# Saturation, Allowed Transitions and Quantum Interference in Laser Cooling of Solids
## 2 Theory
### Improved Cooling Efficiency Using Optical Saturation  
The intensity dependence of the absorption coefficients of cooling ions and background impurities can be exploited to improve the cooling efficiency of ASF cooling. The following analysis of differential absorption saturation reveals its impact on laser cooling efficiency when background impurities saturate more easily than cooling ions. The coolant ions and the background impurities are modeled as 2-level systems [13], with \(N_{1},\ N_{1}^{\prime}\) representing the number of ground state ions and impurities per unit volume. \(N_{2},\ N_{2}^{\prime}\) represent the density of the excited state species, respectively. Absent any internal laser field, the rate equation for the upper state population of the coolant ions is  
\[\frac{dN_{2}}{dt}=\frac{I_{p}\lambda_{p}}{hc}.\big{[}\sigma_{a}\big{(}\lambda_{ p},T\big{)}N_{1}-\sigma_{e}\big{(}\lambda_{p},T\big{)}N_{2}\big{]}-\frac{N_{2}}{ \tau_{f}}. \tag{1}\]  
For unintended impurities, it is  
\[\frac{dN_{2}^{\prime}}{dt}=\frac{\sigma^{\prime}I_{p}\lambda_{p}}{hc}.\big{[} N_{1}^{\prime}-N_{2}^{\prime}\big{]}-\frac{N_{2}^{\prime}}{\tau^{\prime}}. \tag{2}\]  
\(I_{p}\) and \(\lambda_{p}\) denote the intensity and wavelength of the pump light. \(\sigma_{a}\) and \(\sigma_{a}\) are cross-sections for absorption and emission. \(\tau_{f}\) and \(\tau^{\prime}\) are the excited state decay times of the coolant and impurity ions. In this model, the equation for heating power per unit volume is simply  
\[H=I_{p}\big{[}\sigma_{a}\big{(}\lambda_{p},T\big{)}N_{1}-\sigma_{e}\big{(} \lambda_{p},T\big{)}N_{2}\big{]}-\eta_{e}\frac{hc}{\lambda_{f}}\frac{N_{2}}{ \tau_{r}}+\sigma^{\prime}I_{p}\big{[}N_{1}^{\prime}-N_{2}^{\prime}\big{]}. \tag{3}\]  
Because there is no internal laser field to consider in refrigeration with a single pump beam, the population differences for the system under continuous illumination are given by the steady-state solutions of Equations (1) and (2). These expressions are  
\[\Delta N\equiv N_{1}-N_{2}=\frac{N_{T}}{\frac{I_{p}}{T_{at}}+1} \tag{4}\]and  
\[\Delta N^{\prime}\equiv N_{1}^{\prime}-N_{2}^{\prime}=\frac{N_{T}^{\prime}}{ \frac{I_{p}}{I_{sat}^{\prime}}+1}. \tag{5}\]  
In these expressions, we have defined the saturation intensities [17] for the coolant and impurity ions, given by  
\[I_{sat}=\frac{hc}{\lambda_{p}\tau_{f}[\sigma_{a}\big{(}\lambda_{p},T\big{)}+ \sigma_{e}\big{(}\lambda_{p},T\big{)}]}\text{ and }I_{sat}^{\prime}=\frac{hc}{2\sigma^{ \prime}\tau^{\prime}\lambda_{p}}.\]  
The total numbers of coolant and impurity species per unit volume are  
\[N_{T}=N_{1}+N_{2} \tag{6}\]  
and  
\[N_{T}^{\prime}=N_{1}^{\prime}+N_{2}^{\prime}. \tag{7}\]  
The steady-state heating power/volume obtained from Equation (3) is  
\[H=I_{p}\big{[}\sigma_{a}\big{(}\lambda_{p},T\big{)}\Delta N+\sigma^{\prime} \Delta N^{\prime}\big{]}.\Bigg{[}1-\frac{\sigma_{a}\big{(}\lambda_{p},T\big{)} \Delta N}{\sigma_{a}\big{(}\lambda_{p},T\big{)}\Delta N+\sigma^{\prime}\Delta N ^{\prime}}\eta_{ext}\frac{\lambda_{p}}{\lambda_{f}}\Bigg{]}. \tag{8}\]  
The final expression for cooling power in the presence of the optical saturation of the active ions and background impurities is therefore  
\[P_{e}=-P_{abs}\Bigg{[}1-\frac{\sigma_{a}\big{(}\lambda_{p},T\big{)}\Delta N} {\sigma_{a}\big{(}\lambda_{p},T\big{)}\Delta N+\sigma^{\prime}\Delta N^{\prime }}\eta_{ext}\frac{\lambda_{p}}{\lambda_{f}}\Bigg{]}. \tag{9}\]  
The cooling efficiency \(\eta_{c}\) may be derived from Equation (9) as the ratio of the cooling power, \(P_{c}\), to absorbed power, \(P_{abs}\). This yields  
\[\frac{dN_{2}^{\prime}}{dt}=\frac{\sigma^{\prime}I_{p}\lambda_{p}}{hc}.\big{[} N_{1}^{\prime}-N_{2}^{\prime}\big{]}-\frac{N_{2}^{\prime}}{\tau^{\prime}}. \tag{2}\]0  
Here, the external quantum efficiency \(\eta_{ext}\) is the product of the fluorescence escape efficiency \(\eta_{e}\) and the internal quantum efficiency \(\eta_{QE}\) given by the ratio of the fluorescence and radiative lifetimes of the upper state.