the applied field becomes larger. For $^{87}$Sr, the shifts from the AC-Stark shift become of the same order as the AC-Stark splitting due to resonant coupling at relatively high fields, with $I \sim 50$kW/cm$^2$ [@Boydthesis; @yi08]). At higher fields, the potentials $V^{\pm}$ will be modified by these shifts, but can still be made spin-dependent if the detunings and Rabi frequencies of the lattice beams are chosen carefully. This is illustrated in Fig. \[fig:combinedpotentials\], where we show the lower adiabatic potential for each of the two nuclear spin states for a selection of different phases $\phi$ between the potentials. We see that at relative phase $\phi=0$ the potentials for different nuclear spin states are identical, and are given by a combination of the resonant and off-resonant contributions. At phase $\phi=\pi/2$, however, the off-resonant contributions from the two coupling frequencies, which are independent of the nuclear spin state, become spatially homogeneous due to the addition of the two spatially shifted contributions. At this point the sinusoidal form of the lattice potentials is due solely to the resonant contribution. It can be seen that the lattice, also in between, will be modified in such a way that the atoms will be transported through 