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The Z quantum logic gate can be represented as

$$\left( \begin{matrix}1 && 0 \\ 0 &&-1\end{matrix} \right)$$

and can be physically implemented with a rotation around the $Z$ axis by $\pi$ radians. In cold atoms this corresponds to applying a $2\pi$ Rabi pulse.

If I understand this correctly this is straightforward to rotate around all other angles of rotation:

$$\left( \begin{matrix}1 && 0 \\ 0 &&e^{i\phi/2}\end{matrix} \right) \, .$$

However it is not clear to me how to instead rotate around the $X$ and $Y$ axis. If you apply the electric field in the $Y$ direction, doesn't this correspond to a Rabi frequency of 0?

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The answer will depend on how the qubit levels are encoded in the atom. I'll give an example for a possible encoding in a neutral Rubidium atom. You could define the two qubit states as different hyperfine ground state levels, such as $|0\rangle = |F=1, m_F = 0\rangle$ and $|1\rangle = |F=2, m_F = 0\rangle$.

A rotation about the $Z$ axis corresponds to adding a relative phase between the two states $|0\rangle$ and $|1\rangle$. This can be done by shifting the energy of one of the qubit states. While the energy is shifted, that state accumulates phase at a different rate than normal - so by controlling the duration during which you shift the energy, you can control how much extra phase is accumulated. This type of procedure can be performed with a laser.

A rotation about $X$ or $Y$ corresponds to an exchange in population between $|0\rangle$ and $|1\rangle$. For example, a microwave field tuned to the transition from $|0\rangle$ to $|1\rangle$ can implement such a gate. The difference between $X$ and $Y$ corresponds to the phase of the microwave field relative to some 'initial' phase.

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