essence [seen in Sec. 5]. In Ref. [27], the combined effects coming from the SOC and Zeeman splitting on the semi-vortex and mix-mode solitons in quasi-2D spin-1/2 BECs were investigated by the variational method and direct numerical simulation. Here, such kind of the combined effects are generalized to the vector solitons in the quasi-1D spin-1 BECs and the approximate analytical solutions and magnetized properties of the system are especially emphasized.
3. Analysis by multiscale perturbation method We now reduce the system of Eqs. (3)-(5) to a scalar NLSE by using a multiscale perturbation method [46]. Such a method allows us to derive approximate analytical bright and dark soliton solutions with positive or negative mass, depending on the type of the effective dispersion and nonlinear interaction. In order to obtain the solutions of Eqs. (3)-(5), we introduce the ansatz (ψ1 , ψ0 , ψ−1 ) T = ∞ n i (kx−μt ) , where the vectors un = (U n , V n , W n ) T φn are composed by the real coefficients U n , V n and W n , the unknown field n=1 un e envelopes φn ≡ φn ( T , X ). The latter are assumed to be functions of the slow variables T = 2 t and X = (x − v g t ), where v g is the group velocity. Additionally, k is the momentum, μ = ω + 2 ω0 is the chemical potential, ω is the energy in the linear limit, 2 ω0 is a small deviation about this energy ( 1), and ω0 /ω = O (1).
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Fig. 1. Distribution of the excitation spectra of the spin-orbit coupled spin-1 BEC with Zeeman coupling in momentum space: (a, d) ε = 15; (b, e) ε = 8; (c, f) ε = −3. The linear Zeeman coupling δ = 0 in (a)-(c) and δ = 3 in (d)-(f). The other parameters are γ = 4 and R = 1. The red solid, blue long dashed, and green short dashed lines represent ω1 , ω2 , and ω3 , respectively.
Substituting the above ansatz into Eqs. (3)-(5), we obtain the following set of equations at order O ( ), O ( 2 ) and O ( 3 ), respectively:
M1 u1 = 0,
(6)
M1 u2 = iM2 ∂ X u1 ,
(7)
and
M1 u3 = iM2 ∂ X u2 − (i ∂ T +
1 2
The matrices M1 and M3 are
⎛
M1 = ⎝
ω − k 2 /2 − k γ + δ − R 0
and
⎛
∂ X2 − M3 + ω0 )u1 .
(8)
⎞ − R 0 ⎠, ω − k 2 /2 + ε − R 2 − R ω − k /2 + k γ − δ
(9)
⎞
g 0 U 12 + V 12 + W 12 + C 11 g2 V 1 W 1 0 ⎠ |φ1 |2 , ⎝ M3 = (10) g2 V 1 W 1 g 0 U 12 + V 12 + W 12 + C 22 g V 2 1U1 2 2 2 0 g2 V 1 U 1 g 0 U 1 + V 1 + W 1 + C 33 where C 11 = g 2 U 12 + V 12 − W 12 , C 22 = g 2 U 12 + W 12 , and C 33 = g 2 W 12 + V 12 − U 12 , and the matrix M2 = M 1 (where the prime demotes
hereafter differentiation with respect to k). At O ( ), using the solvability condition detM1 = 0, we can obtain the excitation spectra ω1,2,3 ≡ ω1,2,3 (k; γ , , ε , δ), which correspond to three different bands of the system. The spectra can be tuned by the external field parameters. We can calculate the distribution of the energy level in momentum space, and the results for changing ε and δ under given γ and R are shown in Fig. 1. It is showed that the lowest level of the spectra has a single local minimum, two and three degenerate minima depending on the value of ε [Fig. 1 (a)-(c)]. The spectra are symmetric with respect to k = 0 for δ = 0, while the symmetry is broken for δ = 0 [Fig. 1 (d)-(f)]. The energymomentum dispersion for the nonlinear ground states here is very different from the situation for the plane-wave-type ground state of the spin-1 BEC with SOC in Ref. [50], in which the three levels of the energy for the system are independent of each other. In the spin-orbit coupled spin-1/2 BEC, the lowest energy level can only be transformed between the cases of one minimum and two minima by changing the strength ratio between the SOC and Raman coupling or by changing the atomic interactions [34]. The transformation leads to the quantum phase transition between the plane wave phase and stripe phase. The case of three minima of the lowest energy level for spin-orbit coupled spin-1 BEC means there may exist two different types of stripe-like ground states [38,39]. In the following parts, we will focus on nonlinear states, in the form of matter-wave solitons, corresponding to the lowest energy band. At the same order O ( ), it yields:
⎛
⎞
U1 u1 = ⎝ V 1 ⎠ φ1 ( T , X ), W1
(11)
where U 1 = U 1 (k; γ , , ε , δ), V 1 = V 1 (k; γ , , ε , δ) and W 1 = W 1 (k; γ , , ε , δ), which can be found analytically but through cumbersome expressions. That is, the eigenvector of the system at single particle level is a function of atom momentum and determined by
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Fig. 2. Effective dispersion given in Eq. (14) affected by the parameters of the system: (a) ε = 8, R = 1 and δ = 0; (b) δ = 0; (d) γ = 4, R = 1 and ε = 8.
γ = 4, ε = 8 and δ = 0; (c) γ = 4, R = 1 and
the parameters of the system. The compatibility condition of Eq. (7) is (U 1 , V 1 , W 1 )M2 (U 1 , V 1 , W 1 ) T = 0, which enforces a group velocity v g = ∂ ω/∂ k = k + γ (U 12 − W 12 )/(U 12 + V 12 + W 12 ). It means that ω and k should be evaluated at stationary points of the energy spectrum. Furthermore, at the order O ( 2 ), we can obtain the solution
⎛
⎞
P u2 = −i ⎝ Q ⎠ ∂ X φ1 ( T , X ), R
(12)
where P = [2R ( v g − k) V 1 + R ( v g − k + γ ) A 22 W 1 − (2R − A 22 A 33 )( v g − k − γ )U 1 ]/ M, Q = [ R ( v g − k − γ ) A 33 U 1 + ( v g − k + γ )( R A 11 W 1 + 2R W 1 ) + ( v g − k) A 11 A 33 V 1 ]/ M, R = [ R ( v g − k) A 11 V 1 + ( v g − k + γ ) A 11 A 22 W 1 + 2R ( v g − k − γ )U 1 ]/ M, M = A 11 (2R − A 22 A 33 ) + 3R , A 11 = ω − k2 /2 − kγ + δ , A 22 = ω − k2 /2 + ε , and A 33 = ω − k2 /2 + kγ − δ , and the latter three of which are the diagonal elements of M1 . Finally, considering the compatibility condition for Eq. (8) at O ( 3 ), together with Eq. (11) and (12), we obtain the following scalar NLSE for the unknown field φ1 :
i ∂ T φ1 = −
2
∂ X2 φ1 + ge f f |φ1 |2 φ1 − ω0 φ1 ,
(13)
where the coefficient of the dispersion
=1+
2γ ( P U 1 − R W 1 )(U 12 + V 12 + W 12 ) + ( W 12 − U 12 )( P U 1 + Q V 1 + R W 1 )
(U 12 + V 12 + W 12 )2
and the coefficient of the nonlinear interaction
ge f f =
g 0 (U 12
+
V 12
+
W 12 ) +
g 2 U 12 (U 12 + V 12 − W 12 ) + V 12 (U 12 + W 12 ) + W 12 ( W 12 + V 12 − U 12 ) + 4U 1 V 12 W 1 U 12 + V 12 + W 12
(14)
, .
(15)
The sign of the coefficients in the above effective NLSE is crucial in determining the types of solitons that can be supported: for g e f f > 0 the solitons are dark, while for g e f f < 0 the solitons are bright [46,47]. It can be seen from Eq. (14) and Eq. (15) that both of the coefficients of the dispersion and nonlinearity are determined by the parameters of the system. The dispersive features of the NLSE (13) based on Eq. (14) are demonstrated in Fig. 2. The change of the effective dispersive coefficient with momentum are shown by changing the strength of the SOC γ in Fig. 2 (a). It is found that the effective dispersion doesn’t change with the momentum for γ = 0. But for γ = 0, the distribution of the dispersion along the momentum axis becomes nontrivial, and there exist two regions with negative value at both sides of k = 0. The absolute value of the dispersion becomes even more larger for more stronger SOC. The dispersion changes dramatically near k = 0, while for |k| 1 the effective dispersion stays at = 1. The distributions are symmetrical about k = 0 since δ = 0 is considered. The role played by the Raman coupling is given in Fig. 2 (b). Similar distribution of the dispersion is obtained but the absolute value of the dispersion in the negative region becomes even more larger for weaker Raman coupling. In Fig. 2 (c), the change of the effective dispersion with momentum is displayed by changing the strength of the quadratic Zeeman coupling ε . For ε = −3, only one negative region in the distribution at k = 0. If we increase ε ’s value under keeping other parameters unchanged, two such negative regions may occur. The absolute value of the dispersion near k = 0 for ε = −3 is larger than others. The linear Zeeman coupling shifts the positions of the negative dispersion regions, as shown in Fig. 2 (d). This means that the dispersive characteristics of the system can be controlled by the external field parameters. The dispersion is related with the effective mass of the system, it makes the construction of the vector solitons with positive and negative mass possible.
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Fig. 3. Effective nonlinear interaction ge f f given in Eq. (15) affected by the parameters of the system. The values of the atomic interactions are chosen as g 1 = 1.0 and g 2 = −0.2. The other parameters of the system are the same as that in Fig. 2.
Based on Eq. (15), the effective nonlinear interaction in the NLSE (13) can also be demonstrated in Fig. 3. The effective nonlinear interaction is mainly affected by the density-density and spin-exchange interactions between the atoms. The positive or negative nonlinearity can be obtained by adjusting the atomic interactions g 0,2 conveniently. For the given atomic interactions, the effects of the parameters of the system on the effective nonlinear interaction of the NLSE is demonstrated in Fig. 3 (a)-(d). The effective interaction changes dramatically near k = 0, while for |k| 1 it approaches to g 0 + g 2 . The upper limit of the value of the effective interaction is g 0 at k = 0. In Fig. 3 (a), the effective interaction doesn’t change with the momentum for γ = 0. For γ = 0, the distribution of the effective interaction along the momentum axis broadens with increasing the strength of the SOC. The effect of the Raman coupling on the interaction is given in Fig. 3 (b). The maximum value of the effective interaction becomes smaller for larger Raman coupling. The quadratic Zeeman coupling not only broadens the distribution of the effective interaction along the momentum axis, but also enhances the maximum value. The linear Zeeman coupling only shifts the position of the distribution alone the momentum axis. The results are demonstrated in Fig. 3 (c) and (d), respectively. Based on the above results, we will demonstrate that it is not difficult to construct bright-type and dark-type matter-wave solitons in this system. Not only that, the external parameters of the system still have important effects on the properties of the solitons. 4. Solutions for bright and dark solitons We next proceed by presenting various types of exact soliton solutions of Eq. (13) and the corresponding approximate solutions of the original system of Eq. (3)-(5). Having obtained the solutions of Eq. (13), we may write the approximate vector soliton solutions of the Eqs. (3)-(5) for the spin-1 BEC with SOC and Zeeman coupling, in the original coordinates [46,47]. The type of the soliton, i.e., ϕ D (x, t ) or ϕ B (x, t ) is determined by the signs of and ge f f . The product of the effective dispersion and nonlinear interaction ge f f > 0 is related with the dark solitons while g e f f < 0 is for the bright solitons. For the situation of g e f f > 0, we can write the dark soliton solution of Eq. (13), with ω0 > 0, in the form
⎛
⎞ ⎛ ⎞ ⎛ ⎞ ψ1 U1 U1
⎝ ψ0 ⎠ ≈ ⎝ V 1 ⎠ ϕ D (x, t ) ≈ ⎝ V 1 ⎠ ω0 / ge f f (cos θ tanh z D + i sin θ) exp[ikx − i (ω + 2 ω0 )t ], ψ−1 W1 W1
(16)
√
where z D = ω0 / cos θ[ X − X 0 ( T )] and X 0 ( T ) is the soliton center.
√ The phase angle θ (|θ| < π /2) controls the soliton amplitude, ω0 / ge f f cos θ , and soliton velocity through the equation d X 0 /dT = ω0 / sin θ . Note that the above soliton solution is characterized by two free parameters, ω0 and θ . The former is for the background and the latter for the soliton. The limiting case θ = 0 corresponds to a stationary soliton, while θ = 0 give rise to traveling solitons. The stationary dark solitons can be obtained by focusing on the energy minimum (ωmin , kmin ) of the lowest energy level. Here, the moving dark solitons are demonstrated in Fig. 4 by setting the parameters 2 ω0 = 0.01. θ = π /4, > 0 (negative mass) and g e f f are determined by the same value of the parameters with Fig. 1 (b) at k = 0.8 (nearby kmin ). It is found that the amplitudes U 1 , V 1 and W 1 for the three components of the dark vector solitons have different values. Actually, they can also be called as gray solitons with different amplitudes. But the moving velocities for them are same with each other. They move along the positive direction of the x axis for k > 0, the opposite is true for k < 0. For the situation of g e f f < 0, a bright soliton solution of Eq. (13), with ω0 < 0, can be written as
⎛
⎞ ⎛ ⎞ ⎛ ⎞ U1 U1 ψ1 ⎝ ψ0 ⎠ ≈ ⎝ V 1 ⎠ ϕ B (x, t ) ≈ ⎝ V 1 ⎠ η sech z B exp[i (k + κ )x − i (ω − 2 ω0 )t ], ψ−1 W1 W1
where z B = η − g e f f /[ X − X 0 ( T )]. Here,
(17)
η is the soliton amplitude which is connected with the parameter ω0 via the relation ω0 = (κ − η ge f f /)/2, X 0 ( T ) is the soliton center, and the wavenumber κ is connected with the soliton velocity through d X 0 /dT = κ . 2
2
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Fig. 4. Dark vector soliton given in Eq. (16): (a) t = 0; (b) t = 10; (c) t = 20; (d) t = 30. The parameters γ = 4, ε = 8, δ = 0, and R = 1 for all of the cases. The different types of the lines (red-solid, blue-dashed, and green-dot dashed lines) correspond to |1 |2 , |0 |2 , and |−1 |2 , respectively.
Fig. 5. Brigh vector soliton given in Eq. (17): (a) t = 0; (b) t = 10; (c) t = 20; (d) t = 30. The parameters γ = 4, ε = 8, δ = 0, and R = 1 for all of the cases. The different types of the lines (red-solid, blue-dashed, and green-dot dashed lines) correspond to |1 |2 , |0 |2 , and |−1 |2 , respectively.
Fig. 6. (a) Change of the spin polarization S z for the vector soliton with momentum affected by the Zeeman coupling for given other parameters. (b) The width S z of the platform corresponding to S z 0 can be tuned by the SOC, Raman and quadratic Zeeman couplings.
As in the previous case, the above soliton solution also has two free parameters, ω0 and κ . In Fig. 5, the moving bright solitons are demonstrated by adjusting the parameters 2 ω0 = −0.01, θ = π /3, < 0 (negative mass) and g e f f at k = 0.8. The features of the amplitudes and motions of the solitons for the three components almost are same with the situation for the dark solitons. It is noted that we have not found the difference between the bright vector solitons with positive and negative masses in our study since the analytical properties for two cases are same with each other. For the dark and bright vector solitons discussed above, the three components own obvious polarized features. The spin polarization of the vector solitons can be described by S z ≡ |U 1 |2 − | W 1 |2 . Using the spin polarization, we can discuss the magnetized properties of the system. The change of the spin polarization of the system with momentum can be investigated under different values of the parameters of the system. The result affected by the quadratic Zeeman coupling can be seen in the Fig. 6 (a). For smaller ε , the value of the spin polarization changes steeply from 1 to −1 at the two sides of k = 0. | S z | = 1 denotes perfect spin polarization of the vector solitons. When the strength of the quadratic Zeeman coupling is increased, there exists a platform with zero spin polarization near k = 0. The larger the strength of the quadratic Zeeman coupling, the wider the platform with zero spin polarization. The platform with zero spin polarization supports the vector solitons with spin unpolarization. We can extract the width of the platform with zero spin polarization, labeling as S z and consider the effects of the external field parameters on the width. The result is given in Fig. 6 (b). It is found the width S z changes with ε almost linearly under different ratio of the SOC and Raman coupling γ / R . The slope of the oblique line becomes smaller for even larger ratio γ / R . The linear Zeeman coupling δ has no effect on the spin polarization. That is to say, the magnetized features of the vector solitons are easily adjusted by the parameters of the system.
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5. Conclusion In summary, we have studied the matter-wave vector solitons in spin-1 Bose-Einstein condensate with SOC and Zeeman coupling by the multiscale perturbation method. The positive- and negative-mass bright or dark soliton solutions depend on the parameters of the system can be obtained analytically by reducing the cGPEs to a scalar NLSE. The effects of the external field parameters on the effective dispersion and nonlinear interaction of the NLSE are discussed in detail. It is verified that the existence of the positive and negative bright (dark) solitons with positive or negative mass can be supported in this system. The typical bright and dark vector solitons are given in some specific parameters. In addition, it is found that the spin polarization of the vector solitons are easily adjusted by the parameter of the system, which demonstrating the magnetized features of the vector solitons. Our results demonstrated that the different kinds of the vector solitons may exist in the spin-orbit coupled spin-1 BECs with Zeeman coupling, and the adjustability of the external parameters make the system to be a good platform for future investigation of the vector solitons and their dynamics. Acknowledgements This research was supported by the National Natural Science Foundation of China under Grants No. 11875220, No. 11775176 and No. 11405100, the Natural Science Basic Research Plan in Shaanxi Province of China under Grant No. 2019JM-332, and the Doctoral Research Fund of Shaanxi University of Science and Technology, China, under Grant No. 2018BJ-02. References [1] Y.V. Kartashov, B.A. Malomed, L. Torner, Solitons in nonlinear lattices, Rev. Mod. Phys. 83 (2011) 247. [2] A. Chabchoub, O. Kimmoun, H. Branger, N. Hoffmann, D. Proment, M. Onorato, N. Akhmediev, Experimental observation of dark solitons on the surface of water, Phys. Rev. Lett. 110 (2013) 124101. [3] R. Heidemann, S. 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Physics Letters A www.elsevier.com/locate/pla
Magnetized vector solitons in a spin-orbit coupled spin-1 Bose-Einstein condensate with Zeeman coupling Ping Peng a,b , Guan-Qiang Li b , Li-Chen Zhao a , Wen-Li Yang a , Zhan-Ying Yang a a b
School of Physics and Shaanxi Key Laboratory for Theoretical Physics Frontiers, Northwest University, Xi’an 710069, China School of Arts and Sciences, Shaanxi University of Science and Technology, Xi’an 710021, China
a r t i c l e
i n f o
Article history: Received 24 October 2018 Received in revised form 3 June 2019 Accepted 6 June 2019 Available online 6 June 2019 Communicated by R. Wu Keywords: Spin-1 BEC Vector solitons Spin-orbit coupling Zeeman coupling Spin polarization Multiscale perturbation method
a b s t r a c t The property of matter-wave vector solitons in a spin-1 Bose-Einstein condensate with spin-orbit and Zeeman couplings is investigated by multiscale perturbation method. The excitation spectrum and the corresponding state vectors of the system are obtained analytically, and they can be adjusted by the parameters of the system. The bright and dark vector solitons are formulated by reducing the threecomponent coupled Gross-Pitaeviskii equations to a standard nonlinear Schrödinger equation, which has the solutions of the bright and dark solitons with positive or negative mass depending on the product of the effective dispersive and nonlinear coefficients. The moving vector solitons are demonstrated by adjusting specific momentum near the energy minimum. Finally, the magnetized features of the vector solitons are discussed by the spin polarization of the system. © 2019 Elsevier B.V. All rights reserved.
1. Introduction As a self-reinforcing solitary wave, soliton can traverse at a constant velocity without changing its shape due to a balance between the dispersion and nonlinear interaction [1]. The solitary waves have been observed and investigated in a variety of contexts, ranging from water waves [2] and plasmas [3] to nonlinear optics [4] and atomic Bose-Einstein condensates (BECs) [5]. The research progress of the ultracold atomic gases in recent twenty years demonstrated BEC is an ideal platform for studying the solitary waves and their related phenomena [6]. Many achievements have been obtained for matter-wave solitons in BECs, including bright [7–12] and dark [13–15] solitons in single-component system and so-called vector solitons in two-component one [16–18]. The realization of the spin-orbit coupling (SOC) in two-component spin-1/2 BECs makes it possible to generate even more exotic states, such as striped solitons [24–26] and semi-vortex and mixed-mode solitons [27]. Not only that, the critical and supercritical collapse of the self-attractive BECs in higher dimensions can be suppressed by the SOC and the stable higher dimensional solitons in free space without ground state may be realized in spin-orbit coupled BECs [28–30]. Recently, the developments of the three-component spin-1 BECs have further enriched the investigation of solitons in matter waves [19–23]. The spinor-1 BECs introduce more tunable parameters, for example, the interaction between the three species, not only generating new effects, but also bringing in novel nonlinear structures, which have no counterparts in the scalar and two-component BECs [31,32]. The interatomic interactions in spinor BECs lead to inherent nonlinearity under the coupled Gross-Pitaevskii mean-field description, and the interaction strength could also be tuned by magnetic or optical Feshbach resonances [33], which make the investigation of the system complex but necessary. The addition of the SOC in the spin-1 BECs make the study even more interesting, many new results were revealed, such as ground state [34–37], phase transition [38–41], synthetic dimension [42], spin squeezing [43], and so on. The theoretical work on solitary waves in SO-coupled spin-1 BECs has increased significantly [44–47], and many other nonlinear aspects of these systems such as vortex have been also explored [48,49]. In this letter, we aim to investigate the vector solitons in a spin-orbit coupled spin-1 BEC with Zeeman coupling. The paper is organized as follows. Section 2 describes our theoretical models of a spin-1 BEC with spin-orbit and Zeeman couplings. In Sec. 3, we give our
E-mail address: [email protected] (Z.-Y. Yang). https://doi.org/10.1016/j.physleta.2019.06.006 0375-9601/© 2019 Elsevier B.V. All rights reserved.
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analysis for the system by a multiscale perturbation method. The excitation spectrum and the corresponding states of the system without considering the atomic interactions are obtained analytically. Meanwhile, the three-component coupled Gross-Pitaeviskii equations (cGPEs) for the system are reduced to a standard nonlinear Schrödinger equation (NLSE) and the effective dispersive and nonlinear coefficients for the latter equation are also given as two analytical expressions. The information of the physical parameters of the system are included in the coefficients, and their effects on the characteristics of the solution are discussed. In Sec. 4, the bright and dark vector solitons with positive or negative mass depending on the product of the dispersive and effective nonlinear coefficients are constructed. The magnetized features of the vector solitons are also discussed by the spin polarization of the system in this section. These features are suitable not only for bright soliton but also for dark soliton. Finally, a brief discussion and a short conclusion are given in Sec. 5. 2. Theoretical model The spinor BEC with SOC can be realized experimentally by counter-propagating Raman lasers along x to couple the three hyperfine states with momentum transfer of the Raman process 2k R [36,41]. The order parameter of the spin-1 BEC in the mean-field framework is (r, t ) = (1 , 0 , −1 )T , containing all necessary information of the system. The ground-state properties of the system can be obtained by minimizing the energy functional E († , ) =
dr † H 0,3D +
g 03D 2
(† )2 +
g 23D 2
(† F)2
with H 0,3D = H 0 (x) + [¯h2 (k2y + k2z )/2m +
V (r)] 1, where 1 is the 3 × 3 unit matrix. The density of the atomic gas n(r) = † (r)(r) meets the normalized constraint drn(r) = N. The atomic interactions g 03D = 4π h¯ 2 (a0 + 2a2 )/3m and g 23D = 4π h¯ 2 (a2 − a0 )/3m are related to the s-wave scattering lengths in the spin-0 2 (a0 ) and the spin-2 (a2 ) channels. The external harmonic trapping potential is represented by V (r) = 12 m[ωx2 x2 + ω⊥ ( y 2 + z2 )]. When the trapping frequency along the y and z axes is much larger than along the x axis (ω⊥ ωx ), the original mean-field model for the spin-1 BEC can be safely analyzed in a quasi-one-dimensional (quasi-1D) trapping geometry. The dynamics of the system is described by
i
∂ (t , x) = H 0 (x)(t , x) + g 01D |(t , x)|2 (t , x) + g 21D [(t , x)† F(t , x)] · F(t , x), ∂t
(1)
where F = ( F x , F y , F z ) is the spin-1 operator [47,50]. The single-particle Hamiltonian is written as
⎛ ⎜
H 0 (x) = h¯ ⎝
h¯ (kx +2k R )2 2m
R 2
0
−δ
R h¯ k2x 2m
2
− ε R 2
⎞
0
⎟ ⎠,
R 2 h¯ (kx −2k R )2 2m
+δ
(2)
3D 2 the interactions between the atoms are g 01D ¯ /(mω⊥ ). The Raman coupling, represented by R , induces tran,2 = g 0,2 /(2π l⊥ ), and l⊥ = h sitions between the three spin components of the spin-1 BEC. The spinor nature of the system allows to introduce effective linear and quadratic Zeeman terms in external magnetic field, with √strength δ and ε , respectively. 3/ 2 1 j ( ψ Using t = ω⊥ t, x = l− x, and t , x ) = l ( t , x )/ N ( j = −1, 0, 1) as dimensionless scales, the mean-field equation for the spin-1 j ⊥ ⊥ BEC with SOC and Zeeman coupling can be written as follows
∂ψ1 ∂ψ1 ∗ = H (x)ψ1 − i γ − δψ1 + R ψ0 + g 2 |ψ1 |2 + |ψ0 |2 − |ψ−1 |2 ψ1 + g 2 ψ02 ψ− 1, ∂t ∂x ∂ψ0 = H (x)ψ0 − ε ψ0 + R (ψ1 + ψ−1 ) + g 2 |ψ1 |2 + |ψ−1 |2 ψ0 + 2g 2 ψ1 ψ−1 ψ0∗ , i ∂t ∂ψ−1 ∂ψ−1 = H (x)ψ−1 + i γ + δψ−1 + R ψ0 + g 2 |ψ−1 |2 + |ψ0 |2 − |ψ1 |2 ψ−1 + g 2 ψ02 ψ1∗ . i ∂t ∂x i
(3) (4) (5) 2
Here, the tildes overhead the corresponding variables have been omitted for convenience. The Hamiltonian H (x) = − 12 ∂∂x2 + V (x) + 1 2 g0 j =−1 |ψ j | . Eqs. (3)-(5) are called as the coupled Gross-Pitaevskii equations (cGPEs), including complex linear and nonlinear inter−3 actions. The nonlinear interactions are described by g 0,2 = g 01D hω⊥ ), which can be controlled by the external fields [31–33]. The ,2 Nl⊥ /(¯ strength of SOC is defined as γ ≡ 2l⊥ k R , and the other parameters are normalized by the transformations δ/ω⊥ → δ , R /2ω⊥ → R , ε /ω⊥ → ε . These external field parameters altogether determine the properties of the solutions for the system. The trapping potential will be ignored in the next sections for analyzing the motions of the solitons since it doesn’t affect the physics in essence [seen in Sec. 5]. In Ref. [27], the combined effects coming from the SOC and Zeeman splitting on the semi-vortex and mix-mode solitons in quasi-2D spin-1/2 BECs were investigated by the variational method and direct numerical simulation. Here, such kind of the combined effects are generalized to the vector solitons in the quasi-1D spin-1 BECs and the approximate analytical solutions and magnetized properties of the system are especially emphasized.
3. Analysis by multiscale perturbation method We now reduce the system of Eqs. (3)-(5) to a scalar NLSE by using a multiscale perturbation method [46]. Such a method allows us to derive approximate analytical bright and dark soliton solutions with positive or negative mass, depending on the type of the effective dispersion and nonlinear interaction. In order to obtain the solutions of Eqs. (3)-(5), we introduce the ansatz (ψ1 , ψ0 , ψ−1 ) T = ∞ n i (kx−μt ) , where the vectors un = (U n , V n , W n ) T φn are composed by the real coefficients U n , V n and W n , the unknown field n=1 un e envelopes φn ≡ φn ( T , X ). The latter are assumed to be functions of the slow variables T = 2 t and X = (x − v g t ), where v g is the group velocity. Additionally, k is the momentum, μ = ω + 2 ω0 is the chemical potential, ω is the energy in the linear limit, 2 ω0 is a small deviation about this energy ( 1), and ω0 /ω = O (1).
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Fig. 1. Distribution of the excitation spectra of the spin-orbit coupled spin-1 BEC with Zeeman coupling in momentum space: (a, d) ε = 15; (b, e) ε = 8; (c, f) ε = −3. The linear Zeeman coupling δ = 0 in (a)-(c) and δ = 3 in (d)-(f). The other parameters are γ = 4 and R = 1. The red solid, blue long dashed, and green short dashed lines represent ω1 , ω2 , and ω3 , respectively.
Substituting the above ansatz into Eqs. (3)-(5), we obtain the following set of equations at order O ( ), O ( 2 ) and O ( 3 ), respectively:
M1 u1 = 0,
(6)
M1 u2 = iM2 ∂ X u1 ,
(7)
and
M1 u3 = iM2 ∂ X u2 − (i ∂ T +
1 2
The matrices M1 and M3 are
⎛
M1 = ⎝
ω − k 2 /2 − k γ + δ − R 0
and
⎛
∂ X2 − M3 + ω0 )u1 .
(8)
⎞ − R 0 ⎠, ω − k 2 /2 + ε − R 2 − R ω − k /2 + k γ − δ
(9)
⎞
g 0 U 12 + V 12 + W 12 + C 11 g2 V 1 W 1 0 ⎠ |φ1 |2 , ⎝ M3 = (10) g2 V 1 W 1 g 0 U 12 + V 12 + W 12 + C 22 g V 2 1U1 2 2 2 0 g2 V 1 U 1 g 0 U 1 + V 1 + W 1 + C 33 where C 11 = g 2 U 12 + V 12 − W 12 , C 22 = g 2 U 12 + W 12 , and C 33 = g 2 W 12 + V 12 − U 12 , and the matrix M2 = M 1 (where the prime demotes
hereafter differentiation with respect to k). At O ( ), using the solvability condition detM1 = 0, we can obtain the excitation spectra ω1,2,3 ≡ ω1,2,3 (k; γ , , ε , δ), which correspond to three different bands of the system. The spectra can be tuned by the external field parameters. We can calculate the distribution of the energy level in momentum space, and the results for changing ε and δ under given γ and R are shown in Fig. 1. It is showed that the lowest level of the spectra has a single local minimum, two and three degenerate minima depending on the value of ε [Fig. 1 (a)-(c)]. The spectra are symmetric with respect to k = 0 for δ = 0, while the symmetry is broken for δ = 0 [Fig. 1 (d)-(f)]. The energymomentum dispersion for the nonlinear ground states here is very different from the situation for the plane-wave-type ground state of the spin-1 BEC with SOC in Ref. [50], in which the three levels of the energy for the system are independent of each other. In the spin-orbit coupled spin-1/2 BEC, the lowest energy level can only be transformed between the cases of one minimum and two minima by changing the strength ratio between the SOC and Raman coupling or by changing the atomic interactions [34]. The transformation leads to the quantum phase transition between the plane wave phase and stripe phase. The case of three minima of the lowest energy level for spin-orbit coupled spin-1 BEC means there may exist two different types of stripe-like ground states [38,39]. In the following parts, we will focus on nonlinear states, in the form of matter-wave solitons, corresponding to the lowest energy band. At the same order O ( ), it yields:
⎛
⎞
U1 u1 = ⎝ V 1 ⎠ φ1 ( T , X ), W1
(11)
where U 1 = U 1 (k; γ , , ε , δ), V 1 = V 1 (k; γ , , ε , δ) and W 1 = W 1 (k; γ , , ε , δ), which can be found analytically but through cumbersome expressions. That is, the eigenvector of the system at single particle level is a function of atom momentum and determined by
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Fig. 2. Effective dispersion given in Eq. (14) affected by the parameters of the system: (a) ε = 8, R = 1 and δ = 0; (b) δ = 0; (d) γ = 4, R = 1 and ε = 8.
γ = 4, ε = 8 and δ = 0; (c) γ = 4, R = 1 and
the parameters of the system. The compatibility condition of Eq. (7) is (U 1 , V 1 , W 1 )M2 (U 1 , V 1 , W 1 ) T = 0, which enforces a group velocity v g = ∂ ω/∂ k = k + γ (U 12 − W 12 )/(U 12 + V 12 + W 12 ). It means that ω and k should be evaluated at stationary points of the energy spectrum. Furthermore, at the order O ( 2 ), we can obtain the solution
⎛
⎞
P u2 = −i ⎝ Q ⎠ ∂ X φ1 ( T , X ), R
(12)
where P = [2R ( v g − k) V 1 + R ( v g − k + γ ) A 22 W 1 − (2R − A 22 A 33 )( v g − k − γ )U 1 ]/ M, Q = [ R ( v g − k − γ ) A 33 U 1 + ( v g − k + γ )( R A 11 W 1 + 2R W 1 ) + ( v g − k) A 11 A 33 V 1 ]/ M, R = [ R ( v g − k) A 11 V 1 + ( v g − k + γ ) A 11 A 22 W 1 + 2R ( v g − k − γ )U 1 ]/ M, M = A 11 (2R − A 22 A 33 ) + 3R , A 11 = ω − k2 /2 − kγ + δ , A 22 = ω − k2 /2 + ε , and A 33 = ω − k2 /2 + kγ − δ , and the latter three of which are the diagonal elements of M1 . Finally, considering the compatibility condition for Eq. (8) at O ( 3 ), together with Eq. (11) and (12), we obtain the following scalar NLSE for the unknown field φ1 :
i ∂ T φ1 = −
2
∂ X2 φ1 + ge f f |φ1 |2 φ1 − ω0 φ1 ,
(13)
where the coefficient of the dispersion
=1+
2γ ( P U 1 − R W 1 )(U 12 + V 12 + W 12 ) + ( W 12 − U 12 )( P U 1 + Q V 1 + R W 1 )
(U 12 + V 12 + W 12 )2
and the coefficient of the nonlinear interaction
ge f f =
g 0 (U 12
+
V 12
+
W 12 ) +
g 2 U 12 (U 12 + V 12 − W 12 ) + V 12 (U 12 + W 12 ) + W 12 ( W 12 + V 12 − U 12 ) + 4U 1 V 12 W 1 U 12 + V 12 + W 12
(14)
, .
(15)
The sign of the coefficients in the above effective NLSE is crucial in determining the types of solitons that can be supported: for g e f f > 0 the solitons are dark, while for g e f f < 0 the solitons are bright [46,47]. It can be seen from Eq. (14) and Eq. (15) that both of the coefficients of the dispersion and nonlinearity are determined by the parameters of the system. The dispersive features of the NLSE (13) based on Eq. (14) are demonstrated in Fig. 2. The change of the effective dispersive coefficient with momentum are shown by changing the strength of the SOC γ in Fig. 2 (a). It is found that the effective dispersion doesn’t change with the momentum for γ = 0. But for γ = 0, the distribution of the dispersion along the momentum axis becomes nontrivial, and there exist two regions with negative value at both sides of k = 0. The absolute value of the dispersion becomes even more larger for more stronger SOC. The dispersion changes dramatically near k = 0, while for |k| 1 the effective dispersion stays at = 1. The distributions are symmetrical about k = 0 since δ = 0 is considered. The role played by the Raman coupling is given in Fig. 2 (b). Similar distribution of the dispersion is obtained but the absolute value of the dispersion in the negative region becomes even more larger for weaker Raman coupling. In Fig. 2 (c), the change of the effective dispersion with momentum is displayed by changing the strength of the quadratic Zeeman coupling ε . For ε = −3, only one negative region in the distribution at k = 0. If we increase ε ’s value under keeping other parameters unchanged, two such negative regions may occur. The absolute value of the dispersion near k = 0 for ε = −3 is larger than others. The linear Zeeman coupling shifts the positions of the negative dispersion regions, as shown in Fig. 2 (d). This means that the dispersive characteristics of the system can be controlled by the external field parameters. The dispersion is related with the effective mass of the system, it makes the construction of the vector solitons with positive and negative mass possible.
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Fig. 3. Effective nonlinear interaction ge f f given in Eq. (15) affected by the parameters of the system. The values of the atomic interactions are chosen as g 1 = 1.0 and g 2 = −0.2. The other parameters of the system are the same as that in Fig. 2.
Based on Eq. (15), the effective nonlinear interaction in the NLSE (13) can also be demonstrated in Fig. 3. The effective nonlinear interaction is mainly affected by the density-density and spin-exchange interactions between the atoms. The positive or negative nonlinearity can be obtained by adjusting the atomic interactions g 0,2 conveniently. For the given atomic interactions, the effects of the parameters of the system on the effective nonlinear interaction of the NLSE is demonstrated in Fig. 3 (a)-(d). The effective interaction changes dramatically near k = 0, while for |k| 1 it approaches to g 0 + g 2 . The upper limit of the value of the effective interaction is g 0 at k = 0. In Fig. 3 (a), the effective interaction doesn’t change with the momentum for γ = 0. For γ = 0, the distribution of the effective interaction along the momentum axis broadens with increasing the strength of the SOC. The effect of the Raman coupling on the interaction is given in Fig. 3 (b). The maximum value of the effective interaction becomes smaller for larger Raman coupling. The quadratic Zeeman coupling not only broadens the distribution of the effective interaction along the momentum axis, but also enhances the maximum value. The linear Zeeman coupling only shifts the position of the distribution alone the momentum axis. The results are demonstrated in Fig. 3 (c) and (d), respectively. Based on the above results, we will demonstrate that it is not difficult to construct bright-type and dark-type matter-wave solitons in this system. Not only that, the external parameters of the system still have important effects on the properties of the solitons. 4. Solutions for bright and dark solitons We next proceed by presenting various types of exact soliton solutions of Eq. (13) and the corresponding approximate solutions of the original system of Eq. (3)-(5). Having obtained the solutions of Eq. (13), we may write the approximate vector soliton solutions of the Eqs. (3)-(5) for the spin-1 BEC with SOC and Zeeman coupling, in the original coordinates [46,47]. The type of the soliton, i.e., ϕ D (x, t ) or ϕ B (x, t ) is determined by the signs of and ge f f . The product of the effective dispersion and nonlinear interaction ge f f > 0 is related with the dark solitons while g e f f < 0 is for the bright solitons. For the situation of g e f f > 0, we can write the dark soliton solution of Eq. (13), with ω0 > 0, in the form
⎛
⎞ ⎛ ⎞ ⎛ ⎞ ψ1 U1 U1
⎝ ψ0 ⎠ ≈ ⎝ V 1 ⎠ ϕ D (x, t ) ≈ ⎝ V 1 ⎠ ω0 / ge f f (cos θ tanh z D + i sin θ) exp[ikx − i (ω + 2 ω0 )t ], ψ−1 W1 W1
(16)
√
where z D = ω0 / cos θ[ X − X 0 ( T )] and X 0 ( T ) is the soliton center.
√ The phase angle θ (|θ| < π /2) controls the soliton amplitude, ω0 / ge f f cos θ , and soliton velocity through the equation d X 0 /dT = ω0 / sin θ . Note that the above soliton solution is characterized by two free parameters, ω0 and θ . The former is for the background and the latter for the soliton. The limiting case θ = 0 corresponds to a stationary soliton, while θ = 0 give rise to traveling solitons. The stationary dark solitons can be obtained by focusing on the energy minimum (ωmin , kmin ) of the lowest energy level. Here, the moving dark solitons are demonstrated in Fig. 4 by setting the parameters 2 ω0 = 0.01. θ = π /4, > 0 (negative mass) and g e f f are determined by the same value of the parameters with Fig. 1 (b) at k = 0.8 (nearby kmin ). It is found that the amplitudes U 1 , V 1 and W 1 for the three components of the dark vector solitons have different values. Actually, they can also be called as gray solitons with different amplitudes. But the moving velocities for them are same with each other. They move along the positive direction of the x axis for k > 0, the opposite is true for k < 0. For the situation of g e f f < 0, a bright soliton solution of Eq. (13), with ω0 < 0, can be written as
⎛
⎞ ⎛ ⎞ ⎛ ⎞ U1 U1 ψ1 ⎝ ψ0 ⎠ ≈ ⎝ V 1 ⎠ ϕ B (x, t ) ≈ ⎝ V 1 ⎠ η sech z B exp[i (k + κ )x − i (ω − 2 ω0 )t ], ψ−1 W1 W1
where z B = η − g e f f /[ X − X 0 ( T )]. Here,
(17)
η is the soliton amplitude which is connected with the parameter ω0 via the relation ω0 = (κ − η ge f f /)/2, X 0 ( T ) is the soliton center, and the wavenumber κ is connected with the soliton velocity through d X 0 /dT = κ . 2
2
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Fig. 4. Dark vector soliton given in Eq. (16): (a) t = 0; (b) t = 10; (c) t = 20; (d) t = 30. The parameters γ = 4, ε = 8, δ = 0, and R = 1 for all of the cases. The different types of the lines (red-solid, blue-dashed, and green-dot dashed lines) correspond to |1 |2 , |0 |2 , and |−1 |2 , respectively.
Fig. 5. Brigh vector soliton given in Eq. (17): (a) t = 0; (b) t = 10; (c) t = 20; (d) t = 30. The parameters γ = 4, ε = 8, δ = 0, and R = 1 for all of the cases. The different types of the lines (red-solid, blue-dashed, and green-dot dashed lines) correspond to |1 |2 , |0 |2 , and |−1 |2 , respectively.
Fig. 6. (a) Change of the spin polarization S z for the vector soliton with momentum affected by the Zeeman coupling for given other parameters. (b) The width S z of the platform corresponding to S z 0 can be tuned by the SOC, Raman and quadratic Zeeman couplings.
As in the previous case, the above soliton solution also has two free parameters, ω0 and κ . In Fig. 5, the moving bright solitons are demonstrated by adjusting the parameters 2 ω0 = −0.01, θ = π /3, < 0 (negative mass) and g e f f at k = 0.8. The features of the amplitudes and motions of the solitons for the three components almost are same with the situation for the dark solitons. It is noted that we have not found the difference between the bright vector solitons with positive and negative masses in our study since the analytical properties for two cases are same with each other. For the dark and bright vector solitons discussed above, the three components own obvious polarized features. The spin polarization of the vector solitons can be described by S z ≡ |U 1 |2 − | W 1 |2 . Using the spin polarization, we can discuss the magnetized properties of the system. The change of the spin polarization of the system with momentum can be investigated under different values of the parameters of the system. The result affected by the quadratic Zeeman coupling can be seen in the Fig. 6 (a). For smaller ε , the value of the spin polarization changes steeply from 1 to −1 at the two sides of k = 0. | S z | = 1 denotes perfect spin polarization of the vector solitons. When the strength of the quadratic Zeeman coupling is increased, there exists a platform with zero spin polarization near k = 0. The larger the strength of the quadratic Zeeman coupling, the wider the platform with zero spin polarization. The platform with zero spin polarization supports the vector solitons with spin unpolarization. We can extract the width of the platform with zero spin polarization, labeling as S z and consider the effects of the external field parameters on the width. The result is given in Fig. 6 (b). It is found the width S z changes with ε almost linearly under different ratio of the SOC and Raman coupling γ / R . The slope of the oblique line becomes smaller for even larger ratio γ / R . The linear Zeeman coupling δ has no effect on the spin polarization. That is to say, the magnetized features of the vector solitons are easily adjusted by the parameters of the system.
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5. Conclusion In summary, we have studied the matter-wave vector solitons in spin-1 Bose-Einstein condensate with SOC and Zeeman coupling by the multiscale perturbation method. The positive- and negative-mass bright or dark soliton solutions depend on the parameters of the system can be obtained analytically by reducing the cGPEs to a scalar NLSE. The effects of the external field parameters on the effective dispersion and nonlinear interaction of the NLSE are discussed in detail. It is verified that the existence of the positive and negative bright (dark) solitons with positive or negative mass can be supported in this system. The typical bright and dark vector solitons are given in some specific parameters. In addition, it is found that the spin polarization of the vector solitons are easily adjusted by the parameter of the system, which demonstrating the magnetized features of the vector solitons. 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