= Bright and dark state reduction of a star-coupled Hamiltonian
Suppose one state $|0\rangle$ couples to degenerate states $|1\rangle,\ldots,|m\rangle$ with amplitudes $v_1,\ldots,v_m$, while the degenerate states do not couple to each other. The normalized bright state proportional to $\sum_jv_j|j\rangle$ is the only combination coupled to $|0\rangle$. Every orthogonal combination is a dark state and remains an exact eigenstate at the degenerate energy. The remaining spectrum comes from a two-dimensional Hamiltonian whose off-diagonal entry is $(\sum_j|v_j|^2)^{1/2}$.
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