On the damping of the angular momentum of three harmonic oscillators

Core claim

The paper's central result is a closed-form description of angular momentum damping in a specific open quantum system. Starting from the Lindblad generator in the Heisenberg picture for the six phase-space observables $q_k$ and $p_k$, the authors specialize to the case where all off-diagonal opening coefficients vanish ($\alpha_{kl}=\beta_{kl}=\lambda_{kl}=0$ for $k\neq l$) and the three oscillators are identical. In that case the evolution matrix $\hat{M}(t)$ is block diagonal, the secular equation factors, and the covariance matrix follows from $\hat{\sigma}(t)=\hat{M}(t)(\hat{\sigma}(0)-\hat{\sigma}(\infty))\hat{M}^{T}(t)+\hat{\sigma}(\infty)$. The angular momentum projection $\langle L_3(t)\rangle = \sigma_{q_1p_2}(t)-\sigma_{q_2p_1}(t)$ then reduces to the pure damped exponential $\langle L_3(0)\rangle e^{-2\lambda t}$, because the oscillatory parts cancel under the symmetry and equal-frequency assumptions. For the squared angular momentum the paper obtains $\langle L^2(t)\rangle = \beta e^{-4\lambda t} + e^{-2\lambda t}\{\beta_1\sin^2\omega t + \beta_2\cos^2\omega t + \beta_3\sin\omega t + \beta_4\cos\omega t + \beta_5\sin\omega t\cos\omega t\} + \langle L^2(\infty)\rangle$, with the asymptotic value written in terms of stationary diffusion variances.