Realizing exactly solvable $\mathrm{SU}\left(N\right)$ magnets with thermal atoms

We show that $n$ thermal fermionic alkaline-earth-metal atoms in a flat-bottom trap allow one to robustly implement a spin model displaying two symmetries: the $S_n$ symmetry that permutes atoms occupying different vibrational levels of the trap and the $\mathrm{SU}\left(N\right)$ symmetry associated with $N$ nuclear spin states. The symmetries make the model exactly solvable, which, in turn, enables the analytic study of dynamical processes such as spin diffusion in this $\mathrm{SU}(N$) system. We also show how to use this system to generate entangled states that allow for Heisenberg-limited metrology. This highly symmetric spin model should be experimentally realizable even when the vibrational levels are occupied according to a high-temperature thermal or an arbitrary nonthermal distribution.

Figure 1

(a) Contact interactions between atoms in the orbitals of a one-dimensional infinite square well of width $L$ are all-to-all with equal strength. (b) With nuclear spin $I$, each of the electronic clock states $g$ and $e$ of fermionic alkaline-earth-metal atoms can offer $N$ degenerate states, with $N\le 2I+1$.

Figure 2

(a) Young diagram $\vec{\lambda }=(4,2,1)$ [with $\vec{\gamma }=(3,2,1,1)]$ for $n=7,N=3$. (b) A labeling of boxes in $\vec{\lambda }$ from 1 to $n$, increasing down columns, starting at the left. (c) Orbitals associated with boxes in row $p$ are put in spin state $|p\rangle$ to form basis state $|T\rangle =|1231211\rangle$ [spins ordered as in (b)], used to construct eigenstate $|\vec{\lambda }\rangle =|\mathcal{A}\left\{123\right\}\rangle |\mathcal{A}\left\{12\right\}\rangle |11\rangle$ with $E\left(\vec{\lambda }\right)/(-U)={\sum }_i\left(\genfrac{}{}{0pt}{}{{\lambda }_i}{2}\right)-{\sum }_j\left(\genfrac{}{}{0pt}{}{{\gamma }_j}{2}\right)=6+1+0-3-1-0-0=3$. (d) The set of all Young diagrams for $n=4$ and $N=3$, with energies above. Below, eigenstates are represented by colored boxes: rotations in $\mathrm{SU}\left(N\right)$ transform between eigenstates in the same colored column, while permutations in $S_n$ transform between eigenstates in the same colored row. Representative states are found using the prescribed construction to be $|1111\rangle ,(|12\rangle -|21\rangle )|11\rangle ,(|12\rangle -|21\rangle )(|12\rangle -|21\rangle )$, and $(|123\rangle +|312\rangle +|231\rangle -|132\rangle -|213\rangle -|321\rangle )|1\rangle$, respectively. (e) Spectrum for $n=30$ with $N=2$ (red), and $N=3$ (blue).

Figure 3

Exact time evolution of $\widehat{Q}={\sum }_{j=1}^{10}{|1\rangle }_j{\langle 1|}_j$, which counts the number of the first ten orbitals in spin state $|1\rangle$. Two initial states are compared: ${|1\rangle }^{\otimes 10}{|2\rangle }^{\otimes 20}$ for $\text{SU}\left(2\right)$ and ${|1\rangle }^{\otimes 10}{|2\rangle }^{\otimes 10}{|3\rangle }^{\otimes 10}$ for $\text{SU}\left(3\right)$. The initial evolution is similar, but more $|1\rangle$ states diffuse out of the first ten orbitals for $\text{SU}\left(3\right)$ later on. Since all $E\left(\vec{\lambda }\right)$ are integer multiples of $U$, complete revival occurs at $Ut=2\pi$. In the $\text{SU}\left(2\right)$ case, the oscillation is dominated by the smallest $S$ in Eq. (4). This is consistent with the fact that for fixed $S_z$, the size of the eigenspaces decreases with $S$, causing overlap to be larger with subspaces of small $S$ generically.

Figure 4

(a) System prepared in $|1g1g\cdots 1g\rangle$. Spatially inhomogeneous pulse (1) results in equal superposition of this state and $\left|\right\{1e1g\cdots 1g\}\rangle$, containing one $e$ atom. An interaction blockade prevents coupling to states with two $e$ atoms. Pulse (2) flips the spins of the all-$g$ state. The initial pulse is reversed in pulse (3), resulting in the GHZ state. (b) Relevant energy levels of the Hamiltonian with $e$ and $g$ states and the magnetic field. Note that pulses (1) and (3), which involve states $|1g1g\cdots 1g\rangle$ and $\left|\right\{1e1g\cdots 1g\}\rangle$, do not couple to state $\left|\right\{1e1e\cdots 1g\}\rangle$ since there is a blockade of $2U_{1e1g}$. Similarly, during pulse (2), blockade prevents excitation of $\left|\right\{1e1g\cdots 1g\}\rangle$.

Figure 5

Layout of suggested experimental implementation. (a) A bow-tie beam arrangement of two pairs of beams aimed at a vacuum chamber. In each pair, the two beams have different $k$ vector directions of $\theta ={30}^{\text{o}}$, forming an in-plane standing wave perpendicular to that pair's net $k$ vector direction. The pair of perpendicular standing waves forms an attractive lattice. (b) The two-dimensional lattice of attractive-potential tubes forms with transverse vibrational frequency ${\omega }_{\perp }$ and lattice constant $\mathrm{\Delta }x$. The finite beam width results in a weak potential in the $z$ direction with vibrational frequency ${\omega }_z$. Gravity is in the beam plane to avoid a potential gradient along the tubes. Blue-detuned light outside the central region of width $L$ forms caps for the tubes. Following the Supplemental Material [52], we obtain ${\omega }_{\perp }\simeq 2\pi \times 10$ kHz, $\mathrm{\Delta }x\simeq 3\mu \mathrm{m}, {\omega }_z\simeq 2\pi \times 100$ Hz, and $L\simeq 10\mu \mathrm{m}$.