C079 derivation: repump jump operators

Per-ion basis $0=\ket{\downarrow},1=\ket{\uparrow},2=\ket{e}$. The two single-ion jump operators (Eq. S10) are $$ L_{e\to\downarrow} = \sqrt{p_{e\to\downarrow}\,\kappa}\,\ket{\downarrow}\bra{e}, \qquad L_{e\to\uparrow} = \sqrt{p_{e\to\uparrow}\,\kappa}\,\ket{\uparrow}\bra{e}, $$ with $p_{e\to\downarrow}=\sin^2\gamma$, $p_{e\to\uparrow}=\cos^2\gamma$ (same branching parametrisation as the discrete model, P.branching).

Single-ion decay from $\ket{e}$. Both operators contain $\bra{e}$, so they act only on the excited state: $L_{e\to\downarrow}\ket{e}=\sqrt{p_{e\to\downarrow}\kappa}\ket{\downarrow}$, $L_{e\to\uparrow}\ket{e}=\sqrt{p_{e\to\uparrow}\kappa}\ket{\uparrow}$, and both annihilate $\ket{\downarrow},\ket{\uparrow}$. Hence they implement incoherent single-ion decay $\ket{e}\to\{\ket{\downarrow},\ket{\uparrow}\}$, as required for a repump.

Total rate $\kappa$. The decay rate out of $\ket{e}$ is $$ \bra{e}\big(L_{e\to\downarrow}^{\dagger}L_{e\to\downarrow} + L_{e\to\uparrow}^{\dagger}L_{e\to\uparrow}\big)\ket{e} = p_{e\to\downarrow}\kappa + p_{e\to\uparrow}\kappa = (\sin^2\gamma+\cos^2\gamma)\,\kappa = \kappa. $$

Branching. $p_{e\to\downarrow}/p_{e\to\uparrow}=\sin^2\gamma/\cos^2\gamma=\tan^2\gamma$, and $p_{e\to\downarrow}+p_{e\to\uparrow}=1$, so $\kappa$ is the total repump rate and $\gamma$ controls only the branching, exactly as claimed.

The two-ion model uses the same operators on each ion (4 jump operators total, $L_1,\dots,L_4$ in continuous.jump_ops). The symbolic check sympy_check.py confirms all three properties. Verdict: verified.