$q_2$ is the state of the second qubit in the pair, then atoms in the state ${|01 \rangle}$ are brought together, whilst ${|00 \rangle}$, ${|11 \rangle}$ and ${|10 \rangle}$ remain separated (see Fig. \[fig:spindep\]).

2.  A phase shift is generated conditioned on whether two atoms are on the same site or not.

3.  The atoms are returned to their initial positions.

The phase in step two can be generated in a number of different methods, including via direct collisional phase shifts, or the use of blockade mechanisms. These different mechanisms are discussed in the following two subsections.

Phase for two-qubit gates: controlled collisions
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![If the qubits are trapped in a spin-dependent lattice, it is possible to shift the lattice for one qubit state by one site, so that neighbouring atoms are brought together only if the qubit to the left was in state ${|0 \rangle}$ and the qubit to the right was in state ${|1 \rangle}$. This can be used to aid in producing two-qubit quantum gates (see text for details).[]{data-label="fig:spindep"}](spindep){width="8.5cm"}

For alkali atoms, the phase in step 2 is generated by collisional interactions between atoms. This could be performed directly if the atoms used have a relatively large 