loss of atoms from the potential [@yi08]. Alternatively, an additional standing wave at a different frequency could be added to trap states in $^3$P$_2$ manifold via an additional AC-Stark shift.

An alternative to using magnetic field gradients for addressing would be to apply a laser with spatially varying intensity at the magic wavelength (for equal shifts of the $^3$P$_0$ and $^1$S$_0$ levels. This would provide a position-dependent differential AC-Stark shift between the qubit states and the $^3$P$_2$ level, without affecting the relative energy of the $^3$P$_0$ and $^1$S$_0$ levels, and thus the dressed lattice.

![AC-Polarisabilities for $^{87}$Sr in the $^3$P$_2$ manifold with $F=13/2$ at the frequency of the clock transition.[]{data-label="fig:starkshifts"}](starkshifts){width="8.5cm"}

\[fig:polarisability\]

Quantum Gates in spin-dependent potentials
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Single-qubit gates can be performed in one of two ways in this scheme. The simplest means to obtain a global rotation of many qubits is to directly couple the dressed states for two nuclear spins via a Raman process. Alternatively, different nuclear spin (qubit) states can be alternately coupled to auxiliary states in the $^3$P$_2$ level in order to provide individual addressing for single-qubit rotations using the techniques described in the previous section. Such coupling requires the use of a trapped state in 