= Solution
Model the environment's vacuum and one-photon states by $|0_E\rangle,|1_E\rangle$. A <unitary dilation of amplitude damping> acts on the initially vacant environment as
$$
U|0,0_E\rangle=|0,0_E\rangle,\qquad U|1,0_E\rangle=\sqrt{1-p}|1,0_E\rangle+\sqrt p|0,1_E\rangle.
$$
These images are orthonormal. One unitary completion takes $U|0,1_E\rangle=-\sqrt p|1,0_E\rangle+\sqrt{1-p}|0,1_E\rangle$ and fixes $|1,1_E\rangle$. Taking environment matrix elements $\langle k_E|U|0_E\rangle$ gives $K_0,K_1$.
Evolve $\rho\otimes|0_E\rangle\langle0_E|$ by $U$ and take the <partial trace> over the environment. Orthogonality of the two environment records removes the cross-branch terms, giving $\mathcal A_p(\rho)=K_0\rho K_0^\dagger+K_1\rho K_1^\dagger$. Entrywise,
$$
\boxed{\mathcal A_p(\rho)=\begin{pmatrix}\rho_{00}+p\rho_{11}&\sqrt{1-p}\,\rho_{01}\\\sqrt{1-p}\,\rho_{10}&(1-p)\rho_{11}\end{pmatrix}.}
$$
Excited population is transferred to the ground state, while coherence is reduced by the square root of the survival probability.
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