A note on the Fröhlich dynamics in the strong coupling limit

We revise a previous result about the Fr\"ohlich dynamics in strong coupling limit obtained [Gri17]. In latter it was shown that time evolution applied to initial state $\varphi_0 \otimes \xi_\alpha$, where $\varphi_0$ is electron ground of Pekar energy functional and $\xi_\alpha$ associated coherent phonons, can be approximated by global phase for times small compared $\alpha^2$. present note we prove similar approximation holds $t=O(\alpha^2)$ if one includes nontrivial effective phonons generated an operator proportional $\alpha^{-2}$ quadratic creation annihilation operators. Our implies remains close its order $\alpha^2$ while phonon fluctuations around described time-dependent Bogoliubov transformation.