Let $|s\rangle=|\psi^-\rangle$ be the <spin-one-half singlet state>, complete it to an orthonormal basis $\{|s\rangle,|e_1\rangle,|e_2\rangle,|e_3\rangle\}$, and choose
$$
H=0,
\qquad
L_j=\sqrt\gamma\,|s\rangle\langle e_j|,
\quad j=1,2,3.
$$
Each excited population decays into the singlet. On the matrix-unit basis, $|e_j\rangle\langle e_k|$ has decay eigenvalue $-\gamma$ after separating off its contribution to the stationary population, while $|s\rangle\langle e_j|$ and its adjoint have eigenvalue $-\gamma/2$. The sole zero mode is $|s\rangle\langle s|$, so every initial state converges to
$$
\rho(\infty)=|\psi^-\rangle\langle\psi^-|.
$$
The nonzero eigenvalue closest to the origin is therefore
$$
\boxed{\lambda_{\rm slow}=-\frac\gamma2},
\qquad
\boxed{\Delta_{\mathcal L}=\frac\gamma2}.
$$

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-343/1/c/solution