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Defect matter-wave gap solitons in spin–orbit-coupled Bose–Einstein condensates in Zeeman lattices
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Discussion
Defect matter-wave gap solitons in spin–orbit-coupled Bose–Einstein condensates in Zeeman lattices Xing Zhu a , Huagang Li a,∗ , Zhiwei Shi b a b
Department of Physics and Information Engineering, Guangdong University of Education, Guangzhou 510303, China School of Electro-Mechanical Engineering, Guangdong University of Technology, Guangzhou 510006, China
a r t i c l e
i n f o
Article history: Received 23 March 2016 Received in revised form 25 July 2016 Accepted 25 July 2016 Available online xxxx Communicated by V.A. Markel Keywords: Spin–orbit-coupled Bose–Einstein condensates Zeeman lattices Defect gap solitons
a b s t r a c t We report on the properties of fundamental defect matter-wave gap solitons in spin–orbit-coupled Bose– Einstein condensates in one-dimensional Zeeman lattices with attractive nonlinearity. One component of these solitons is a real even function, and the other is an imaginary odd function. When the defect is repulsive, these solitons can be stable in the semi-infinite, first, and second gaps. Increasing the strength of spin–orbit coupling, stable defect gap-stripe solitons in the semi-infinite and first gaps are found. However, for an attractive defect, the solitons only stably exist in the semi-infinite gap and cannot be close to the lower edge of the first Bloch band. © 2016 Elsevier B.V. All rights reserved.
1. Introduction In BECs, atomic interactions can produce nonlinearities [1], thus enabling matter-wave solitons to be formed. In 2011, Spielman’s group realized spin–orbit-coupled Bose–Einstein condensates (SOC-BECs) experimentally [2], and the solitons in SOC-BECs have been widely investigated in recent years [3–10,14,16,17]. In [3–10], the Zeeman fields are constants (including zeros). Stable matter-wave bright solitons can exist in SOC-BECs [3], and moving bright solitons were also found in SOC-BECs [4]. Zeeman splitting can cause even nonlinear modes to transform into asymmetric ones [5]. The interplay between the localized spin–orbit (SO) coupling and the Zeeman splitting led to the Gross–Pitaevskii (GP) equations that are nonintegrable and numerous of nontrivial soliton properties [6]. The exact static and moving soliton solutions in the three components SOC-BECs were found [7]. Three stable types of dark solitons can exist in quasi-one-dimensional repulsive SOC-BECs [8] and the dynamics and interaction properties of dark solitons in SOC-BECs were also investigated [9]. Recently, two-dimensional (2D) vortex solitons in dipolar SOC-BECs were reported [10]. Optical lattices (periodic potentials) can act as trapping potentials for each spinor component in SOC-BECs [11,12], and these lattices have been created experimentally [13]. Stable gap solitons in SOC-BECs in one-dimensional (1D) optical lattices have also
*
Corresponding author. E-mail address: [email protected] (H. Li).
http://dx.doi.org/10.1016/j.physleta.2016.07.060 0375-9601/© 2016 Elsevier B.V. All rights reserved.
been reported [14]. In contrast, while the Zeeman lattices are also periodic potentials, they are exactly inverted in the two coupled GP equations; they have also been created successfully in BECs in experiments, the combination of the radio-frequency and Raman fields create the 1D Zeeman lattices [15]. Later, the gap solitons in SOC-BECs in Zeeman lattices were first studied in [16]. The existence and the stability of two types of gap solitons were carefully investigated. The first soliton is parity-time (PT) symmetric and the second is pseudocharge (C)-parity-time (CPT) symmetric. For attractive nonlinearity, the PT-symmetric gap-stripe solitons exist in the semi-infinite gap, and they are all stable. They can also be found in the finite gap for repulsive nonlinearity, and for the parameter of the potential depth in [16], these gap-stripe solitons are generically unstable. Moreover, the 2D Zeeman lattices can support fundamental, multipole, and half-vortex gap solitons in SOC-BECs [17]. For simplicity, in Refs. [4,6,14,16,17], the inter- and intraspecies scattering lengths are equal. Matter-wave dark solitons with repulsive interactions [18,19] and bright solitons with attractive interactions [20,21] in BECs were observed in real experiments by using phase imprinting method and Feshbach resonance [22], respectively. We believe that the matter-wave gap solitons in SOCBECs will also be observed in experiments in the future. Moreover, unstable matter-wave gap solitons in the presence of defect in polariton condensation were also observed experimentally [23]. Defect optical solitons in 1D [24] and 2D [25,26] real optical lattices with saturable nonlinearity and in 1D [27,28] PT-symmetric optical lattices with Kerr nonlinearity have been reported. Moreover, the optical solitons in PT-symmetric lattices were observed
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in experiment [29]. In [24–26,28], for repulsive defects, fundamental defect solitons are stable in the semi-infinite and first gaps. The out-of-phase dipole solitons in the first gap for a repulsive defect were also studied in [27]. However, these solitons are all unstable. In the optical media, the modulation of the amplitude of the laser beam can produce the optical lattices with the defects [30]. Similarly, the Zeeman lattices with defects in BECs can also be created by the modulation of the amplitude of the radio-frequency and the Raman fields. Defect optical solitons in periodic potentials in optical media were widely studied. However, defect matter-wave gap solitons in SOC-BECs in Zeeman lattices have not been studied yet. We wonder whether the defect gap solitons in SOC-BECs in Zeeman lattices can exhibit new properties. Activated by the works [16,17,24–28], in this paper, we investigate the fundamental defect matter-wave gap solitons in SOC-BECs in 1D Zeeman lattices with attractive nonlinearity. The new results demonstrate that these gap solitons can reach the lower edge of the first Bloch band and can be stable in the semi-infinite, first, and second gaps for a repulsive defect. Moreover, by increasing the strength of SO coupling, stable defect gap-stripe solitons can stably exist in the semi-infinite and first gaps simultaneously, but these defect gapstripe solitons cannot be close to the lower edge of the first Bloch band. For an attractive defect, fundamental solitons only stably exist in the semi-infinite gap. We also perform stability analyses for these solitons in SOC-BECs, and they are confirmed by the simulations of soliton evolution.
are n1,2 = φ1,2 , respectively. For 87 Rb atoms confined by a harmonic trap with the frequency ω0 = 140 Hz [17] and the scattering length is 5.29 nm [32], if λRam = 790.33 nm [15], the real total number of atoms is ≈ 3931 × N, where N = N 1 + N 2 , and 2 +∞ N 1,2 = −∞ φ1,2 dx [5,16]. The total magnetization is given by M = ( N 1 − N 2 )/ N [16]. We can obtain the band structures using the plane wave expansion method [33]. The corresponding band structures for κ = 2, κ = 4 and ε = 0 are shown in Figs. 1(a) and 1(b), respectively. For κ = 2, the semi-infinite gap is μ ≤ −5.07, the first gap is −5.05 ≤ μ ≤ −3.42, and the second gap is −3.15 ≤ μ ≤ −2.23. For κ = 4, the semi-infinite gap and the first gap are μ ≤ −8.59 and −8.31 ≤ μ ≤ −7.73, respectively.
2. The theory model
3. The numerical results
The nonlinear dynamic behavior of SOC-BECs can be described using the following coupled GP equations [16,17]:
First, we take κ = 2 and ε = −0.5. The fundamental defect gap solitons can exist in the semi-infinite, first, and second gaps simultaneously, as depicted in Figs. 2(a), 2(b), and 2(c), respectively. In the semi-infinite gap, defect solitons (DSs) can reach the lower edge of the first Bloch band, so band structure is almost not affected by the defect. The three figures show that the total magnetizations change very slowly with increasing chemical potential (μ). We take μ = −6, μ = −4.3, and μ = −3.05 to be the stable cases in the three gaps. Figs. 2(e) and 2(f) show the stable evolution characteristics of the perturbed components (random noises characteristics with 5% of the amplitudes of the two components are added) of the defect soliton for μ = −6. When μ = −4.3, the soliton profile is depicted in Fig. 2(g). In the first gap, when approaching the lower edge of the second Bloch band, the total number of atoms increases slightly, as shown in Fig. 2(b), and the DSs become unstable. Fig. 2(i) shows the maximum value, max[Re(λ)] (λ is growth rate [34], we obtain the max[Re(λ)] by the Fourier collocation method [35]), in the region. Figs. 3(a) and 3(b) show the mildly unstable evolutions of the two perturbed components when μ = −3.44. Figs. 3(c) and 3(d) show the concentration distributions of the two components when μ = −3.05. With an increase of the strength of SO coupling, the fundamental DSs will become less localized. For κ = 4 and ε = −0.5, defect gap-stripe solitons are found in the semi-infinite and first gaps. We take μ = −8.95 and μ = −8.2 to be the stable cases of gap-stripe solitons in the two gaps. Figs. 4(c), 4(h) and 4(i) depict the concentration profiles of the two defect gap-stripe solitons. In the semi-infinite gap, at the end of this family of gap-stripe solitons, the total number of atoms also increases slightly, as shown in Fig. 4(a). The defect gap-stripe solitons also become unstable, and max[Re(λ)] > 0 in the region [Fig. 4(d)] also indicates that the gap-stripe solitons cannot be stable. We take μ = −8.78 as the unstable case in the semi-infinite gap. The gap-stripe soliton concentration profile is shown in Fig. 4(e), and the unstable evolutions of the two perturbed components are depicted in Figs. 4(f) and 4(g). When μ ≥ −8.75, gap-stripe solitons in this
i
∂ψ1 1 ∂ 2 ψ1 1 ∂ψ2 =− − (x)ψ1 + i κ − |ψ1 |2 + |ψ2 |2 ψ1 , 2 ∂t 2 ∂x 2 ∂x
i
∂ψ2 1 ∂ ψ2 1 ∂ψ1 =− + (x)ψ2 + i κ − |ψ1 |2 + |ψ2 |2 ψ2 . ∂t 2 ∂ x2 2 ∂x
(1a) 2
(1b) Here, (x) and κ are the Zeeman field and the strength of SO coupling and the units of length, time, and energy are 1/kRam , m/(¯hk2Ram ), and h¯ 2 k2Ram /m, respectively. Where m is the atomic mass and k Ram is the wavenumber of the Raman laser beams. Artificial SO coupling can be generated by imparting momentum to the atoms from the Raman laser beams [14,15]. The stationary soliton solutions are sought in the forms of ψ1,2 (x, t ) = φ1,2 (x)e −i μt , where μ is the chemical potential, and φ1,2 satisfy the following relationships:
1 ∂ 2 φ1 2
∂ x2
1
+ (x)φ1 − i κ 2
∂φ2 + |φ1 |2 + |φ2 |2 φ1 + μφ1 = 0, ∂x (2a)
1 ∂ 2 φ2 2 ∂ x2
∂φ1 − (x)φ2 − i κ + |φ1 |2 + |φ2 |2 φ2 + μφ2 = 0. 2 ∂x 1
(2b) We consider the periodic Zeeman field with a defect [24]:
8 (x) = 12 cos2 (x) 1 + εe −x /128 .
(3)
The defect strength is controlled by ε . We use a method that was developed from the squared-operator iteration method [31] to numerically solve Eqs. (2) and we can thus obtain the PTsymmetric soliton solutions. Component φ1 is the real even function, while component φ2 is the imaginary odd function. We set φ1 = q1 and φ2 = iq2 . The concentrations of the two components
Fig. 1. (Color online.) Band structures for (a) eter is ε = 0.
κ = 2 and (b) κ = 4. The other param-
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Fig. 2. (Color online.) (a), (b), and (c) show the total numbers of atoms and the total magnetizations versus the chemical potential (μ). (d) Soliton profile for μ = −6. (e) and (f) show the corresponding stable evolutions of the two perturbed components. (g) and (h) show the soliton profiles for μ = −4.3 and μ = −3.44, respectively. (i) shows the max[Re(λ)] in the first gap. The other parameters are κ = 2 and ε = −0.5.
For uniform Zeeman lattices (ε = 0), when κ = 2 and κ = 4, fundamental gap solitons also only exist in the semi-infinite gap, but they can reach the lower edge of the first Bloch band [Figs. 6(a) and 6(b)]. For κ = 2 and κ = 4, when μ = −9, the two solitons can also be stable. The profiles of the two solitons are displayed in Figs. 6(c) and 6(d), respectively. If the fundamental gap solitons are localized in the defect site, the defect will significantly affect the existence of the solitons. For repulsive defect, the defect can broaden the matter-wave solitons, so the defect can help to balance the attractive nonlinearity (originating from the attractive atomic interactions) to form the matter-wave solitons. Thus fundamental solitons can exist in different gaps. But for attractive defect, both defect and attractive nonlinearity will narrow the matter-wave solitons, so fundamental solitons only exist in the semi-infinite gap and cannot be close to the lower edge of the first Bloch band. Moreover, the sign of the Zeeman field [(x)] can be reversed by swapping even and odd components of the defect solitons [see Eqs. (2)]. Fig. 3. (Color online.) When κ = 2 and ε = −0.5. The unstable evolutions of the two perturbed components for μ = −3.44 are shown in (a) and (b). (c) and (d) depict the concentration distributions of the two components for μ = −3.05.
4. Conclusions
family cannot exist in the semi-infinite gap any more. In this case (κ = 4), the defect gap-stripe solitons in the semi-infinite gap cannot be close to the lower edge of the first Bloch band, because the repulsive defect changes the band structure. When ε = 0.5, DSs only stably exist in the semi-infinite gap for κ = 2 and κ = 4, and DSs cannot approach the lower edge of the first Bloch band. Fig. 5(c) shows the soliton profile for μ = −16 and κ = 2 and the soliton profile for μ = −16 and κ = 4 is depicted in Fig. 5(d). The two solitons are stable.
In conclusion, we have investigated the properties of PTsymmetric fundamental defect matter-wave gap solitons in SOCBECs in 1D Zeeman lattices with attractive nonlinearity. For a repulsive defect, the DSs can stably exist in the semi-infinite, first, and second gaps. The stable defect gap-stripe solitons in the semiinfinite and first gap were also found by increasing the strength of SO coupling. For an attractive defect, the DSs only stably exist in the semi-infinite gap and cannot approach the lower edge of the first Bloch band. Moreover, the stability analyses are verified by the evolutions of these defect gap solitons.
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Fig. 4. (Color online.) (a), (b) Total numbers of atoms and total magnetizations versus μ. (c) Gap-stripe soliton concentration profile for μ = −8.95. (d) shows the max[Re(λ)] in the semi-infinite gap. (e) Gap-stripe soliton concentration profile, and (f), (g) the unstable evolutions of the two perturbed components for μ = −8.78. (h) and (i) show the concentration distributions of the two components of the gap-stripe soliton for μ = −8.2. The other parameters are κ = 4 and ε = −0.5.
Fig. 5. (Color online.) (a) and (b) show the total numbers of atoms and the total magnetizations versus μ for κ = 2 and κ = 4, respectively. (c) depicts the soliton profile for μ = −16 and κ = 2 and (d) is the soliton profile when μ = −16 and κ = 4. The other parameter is ε = 0.5.
Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant Nos. 11547212, 11575063, 61575223, and 61475049) and the Appropriative Researching Fund for Professors and Doctors, Guangdong University of Education (Grant No. 2014ARF01).
Fig. 6. (Color online.) When ε = 0 and μ = −9. The total numbers of atoms and the total magnetizations versus μ for κ = 2 and κ = 4 are shown in (a) and (b), respectively. (c) and (d) depict the solitons profile for κ = 2 and κ = 4, respectively.
References [1] C.J. Pethick, H. Smith, Bose–Einstein Condensation in Dilute Gases, Cambridge University Press, Cambridge, 2002. [2] Y.–J. Lin, K. Jiménez-García, I.B. Spielman, Spin–orbit-coupled Bose–Einstein condensates, Nature 471 (2011) 83. [3] V. Achilleos, D.J. Frantzeskakis, P.G. Kevrekidis, D.E. Pelinovsky, Matter-wave bright solitons in spin–orbit coupled Bose–Einstein condensates, Phys. Rev. Lett. 110 (2013) 264101.
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[4] Y. Xu, Y. Zhang, B. Wu, Bright solitons in spin–orbit-coupled Bose–Einstein condensates, Phys. Rev. A 87 (2013) 013614. [5] D.A. Zezyulin, R. Driben, V.V. Konotop, B.A. Malomed, Nonlinear modes in binary bosonic condensates with pseudo-spin–orbital coupling, Phys. Rev. A 88 (2013) 013607. [6] Y.V. Kartashov, V.V. Konotop, D.A. Zezyulin, Bose–Einstein condensates with localized spin–orbit coupling: soliton complexes and spinor dynamics, Phys. Rev. A 90 (2014) 063621. [7] Y. Liu, S. Yang, Exact solitons and manifold mixing dynamics in the spin–orbitcoupled spinor condensates, Europhys. Lett. 108 (2014) 30004. [8] V. Achilleos, J. Stockhofe, P.G. Kevrekidis, D.J. Frantzeskakis, P. Schmelcher, Matter-wave dark solitons and their excitation spectra in spin–orbit coupled Bose–Einstein condensates, Europhys. Lett. 103 (2013) 20002. [9] S. Cao, C. Shan, D. Zhang, X. Qin, J. Xu, Dynamical generation of dark solitons in spin–orbit-coupled Bose–Einstein condensates, J. Opt. Soc. Am. B 32 (2015) 201. [10] X. Jiang, Z. Fan, Z. Chen, W. Pang, Y. Li, B.A. Malomed, Two-dimensional solitons in dipolar Bose–Einstein condensates with spin–orbit coupling, Phys. Rev. A 93 (2016) 023633. [11] H. Sakaguchi, B. Li, Vortex lattice solutions to the Gross–Pitaevskii equation with spin–orbit coupling in optical lattices, Phys. Rev. A 87 (2013) 015602. [12] Y. Zhang, C. Zhang, Bose–Einstein condensates in spin–orbit-coupled optical lattices: flat bands and superfluidity, Phys. Rev. A 87 (2013) 023611. [13] C. Hamner, Y. Zhang, M.A. Khamehchi, M.J. Davis, P. Engels, Spin–orbit-coupled Bose–Einstein condensates in a one-dimensional optical lattice, Phys. Rev. Lett. 114 (2015) 070401. [14] Y. Zhang, Y. Xu, T. Busch, Gap solitons in spin–orbit-coupled Bose–Einstein condensates in optical lattices, Phys. Rev. A 91 (2015) 043629. [15] K. Jiménez-García, L.J. LeBlanc, R.A. Williams, M.C. Beeler, A.R. Perry, I.B. Spielman, Peierls substitution in an engineered lattice potential, Phys. Rev. Lett. 108 (2012) 225303. [16] Y.V. Kartashov, V.V. Konotop, F.Kh. Abdullaev, Gap solitons in a spin–orbitcoupled Bose–Einstein condensate, Phys. Rev. Lett. 111 (2013) 060402. [17] V.E. Lobanov, Y.V. Kartashov, V.V. Konotop, Fundamental, multipole, and halfvortex gap solitons in spin–orbit coupled Bose–Einstein condensates, Phys. Rev. Lett. 112 (2014) 180403. [18] S. Burger, K. Bongs, S. Dettmer, W. Ertmer, K. Sengstock, A. Sanpera, G.V. Shlyapnikov, M. Lewenstein, Dark solitons in Bose–Einstein condensates, Phys. Rev. Lett. 83 (1999) 5198. [19] J. Denschlag, J.E. Simsarian, D.L. Feder, C.W. Clark, L.A. Collins, J. Cubizolles, L. Deng, E.W. Hagley, K. Helmerson, W.P. Reinhardt, S.L. Rolston, B.I. Schneider, W.D. Phillips, Generating solitons by phase engineering of a Bose–Einstein condensate, Science 287 (2000) 97.
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[20] K.E. Strecker, G.B. Partridge, A.G. Truscott, R.G. Hulet, Formation and propagation of matter-wave soliton trains, Nature 417 (2002) 150. [21] L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L.D. Carr, Y. Castin, C. Salomon, Formation of a matter-wave bright soliton, Science 296 (2002) 1290. [22] S. Inouye, M.R. Andrews, J. Stenger, H.-J. Miesner, D.M. Stamper-Kurn, W. Ketterle, Observation of feshbach resonances in a Bose–Einstein condensate, Nature 392 (1998) 151. [23] D. Tanese, H. Flayac, D. Solnyshkov, A. Amo, A. Lemaître, E. Galopin, R. Braive, P. Senellart, I. Sagnes, G. Malpuech, J. Bloch, Polariton condensation in solitonic gap states in a one-dimensional periodic potential, Nat. Commun. 4 (2013) 1749. [24] J. Yang, Z. Chen, Defect solitons in photonic lattices, Phys. Rev. E 73 (2006) 026609. [25] W. Chen, X. Zhu, T. Wu, R. Li, Defect solitons in two-dimensional optical lattices, Opt. Express 18 (2010) 10956. [26] X. Zhu, H. Wang, L. Zheng, Defect solitons in kagome optical lattices, Opt. Express 18 (2010) 20786. [27] H. Wang, J. Wang, Defect solitons in parity-time periodic potentials, Opt. Express 19 (2011) 4030. [28] Z. Lu, Z. Zhang, Defect solitons in parity-time symmetric superlattices, Opt. Express 19 (2011) 11457. [29] M. Wimmer, A. Regensburger, M.-A. Miri, C. Bersch, D.N. Christodoulides, U. Peschel, Observation of optical solitons in PT-symmetric lattices, Nat. Commun. 6 (2015) 7782. [30] J. Yang, X. Wang, J. Wang, Z. Chen, Light localization by defects in optically induced photonic structures, in: C. Denz, S. Flach, Y.S. Kivshar (Eds.), Nonlinearities in Periodic Structures and Metamaterials, Springer, Berlin, 2009, pp. 127–143. [31] J. Yang, T.I. Lakoba, Universally-convergent squared-operator iteration methods for solitary waves in general nonlinear wave equations, Stud. Appl. Math. 118 (2007) 153. [32] P.S. Julienne, F.H. Mies, E. Tiesinga, C.J. Williams, Collisional stability of double Bose condensates, Phys. Rev. Lett. 78 (1997) 1880. [33] D.R. Smith, R. Dalichaouch, N. Kroll, S. Schultz, S.L. McCall, P.M. Platzman, Photonic band structure and defects in one and two dimensions, J. Opt. Soc. Am. B 10 (1993) 314. [34] S. Nixon, L. Ge, J. Yang, Stability analysis for solitons in PT-symmetric optical lattices, Phys. Rev. A 85 (2012) 023822. [35] J. Yang, Nonlinear Waves in Integrable and Nonintegrable Systems, SIAM, Philadelphia, 2010.
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Physics Letters A ••• (••••) •••–•••
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Discussion
Defect matter-wave gap solitons in spin–orbit-coupled Bose–Einstein condensates in Zeeman lattices Xing Zhu a , Huagang Li a,∗ , Zhiwei Shi b a b
Department of Physics and Information Engineering, Guangdong University of Education, Guangzhou 510303, China School of Electro-Mechanical Engineering, Guangdong University of Technology, Guangzhou 510006, China
a r t i c l e
i n f o
Article history: Received 23 March 2016 Received in revised form 25 July 2016 Accepted 25 July 2016 Available online xxxx Communicated by V.A. Markel Keywords: Spin–orbit-coupled Bose–Einstein condensates Zeeman lattices Defect gap solitons
a b s t r a c t We report on the properties of fundamental defect matter-wave gap solitons in spin–orbit-coupled Bose– Einstein condensates in one-dimensional Zeeman lattices with attractive nonlinearity. One component of these solitons is a real even function, and the other is an imaginary odd function. When the defect is repulsive, these solitons can be stable in the semi-infinite, first, and second gaps. Increasing the strength of spin–orbit coupling, stable defect gap-stripe solitons in the semi-infinite and first gaps are found. However, for an attractive defect, the solitons only stably exist in the semi-infinite gap and cannot be close to the lower edge of the first Bloch band. © 2016 Elsevier B.V. All rights reserved.
1. Introduction In BECs, atomic interactions can produce nonlinearities [1], thus enabling matter-wave solitons to be formed. In 2011, Spielman’s group realized spin–orbit-coupled Bose–Einstein condensates (SOC-BECs) experimentally [2], and the solitons in SOC-BECs have been widely investigated in recent years [3–10,14,16,17]. In [3–10], the Zeeman fields are constants (including zeros). Stable matter-wave bright solitons can exist in SOC-BECs [3], and moving bright solitons were also found in SOC-BECs [4]. Zeeman splitting can cause even nonlinear modes to transform into asymmetric ones [5]. The interplay between the localized spin–orbit (SO) coupling and the Zeeman splitting led to the Gross–Pitaevskii (GP) equations that are nonintegrable and numerous of nontrivial soliton properties [6]. The exact static and moving soliton solutions in the three components SOC-BECs were found [7]. Three stable types of dark solitons can exist in quasi-one-dimensional repulsive SOC-BECs [8] and the dynamics and interaction properties of dark solitons in SOC-BECs were also investigated [9]. Recently, two-dimensional (2D) vortex solitons in dipolar SOC-BECs were reported [10]. Optical lattices (periodic potentials) can act as trapping potentials for each spinor component in SOC-BECs [11,12], and these lattices have been created experimentally [13]. Stable gap solitons in SOC-BECs in one-dimensional (1D) optical lattices have also
*
Corresponding author. E-mail address: [email protected] (H. Li).
http://dx.doi.org/10.1016/j.physleta.2016.07.060 0375-9601/© 2016 Elsevier B.V. All rights reserved.
been reported [14]. In contrast, while the Zeeman lattices are also periodic potentials, they are exactly inverted in the two coupled GP equations; they have also been created successfully in BECs in experiments, the combination of the radio-frequency and Raman fields create the 1D Zeeman lattices [15]. Later, the gap solitons in SOC-BECs in Zeeman lattices were first studied in [16]. The existence and the stability of two types of gap solitons were carefully investigated. The first soliton is parity-time (PT) symmetric and the second is pseudocharge (C)-parity-time (CPT) symmetric. For attractive nonlinearity, the PT-symmetric gap-stripe solitons exist in the semi-infinite gap, and they are all stable. They can also be found in the finite gap for repulsive nonlinearity, and for the parameter of the potential depth in [16], these gap-stripe solitons are generically unstable. Moreover, the 2D Zeeman lattices can support fundamental, multipole, and half-vortex gap solitons in SOC-BECs [17]. For simplicity, in Refs. [4,6,14,16,17], the inter- and intraspecies scattering lengths are equal. Matter-wave dark solitons with repulsive interactions [18,19] and bright solitons with attractive interactions [20,21] in BECs were observed in real experiments by using phase imprinting method and Feshbach resonance [22], respectively. We believe that the matter-wave gap solitons in SOCBECs will also be observed in experiments in the future. Moreover, unstable matter-wave gap solitons in the presence of defect in polariton condensation were also observed experimentally [23]. Defect optical solitons in 1D [24] and 2D [25,26] real optical lattices with saturable nonlinearity and in 1D [27,28] PT-symmetric optical lattices with Kerr nonlinearity have been reported. Moreover, the optical solitons in PT-symmetric lattices were observed
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in experiment [29]. In [24–26,28], for repulsive defects, fundamental defect solitons are stable in the semi-infinite and first gaps. The out-of-phase dipole solitons in the first gap for a repulsive defect were also studied in [27]. However, these solitons are all unstable. In the optical media, the modulation of the amplitude of the laser beam can produce the optical lattices with the defects [30]. Similarly, the Zeeman lattices with defects in BECs can also be created by the modulation of the amplitude of the radio-frequency and the Raman fields. Defect optical solitons in periodic potentials in optical media were widely studied. However, defect matter-wave gap solitons in SOC-BECs in Zeeman lattices have not been studied yet. We wonder whether the defect gap solitons in SOC-BECs in Zeeman lattices can exhibit new properties. Activated by the works [16,17,24–28], in this paper, we investigate the fundamental defect matter-wave gap solitons in SOC-BECs in 1D Zeeman lattices with attractive nonlinearity. The new results demonstrate that these gap solitons can reach the lower edge of the first Bloch band and can be stable in the semi-infinite, first, and second gaps for a repulsive defect. Moreover, by increasing the strength of SO coupling, stable defect gap-stripe solitons can stably exist in the semi-infinite and first gaps simultaneously, but these defect gapstripe solitons cannot be close to the lower edge of the first Bloch band. For an attractive defect, fundamental solitons only stably exist in the semi-infinite gap. We also perform stability analyses for these solitons in SOC-BECs, and they are confirmed by the simulations of soliton evolution.
are n1,2 = φ1,2 , respectively. For 87 Rb atoms confined by a harmonic trap with the frequency ω0 = 140 Hz [17] and the scattering length is 5.29 nm [32], if λRam = 790.33 nm [15], the real total number of atoms is ≈ 3931 × N, where N = N 1 + N 2 , and 2 +∞ N 1,2 = −∞ φ1,2 dx [5,16]. The total magnetization is given by M = ( N 1 − N 2 )/ N [16]. We can obtain the band structures using the plane wave expansion method [33]. The corresponding band structures for κ = 2, κ = 4 and ε = 0 are shown in Figs. 1(a) and 1(b), respectively. For κ = 2, the semi-infinite gap is μ ≤ −5.07, the first gap is −5.05 ≤ μ ≤ −3.42, and the second gap is −3.15 ≤ μ ≤ −2.23. For κ = 4, the semi-infinite gap and the first gap are μ ≤ −8.59 and −8.31 ≤ μ ≤ −7.73, respectively.
2. The theory model
3. The numerical results
The nonlinear dynamic behavior of SOC-BECs can be described using the following coupled GP equations [16,17]:
First, we take κ = 2 and ε = −0.5. The fundamental defect gap solitons can exist in the semi-infinite, first, and second gaps simultaneously, as depicted in Figs. 2(a), 2(b), and 2(c), respectively. In the semi-infinite gap, defect solitons (DSs) can reach the lower edge of the first Bloch band, so band structure is almost not affected by the defect. The three figures show that the total magnetizations change very slowly with increasing chemical potential (μ). We take μ = −6, μ = −4.3, and μ = −3.05 to be the stable cases in the three gaps. Figs. 2(e) and 2(f) show the stable evolution characteristics of the perturbed components (random noises characteristics with 5% of the amplitudes of the two components are added) of the defect soliton for μ = −6. When μ = −4.3, the soliton profile is depicted in Fig. 2(g). In the first gap, when approaching the lower edge of the second Bloch band, the total number of atoms increases slightly, as shown in Fig. 2(b), and the DSs become unstable. Fig. 2(i) shows the maximum value, max[Re(λ)] (λ is growth rate [34], we obtain the max[Re(λ)] by the Fourier collocation method [35]), in the region. Figs. 3(a) and 3(b) show the mildly unstable evolutions of the two perturbed components when μ = −3.44. Figs. 3(c) and 3(d) show the concentration distributions of the two components when μ = −3.05. With an increase of the strength of SO coupling, the fundamental DSs will become less localized. For κ = 4 and ε = −0.5, defect gap-stripe solitons are found in the semi-infinite and first gaps. We take μ = −8.95 and μ = −8.2 to be the stable cases of gap-stripe solitons in the two gaps. Figs. 4(c), 4(h) and 4(i) depict the concentration profiles of the two defect gap-stripe solitons. In the semi-infinite gap, at the end of this family of gap-stripe solitons, the total number of atoms also increases slightly, as shown in Fig. 4(a). The defect gap-stripe solitons also become unstable, and max[Re(λ)] > 0 in the region [Fig. 4(d)] also indicates that the gap-stripe solitons cannot be stable. We take μ = −8.78 as the unstable case in the semi-infinite gap. The gap-stripe soliton concentration profile is shown in Fig. 4(e), and the unstable evolutions of the two perturbed components are depicted in Figs. 4(f) and 4(g). When μ ≥ −8.75, gap-stripe solitons in this
i
∂ψ1 1 ∂ 2 ψ1 1 ∂ψ2 =− − (x)ψ1 + i κ − |ψ1 |2 + |ψ2 |2 ψ1 , 2 ∂t 2 ∂x 2 ∂x
i
∂ψ2 1 ∂ ψ2 1 ∂ψ1 =− + (x)ψ2 + i κ − |ψ1 |2 + |ψ2 |2 ψ2 . ∂t 2 ∂ x2 2 ∂x
(1a) 2
(1b) Here, (x) and κ are the Zeeman field and the strength of SO coupling and the units of length, time, and energy are 1/kRam , m/(¯hk2Ram ), and h¯ 2 k2Ram /m, respectively. Where m is the atomic mass and k Ram is the wavenumber of the Raman laser beams. Artificial SO coupling can be generated by imparting momentum to the atoms from the Raman laser beams [14,15]. The stationary soliton solutions are sought in the forms of ψ1,2 (x, t ) = φ1,2 (x)e −i μt , where μ is the chemical potential, and φ1,2 satisfy the following relationships:
1 ∂ 2 φ1 2
∂ x2
1
+ (x)φ1 − i κ 2
∂φ2 + |φ1 |2 + |φ2 |2 φ1 + μφ1 = 0, ∂x (2a)
1 ∂ 2 φ2 2 ∂ x2
∂φ1 − (x)φ2 − i κ + |φ1 |2 + |φ2 |2 φ2 + μφ2 = 0. 2 ∂x 1
(2b) We consider the periodic Zeeman field with a defect [24]:
8 (x) = 12 cos2 (x) 1 + εe −x /128 .
(3)
The defect strength is controlled by ε . We use a method that was developed from the squared-operator iteration method [31] to numerically solve Eqs. (2) and we can thus obtain the PTsymmetric soliton solutions. Component φ1 is the real even function, while component φ2 is the imaginary odd function. We set φ1 = q1 and φ2 = iq2 . The concentrations of the two components
Fig. 1. (Color online.) Band structures for (a) eter is ε = 0.
κ = 2 and (b) κ = 4. The other param-
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Fig. 2. (Color online.) (a), (b), and (c) show the total numbers of atoms and the total magnetizations versus the chemical potential (μ). (d) Soliton profile for μ = −6. (e) and (f) show the corresponding stable evolutions of the two perturbed components. (g) and (h) show the soliton profiles for μ = −4.3 and μ = −3.44, respectively. (i) shows the max[Re(λ)] in the first gap. The other parameters are κ = 2 and ε = −0.5.
For uniform Zeeman lattices (ε = 0), when κ = 2 and κ = 4, fundamental gap solitons also only exist in the semi-infinite gap, but they can reach the lower edge of the first Bloch band [Figs. 6(a) and 6(b)]. For κ = 2 and κ = 4, when μ = −9, the two solitons can also be stable. The profiles of the two solitons are displayed in Figs. 6(c) and 6(d), respectively. If the fundamental gap solitons are localized in the defect site, the defect will significantly affect the existence of the solitons. For repulsive defect, the defect can broaden the matter-wave solitons, so the defect can help to balance the attractive nonlinearity (originating from the attractive atomic interactions) to form the matter-wave solitons. Thus fundamental solitons can exist in different gaps. But for attractive defect, both defect and attractive nonlinearity will narrow the matter-wave solitons, so fundamental solitons only exist in the semi-infinite gap and cannot be close to the lower edge of the first Bloch band. Moreover, the sign of the Zeeman field [(x)] can be reversed by swapping even and odd components of the defect solitons [see Eqs. (2)]. Fig. 3. (Color online.) When κ = 2 and ε = −0.5. The unstable evolutions of the two perturbed components for μ = −3.44 are shown in (a) and (b). (c) and (d) depict the concentration distributions of the two components for μ = −3.05.
4. Conclusions
family cannot exist in the semi-infinite gap any more. In this case (κ = 4), the defect gap-stripe solitons in the semi-infinite gap cannot be close to the lower edge of the first Bloch band, because the repulsive defect changes the band structure. When ε = 0.5, DSs only stably exist in the semi-infinite gap for κ = 2 and κ = 4, and DSs cannot approach the lower edge of the first Bloch band. Fig. 5(c) shows the soliton profile for μ = −16 and κ = 2 and the soliton profile for μ = −16 and κ = 4 is depicted in Fig. 5(d). The two solitons are stable.
In conclusion, we have investigated the properties of PTsymmetric fundamental defect matter-wave gap solitons in SOCBECs in 1D Zeeman lattices with attractive nonlinearity. For a repulsive defect, the DSs can stably exist in the semi-infinite, first, and second gaps. The stable defect gap-stripe solitons in the semiinfinite and first gap were also found by increasing the strength of SO coupling. For an attractive defect, the DSs only stably exist in the semi-infinite gap and cannot approach the lower edge of the first Bloch band. Moreover, the stability analyses are verified by the evolutions of these defect gap solitons.
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Fig. 4. (Color online.) (a), (b) Total numbers of atoms and total magnetizations versus μ. (c) Gap-stripe soliton concentration profile for μ = −8.95. (d) shows the max[Re(λ)] in the semi-infinite gap. (e) Gap-stripe soliton concentration profile, and (f), (g) the unstable evolutions of the two perturbed components for μ = −8.78. (h) and (i) show the concentration distributions of the two components of the gap-stripe soliton for μ = −8.2. The other parameters are κ = 4 and ε = −0.5.
Fig. 5. (Color online.) (a) and (b) show the total numbers of atoms and the total magnetizations versus μ for κ = 2 and κ = 4, respectively. (c) depicts the soliton profile for μ = −16 and κ = 2 and (d) is the soliton profile when μ = −16 and κ = 4. The other parameter is ε = 0.5.
Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant Nos. 11547212, 11575063, 61575223, and 61475049) and the Appropriative Researching Fund for Professors and Doctors, Guangdong University of Education (Grant No. 2014ARF01).
Fig. 6. (Color online.) When ε = 0 and μ = −9. The total numbers of atoms and the total magnetizations versus μ for κ = 2 and κ = 4 are shown in (a) and (b), respectively. (c) and (d) depict the solitons profile for κ = 2 and κ = 4, respectively.
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