 Solution

Model the environment's vacuum and one-photon states by $|0_E⟩,|1_E⟩$. A unitary dilation of amplitude damping acts on the initially vacant environment asThese images are orthonormal. One unitary completion takes $U|0,1_E⟩=-\sqrt{p}|1,0_E⟩+\sqrt{1-p}|0,1_E⟩$ and fixes $|1,1_E⟩$. Taking environment matrix elements $⟨k_E|U|0_E⟩$ gives $K_0,K_1$.

Evolve $\rho \otimes |0_E⟩⟨0_E|$ by $U$ and take the partial trace over the environment. Orthogonality of the two environment records removes the cross-branch terms, giving ${\mathcal{A}}_p(\rho )=K_0\rho K_0^{†}+K_1\rho K_1^{†}$. Entrywise,Excited population is transferred to the ground state, while coherence is reduced by the square root of the survival probability.