THE EUROPEAN PHYSICAL JOURNAL D Families of matter-waves in two-component Bose-Einstein condensates

We produce several families of solutions for two-component nonlinear Schrödinger/GrossPitaevskii equations. These include domain walls and the first example of an antidark or gray soliton in one component, bound to a bright or dark soliton in the other. Most of these solutions are linearly stable in their entire domain of existence. Some of them are relevant to nonlinear optics, and all to BoseEinstein condensates (BECs). In the latter context, we demonstrate robustness of the structures in the presence of parabolic and periodic potentials (corresponding, respectively, to the magnetic trap and optical lattices in BECs). PACS. 03.75.-b Matter waves – 52.35.Mw Nonlinear phenomena: waves, wave propagation, and other interactions (including parametric effects, mode coupling, ponderomotive effects, etc.) The recent progress in experimental and theoretical studies of Bose-Einstein condensates (BECs) [1] has made matter-wave solitons physically relevant objects. One dimensional (1D) dark [2] and bright [3] solitons have been observed in experiments, and possibilities for the observation of their multidimensional counterparts were predicted [4]. Further study of matter-wave solitons is a subject of profound interest, not only from a theoretical perspective, but also for applications, as there are possibilities to coherently manipulate such robust structures in matter-wave devices, e.g., atom chips, which are analogs of the existing optical ones [5]. On the other hand, many results obtained for optical solitons as fundamental nonlinear excitations in optical fibers and waveguides (see, e.g., recent reviews [6,7]) suggest the possibility to search for similar effects in BECs. A class of physically important generalizations of the nonlinear Schrödinger (NLS) equation for optical media, or its BEC counterpart, the Gross-Pitaevskii equation (GP), is based on their multi-component versions. In particular, the theoretical work has already gone into studying ground-state solutions [8,9] and smallamplitude excitations [10] of the order parameters in multi-component BECs. Additionally, the structure of binary BECs [11], including the formation of domain walls in the case of immiscible species, has also been studa e-mail: [email protected] b http://nlds.sdsu.edu/ ied [11,12]; 1D bound dark-dark [13] and dark-bright [14] soliton complexes, as well as spatially periodic states [15], were predicted too. Experimental results have been reported for mixtures of different spin states of Rb [16] and mixed condensates [17]. Efforts were also made to create two-component BECs with different atomic species, such as K–Rb [18] and Li–Cs [19]. In this work, we report novel solitons in the context of coupled two-component GP equations. These solutions correspond to new families of solitons even for the NLS equations per se, hence they are also interesting as nonlinear waves in their own right. We start by demonstrating their existence in the context of two coupled NLS equations. Some of them are relevant as new solitons in nonlinear-optical models as well. We also demonstrate that all the solutions proposed herein persist in the presence of the magnetic trap and optical lattice (OL), i.e., parabolic and sinusoidal potentials [20], which are important ingredients of experimental BEC setups. Assuming that the nonlinear interactions are weak relative to the confinement in the transverse dimensions, the transverse size of the condensate is much smaller than its lengths. In this case, the BEC is a “cigar-shaped” one, and the GP equations take an effectively 1D form [21]: i ∂uj ∂t = −1 mj ∂uj ∂x2 + 2 ∑ k=1 ajk|uk|uj+Vj(x)uj , (j = 1, 2) (1) 182 The European Physical Journal D where uj(x, t) are the mean-field wave functions of the two species, t and x are, respectively, measured in units of 2/ω1⊥ and the transverse harmonic-oscillator length a1⊥ ≡ √ /(m1ω1) (mj and ωj⊥ are the mass and transverse confining frequency of each species), while ω1⊥/2 is the energy unit. The coefficients ajk in equation (1), related to the three scattering lengths αjk (note that α12 = α21) through ajk = 4πm1(αjk/a1⊥)(mj + mk)/(mjmk), account for collisions between atoms belonging to the same (ajj) and different (ajk, j = k) species; they are counterparts of the, respectively, self-phase and cross-phase modulation in nonlinear optics. While in optics only specific ratios of the nonlinear coefficients are relevant (such as aij/aii = 2 or aij/aii = 2/3 [6]), in the BEC context the interactions are tunable [9,15], especially because they can be modified by means of the Feshbach resonance (i.e., by magnetic field affecting the sign and magnitude of the scattering length of the interatomic collisions) [22]. The Feshbach resonance allows one to switch between attractive and repulsive interaction [23], and even to switch it periodically in time, by means of an ac magnetic field, which allows one to create a self-confined 2D BEC without the magnetic trap [24]. In this work, we consider the case with m1 = m2 ≡ m and a11 = a22, which corresponds to the most experimentally feasible mixture of two different hyperfine states of the same atom species, or, approximately, to different isotopes of the same alkali metal, trapped in the potential including the magnetic trapping and OL components: V1(x) = V2(x) ≡ V (x) = ( Ω/2 ) x+V0 sin(kx+φ). (2) In equation (2), Ω ≡ 2ω x/ω ⊥ (ω1x = ω2x ≡ ωx are the confining frequencies in the axial direction) and V0 (measured in units of the recoil energy [20]) set the respective potential strengths, k is the wavenumber of the interference pattern of the laser beams forming the OL, and φ is an adjustable phase parameter (φ ∈ {0, π/2}). To estimate physical parameters, we resort to a mixture of two different spin states of Rb, confined in a trap with the transverse frequency ω1⊥ = 183 rad/s, which implies that the length and time units are 2 μm and 5.46 ms, respectively. We consider a rather general case, in which the interatomic interactions in the first species are repulsive [therefore, we will use the normalization a11 ≡ +1 in Eqs. (1)], while in the other species they may be either attractive or repulsive. As concerns the interactions between the different species, they are, typically, repulsive. Nevertheless, in the case of two different spin states of the same atom species, the Feshbach resonance between such states is possible too (experimental studies of the Feshbach resonance in this case are currently in progress [25]), therefore attractive inter-species interactions may be relevant, and this case is also considered below. The solutions reported herein, and their existence and stability regimes are summarized in Table 1. In most cases the existence and stability of the solution families is investigated numerically. The numerical method was implemented as follows: we first seek staTable 1. Existence and stability of structures in the binary BEC. In the “existence” column, +/− indicates the repulsive/attractive character of the respective inter-atomic interaction which is necessary for the solution to exist. The “stability” column indicates the sign of the coefficient a22 (we normalize a11 ≡ +1, and set a22 = ±1) and an interval of the values of a12 for which the solution is stable. Types of solitons Existence Stability a22 a12 a22 a12 Domain wall + + +1 > 1 Dark-antidark + + +1 (0, 0.7] Dark-gray + − +1 [−0.83, 0) Bright-antidark − − −1 (−1, 0) Bright-gray − + −1 > 0 tionary solutions by means of Newton iterations which are applied to the steady-state equations μuj = −uj,xx + ∑2 k=1 ajk|uk|2uj + V (x)uj (μ is the chemical potential). Subsequently, we perform the linear-stability analysis of the obtained soliton solutions u j (x), setting the perturbed solution to be uj = e−iμt [ u (0) j (x) + ( bje −iωt + cje ∗t )] , where ω ≡ ωr + iωi is a (generally, complex) perturbation eigenfrequency. Then, the ensuing linear stability problem [26] is solved for the eigenfrequencies and eigenfunctions {bj , cj}. Whenever the solution is unstable, we also examine its evolution in direct simulations of the full equations (1), using a fourth-order Runge-Kutta time integrator with the time step dt = 0.005 (27.3 μs in physical units). To initiate the instability development, a uniformly distributed random perturbation of amplitude ∼ 10−4 was typically added to an unstable solution. We now examine in detail the solutions shown in Table 1. First, in the absence of the external potential, a family of domain walls can be found in an exact form for the special case, a12 = 3a11 = 3a22: uj(x, t) = Ae−iμt [ 1 + (−1)j tanh(ηx)] , (3) where the chemical potential is μ = 4a12A, and η = 2a12A (they follow the pattern of domain-wall solutions found long ago in the context of coupled Ginzburg-Landau equations [27]). These solutions exist only if a12 > 0 and μ > 0. Similar patterns were found in reference [12] and other related structures were also predicted to occur in higher dimensions [28]. We have confirmed the existence and stability of the domain walls by direct numerical simulations (not shown here), using numerical continuation to extend them to the case a12 = 3a11, where the analytical solution is not available. We have thus found that the domain walls exist and are stable for values of a12 down to a12 = 1. The case a12 = 2 is relevant to nonlinear optics; stability of the domain wall family for this case was suggested by recent numerical results obtained for a similar discrete coupledNLS model [29]. Here, we find that these solutions are robust as well for other values of a12, and, as will be shown P.G. Kevrekidis et al.: Matter-waves for two-component BECs 183 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 35 40 45 50 55 60