Consider the interaction of a single surface plasmon mode in a metal nanoparticle with a dipole emitter(atom) under the classical driving field $ E_i = E_0e^{-i\omega t} + c.c $ , with the following Hamiltonian and master equation:

(Let $\textbf{a}$ and $\textbf{a}^\dagger$ be the annihilation and creation operators of the surface plasmon mode, $\sigma$ and $\sigma^\dagger$ be the lowering and raising operators of the atom and $\omega_{sp}$ and $\omega_x$ be the plasmon and atomic resonance frequencies respectively. $\rho$ is the density matrix of the combined system. Coupling g and nanoparticle property $\chi$ are to be determined)

$$\begin{align}H_0 &= \hbar\omega_{sp}\textbf{a}^\dagger\textbf{a} + \hbar\omega_{x}\sigma^\dagger\sigma\\H_\textrm{int} &= i\hbar g\left(\textbf{a}^\dagger \sigma-\textbf{a} \sigma^\dagger\right)\\ H_\textrm{drive} &=-~E_o\left(\chi \textbf{a}^\dagger+\chi^* \textbf{a}\right)-\mu E_0 \left(\sigma ^\dagger+\sigma\right)\\ H_S &=H_0+H_\textrm{int}+H_\textrm{drive}\end{align}$$

The Markovian interaction with reservoirs determining the decay rates $\gamma_x$ and $\gamma_{sp}$ for the atom and surface plasmon mode respectively are given by the following Liouvillian terms;

$$\begin{align}\mathscr{L}_{sp} &= \frac{\gamma_{sp}}{2} \left(2 \textbf{a}\rho\textbf{a}^\dagger - \textbf{a}^\dagger\textbf{a}\rho - \rho\textbf{a}^\dagger\textbf{a}\right)\\ \mathscr{L}_{x} &= \frac{\gamma_{x}}{2} \left(2 \sigma\rho\sigma^\dagger - \sigma^\dagger\sigma\rho - \rho\sigma^\dagger \sigma\right)\end{align}$$

The master equation for the density operator reads: $$ \dot{\rho}=\frac{\mathrm i}{\hbar}[\rho,H_s]+\mathscr{L}_{sp}+\mathscr{L}_{x} $$

I am trying to derive equations motion of the following form for the operators, but am not sure of how the $\omega$ term in this equation appears:

$$ \frac{\mathrm d}{\mathrm dt} \langle\sigma\rangle = -\left[\mathrm i(\omega_x-\omega)+\frac{\gamma_x}{2}\right]\langle \sigma \rangle - g \langle \textbf{a} \rangle + 2g \langle \textbf{a} \sigma^\dagger\sigma \rangle + \frac{\mathrm i\mu E_0}{\hbar}\langle\left(1-2\langle \sigma^\dagger\sigma\rangle\right) $$

Should I be considering the effective Hamiltonian to arrive at this equation of motion? Guidance on when and how to use the effective Hamiltonian to arrive at this equation is much valued.


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