Modulational instability of spinor condensates

We demonstrate, analytically and numerically, that the ferromagnetic phase of the spinor Bose-Einstein condenstate may experience modulational instability of the ground state leading to a fragmentation of the spin domains. Together with other nonlinear effects in the atomic optics of ultra-cold gases (such as coherent photoassociation and four-wave mixing) this effect provides one more analogy between coherent matter waves and light waves in nonlinear optics. Typeset using REVTEX 1 Recent experimental studies of Bose-Einstein condensation (BEC) in an optical trap [1] have opened a new direction in the research of ultra-cold atomic clouds associated with their spin degree of freedom, or spinor BEC. An optical trap does not force atoms to align along the orientation of the strong confining magnetic field, as happens in the case of a magnetic trap, allowing the study of atoms confined in all hyperfine states. Several recent theoretical studies have predicted a variety of novel phenomena that may occur in the spinor BEC, such as the propagation of spin waves and the existence of topological states -skyrmions, vortex states without a core [2–8]. In the case of spin-1 bosons such as Na, K and Rb, the dynamics of the spinor BEC is described by the three spin degrees of freedom (mF = 1, 0,−1 of the F = 1 atomic hyperfine state) which are coupled parametrically. Depending on the parameters, such as scattering lengths, in an optical trap the ground state of the spinor BEC can be either ferromagnetic or antiferromagnetic (”polar”). In this paper, we demonstrate that the parametric coupling between the spin degrees of freedom provides a physical mechanism for the modulational instability of the ferromagnetic ground state, for large enough densities of spinor BEC. In contrast, the antiferromagnetic state is always modulationally stable. This effect is reminiscent of the quasiparticle instabilities in two-component homogeneous condensates that are known to occur only for certain ratios of the interand intra-component interaction strengths [9]. It also suggests one more example of a deep analogy between the coherent matter waves and light waves in nonlinear optics, along with the already studied cases of four-wave mixing [10] and atomic-molecular photoassociation [11]. Model. We consider a spinor (atomic) BEC trapped in an optical trap in the magneticfield-free case. The Hamiltonian for the spinor BEC in the optical trap has the form [2]: H = ∫