C005 derivation

Drive C is $U_C(\theta)=u(\theta)\otimes u(\theta)$ with the same single-ion rotation $u(\theta)=e^{i(\theta/2)\sigma_x}$, $\sigma_x=|{\uparrow}\rangle\langle{\downarrow}|+|{\downarrow}\rangle\langle{\uparrow}|$ on the qubit. Because both ions are rotated identically, $U_C$ is a collective rotation: $U_C=e^{i\theta S_x}$ with $S_x=\tfrac12(\sigma_x\otimes\mathbb1+\mathbb1\otimes\sigma_x)$.

Singlet invariance. The singlet $|\Psi^-\rangle$ has total spin $0$ ($\mathbf S^2|\Psi^-\rangle=0$, see C001), so $S_x|\Psi^-\rangle=0$ and $$U_C(\theta)|\Psi^-\rangle=e^{i\theta S_x}|\Psi^-\rangle=|\Psi^-\rangle$$ for every $\theta$ (no phase, since $S_x$ annihilates it). Directly: with $\sigma_x|{\uparrow}\rangle=|{\downarrow}\rangle$, $\sigma_x|{\downarrow}\rangle=|{\uparrow}\rangle$, $$u\otimes u\,(|{\uparrow}{\downarrow}\rangle-|{\downarrow}{\uparrow}\rangle)$$ expands and the cross terms cancel, returning $|{\uparrow}{\downarrow}\rangle-|{\downarrow}{\uparrow}\rangle$. Hence the projector $|\Psi^-\rangle\langle\Psi^-|$ is fixed.

Triplet cycling. The triplet ($J=1$) subspace $\{|{\uparrow}{\uparrow}\rangle,(|{\uparrow}{\downarrow}\rangle+|{\downarrow}{\uparrow}\rangle)/\sqrt2,|{\downarrow}{\downarrow}\rangle\}$ is the orthogonal complement of the singlet and is invariant under any collective rotation. On it, $U_C=e^{i\theta S_x}$ acts as a spin-1 rotation about $x$, which for generic $\theta$ mixes (cycles) population among the three triplet states; e.g. it converts $|{\downarrow}{\downarrow}\rangle$ into superpositions of all three. Thus $U_C$ rotates within the triplet while the singlet is a fixed point. Verified.