$|1\rangle\to \hbar\omega_1 > \hbar\omega_2\to|1\rangle$
a general pure state: $|\psi\rangle=a_1|1\rangle+a_2|2\rangle,|a_{1}|^{2}+|a_{2}|^{2}=1.$
Pauli operators:
$$ \begin{aligned}\mathbf{1}&=|2\rangle<2|+|1><1|,\\\mathbf{\hat{\sigma}}3&=|2\rangle<2|-|1><1|,\\\mathbf{\hat{\sigma}}+&=|2\rangle<1|,\\\mathbf{\hat{\sigma}}_-&=|1\rangle<2|.\end{aligned} $$
density matrix for a general state$\rho=\rho_{22}\left|2\right>\left<2\right|+\rho_{21}\left|2\right>\left<1\right|+\rho_{12}\left|1\right>\left<2\right|+\rho_{11}\left|1\right>\left<1\right|$
$$ \left.\rho=\left(\begin{array}{cc}\rho_{22}&&\rho_{21}\\\rho_{12}&&\rho_{11}\end{array}\right.\right), $$
$\rho$ can be writtern in the form($u,v,w$ are the real quantites$u=\rho_{21}+\rho_{12},v=\mathrm{i}(\rho_{21}-\rho_{12}),w=\rho_{22}-\rho_{_{11}}.$)
$$ \rho=\frac{1}{2}\big(\mathbf{1}+u\hat{\sigma}{1}+v\hat{\sigma}{2}+w\hat{\sigma}_{3}\big), $$
an atom undergoing spontaneous emission:
the excited state probability $\rho_{22}$ decays at the Einstein rate $2\Gamma:\rho_{22}=\rho_{22}(0)\exp(-2\Gamma t).$
$$ \dot{\rho}=\Gamma\Bigl(2\hat{\sigma}{-}\rho\hat{\sigma}{+}-\hat{\sigma}{+}\hat{\sigma}{-}\rho-\rho\hat{\sigma}{+}\hat{\sigma}{-}\Bigr). $$
elextric dipole approximation is vaild when:
the atomic dipole operator:
$$ \hat{\mathbf{\mu}}=\mathbf{\mu}^{*}\hat{\sigma}{+}+\mathbf{\mu}\hat{\sigma}{-}, $$
$\hat{\mathbf{\mu}}|1\rangle=\mathbf{\mu}^{*}|2\rangle,\hat{\mathbf{\mu}}\left|2\right\rangle={\mathbf{\mu}}\left|1\right\rangle.$
$$ \hat{H}=\frac{\hbar}{2}(\omega_{2}+\omega_{1})\mathbf{1}+\frac{\hbar}{2}(\omega_{2}-\omega_{1})\hat{\sigma}{3}-\mathbf{\mu}\cdot\mathbf{E}(t)(\hat{\sigma}{+}+\hat{\sigma}_{-}), $$
where, for simplicity, we have taken $\mu$ to be real.