I am interested in a scenario where a three level atom (cascade/ladder type) is interacting with a spherical metal nanoparticle (MNP) with single plasmonic excitations. Let us assume that the ground, first excited and second excited states of the atom are given by $|S1\rangle$, $|S2\rangle$ and $|S3\rangle$ forming a complete basis for the atom's Hilbert space, and the creation and annihilation operators of the MNP plasmonic mode is given by $\hat{a}^\dagger = |N\rangle \langle0|$ and $\hat{a}=|0\rangle \langle N|$. An incoherent pumping field ($P_{13}$) is applied between levels $|S1\rangle$ and $|S3\rangle$ of the atom. A coherent probe field $E_{pr}=E_{0pr}(e^{i\omega_{pr}t}+e^{-i\omega_{pr}t})$ is applied between the levels $|S2\rangle$ and $|S3\rangle$. The levels $|S1\rangle$ and $|S2\rangle$ undergo dipole-dipole coupling with the plasmonic field. The interaction picture Hamiltonian of the system was found to be $$ H_{sys}=\hbar\Delta_1\hat{a}^\dagger\hat{a} + \hbar\Delta_1|S1\rangle \langle S1| +\hbar\Delta_2|S3\rangle \langle S3|+i\hbar g(\hat{\sigma}_{12}\hat{a}^\dagger - \hat{\sigma}_{12}^\dagger\hat{a})-E_{0pr}\mu_{23}(\hat{\sigma}^{\dagger}_{23}+\hat{\sigma}_{23}) $$ where $\Delta_1=\omega_{plasmon}-\omega_{(\langle S2|-\langle S1|)}$ and $\Delta_2=\omega_{(\langle S3|-\langle S2|)}-\omega_{pr}$, $g$ is a coupling constant, $\mu_{23}$ is the dipole moment between $|S2\rangle$,$|S3\rangle$, $\hat{\sigma}$ and $\hat{\sigma}^{\dagger}$ represent the atomic lowering and raising operators between the corresponding levels.
The full quantum dynamics of the coupled nano system were derived using the master equation for the interaction picture density operator as:
$$ \frac{\partial\hat{\rho}}{\partial t}=\frac{i}{\hbar}\left[\hat{\rho},H_{sys}\right]+\mathcal{L}_m+\mathcal{L}_{12}+\mathcal{L}_{23}+\mathcal{L}_p $$
where $\mathcal{L}_m, \mathcal{L}_{12},\mathcal{L}_{23}$ are the Lindblad terms for the decay of plasmon mode and atomic levels respectively and $\mathcal{L}_p$ is the Lindblad term for the incoherent pump.
$\textbf{Question:}$ I intend to find the equations of motion for the expectation values of the plasmon and atomic annihilation operators ($\langle \hat{a} \rangle$, $\langle \hat{\sigma}_{12} \rangle$, $\langle \hat{\sigma}_{23} \rangle$ ). With the atom being a three level system and the plasmon operators being two level, is it possible to use the trace as follows for this purpose?
Eg:$\frac{\partial \langle \hat{a} \rangle}{\partial t} = Tr\left[ \langle \hat{a} \rangle \frac{\partial \langle \hat{\rho} \rangle}{\partial t}\right] $, $\frac{\partial \langle \hat{\sigma}_{12} \rangle}{\partial t} = Tr\left[ \langle \hat{\sigma}_{12} \rangle \frac{\partial \langle \hat{\rho} \rangle}{\partial t}\right] $
(I intend to use the cyclic property of trace, bosonic commutator relations and orthogonality of atomic states to simplify the above.) If this doesn't seem like a valid approach, it would be much helpful if you can kindly mention the approach I should take or guide me to a similar calculation available online. Thank you