Formula_758

\[\begin{gathered}\frac{\delta}{\delta t}\langle 0\rangle_{t}+i\left\langle\left[O, H_{M}\right]\right\rangle_{t}= \\=\sum_{k} \int_{-\infty}^{t} d \tau \exp \left\{i \omega_{k}(\tau-t)\right\}\left\{N_{k}\left\langle M_{k}^{+}(\tau)\left[O(t), M_{k}(t)\right]\right\rangle_{t}+\right. \\\left.+\left(1+N_{k}\right)\left\langle\left[M_{k}(t), O(t)\right] M_{k}^{+}(\tau)\right\rangle_{t}\right\}+ \\+\sum_{k} \int_{-\infty}^{t} d \tau \exp \left\{-i \omega_{k}(\tau-t)\right\}\left\{N_{k}\left\langle\left[M_{k}^{+}(t), O(t)\right] M_{k}(\tau)\right\rangle_{t}^{+}\right. \\\left.+\left(1+N_{k}\right)\left\langle M_{k}(\tau)\left[O(t), M_{k}^{+}(t)\right]\right\rangle_{t}\right\} .\end{gathered}\]