15 kHz between atoms in neighbouring sites for a field gradient of 100 Gauss/cm. Atoms in the dressed lattice can then be selectively transferred via a Raman process connecting off-resonantly via the $^3$S$_1$ manifold to the $^3$P$_2$ manifold, on a timescale limited by the frequency shift between neighbouring sites.

This assumes, of course, that the state in the $^3$P$_2$ manifold to which we couple, ${|0x \rangle}$ is trapped in a lattice, preferably in a lattice at the same position as our qubit states ${|0 \rangle}$ and ${|1 \rangle}$. Thus, the most favourable states are those with a significant negative $AC$-polarisability $\alpha$ at the wavelength of the clock transition, as the potential they experience due to the AC-Stark shift will have minima in the same places as the lower dressed state generated by the same lattice laser. We have computed the polarisability from known data of the states in the $^3$P$_2$ manifold of $^{87}$Sr, and have found that they vary substantially due to a large tensor shift. We write the shift $\Delta_E$ from linearly polarised light as $$\begin{aligned}
h\Delta_E&=&-\frac{1}{2}\alpha E^2,\\
&=&-\left[\alpha^{\rm scalar} +\alpha^{\rm tensor}\frac{3m_F^2-F(F+1)}{F(2F-1)}\right]\frac{E^2}{2},\end{aligned}$$ where we have separated the coefficients of the scalar and tensor shifts [@boyd07; @Boydthesis], and we obtain 