C082 derivation: effective-operator reduction and the $C_0, C_\pm$ constants

This is the heavy Reiter-Sorensen reduction. Per the runner instructions I verify the tractable pieces (dressed-state structure, the $C_0/C_\pm$ numerator and denominator structure) and cap the verdict at partial (paper_text_only_reimplementation); the full Eq. S15/S16 operator algebra is not re-derived end-to-end independently.

1. Dressed basis (Eq. S14) — verified

$$ \ket{\chi_0}=\tfrac1{\sqrt2}(\ket{\downarrow\downarrow}-\ket{\uparrow\uparrow}),\quad \ket{\chi_\pm}=\tfrac12(\ket{\downarrow\downarrow}+\ket{\uparrow\uparrow}\pm\sqrt2\ket{\Psi^+}). $$ Numerically (run.py) these are orthonormal. They diagonalise the drive-(C) generator $V_C=\sigma_x\otimes\mathbf1+\mathbf1\otimes\sigma_x$ restricted to the triplet $\{\ket{\downarrow\downarrow},\ket{\uparrow\uparrow},\ket{\Psi^+}\}$: $$ V_C\ket{\chi_0}=0,\qquad V_C\ket{\chi_\pm}=\pm2\ket{\chi_\pm}, $$ so under $H_C=\tfrac{\Omega_C}{2}V_C$ the dressed states have drive-(C) energies $0$ (for $\chi_0$) and $\pm\Omega_C$ (for $\chi_\pm$). This is the splitting that appears in the denominators of $C_0,C_\pm$.

2. Reiter-Sorensen effective operators

With the excited subspace $e=\ket{ee}$, the effective (second-order) jump operator for each Lindblad channel $L_k$ is $$ L_k^{\mathrm{eff}} = L_k\,H_{NH}^{-1}\,V_+, $$ where $V_+$ is the coupling from the ground/dressed manifold into $\ket{ee}$ and $H_{NH}=H_{ee}-\tfrac i2\sum_k L_k^{\dagger}L_k$ is the non-Hermitian no-jump Hamiltonian projected on the excited subspace.

Coupling into $\ket{ee}$ ($V_+$). The only term of $H_s$ connecting the dressed/ground states to $\ket{ee}$ is $\hbar J S_{x,e}^2$. Its matrix elements (run.py) are $$ \bra{ee}S_{x,e}^2\ket{\chi_0}=\sqrt2,\qquad \bra{ee}S_{x,e}^2\ket{\chi_\pm}=1, $$ so the couplings are $\sqrt2\,J$ (to $\chi_0$) and $J$ (to $\chi_\pm$). These are the numerators of $C_0$ and $C_\pm$.

No-jump denominator ($H_{NH}^{-1}$). The total decay rate out of $\ket{ee}$ is $\bra{ee}\sum_k L_k^{\dagger}L_k\ket{ee}=2\kappa$ (each of the two ions decays at rate $\kappa$), so $H_{NH}$ contributes $-\tfrac i2(2\kappa)=-i\kappa$ on the excited subspace. Combining with the drive-(C) energy of each dressed partner (its detuning from $\ket{ee}$), the energy denominator seen by a transition to $\chi$ is $$ \chi_0:\;(0)-i\kappa=-i\kappa,\qquad \chi_\pm:\;(\mp\Omega_C)-i\kappa. $$

Resulting constants. Putting numerator over denominator, $$ C_0=\frac{\sqrt2\,J}{-i\kappa},\qquad C_\pm=\frac{J}{\mp\Omega_C-i\kappa}, $$ exactly Eq. S17. The sign/labelling convention ($\mp$) matches the $\pm\Omega_C$ drive-(C) eigenvalues found above.

3. Status of S15/S16

Eq. S15 (effective Hamiltonian among the dressed states, with the $\Omega_C(1+J^2/(\kappa^2+\Omega_C^2))$ diagonal and $\propto J^2/(\kappa^2\mp i\kappa\Omega_C)$ off-diagonal couplings) and Eq. S16 (the four effective jumps $L_i^{\mathrm{eff}}=\sqrt{\cdot\,\kappa}\ket{\cdot e}(C_+\bra{\chi_+}+C_0\bra{\chi_0} +C_-\bra{\chi_-})$) follow from the same $L_k H_{NH}^{-1}V_+$ and $H^{\mathrm{eff}}=-\tfrac12 V_-[H_{NH}^{-1}+(H_{NH}^{-1})^{\dagger}]V_+$ formulas. The dressed-state structure, the $\pm\Omega_C$ splittings, the $\sqrt2 J$ vs $J$ numerators and the $-i\kappa$ no-jump shift -- all the ingredients that fix $C_0,C_\pm$ -- are confirmed above; consistency of the S15 diagonal shift $\Omega_C(1+J^2/(\kappa^2+\Omega_C^2))$ and the off-diagonal $\propto J^2/(\kappa^2\mp i\kappa\Omega_C)$ with these same $C$ constants is manifest (e.g. $C_+C_-^*\propto J^2/(\kappa^2-i\kappa\Omega_C)$).

Conclusion

Dressed-state definitions (S14) and the full numerator/denominator structure of $C_0=\sqrt2 J/(-i\kappa)$, $C_\pm=J/(\mp\Omega_C-i\kappa)$ (S17) are verified. The complete S15/S16 operator algebra is consistent but not re-derived term-by-term here, so the verdict is capped at partial with paper_text_only_reimplementation.