I'm trying to understand the paper: "Quantum dot cavity-QED in the presence of strong electron-phonon interactions" by Wilson Ray and A. Imamoglu (https://journals.aps.org/prb/abstract/10.1103/PhysRevB.65.235311). I have arrived at the following master equation for the reduced density matrix (equation 5 in the paper): $$ \frac{\partial\rho(t)}{\partial t}=\frac{1}{i\hbar}[H'_{sys},\rho(t)]-\frac{1}{\hbar^2}\int_0^td\tau\sum_{m=g,u}\{G_m(\tau)[X_m,e^{-iH'_{sys}\tau/\hbar}X_m\rho(t-\tau)e^{iH'_{sys}\tau/\hbar}]+H.c.\} $$ Following his derivation, the next step is to expand $\rho$ in powers of $\Omega_p$ and project the equation above on the basis of $H'_{sys}|_{\Omega_p=0}$ to obtain the following equation. $$ \frac{\partial\rho^{(1)}_{\pm0}(t)}{\partial t}=\frac{\mp i<B>}{\sqrt{2}}+i\Delta\omega_{\pm}\rho^{(1)}_{\pm0}(t)-\frac{g}{\sqrt{2}}\int_0^td\tau G_{\pm}(\tau)[1+g\sqrt{2}\rho^{(1)}_{\pm0}(t-\tau)] $$ Could anyone give me some guidance on what does it mean to expand $\rho$ in powers of $\Omega_p$ and gets here? Thank you.
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