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I am looking at the derivation for EIT susceptibility (Equation 6.70) in the textbook Quantum and Atom Optics by Steck. If someone has a better resource then please recommend it, or just how to follow his math.

First he gives a master equation (6.63)

\begin{equation} \partial_{t} \tilde{\rho}=-\frac{i}{\hbar}\left[\tilde{H}_{\mathrm{A}}+\tilde{H}_{\mathrm{AF}}, \tilde{\rho}\right]+\Gamma_{1} \mathcal{D}\left[\sigma_{1}\right] \tilde{\rho}+\Gamma_{2} \mathcal{D}\left[\sigma_{2}\right] \tilde{\rho}+\gamma_{\mathrm{g}} \mathcal{D}\left[\sigma_{\mathrm{g}}\right] \tilde{\rho} \end{equation} where the two Hamiltonian terms are \begin{equation} \tilde{H}_{\mathrm{A}}=\hbar \Delta_{1}\left|\mathrm{g}_{1}\right\rangle\left\langle\mathrm{g}_{1}\left|+\hbar \Delta_{2}\right| \mathrm{g}_{2}\right\rangle\left\langle\mathrm{g}_{2}\right| \end{equation} and \begin{equation} \tilde{H}_{\mathrm{AF}}=\frac{\hbar \Omega_{1}}{2}\left(\sigma_{1}+\sigma_{1}^{\dagger}\right)+\frac{\hbar \Omega_{2}}{2}\left(\sigma_{2}+\sigma_{2}^{\dagger}\right) \end{equation} (6.45) and (6.46). These are simple qubit / quantized field Hamiltonians. The rest of the master equation are dissipation operators which I also understand. But then his next lines are like partial traces of the density operator, given by

\begin{equation} \begin{aligned} \partial_{t} \tilde{\rho}_{\mathrm{eg}_{2}} &=\left(-\frac{\Gamma_{2}}{2}+i \Delta_{2}\right) \tilde{\rho}_{\mathrm{eg}_{2}}+\frac{i \Omega_{2}}{2}\left(\rho_{\mathrm{ee}}-\rho_{\mathrm{g} 2 \mathrm{g}_{2}}\right)-\frac{i \Omega_{1}}{2} \tilde{\rho}_{\mathrm{g} 1 \mathrm{g} 2} \\ & \approx\left(-\frac{\Gamma_{2}}{2}+i \Delta_{2}\right) \tilde{\rho}_{\mathrm{eg}_{2}}-\frac{i \Omega_{2}}{2}-\frac{i \Omega_{1}}{2} \tilde{\rho}_{\mathrm{g}_{1} \mathrm{g}_{2}} \end{aligned} \end{equation} and \begin{equation} \begin{aligned} \partial_{t} \tilde{\rho}_{\mathrm{g}_{1} \mathrm{g}_{2}} &=i\left(\Delta_{2}-\Delta_{1}\right) \tilde{\rho}_{\mathrm{g}_{1} \mathrm{B}_{2}}-\gamma_{\mathrm{g}} \tilde{\rho}_{\mathrm{g}_{1} \mathrm{g} 2}-\frac{i \Omega_{1}}{2} \tilde{\rho}_{\mathrm{eg}_{2}}+\frac{i \Omega_{2}}{2} \tilde{\rho}_{\mathrm{g}_{1} \mathrm{e}} \\ & \approx i\left[\left(\Delta_{2}-\Delta_{1}\right)+i \gamma_{\mathrm{g}}\right] \tilde{\rho}_{\mathrm{g}_{1} \mathrm{g}_{2}}-\frac{i \Omega_{1}}{2} \tilde{\rho}_{\mathrm{eg}_{2}} \end{aligned} \end{equation}

How is the commutator equal to this? I don't see how the operators in the Hamiltonian basically disappear? Why do I no longer have pauli matrices but I do have their coefficients?

He also sets density operators equal to 1 or 0 in his approximation? These are matrices though?

I understand the algebra after this to arrive at 6.70 but could someone explain how to get these density matrix equations? Thank you in advance.

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