and the trapping frequency in the lattice, $\Gamma_l\sim \tilde \Gamma_0 \exp(-\omega_{\rm diff} / \omega)$ [@yi08], where $\tilde \Gamma_0$ is a prefactor [@yi08] $\omega_{\rm diff}$ is the frequency difference of the lattices for the two qubit states. If we operate in a field $\sim 5000$ G, then $\omega_{\rm diff}\sim 2\pi \times 550$kHz, and if we choose the Rabi Frequency $\Omega \sim 120$ kHz, then $\omega\sim 15$ kHz. We can also reduce the required field strength by choosing $m_I$ levels that are further separated (in $^{87}$Sr we can reduce the required field strength by a factor of 9 by choosing $m_l=-9/2$ and $m_l=+9/2$ as the two trapped states).

Combining resonant and off-resonant potentials {#sec:combining}
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In practice, strong coupling at intensity $I$ on the clock transition will also give rise to off-resonant AC-Stark shifts $\Delta E_{AC}^e$ and $\Delta E_{AC}^g$ of the states ${|e,i \rangle}$ and ${|g,i \rangle}$ from coupling to other manifolds (e.g., $^1$P$_1$ and $^3$S$_1$) in addition to the resonant couplings between the two levels. These must be added to the Hamiltonian, as $H=H_M+H_0+\Delta E_{AC}^e |e,i\rangle \langle e,i|+\Delta E_{AC}^g |g,i\rangle \langle g,i|$. As $\Delta E_{AC}^{e,g} \propto I$ and $\Omega\propto \sqrt{I}$, the off-resonant contributions will become more important as the intensity of 