Atomic Quantum Simulation of $\mathbf{U}(N)$ and $\mathrm{SU}(N)$ Non-Abelian Lattice Gauge Theories

Using ultracold alkaline-earth atoms in optical lattices, we construct a quantum simulator for $\mathrm{U}(N)$ and $\mathrm{SU}(N)$ lattice gauge theories with fermionic matter based on quantum link models. These systems share qualitative features with QCD, including chiral symmetry breaking and restoration at nonzero temperature or baryon density. Unlike classical simulations, a quantum simulator does not suffer from sign problems and can address the corresponding chiral dynamics in real time.

Figure 1

(a) (upper panel) $\mathrm{U}(N)$ QLM in $(1+1)\mathrm{D}$ with quark fields ${\psi }_x^i$ on lattice sites and gauge fields $U_{x,x+1}^{ij}$ on links; (lower panel) hopping of AE atoms between quark and rishon sites of the same shading. (b) Implementation of the QLM in rishon representation with fermionic atoms in $(2+1)\mathrm{D}$. (c) Encoding of the color degrees of freedom for $N=2$ ($\uparrow$, $\downarrow$) in Zeeman states of a fermionic AE atom with $I=3/2$. (d) Lattice structure to avoid the interaction in fermionic matter sites using a species-dependent optical lattice (for an alternative method using site-dependent optical Feshbach resonances, see the main text). (e) Initial state loaded in the optical lattice with a staggered distribution of doubly occupied sites for a U(2) QLM with $\mathcal{N}=2$.

Figure 2

Dynamical processes in U(2) QLMs with $\mathcal{N}=2$. (a) Matter-gauge interaction as correlated hopping of quarks and rishons. Starting with a configuration of site singlets, the matter-gauge interaction converts them into nearest-neighbor singlets, keeping the rishon number per link constant. (b) The determinant term corresponds to two-body hopping of both rishons on the link.

Figure 3

(a) Size $L$ dependence of the energy splitting between the lowest energy eigenstates of a U(2) QLM with $m=0$ and $V=-6t$. (b) Real-time evolution of the order parameter profile $(\overline{\psi }\psi {)}_x(\tau )$ for $L=12$, mimicking the expansion of a hot quark-gluon plasma. Here, circles (thin line), diamonds (dashed line), and squares (dotted line) correspond to $\tau /t=0$, 1, 10, respectively.