= Solution
Choose $|0\rangle$ as the ground state and $|1\rangle$ as the excited state. The relevant <quantum channel> is the <amplitude damping channel>. Its <Kraus operators> are
$$
\boxed{K_0=\begin{pmatrix}1&0\\0&\sqrt{1-p}\end{pmatrix},\qquad K_1=\begin{pmatrix}0&\sqrt p\\0&0\end{pmatrix}=\sqrt p\,|0\rangle\langle1|.}
$$
They satisfy $K_0^\dagger K_0+K_1^\dagger K_1=I$. The $K_0$ branch corresponds to no emitted photon: the ground-state amplitude is unchanged and the excited-state amplitude is attenuated. The $K_1$ branch corresponds to photon emission and transition to the ground state. These are unnormalized branch states; the squared norm or <matrix trace> of each branch gives its probability.
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