Exploring Ground States and Excited States of Spin-1 Bose-Einstein Condensates

Start: 01/23/2013 - 1:15pm
End  : 01/23/2013 - 2:15pm

Applied Math Seminar

I-Liang Chern (National Taiwan University)


In this talk, I will first give a brief introduction to the spinor Bose-Einstein condensates. Then I will present two recent results, one is numerical, the other is analytical. In 1925, Bose and Einstein predicted that massive bosons could occupy the same lowest-energy state at low temperature and formed the so-called Bose-Einstein condensates (BECs). It was realized on several alkali atomic gases in 1995 by laser cooling technique. In the numerical study of spinor BEC, a pseudo-arclength continuation method (PACM) was proposed and employed to compute the ground state and excited state solutions of spin-$ 1 $ BEC. Numerical results on the wave functions and their corresponding energies of spin-1 BEC with repulsive/attractive and ferromagnetic/antiferromagnetic interactions are presented. Furthermore, it is found that the component separation and population transfer between the different hyperfine states can only occur in excited states due to the spin-exchange interactions. In the analytical study, the ground states of spin-1 BEC are characterized. For ferromagnetic systems, we show the validity of the so-called single-mode approximation (SMA). For antiferromagnetic systems, there are two subcases. When the total magnetization $ M \ne 0 $, the corresponding ground states have vanishing zeroth ($ m_F = 0 $) components, thus are reduced to two-component systems. When $ M = 0 $, the ground states are also reduced to the SMA, and there are one-parameter families of such ground states. The key idea is a redistribution of masses among different components, which reduces kinetic energy in all situations, and makes our proofs simple and unified. Finally, a fast algorithm based on the above analytic result is provided. It is shown that the new method is about 10 to 20 faster than an old method of Bao-Lim's continuous normalized gradient method. The numerical part is a joint work with Jen-Hao Chen and Weichung Wang, the fast algorithm part is with Weizhu Bao and Yanzhi Zhang, whereas the analytical part is jointly with Liren Lin.

RS 105