# Saturation, Allowed Transitions and Quantum Interference in Laser Cooling of Solids
## Appendix A  
In this Appendix, with reference to the 3-level system of Figure 7 in the text, density matrix theory is applied to calculate the third-order susceptibility of Ce\({}^{3+}\) at the pump frequency \(\omega\). Two limits are considered to simplify the calculation by choosing frequencies with small detunings from the transition at \(\omega_{12}\) or the transition at \(\omega_{13}\), respectively. This permits the rotating wave approximation (RWA) to be made for off-diagonal matrix elements corresponding to transitions at small detunings from the optical frequency. Transitions with large detunings must retain non-rotating wave terms, which make the calculation more cumbersome. Explicit results are presented only for the case of optical frequencies close to \(\omega_{12}\). In this limit, the RWA is appropriate for the 12 and 23 transitions but not the 13 transition. Hence, the interaction Hamiltonian has the form  
\[ \begin{array}{rl}V&=\left\{\left(\widetilde{V}_{12}^{(1)}e^{i\omega t}+ \widetilde{V}_{32}^{(1)}e^{i\omega t}+\widetilde{V}_{13}^{(1+)}e^{i\omega t}+ \widetilde{V}_{13}^{(1-)}e^{-i\omega t}\right)+c.c.\right\}+\widetilde{V}_{3 2}^{(0)}\\ &=\left\{\left(-\frac{\hbar\Omega_{12}}{2}e^{i\omega t}-\frac{\hbar\Omega_{2 }}{2}e^{i\omega t}-\frac{\hbar\Omega_{13}}{2}e^{i\omega t}-\frac{\hbar\Omega_{ 13}}{2}\right)+c.c.\right\}+\widetilde{V}_{32}^{(0)},\end{array}  \tag{10}\]  
where \(\widetilde{V}_{32}^{(0)}\) is a static mixing interaction of states 2 and 3, and the Rabi frequencies of the optical interaction terms between states \(i\) and \(j\) are defined by \(\Omega_{ij}\equiv\mu_{ij}E_{0}/\hbar\). The equation of motion for the density matrix is  
\[i\hbar\dot{\rho}=[(H_{0}+V),\rho]. \tag{11}\]  
Because the 13 transition is forbidden, the dipole moment is negligible compared to moments on the allowed interactions with small detunings. Hence, we set \(V_{13}\approx 0\) and have the following equations for individual matrix elements:  
\[i\hbar\dot{\rho}_{11}=(V_{12}\rho_{21}-\rho_{12}V_{21})+i\hbar\gamma_{21}\rho _{22}+i\hbar\gamma_{31}\rho_{33} \tag{12}\]\[i\hbar\dot{\rho}_{22}=-(V_{12}\rho_{21}-\rho_{12}V_{21})-(V_{32}\rho_{23}-\rho_{3 2}V_{23})-i\hbar\dot{\gamma}_{21}\rho_{22}-i\hbar\dot{\gamma}_{23}\rho_{22} \tag{111}\]  
\[i\hbar\dot{\rho}_{33}=(V_{32}\rho_{23}-\rho_{32}V_{23})+i\hbar\dot{\gamma}_{23} \rho_{22}-i\hbar\dot{\gamma}_{31}\rho_{33} \tag{112}\]  
\[\dot{\rho}_{12}=i\omega_{12}\rho_{12}+\frac{1}{i\hbar}V_{12}(\rho_{22}-\rho_{1 1})-\frac{1}{i\hbar}\rho_{13}V_{32}-\Gamma_{21}\rho_{12} \tag{113}\]  
\[\dot{\rho}_{13}=i\omega_{13}\rho_{13}+\frac{1}{i\hbar}(V_{12}\rho_{23}-\rho_{1 2}V_{23})-\Gamma_{31}\rho_{13} \tag{114}\]  
\[\dot{\rho}_{23}=-i\omega_{32}\rho_{23}+\frac{1}{i\hbar}V_{23}(\rho_{33}-\rho_{ 22})+\frac{1}{i\hbar}V_{21}\rho_{13}-\Gamma_{32}\rho_{23} \tag{115}\]  
\[\rho_{ji}=\rho_{ij}^{*}. \tag{116}\]  
As the RWA is not valid on the 13 transition in the present case, the off-diagonal term \(\rho_{31}\) must include positive and negative frequency terms. Thus, we assume that  
\[\rho_{13}=\widetilde{\rho}_{13}^{(+)}e^{i\omega t}+\widetilde{\rho}_{13}^{(-) }e^{-i\omega t} \rightarrow \dot{\rho}_{13}=i\omega\widetilde{\rho}_{13}^{(+)}e^{i\omega t}- i\omega\widetilde{\rho}_{13}^{(-)}e^{-i\omega t} \tag{117}\]  
\[\rho_{31}=\widetilde{\rho}_{31}^{(+)}e^{i\omega t}+\widetilde{\rho}_{31}^{(-) }e^{-i\omega t} \rightarrow \dot{\rho}_{31}=i\omega\widetilde{\rho}_{31}^{(+)}e^{i\omega t}- i\omega\widetilde{\rho}_{31}^{(-)}e^{-i\omega t}. \tag{118}\]  
The tilde in these expressions indicates a slowly varying amplitude (whose time derivative may be set to zero). The other coherences in the system have simpler forms and obey the RWA.