We show that $n$ thermal fermionic alkaline-earth atoms in a flat-bottom trapallow one to robustly implement a spin model displaying two symmetries: the$S_n$ symmetry that permutes atoms occupying different vibrational levels ofthe trap and the SU($N$) symmetry associated with $N$ nuclear spin states. Thehigh symmetry makes the model exactly solvable, which, in turn, enables theanalytic study of dynamical processes such as spin diffusion in this SU($N$)system. We also show how to use this system to generate entangled states thatallow for Heisenberg-limited metrology. This highly symmetric spin model shouldbe experimentally realizable even when the vibrational levels are occupiedaccording to a high-temperature thermal or an arbitrary non-thermaldistribution.
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机译：我们显示平底陷井中的$ n $个热铁离子碱土原子能够稳健地实现显示两个对称性的自旋模型：$ S_n $对称性，该对称性置换了占据陷阱的不同振动水平的原子和SU（$ N $ ）与$ N $核自旋态相关的对称性。高对称性使模型可以完全求解，从而可以对该SU（$ N $）系统中的自旋扩散等动力学过程进行解析研究。我们还将展示如何使用该系统生成纠缠态，以实现海森堡有限的计量。即使当振动水平根据高温热分布或任意非热分布占据时，该高度对称的自旋模型也应可通过实验实现。
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