Liu Y M, Bao C G
Department of physics, Shaoguan University, Shaoguan, 510205, People's Republic of China.
School of Physics, Sun Yat-Sen University, Guangzhou, 510275, People's Republic of China.
Sci Rep. 2021 Sep 9;11(1):17999. doi: 10.1038/s41598-021-97521-y.
We consider the case that four spin-3 atoms are confined in an optical trap. The temperature is so low that the spatial degrees of freedom have been frozen. Exact numerical and analytical solutions for the spin-states have been both obtained. Two kinds of phase-diagrams for the ground states (g.s.) have been plotted. In general, the eigen-states with the total-spin S (a good quantum number) can be expanded in terms of a few basis-states [Formula: see text]. Let [Formula: see text] be the probability of a pair of spins coupled to [Formula: see text], and 6 in the [Formula: see text] state. Obviously, when the strength [Formula: see text] of the [Formula: see text]-channel is more negative, the basis-state with the largest [Formula: see text] would be more preferred by the g.s.. When two strengths are more negative, the two basis-states with the two largest probabilities would be more important components. Thus, based on the probabilities, the spin-structures (described via the basis-states) can be understood. Furthermore, all the details in the phase-diagrams, say, the critical points of transition, can also be explained. Note that, for [Formula: see text], [Formula: see text] is completely determined by symmetry. Thus, symmetry plays a very important role in determining the spin-structure of the g.s..
我们考虑四个自旋为3的原子被限制在一个光学阱中的情况。温度极低，以至于空间自由度已被冻结。已经获得了自旋态的精确数值解和解析解。绘制了两种基态的相图。一般来说，总自旋为S（一个好的量子数）的本征态可以用几个基态展开[公式：见正文]。设[公式：见正文]为一对自旋耦合到[公式：见正文]且处于[公式：见正文]态的概率。显然，当[公式：见正文]通道的强度[公式：见正文]更负时，总自旋最大的基态更受基态青睐。当两个强度都更负时，概率最大的两个基态将是更重要的组成部分。因此，基于概率，可以理解自旋结构（通过基态描述）。此外，相图中的所有细节，比如转变的临界点，也都可以得到解释。注意，对于[公式：见正文]，[公式：见正文]完全由对称性决定。因此，对称性在确定基态的自旋结构中起着非常重要的作用。
