How to get Heisenberg equation of motion? A system Hamiltonian is given by
$$
H=\hbar\omega_{1}\hat{a}^{\dagger}\hat{a}+\Sigma_{i=1}^{N}\left(\hbar\omega_{se}\hat{\sigma}_{ss}^{i}+\hbar\omega_{ge}\hat{\sigma}_{ee}^{i}\right)-\hbar\Sigma_{i=1}^{N}\left[\hat{a}ge^{i\omega_{1}z_{i}/c}\hat{\sigma}_{eg}^{i}+\Omega e^{-i\omega_{2}(t-z_{i}/c)}\hat{\sigma}_{es}^{i}\right]+\textrm{H.c.}
$$
After transformation, we got the effective rotating frame Hamiltonian,
$$\tilde{H}=\hbar\Delta\hat{\sigma}_{ee}-\left(\hbar\Omega\hat{\sigma}_{es}+\hbar g\hat{\varepsilon}\hat{\sigma}_{eg}+\textrm{H.c.}\right)$$
And from this result, how can I derive the Heisenberg Equations of Motion?
Note: The Hamiltonian is given in this paper, Photon Storage in $\Lambda$-type Optically Dense Atomic Media. I. Cavity Mode (Physical Review A 76, 033804 (2007))
 A: The Heisenberg equation of motion for a general system operator $\hat{A}$ (that has no explicit dependence upon $t$) where the full system is specified by the Hamiltonian $\hat{H}$ is
\begin{equation}
\frac{d \hat{A}}{dt} = \frac{i}{\hbar} [\hat{H}, \hat{A}] .
\end{equation}
For example (from the paper you referenced), let's take the Hamiltonian $\hat{\tilde{H}} = \hbar \Delta \hat{\sigma}_{ee} - (\hbar \Omega(t) \hat{\sigma}_{es} + \hbar g \hat{\mathcal{E}} \hat{\sigma}_{eg} + \text{H.c.})$ and calculate the Heisenberg equation of motion for $\hat{\mathcal{E}}$:
\begin{equation}
\frac{d\hat{\mathcal{E}}}{dt} = \frac{i}{\hbar} [\hbar \Delta \hat{\sigma}_{ee} - (\hbar \Omega(t) \hat{\sigma}_{es} + \hbar g \hat{\mathcal{E}} \hat{\sigma}_{eg} + \text{H.c.}), \hat{\mathcal{E}}] = - i g \hat{\sigma}_{ge} [\hat{\mathcal{E}}^\dagger, \hat{\mathcal{E}}] = i g \hat{\sigma}_{ge} .
\end{equation}
Note that, in the paper you see the extra terms $- \kappa \hat{\mathcal{E}}$ and $\sqrt{2 \kappa} \hat{\mathcal{E}}_{\text{in}}$ in the equation of motion for $\hat{\mathcal{E}}$; these terms are due to a coupling with a bath that has been traced out.
