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When implementing the Cirac-Zoller CNOT gate between two ions $A$ and $B$, the second step of the procedure changes the sign of the second ion when the ion is in its ground state $|0_B\rangle$ and the ion lattice is in its first excited state $|1_z\rangle$ and leave any of the other possible states unchanged. This step is called the $2\pi$-pulse. I know that a rotation gate $R_x(2\pi)$ changes the global sign of the qubit state. However, I can't see how it can take into account the state of the lattice vibration.

The first step has a similar procedure, acting on qubit $A$. In that step, it is easy to see why the other states are not affected because the red-detuned laser only allows for the specific transition $|1_A,0_z\rangle \leftrightarrow |0_A,1_z\rangle$.

How is this sort of controlled $R_x(2\pi)$ doing the "controlled" part?

PS: Apparently this can be done using a third qubit state in a way that is not really clear to me which was explained in this question. Can it be done in a different way?

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