at substantially different frequencies for different nuclear spin states (see Fig. \[fig:levelstructure\]c). We can then drive each transition independently with Rabi frequencies $\Omega_0=\Omega_{\pm}$, as shown in Fig. \[fig:levelstructure\]b. Provided that the shift $\Delta E_Z \gg \Omega_{\pm}$, we will then obtain independent two-level systems for each $m_I$ state for which we apply the appropriate coupling frequency. For example, if we choose $\Omega_0\sim 100$ kHz, then for $^{87}$Sr, we would like to apply a field $\gtrsim 1000$ G in order to obtain shifts between neighbouring states $\gtrsim 100$ kHz (however, states separated further in $m_I$ could also be used to reduce the required field - see below). In this way, we can choose, e.g., two $m_I$ states as our two qubit states, ${|0 \rangle}$ and ${|1 \rangle}$, and create independent potentials for these two states. At the same time, because the frequency differences between lattices for different $m_I$ states will of the order of 1 MHz, the lattice laser wavenumber $k_l$ is approximately the same for the two species - in fact the resulting lattice potentials will overlap for the order of millions of periods.

Dressed potentials for a two-level system
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We will now discuss the form of the dressed potentials 