Presented is a study of a spin-2 Bose-Einstein condensate (BEC) by unitary quantum simulations of time-dependent soliton-soliton scattering. The quantum simulation method is based on a quantum lattice algorithm which is designed for implementation on a future digital quantum computer but is tested today using a parallel computing architecture based on graphical processing units (GPUs). We analytically solve the spin-2 BEC equations of motion, a nonlinear system of 5 coupled Gross-Pitiaevskii (GP) equations, in one- and two-spatial dimensions. In 1D there are 16 bright soliton and 16 dark soliton solutions. In 2D there are 3 dark soliton solutions Pade approximation solutions, for m_f=+/-2, m_f=+/-1 and m_f=0, corresponding to quantum vortices. We report on the implementation the unitary quantum lattice gas algorithm for spinor superfluid and establish its efficacy by validating the stability of the 1D and 2D energy eigenstate solutions of the spin-2 BEC Hamiltonian. Using the calibrated quantum lattice gas algorithm, the highly nonlinear physics in the nonintegrable regime of the spin-2 BEC is studied by performing soliton-soliton scattering experiments. The scattering of topological solitons produces breathers and complex quantum vortices characterized by local entanglement across multiple m_f-hyperfine states of the Zeeman manifold.
In this short note, we construct mappings from one-dimensional integrable spinor BECs to matrix nonlinear Schrödinger equation, and solve the Bogoliubov equation of these systems. A map of spin-
n BEC is constructed from the
2n-dimensional spinor representation of irreducible tensor operators of
so(2n+1). Solutions of Bogoliubov equation are obtained with the aid of the theory of squared Jost functions.
As a first approximation beyond linearity, the nonlinear Schrodinger equation reliably describes a broad class of physical systems. Though numerical solutions of this model are well-established, these methods can be computationally complex, especially when system-specific details are incorporated. In this paper, we demonstrate how numerical computations that exploit the features of a graphics processing unit (GPU) result in 40-70
× reduction in the time for solutions (depending on hardware details). As a specific case study, we investigate the Gross-Pitaevskii equation, a specific version of the nonlinear Schrodinger model, as it describes a trapped, interacting two-component Bose-Einstein condensate subject to spatially dependent interspin coupling, resulting in an analog to a spin-Hall system. This computational method allows us to probe high-resolution spatial features -- revealing an interaction dependent phase transition -- all in a reasonable amount of time with readily available hardware.