Exact periodic and solitonic states in the spinor condensates
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] M a y Exact periodic and solitonic states in the spinor condensates
Zhi-Hai Zhang and Shi-Jie Yang ∗
1, 2 Department of Physics, Beijing Normal University, Beijing 100875, China State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing, 100190
We propose a method to analytically solve the one-dimensional coupled nonlinear Gross-Pitaevskiiequations which govern the motion of the spinor Bose-Einstein condensates. In a uniform externalpotential, the Hamiltonian comprises the kinetic energy, the linear and the quadratic Zeeman en-ergies. Several classes of exact periodic and solitonic solutions, either in real or in complex forms,are obtained for both the F = 1 and F = 2 condensates. These solutions are general that containneither approximations nor constraints on the system parameters. PACS numbers: 03.75.Mn, 67.85.Fg, 03.75.Lm, 03.75.Hh
I. INTRODUCTION
The experimental achievement of Stenger et al . intrapping sodium atoms by optical means in 1998[1], trig-gered the study of the magnetism in quantum degenerateatomic gases. Since the atom spins are not frozen, thedirection of the spin can change dynamically through col-lisions between the atoms[2–4]. In contrast to the scalargases, spinor gases can host a wide variety of complexstructures at zero temperature, from spin textures, mag-netic crystallization, to fractional vortices et al. [5, 6]. Inthe ground state, the symmetry is spontaneously brokenin several different ways, leading to a number of possiblephases[7–10]. There exists an interplay between superflu-idity and magnetism due to the spin-gauge symmetry. Aferromagnetic Bose-Einstein condensates (BECs) spon-taneously creates a supercurrent as the spin is locallyrotated[11, 12]. The study of ultracold spinor is of pri-mordial importance in deepening our understanding ofcondensed matter related issues.The motion of the dilute spinor condensates is gov-erned by the coupled Gross-Pitaevskii equations (GPEs).There is a large amount of works that numericallysolve GPEs[13–15]. Analytically, some solitonic solutionsare obtained by means of variable functions or similartransformation for time or spatial modulated couplingconstants[16–19]. Various approximations are employedto study the solitons such as bright and dark solitonsin the F = 1 spinor BECs[20–23]. Exact solutions areusually difficult to obtain due to the complexity of thecoupled nonlinear GPEs. The challenges are two-fold:one is the nonlinear density-density interactions, whilethe other is the spin-exchange couplings between the hy-perfine states. In our previous publications we have givenexact solutions for the F = 1 and F = 2 spinor BECsfor some special cases[24, 25]. In this paper, we pro-pose a general method which simultaneously decouplesthe nonlinear density-density interactions and the spin-spin interactions in the GPEs. Classes of the exact so- ∗ Corresponding author: [email protected] lutions, either in real or in complex forms, are system-atically constructed for the Hamiltonian containing thelinear and quadratic Zeeman energies. The solutions areexpressed by combinations of the Jacobi elliptical func-tions for periodic states or the hyperbolic functions forsolitonic states. The latter are identified as vector soli-tons or scalar solitons, respectively.The paper is organized as follows: In Sec.II we de-scribed the method and systematically present solutionsfor the spin-1 condensates. In Sec.III we give a solutionto the spin-2 condensates as an example. Section IV con-tains a brief summary.
II. SPIN-1 CONDENSATES
We are concerned with the quasi-one-dimensional (1D)spinor system in a uniform external potential ( V ( r ) = 0).In this section, we deal with the F = 1 condensates inwhich the meanfield order parameters are described bya macroscopic wavefunction with three hyperfine statesΨ = ( ψ +1 , ψ , ψ − ) T . The Hamiltonian that contains thelinear and the quadratic Zeeman effects reads[7, 8] H = Z d r { X m = − ψ ∗ m [ − ¯ h M ▽ + V ( r ) − pm + qm ] ψ m + ¯ c n tot + ¯ c | F | } , (1)where the spin-polarization vector F = ψ ∗ m ˆ F imn ψ n withˆ F i ( i = x, y, z ) the spin matrices. The terms withcoefficients c and c describe respectively the spin-independent and the spin-dependent binary elastic col-lisions in the combined symmetric channels of total spin0 and 2. They are expressed in terms of the s -wave scat-tering lengths a and a as: ¯ c = 4 π ¯ h ( a + 2 a ) / M and ¯ c = 4 π ¯ h ( a − a ) / M . p and q are linearand quadratic Zeeman coupling coefficients, respectively. n tot = | ψ | + ψ | + | ψ − | and V ( r ) is the externalpotential.The dynamical motion of the F = 1 spinor wavefunc-tions are governed by i∂ t ψ m = δH/δψ ∗ m , which are ex-plicitly written as the coupled GPEs, i∂ t ψ m = [ − ¯ h M ∂ x − pm + qm + c n tot ] ψ m (2)+ c X n = − F · ˆ F i ψ n , ( m = 1 , , − c = ¯ c / a ⊥ and c = ¯ c / a ⊥ are the reducedcoupling constants with a ⊥ the transverse width of the quasi-1D system.Below we choose ¯ h = M = 1 as the units for conve-nience. By substituting the wavefunction Ψ( x, t ) with ψ ( x, t ) ψ ( x, t ) ψ − ( x, t ) → ψ ( x ) e − i ( µ + µ ) t ψ ( x ) e − iµt ψ − ( x ) e − i ( µ − µ ) t , (3)we obtain the stationary GPEs as,( µ + µ ) ψ = [ − ∂ x + ( c + c )( | ψ | + | ψ | ) + ( c − c ) | ψ − | − p + q ] ψ + c ψ ψ ∗− , (4) µψ = [ − ∂ x + ( c + c )( | ψ | + | ψ − | ) + c | ψ | ] ψ + 2 c ψ ∗ ψ ψ − , ( µ − µ ) ψ − = [ − ∂ x + ( c + c )( | ψ − | + | ψ | ) + ( c − c ) | ψ | + p + q ] ψ − + c ψ ψ ∗ . Since the chemical potential of each hyperfine state is dif-ferent, Eqs.(4) are not really stationary equations given µ = 0. It comprises a ”Lamor precession” between thehyperfine states. However, the density profile of eachstate are time-invariant so we simply call them the sta-tionary states. The periodic boundary conditions, ψ m (1) = ψ m (0) , ψ ′ m (1) = ψ ′ m (0) . (5)is adopted in our calculations. We consider several typesof real and complex solutions for F = 1 condensates.In order to seek the analytical solutions we decou-ple spin-spin interactions in the Eqs.(4) by requiring ψ ∗− m = ± ψ m and ψ ∗ = ψ . It is directly to checkthat ψ ∗− m = ψ m corresponds to (partially) spin-polarizedstates ( | F | 6 = 0) whereas ψ ∗− m = − ψ m corresponds tospin-unpolarized or polar states ( | F | = 0). On the otherhand, the nonlinear density-density couplings betweenthe hyperfine states are decoupled by making use of theproperties of the Jacobi elliptical functions or the hy-perbolic functions. There are real and complex forms ofsolutions which are explicitly described as follows. A. Real solutions
We first consider the following sn-cn-sn form of solutionto the nonlinear Eq.(4), ψ ( x ) ψ ( x ) ψ − ( x ) = A sn( kx, m ) D cn( kx, m ) − A sn( kx, m ) , (6)where sn and cn are the Jacobi elliptical functions ofmodulus m . The period is k = 4 jK ( m ) with j the num-ber of periods which will always be set to j = 2 in the figures). A and D are the real constants. From the rela-tion sn + cn = 1, one has | ψ − | = | ψ | , | ψ | = D − D A | ψ | . (7)By substituting (7) into the equations (4), we obtainthree decoupled equations˜ µ m ψ m = − ψ ′′ m + ˜ γ m | ψ m | ψ m , ( m = 0 , ±
1) (8)where the effective chemical potentials ˜ µ m and interact-ing constants ˜ γ m are defined as˜ µ ± = µ − q − c D , ˜ µ = µ − c A , ˜ γ ± = 2 c − c D A , ˜ γ = c − c A D . (9)In order to obtain self-consistent solutions, one shouldtake p = − µ . Therefore, the linear Zeeman energy playsthe role of balancing the chemical potentials between thehyperfine states ψ and ψ − . The effective chemical po-tentials ˜ µ m and amplitudes A , D can be obtained as,˜ µ ± = k (1 + m ) , ˜ µ = k (1 − m ) ,A = m k ˜ γ ± ,D = − m k ˜ γ . (10)From relations (9) and (10) we conclude that the effectiveintra-species interactions for hyperfine states ψ ± shouldbe repulsive (˜ γ ± >
0) while for ψ be attractive (˜ γ < c , q and µ . x | Ψ | ψ ψ −4 −2 0 2 40123 x | Ψ | ψ ψ (b)(a) FIG. 1: (a) The density profiles of the solution (6). c = 30, q = − . m = 0 . µ = 300. (b) The density profilesfor the single soliton solution (11). c = 1 . k = 2, q = − µ = 5. Figure 1(a) illustrates the density profiles of each hy-perfine states for solution (6). The parameters are cho-sen as c = 30, q = − . A = 2 . D = 2 . m = 0 . µ = 300. This state has vanishing spin-polarization | F | = 0. The modulus of the Jacobi ellipticalfunctions is a free parameter. As m →
1, we naturallyobtain the periodic soliton train solution. The single soli-ton solution can be obtained by directly substituting theJacobi elliptical functions with the hyperbolic functionsin (6). Namely, ψ ( x ) ψ ( x ) ψ − ( x ) = A tanh( kx ) D sech( kx ) − A tanh( kx ) , (11)This solution has been addressed in literatures[13, 15].The density profile is displayed in Fig.1(b), with theparameters c = 1 . k = 2, q = − A = 1 . D = 1 . µ = 5. It is a dark-bright-dark com-posite soliton.Other forms of real solutions can also be constructedby the same way. For example, we seek the cn-sn-cn formof solution to Eq.(4), ψ ( x ) ψ ( x ) ψ − ( x ) = A cn( kx, m ) D sn( kx, m ) − A cn( kx, m ) . (12)One has | ψ − | = | ψ | , | ψ | = D − D A | ψ | . (13)Eq.(4) are again decoupled into (8). It follows that the ef-fective chemical potentials ˜ µ m and interacting constants x | Ψ | ψ ψ −4 −2 0 2 40246 x | Ψ | ψ ψ (b)(a) FIG. 2: (a) The density profiles of the solution (12) with c = − q = 22 . m = 0 . µ = − c = − k = 3, q = 4 . µ = − . ˜ γ m , ˜ µ ± = µ − q − c D , ˜ µ = µ − c A , ˜ γ ± = 2 c − c D A , ˜ γ = c − c A D . (14)The effective chemical potentials ˜ µ m and amplitudes A , D are self-consistently calculated as,˜ µ ± = k (1 − m ) , ˜ µ = k (1 + m ) ,A = − m k ˜ γ ± ,D = m k ˜ γ . (15)Relations (14) and (15) reveal that the effective intra-species interactions for the hyperfine states ψ ± should beattractive (˜ γ ± <
0) while for ψ be repulsive (˜ γ > c = − q = 22 . A = 3 . D = 5 . m = 0 . µ = − | F | = 0.Similarly, the single solitonic solution is constructed bysubstitution sn −→ tanh and cn −→ sech. Figure 2(b) isthe density profiles of a typical bright-dark-bright solitonwith c = − k = 3, q = 4 . A = 2 . D = 1 . µ = − . B. Complex solutions
We seek complex forms of periodic solution to the non-linear Eq.(4) as, ψ ψ ψ − = f ( x ) e iθ ( x ) D sn( kx, m ) f ( x ) e − iθ ( x ) , (16)where f ( x ) = p A + B cn ( kx, m ). A , B and D are realconstants. One has | ψ − | = | ψ | , | ψ | = D (1 + AB ) − D B | ψ | . (17)By substituting (17) into the coupled GPEs (4), weagain obtain the decoupled equations (8) with the effec-tive chemical potentials and intr-species interaction con-stants, ˜ µ ± = µ − q − ( c + 2 c ) D B ( A + B ) , ˜ µ = µ − c + 2 c )( A + B ) , ˜ γ ± = 2 c − ( c + 2 c ) D B , ˜ γ = c − c + 2 c ) BD . (18)In order to obtain the self-consistent solution, we set µ = − p which yields to B = − m k ˜ γ ,A = µ ± − (1 − m ) k γ ± ,D = m k ˜ γ . (19)We note that the effective interactions ˜ γ >
0. Thephase is θ ( x ) = Z x αf ( ξ ) dξ, (20)where α = ± (2˜ µ ± A − γ ± A + k AB (1 − m )) is anintegral constant. The periodic boundary conditions (5)require that the amplitude and phase satisfy, respectively, f (1) = f (0) , θ (1) − θ (0) = 2 jπ × n, (21)where n is an integer. The periodic condition for thephase can be fulfilled by properly adjusting the mod-ulus m of the Jacobi elliptical functions. Figure 3(a)and (b) display the phase and density profiles of thecomplex solution (16), respectively. The parameters aretaken as n = 2, c = 34, c = 43, q = 17, A = 1 . B = − . D = 1 . m = 0 .
82 and µ = 448 . m →
1, it results into the soliton train state.The single soliton solution is obtained by substitutingthe Jacobi elliptical functions with the hyperbolic func-tions as ψ ψ ψ − = f ( x ) e iθ ( x ) D tanh( kx ) f ( x ) e − iθ ( x ) , (22) θ ( x )2 π | Ψ | ψ ψ −4 −2 0 2 40612 x θ ( x )2 π −4 −2 0 2 4048 x | Ψ | ψ ψ FIG. 3: The phase profile (a) and the density profile (b) forthe complex solutions (16). The parameters are n = 2, m =0 . q = 17, and µ = 448 . n = 6, k = 8 . q = 5, and µ = 130 . where f ( x ) = q A + B sech ( kx ). Figure 3(c) and (d)plots a typical grey-dark-grey soliton for the solution (22)with n = 6, c = 3, c = 4, q = 5, k = 8 . A =5 . B = − . D = 2 . µ = 130 .
69. Herethe phases θ ( x ) should satisfy the periodic condition (5)by properly adjusting the width of the soliton k . Since( | F | 6 = 0), this type of composite soliton may be properlycalled the polarized or vector soliton.The spin-unpolarized ( F = 0) complex solution can beconstructed as, ψ ψ ψ − = f ( x ) e iθ ( x ) D cn( kx, m ) − f ( x ) e − iθ ( x ) , (23)where f ( x ) = p A + B sn ( kx, m ). A , B and D are realconstants. One has | ψ − | = | ψ | , | ψ | = D (1 + AB ) − D B | ψ | . (24)The effective chemical potentials and interaction con-stants are, ˜ µ ± = µ − q − c D B ( A + B ) , ˜ µ = µ − c ( A + B ) , ˜ γ ± = 2 c − c D B , ˜ γ = c − c BD (25)One obtains B = m k ˜ γ ,A = µ ± − (1+ m ) k γ ± ,D = − m k ˜ γ , (26) θ ( x )2 π | Ψ | ψ ψ −4 −2 0 2 4024 x θ ( x )2 π −4 −2 0 2 4024 x | Ψ | ψ ψ (b)(d)(a)(c) FIG. 4: The phase (a) and the density (b) distributions ofcomplex solutions (23) for the F = 1 condensates with n = 2, m = 0 . q = − µ = 4. (c) and (d) are the phase anddensity distributions for the single soliton solutions of (23)with n = 2, k = 3, q = −
8, and µ = − which require that the effective intra-species interactionsbe attractive (˜ γ < n = 2, c = − q = − A = 4 . B = 0 . D = 2 . m = 0 .
44 and µ = 4. Figure 4(c) and (d) are the corre-sponding single soliton solution with parameters n = 2, c = − q = − k = 3, A = 0 . B = 0 . D = 1 . µ = −
20. It is a typical grey-bright-greycomposite soliton. In comparison to the polarized soliton(22), we may call this spin-unpolarized soliton the polaror scalar solitons.
III. SPIN-2 CONDENSATES
Our method is also applicable to the F = 2 conden-sates. For illustration, we give an example in this section.By substituting the wavefunction with[11, 25] ψ ( x, t ) ψ ( x, t ) ψ ( x, t ) ψ − ( x, t ) ψ − ( x, t ) → ψ ( x ) e − i ( µ + µ ) t ψ ( x ) e − i ( µ + µ ) t ψ ( x ) e − iµt ψ − ( x ) e − i ( µ − µ ) t ψ − ( x ) e − i ( µ − µ ) t , (27)we obtain the generalized stationary GPEs as,( µ ± µ ) ψ ± = [ − ∂ x + c n tot ± c F z ∓ p + 4 q ] ψ ± + c F ∓ ψ ± + c √ Aψ ∗∓ , ( µ ± µ ) ψ ± = [ − ∂ x + c n tot ± c F z ∓ p + q ] ψ ± θ ( x )2 π | Ψ | ψ ψ ψ −4 −2 0 2 40510 x θ ( x )2 π −4 −2 0 2 4048 x | Ψ | ψ ψ ψ (a) (b)(c) (d) FIG. 5: The profiles of phase (a) and the density (b) for F = 2BEC solution (29). The parameters are n = 2, m = 0 . q = − . µ = 210. The profiles of phase (c) and density(d) for the single soliton solution corresponding to (29) with n = 5, k = 4 . q = − . µ = 200. + c ( √ F ∓ ψ + F ± ψ ± ) − c √ Aψ ∗∓ ,µψ = [ − ∂ x + c n tot ] ψ + √ c ( F + ψ + F − ψ − )+ c √ Aψ ∗ . (28)We seek the following form of solution which satisfies ψ ∗− m = ( − m ψ m and ψ ∗ = ψ , ψ ψ ψ ψ − ψ − = f ( x ) e iθ ( x ) C sn( kx, m ) D cn( kx, m ) − C sn( kx, m ) f ( x ) e − iθ ( x ) , (29)where f ( x ) = p A + B sn ( kx, m ). A , B , C and D arereal constants. This form of solution has vanishing spin-polarization | F | = 0. The equations (29) can be decou-pled and self-consistently solved in the same way as forthe F = 1 condensates. The linear Zeeman energy satis-fies µ = − p and µ = 2 p so as to balancing the chemicalpotential differences between the hyperfine states. Weskip the calculation details and just illustrate the results.Figure 5(a) and (b) display the phase and density pro-files of the solution (29) for F = 2 condensates. Theparameters are n = 2, c = 10, c = 25, q = − . m = 0 . A = 1 . B = 0 . C = 2 . D = 1 . µ = 210. Figure 5(c) and (d) are results for the singlesoliton solution which corresponding to the periodic so-lution (29) with n = 5, c = 10, c = 20, q = − . k = 4 . A = 3 . B = 1 . C = 1 . D = 2 .
429 and µ = 200. It exhibits a typical grey-dark-bright-dark-greycomposite soliton structure. IV. SUMMARY
We have systematically solved the one-dimensionalcoupled nonlinear GPEs which govern the motion of thespinor BECs exposed in a uniform magnetic field. Bothperiodic and solitonic stationary solutions for the F = 1and F = 2 condensates are constructed. Other forms of solutions with different combinations of the Jacobielliptical functions or hyperbolic functions can also beobtained in the same way. Our method is general andexact, without any approximations or special constraintson the system parameters. It may be extended to othernonlinear systems such as the coupled nonlinear Klein-Gordon equations or the dynamical coupled nonlinearSchrodinger equations.This work is supported by the funds from the Min-istry of Science and Technology of China under GrantNo. 2012CB821403. [1] D. M. Stamper-Kurn, M. P. Andrews, A. P. Chikkatur,S. Inouye, H.-J. Miesner, J. Stenger, and W. Ketterle,Phys. Rev. Ltee. , 2027 (1998).[2] H. Ott, J. Fortagh, G. Schlotterbeck, A. Grossmann, andC. Zimmermann, Phys. Rev. Lett. , 230401 (2001).[3] A. G¨orlitz, T. L. Gustavson, A. E. Leanhardt, R. L¨ow,A. P. Chikkatur, S. Gupta, S. Inouye, D. E. Pritchard,and W. Ketterle, Phys. Rev. Lett. , 090401 (2003).[4] A. E. Leanhardt, Y. Shin, D. Kielpinski, D. E. Pritchard,and W. Ketterle, Phys. Rev. Lett. , 140403 (2003).[5] J. R. Anglin, and W. Ketterle, Nature , 211-218(2002).[6] M. Kobayashi, Y. Kawaguchi, M. Nitta, and M. Ueda,Phys. Rev. Lett. , 115301 (2009).[7] T.-L. Ho, Phys. Rev. Lett. , 742 (1998).[8] M.-S. Chang, C. D. Hamley, M. D. Barrett, J. A. Sauer,K. M. Fortier, W. Zhang, L. You, and M. S. Chapman,Phys. Rev. Lett. , 140403 (2004).[9] K. Murata, H. Saito, and M. Ueda, Phys. Rev. A ,013607 (2007).[10] A. Imambekov, M. Lukin, and E. Demler, Phys. Rev. A , 063602 (2003).[11] M. Ueda and Y. Kawaguchia, Physics Reports , 080405 (2006).[13] L. Li, Z. Li, B. A. Malomed, D. Mihalache, and W. M.Liu, Phys. Rev. A ,033611 (2005).[14] W. X. Zhang, ¨O. E. M¨ustecaplioˇglu, and L. You, Phys. Rev. A , 043601 (2007).[15] H. E. Nistazakis, D. J. Frantzeskakis, P. G. Kevrekidis,B. A. Malomed and R. Carretero-Gonz´alez, Phys. Rev.A ,033612(2008).[16] J. Belmonte-Beitia, V. M. P´erez-Garc´ıa, V. Vekslerchik,and P. J. Torres, Phys. Rev. Lett. , 064102 (2007).[17] G. Theocharis, P. Schmelcher, P. G. Kevrekidis, and D.J. Frantzeskakis, Phys. Rev. A , 033614 (2005).[18] A. T. Avelar, D. Bazeia, and W. B. Cardoso, Phys. Rev.E , 025602(R) (2009).[19] D.-S. Wang, X.-H. Hu, and W. M. Liu, Phys. Rev. A ,023612 (2010).[20] B. J. Dabrowska-W¨uster, E. A. Ostrovskaya, T. J.Alexander, and Y. S. Kivshar, Phys. Rev. A , 023617(2007).[21] R. M. Bradley, B. Deconinck, and J. N. Kutz, J. Phys.A: Math. Gen. , 1901 (2005).[22] L. Li, B. A. Malomed, D. Mihalache and W. M. Liu,Phys. Rev. E , 066610 (2006).[23] D. Yan, J. J. Chang, C. Hamner, P. G. Kevrekidis, P.Engels, V. Achilleos, D. J. Frantzeskakis, R. Carretero-Gonzalez, and P. Schmelcher, Phys. Rew. A , 053630(2011).[24] Z. H. Zhang, C. Zhang, S. J. Yang and S. P. Feng, J.Phys. B: At. Mol. Opt. Phys. , 215302 (2012).[25] Z. H. Zhang, Y. K. Liu, and S. J. Yang, Mod. Phys. Lett.B27