Two Weight Dipolar Bose - Einstein Condensates Numerical Simulation

The macroscopic dipole Bose-Einstein condensate(BEC) phenomenon is described byGross-Pitaevskii (GP) equations with dipole-dipole interaction potential in the mean-fieldtheory. In this paper, we consider a ultracold atom system consisting of metal and transitionmetal. A component of the transition metal has the dipole-dipole interaction while the othercomponent of the metal has only the usual s-wave contact interaction. The ground state andthe dynamic evolution of the system are discussed by establishing of a mathematical modeland constructing numerical methods.Based on the one component physical model, similar to the model of the alkali metalwithout a dipole-dipole interaction and the model of transition metal-dipole interaction, atwo component Bose-Einstein condensate model is established.Since the dipole-dipole interaction is long-range, anisotropic and the different structure,atoms will be partially attract or exclude each other. Namely the controlled equation has aconvolution term and its kernel has a singularity. It will be very difficult to solve theequations for three-dimensional system with the singularity. For ease of calculation, weturn the model into two-component Gross-Pitaevskii equations coupled with Poissonequation without convolution term.Based on imagery-time methods, a backward Euler method in time and sine spectralmethod in space are proposed to compute the ground states. Numerical examples arecarried out for different parameters and initial conditions to show their effect on the groundstate solution. Numerical show that the total energy decreases and converges to the groundstate solution.The fractional method in time and the sine spectral method in space are proposed forcomputing the dynamics of dipolar BEC. And it is shown that the method can alsomaintain particle conservation. Numerical examples are carried out for different parametersand initial conditions to show their effect on the system and they also show that the methodcan maintain energy conservation properties well.The spectral method has spectral-order accuracy, speed and other advantages ofcomputing. A sine-spectral method in this paper is applied to achieve three-dimensionaldipolar Bose-Einstein condensate simulation problem. Experimental results are presentedto demonstrate the efficiency and stable of the numerical methods.