Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 343 / 1 / c / Solution

Let $|s⟩=|{\psi }^{-}⟩$ be the spin-one-half singlet state, complete it to an orthonormal basis $\{|s⟩,|e_1⟩,|e_2⟩,|e_3⟩\}$, and chooseEach excited population decays into the singlet. On the matrix-unit basis, $|e_j⟩⟨e_k|$ has decay eigenvalue $-\gamma$ after separating off its contribution to the stationary population, while $|s⟩⟨e_j|$ and its adjoint have eigenvalue $-\gamma /2$. The sole zero mode is $|s⟩⟨s|$, so every initial state converges toThe nonzero eigenvalue closest to the origin is therefore