the lattice spin-dependently.

![a) Lower dressed potentials for the ${|0 \rangle}$ qubit state (solid lines) and the ${|1 \rangle}$ qubit state (dashed lines) formed by combining resonant and off-resonant contributions. These are plotted as a function of position for varying phase offsets $\phi$ between the coupling fields for the ${|0 \rangle}$ and ${|1 \rangle}$ states. For $^{87}$Sr at the wavelength of the clock transition, we obtain $\Delta E_{AC}^e \approx 3 \Delta E_{AC}^g$, and we choose the intensity so that $\Omega=4 \Delta E_{AC}^g$ (ca. 10 kW/cm$^2$). Here, $\delta=-3\Omega/4$. b) Projection on the excited state of the dressed state corresponding to the lower dressed potential, plotted for the same values of $\Omega$, $\Delta E_{AC}^{g}$, and $\Delta E_{AC}^{e}$ as in (a) as a function of $\phi$. The different lines correspond to varying $\delta$, from top to bottom, $\delta=-\Omega/2$, $-3\Omega/4$, $-\Omega$, and $-5\Omega/4$.[]{data-label="fig:combinedpotentials"}](combined){width="8.5cm"}

We also note that two important characteristics relating to the shape of the lattice and the form of the dressed states changes as a function of $\phi$. Firstly, the lattice depth changes, because for $\phi=0$ the effect of the resonant and off-resonant contributions to the lower dressed potential are summed, whereas for $\phi=\pi/2$, the lattice is formed solely by a resonant 