scattering length in the $^1$S$_0$ manifold (e.g., $^{87}$Sr). For other species and isotopes such as $^{171}$Yb, this could also be achieved using optical Feshbach resonances [@srfeshbach; @ybfeshbach] to enhance the otherwise very weak collisional interaction. The speed of such gates is limited by the strength of the on-site interaction between atoms, which for a single-band model is limited by the trap frequency in each lattice site, $\omega$.

However, the existence of weak collisional interactions for certain isotopes also motivates us to look at other gate schemes, particularly using excitations to states in the $^3$P$_2$ manifold.

Use of $^3$P$_2$ levels
-----------------------

We would again like to make use of states in the $^3$P$_2$ manifold to which our dressed qubit states (for a fixed nuclear spin) can be coupled, and which are trapped in the same locations as our qubits. This time we will assume that we have two such auxiliary levels, ${|0x \rangle}$ and ${|1x \rangle}$, as depicted in Fig. \[fig:readoutlevels\].

### Phase for two-qubit gates: dipole blockade mechanism

For sufficiently large onsite dipole-dipole interactions, which provide a energy shift between $^3$P$_2$-$^3$P$_2$ collisional interactions and $^3$P$_0$/$^1$S$_0$-$^3$P$_2$ corresponding to a large frequency shift $\Delta$, we can use a dipole blockade mechanism to 