Mobile vector soliton in a spin-orbit coupled spin- condensate
We study the formation of bound states and three-component bright vector solitons in a quasi-one-dimensional spin-orbit-coupled hyperfine spin Bose-Einstein condensate using numerical solution and variational approximation of a mean-field model. In the antiferromagnetic domain, the solutions are time-reversal symmetric, and the component densities have multi-peak structure. In the ferromagnetic domain, the solutions violate time-reversal symmetry, and the component densities have single-peak structure. The dynamics of the system is not Galelian invariant. From an analysis of Galelian invariance, we establish that the single-peak ferromagnetic vector solitons are true solitons and can move maintaining constant component densities, whereas the antiferromagnetic solitons cannot move with constant component densities.
pacs:03.75.Mn, 03.75.Hh, 67.85.Bc, 67.85.Fg
4 February 2015
Bright soliton is a self-reinforcing solitary wave that can traverse at a constant velocity without changing its shape due to a cancellation of the non-linear and dispersive interactions. The various systems in which solitons have been studied include water waves, non-linear optics, Bose-Einstein condensates (BECs), etc. . Solitons have been observed by manipulating the non-linear interaction near a Feshbach resonance  in a BEC of Li  and Rb . Solitons have also been studied in binary BECs .
In a neutral spinor BEC with a nonzero hyperfine spin , there is no spin-orbit (SO) coupling between the spin of the atoms and their center-of-mass motion . However, a synthetic SO coupling can be realized in a spinor BEC by controlling the atom-light interaction leading to the generation of artificial Abelian and non-Abelian gauge potentials coupled to the atoms . Solitons have been extensively studied in spinor BECs without SO coupling . An SO coupling with equal Rashba  and Dresselhaus  strengths was realized experimentally by Raman dressing two atomic spin states with a pair of lasers . In that study, the SO coupling between two of the three spin components of the state 5S of Rb the so-called pseudospin- state was considered. There are other experimental studies on SO-coupled spinor BECs . Solitonic structures in SO-coupled pseudospin- [13, 14] and spin- BECs  have also been investigated theoretically.
In this letter, we study two types of three-component vector solitons in an SO-coupled spin- BEC in a quasi-one-dimensional (quasi-1D) trap  with muliti-peak or single-peak structure using a mean-field coupled Gross-Pitaevskii (GP) equation. A spin- spinor BEC is characterized by two interaction strengths, namely and , where and are -wave scattering lengths in total spin and 2 channels respectively . For (antiferromagnetic) the multi-peak structure emerges, whereas for (ferromagnetic) the single-peak structure emerges. We use variational method to determine the bright soliton solutions for the SO-coupled trapless BEC in each of the two domains. The appropriate variational ansatz in each of the domains is constructed using the solutions of the SO-coupled single particle Hamiltonian. The variational analysis provides the necessary and sufficient conditions which and must satisfy to obtain a stable bright soliton. We also compare the variational results with the numerical solution of the GP equation.
In Ref. , only antiferromagnetic multi-peak solitons for were identified as the bright solitons in a three-component spin- SO-coupled BEC. These solitons are time-reversal symmetric, but are not true vector solitons as they cannot propagate maintaining the shape of the individual components. We demonstrate that this system can also support ferromagnetic single-peak solitons for , provided that . These solitons break the time-reversal symmetry of the Hamiltonian. Nevertheless, they are shown to be true vector solitons as they can propagate with a constant velocity maintaining the shape of the individual components.
2 Spin-Orbit-coupled BEC in quasi-1D trap
We consider an SO-coupled spinor condensate in a quasi-1D trap in which the trapping frequencies along the and axes ( and ) are much larger than that along the axis () . The single particle Hamiltonian of the condensate with equal strengths of Rashba  and Dresselhaus  SO couplings in such a quasi-1D trap is 
where is the momentum operator along x axis, is the harmonic trapping potential along axis, and is the irreducible representation of the component of the spin matrix:
Using the single particle model Hamiltonian (1) and considering interactions in the Hartree approximation, a quasi-1D  spin-1 BEC can be described by the following set of three coupled mean-field partial differential equations for the wave-function components [17, 21]
where , , and are the -wave scattering lengths in the total spin and channels, respectively, with are the component densities, is the total density, and with is the oscillator length in the transverse plane. For the sake of simplicity, let us transform (3)-(4) into dimensionless form using
where ) is the oscillator length along axis, and is the total number of atoms:
where , , , , with , and The total density is now normalized to unity, i.e., We present the scaled variables without tildes in the rest of the letter for notational simplicity. For a non-interacting trapless system , there are two linearly independent solutions of the SO-coupled set of equations (6)-(7) with the lowest energy :
where wave functions and are normalized to unity. Hence, the most general solution of Eqs. (6)-(7) for a non-interacting trapless system with a fixed density is given by the linear superposition of and
where to ensure that is normalized to unity.
The energy of the BEC in scaled units is
where is spin density vector, whose three components , and are defined as
Hence, for the SO-coupled Hamiltonian with its general solution given by (9), we get
Also, the magnetization for minimum energy solutions of the single-particle SO-coupled Hamiltonian.
Now, let us switch on the interactions; the interaction energy per particle for the uniform system is 
If , then the BEC is in the antiferromagnetic or polar phase, and the minimum of corresponds to leading to . In this case, the wave function (9) is time-reversal symmetric. On the other hand, for , the BEC is in the ferromagnetic phase, and can be minimized if or , which leads to . This corresponds to the wave functions (8) apart from a multiplying phase factor. These states are degenerate, violate the time-reversal symmetry and are mutually connected by the time-reversal operator. These are the only two distinct structures which emerge as the ground states in the SO-coupled quasi-1D BECs. In a quasi-two-dimensional BEC with Rashba or Dresselhaus SO coupling, there is a circular degeneracy in the energy eigen functions of the single particle Hamiltonian . Hence, depending upon the interaction parameters, more than two plane waves can also superpose resulting in different types of lattice structures in ground state density profiles .
3 Bright solitons
3.1 Stationary bright solitons
Stationary bright solitons can emerge as the ground state of a spinor BEC with attractive interactions [13, 15]. We use variational method to determine the bright soliton solutions of (6)-(7). As has been discussed in Sec. 2, an SO-coupled spinor BEC can have two types of ground states depending upon the sign of . This necessitates the use of two different variational ansatz in these two domains.
Antiferromagnetic phase (): Here we consider the following variational ansatz to determine the shape of the soliton
where is a variational parameter and characterizes the width and the strength of the bright soliton. The ansatz (17) corresponds to the wave function (9) with multiplied by the localized spatial soliton instead of . As two solutions (8) are degenerate, and a mixing between them is allowed, the soliton profile could have a multi-peak structure. Noting that in the domain, for , one can have other choices for the variational ansatz like
The minima of this energy occurs at
provided . Hence, the SO-coupled spin- spinor BEC can support an antiferromagnetic bright soliton defined by (17) [or (18) or (19)] and (21), provided that and . From (17) and (21) it is evident that the wavefunction of the bright soliton is independent of the strength of spin-exchange interactions . This is expected since for , there is no contribution to the energy from the -dependent term of the SO-coupled spinor BEC.
Ferromagnetic phase (): Here we consider the following variational ansatz
where is, again, a variational parameter characterizing the width and the strength of the bright soliton. This variational ansatz corresponds to in (9) multiplied by the localized bright soliton instead of . In this case the soliton will have a single peak. Also, the ansatz like are equally reasonable choices and correspond to and , respectively, in (9). Substituting (22) in (10), the energy of the soliton is
The minima of this energy occurs at
provided . Hence the SO-coupled spinor BEC can have a ferromagnetic soliton defined by (22) and (24), provided and . In this case, unlike in the case of an antiferromagnetic soliton, the bright soliton profile is sensitive to both and .
3.2 Moving bright solitons
where characterizes the width and the strength of the soliton. The breakdown of the Galilean invariance of the SO-coupled equation can be explicitly seen by using the transformation , , where is the velocity of the unprimed coordinate system with respect to primed coordinate system, then the wavefunction of (6)-(7) should transform to as
where the terms proportional to and have been suppressed for the sake of simplicity in addition to a -dependent additive term which does not contribute to the dynamics. The presence of the extra term on the right hand side of (27) shows that the SO-coupled Hamiltonian is no longer Galilean invariant and the SO-coupled soliton solution of the GP equation will depend on its velocity . The SO-coupled equation (27), in the absence of trap and interactions, has the solutions and of Eq. (8) with energies and , respectively. For , the two solutions (8) were degenerate, and this degeneracy has been removed in the case of the SO-coupled moving solutions. In the antiferromagnetic phase, a multi-peak solution was possible through a mixture of two degenerate solutions (8) for . For a nonzero , the degeneracy is removed and such a mixing is not possible. This means that the multi-peak soliton cannot propagate with a constant velocity maintaining its shape and energy. For the moving multi-peak soliton profile, the variational analysis of Sec. 3.1 will no longer be valid. In the ferromagnetic phase, as a mixing between the two degenerate solutions is not allowed, one can only have a single-peak soliton which can propagate with a constant velocity maintaining its shape, and the variational analysis presented in Sec. 3.1 remains valid.
4 Results and conclusions
We numerically solve the coupled equations (6)-(7) using the split-time-step Crank-Nicolson method [24, 25] with real- and imaginary-time propagations. The ground state is determined by solving (6)-(7) using imaginary-time propagation, which neither conserves norm nor magnetization. Both norm and magnetization can be fixed by transforming the wave-function components as
and here . These normalizations ensure simultaneous conservation of norm and magnetization after each iteration in imaginary time. The spatial and time steps used in the present work are and , respectively.
We consider an SO-coupled spin- spinor BEC of Na or Rb atoms trapped in a harmonic trapping potential with Hz and Hz. The oscillator lengths for Na with these parameters are m and m, whereas those for Rb are m and m. We use these values of for writing the dimensionless GP equations (6)-(7) for the trapped states, whereas for solitons m in this letter. The scattering lengths of Na in total spin and channels are nm, nm, respectively , resulting in and . Similarly, the scattering lengths of Rb are nm and nm , leading to and . In imaginary time propagation, we use a real Gaussian function multiplied by the solution of the single-particle SO-coupled Hamiltonian as the initial input for the component wavefunctions, i.e.,
where for Na and for Rb. Hence, by using different values of and in (32), one can obtain different solutions corresponding to the same density distribution and energy. For example, the two ground state solutions with for Na obtained by using and are shown in figures 1(a) and (b), respectively. In figures 1(a) and (b), only the non-zero real () and imaginary () parts of the component wavefunctions are shown. In these two cases, wavefunctions are either purely real or imaginary and not complex. On the other hand, the component wavefunctions in the ground state solution for Na obtained by using are complex with non-zero real and imaginary parts. The real and imaginary parts of the component wavefunctions in this case are shown in figures 1 (c) and (d), respectively. The multi-peak density profile corresponding to these three solutions presented in figures 1(a),(b), and (c) and (d) is the same and is shown in Fig. 1(g). The multi-peak nature of the solution in this case is consistent with analytic results obtained in Sec. 2. The multi-peak solution effectively leads to a weak phase separation between and , here weak phase separation implies that there are no local minima in the total density profile . This is in contrast to the strong phase separation possible with the model of the SO coupling discussed in Refs. [19, 20], where a notch appears in the total density profile at the interface separating the components when exceeds a critical value. The solutions illustrated in figures 1(a),(b), (c) and (d) are time-reversal symmetric. Similarly, the real and imaginary parts of the complex ground state solution with for Rb obtained with in Eq. are shown in figures 1(e) and (f), which lead to the single-peak density distribution of figure 1(h). The solution presented in figures 1(e), (f), and (h) violates time-reversal symmetry, as there are two degenerate solutions in this case connected by the time-reversal operation.
In order to obtain the bright solitons in SO-coupled spinor BECs, we take in (6)-(7) and consider two cases: (a) and (b) . In case (a), we consider . The numerically and variationally obtained bright solitons, defined by (17) and (21) with , are shown in figures 2(a). The multi-peak solution in this case is time-reversal symmetric. In case (b), we consider . The numerical and variational solutions, defined by (22) and (24), in this case are shown in figure 2(b). The single-peak solution in this case breaks time-reversal symmetry of the Hamiltonian. It is evident from figure 2 that there is an excellent agreement between the numerical and variational results.
In order to study the dynamics of the moving solitons, we first generate the stationary solitons numerically using imaginary-time propagation for both antiferromagnetic and ferromagnetic interactions. In order to set these solitons into motion with a constant velocity , we multiply the wavefunction components for the stationary soliton with , and then use real-time propagation to study its evolution. We observe that in the case of the antiferromagnetic soliton, there is spin-mixing dynamics due to which the component densities are not conserved as the soliton moves. This is evident from figure 3(a) and its inset, which show the dynamics of the antiferromagnetic soliton initially located at and the spin-mixing dynamics, respectively; the interaction parameters are the same as those in figure 2(a) . At the soliton is set into motion at a constant velocity. As the soliton moves component densities keep on changing without any change in the total density. On the other hand, if one starts with the ferromagnetic soliton at , the component densities and hence the total density do not change while the soliton is moving. This shown in figure 3(b) for the soliton initially located at and with the same interaction parameters as in figure 2(b). This is consistent with the analytic results of Sec. 3.2.
We study the generation and propagation of a vector soliton with three components in an SO-coupled spin-1 BEC with either antiferromagnetic or ferromagnetic interactions. In the antiferromagnetic case, the solutions are time-reversal symmetric and the component densities have multi-peak structure. In the ferromagnetic case, the solutions violate time-reversal symmetry and the component densities have single-peak structure. The GP equation for this system is not Galelian invariant. From an analysis of the Galelian invariance of this equation, we establish that the single-peak ferromagnetic SO-coupled solitons can move with constant component densities and are true solitons, whereas the multi-peak antiferromagnetic SO-coupled solitons change the component densities during motion.
This work is financed by the Fundação de Amparo à Pesquisa do Estado de São Paulo (Brazil) under Contract Nos. 2013/07213-0, 2012/00451-0 and also by the Conselho Nacional de Desenvolvimento Científico e Tecnológico (Brazil).
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