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In the following, we discuss a quantum register formed by one atom trapped every site of a deep optical lattice, where tunnelling of atoms between sites can be neglected on the timescale of the experiment. As discussed above, we identify hyperfine states with two chosen nuclear spin states, and we would like to create spin-dependent potentials in order to move these qubit states independently. We will make use of these in the two-qubit gate operations that we discuss in Sec. IV.

A novel method of forming optical lattices for alkaline earth atoms is to make use of a near-resonant optical coupling directly on the clock transition, which will produce adiabatic dressed potentials [^1]. In the case that the coupling field is a standing wave, the Rabi frequency, and thus the final dressed potential, will be sinusoidally varying, providing an optical lattice for dressed states that are superposition of states in the $^1$S$_0$ and $^3$P$_0$ levels (as shown in Fig. \[fig:levelstructure\]b). In a large magnetic field, there is a differential Zeeman shift $\Delta E_Z$ between the $^1$S$_0$ and $^3$P$_0$ states (109 Hz/G for $^{87}$Sr [@Boydthesis]), meaning that a direct coupling preserving the nuclear spin (with $\pi$-polarised light) will be resonant 