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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.

Take an environment initially in its vacuum state $|0{⟩}_E$, with $|1{⟩}_E$ recording an emitted photon. In atom-environment order, put $q=\sqrt{1-p}$ and $u=\sqrt{p}$. A unitary dilation of amplitude damping is defined byThe middle two images are orthonormal, and they are orthogonal to the two fixed images. Thus this completion really is a unitary operator. For the occupied initial environmental subspace, it leaves a ground-state atom unchanged and splits an excited-state atom into no-emission and emission alternatives with probabilities $1-p$ and $p$.

Tracing out the environment after this unitary evolution givesThe environment matrix elements are $⟨0{|}_{E}U|0{⟩}_E=A_0$ and $⟨1{|}_{E}U|0{⟩}_E=A_1$, recovering the Kraus operators in part (a). More explicitly, the basis operators transform asApplying linearity to the atomic density operator therefore yieldsThe lost excited population goes to the ground state, the trace stays one, and coherences decrease by $\sqrt{1-p}$. Discarding the distinguishable environmental photon record is precisely what turns the joint unitary evolution into a noisy atomic quantum channel.