the $^3$P$_2$ level can then be determined by fluorescence measurements, e.g., using the cycling transition $^3$P$_2$-$^3$D$_3$, independent of the atoms remaining in the $^1$S$_0$ and $^3$P$_0$ levels. Note that the timescale for this readout process $\tau_{\rm readout}$ is limited by the trapping frequency in the dressed lattice potential, $\tau_{\rm readout} \gg 2\pi /\omega$. This requirement must be fulfilled so that the atom is not coupled to excited Bloch bands of the lattice. It is also desirable for this coupling to have similar trapping frequencies for the lattices trapping ${|0 \rangle}$ and ${|0x \rangle}$, so maximising spatial overlap of the wavefunctions. Again, the $F=13/2$, $m_F=-13/2$ state of the $^3$P$_2$ manifold is favourable for this, as the polarisability indicates that the lattice depth will be around $150$ kHz for $I\sim3$kW/cm$^3$, which is a similar depth to that of the lattice for the dressed levels at the same lattice intensity (assuming that the detuning of the resonant coupling lasers, $\delta_i$ is small).

Note that one could equally use states with $|m_I|<-13/2$ in this process if one stores the qubit states in the upper dressed potential. This is disadvantageous, because a large detuning $\delta$ must be chosen for the lattice lasers to prevent non-adiabatic 