C004 derivation

Drive B is a single-ion repump channel taking $|e\rangle$ to the qubit states with probabilities $p_{e\to\downarrow}$ and $p_{e\to\uparrow}$. The parameterization (Supp. Mat. S1, protocol Kraus operators) is $$p_{e\to\downarrow}=\sin^2\gamma,\qquad p_{e\to\uparrow}=\cos^2\gamma.$$

Normalization. $p_{e\to\downarrow}+p_{e\to\uparrow}=\sin^2\gamma+\cos^2\gamma=1$, so these are valid (complete) branching probabilities. This is exactly the Kraus completeness relation for the channel: with $$E_0=|{\downarrow}\rangle\langle{\downarrow}|+|{\uparrow}\rangle\langle{\uparrow}|,\quad E_1=\sqrt{p_{e\to\downarrow}}\,|{\downarrow}\rangle\langle e|,\quad E_2=\sqrt{p_{e\to\uparrow}}\,|{\uparrow}\rangle\langle e|,$$ one has $\sum_k E_k^\dagger E_k=\mathbb1$ iff $p_{e\to\downarrow}+p_{e\to\uparrow}=1$.

Ratio. $$\frac{p_{e\to\downarrow}}{p_{e\to\uparrow}}=\frac{\sin^2\gamma}{\cos^2\gamma}=\tan^2\gamma,$$ which is the stated parameterization.

Range coverage. As $\gamma$ ranges over $[0,\pi/2)$, $\tan^2\gamma$ ranges monotonically over $[0,\infty)$. Hence every physically admissible branching ratio (from "all population to $|\uparrow\rangle$" at $\gamma=0$, through balanced $1{:}1$ at $\gamma=\pi/4$, to "all to $|\downarrow\rangle$" as $\gamma\to\pi/2$) is realized exactly once. The parameterization is therefore both correct and onto the full range. Verified.