# Laser Cooling and Trapping of Rare-Earth-Doped Particles
## 2 Laser Cooling of Bulk Rare-Earth-Doped Solids
If the quantum efficiency of the anti-Stokes generation process is very high (\(\eta_{q}\geq 99\%\)), laser cooling caused by the removal of phonon energy with anti-Stoke photons from the material takes place in the system, as the probability of nonradiative decay with phonon generation from the excited \({}^{2}F_{5/2}\) to the ground \({}^{2}F_{7/2}\) manifold is very low [16]. Only such samples are suitable for laser cooling. To simulate cooling in a RE-doped sample, one can use a two-level model developed in [17; 18]:  
\[\left\{ \begin{array}{c}\frac{dN_{1}}{dt}=\frac{I_{p}}{h\
u_{P}}\big{[}N_{0} \sigma_{a}\big{(}\
u_{p}\big{)}-N_{1}\sigma_{e}\big{(}\
u_{p}\big{)}\big{]}- \frac{N_{1}}{\tau_{r}}-\frac{N_{1}}{\tau_{nr}}\\ N_{T}=N_{0}+N_{1}\end{array} \right. \tag{4}\]  
Figure 2: Energy levels of Yb\({}^{3+}\) ions doped in a solid. Arrows illustrate the excitation and decay processes for a cooling cycle. where \(N_{T}\) is the RE ions density in the sample, and \(N_{0}\) and \(N_{1}\) are the population density in the ground and excited manifolds, respectively. \(I_{p}\) is the pump intensity at the frequency \(\
u_{p}\). \(\sigma_{a,e}(\
u_{p})\) is the absorption (\(a\)) and emission (\(e\)) cross-sections at the pump frequency \(\
u_{p}\). The cooling power density generated in the sample, \(P_{cool}\), is the difference between the absorbed power density and radiated power density. \[P_{cool}=-I_{p}\big{[}N_{0}\sigma_{a}\big{(}\
u_{p}\big{)}-N_{1}\sigma_{e} \big{(}\
u_{p}\big{)}\big{]}+N_{1}\frac{h\
u_{f}}{\tau_{r}}-N_{1}\frac{h\
u_{f }}{\tau_{nr}} \tag{5}\]  
The cooling power generated in a sample of volume \(V\) when the system is in the steady-state regime (\(dN_{1}/dt=0\)) is [19]  
\[P_{cool}^{V}=\frac{h\widetilde{\
u}_{F}\ -h\
u_{p}}{h\
u_{p}}\ \frac{VN_{T}\sigma_{a} (\
u_{p})I_{p}}{\Big{[}1+\frac{I_{p}}{I_{S}}\Big{(}1+\frac{\sigma_{e}(\
u_{p} )}{\sigma_{a}(\
u_{p})}\Big{)}\Big{]}}\ \ \approx\ \ \frac{h\widetilde{\
u}_{F}\ -h\
u_{p}}{h\
u_{p}}\ VN_{T} \sigma_{a}(\
u_{p})I_{p} \tag{6}\]  
where \(I_{S}=h\
u_{p}/\big{[}\eta_{q}\tau_{r}\sigma_{a}(\
u_{p})\big{]}\) is the saturation intensity and \(\widetilde{\
u}_{F}\ =\big{(}2\eta_{q}-1\big{)}\
u_{F}\approx\eta_{q}\
u_{F}\) is the effective mean fluorescence frequency. If the sample is placed on a low-contact and low-thermal conductivity support in a vacuum chamber, only radiative heat load may take place in the system. The final equilibrium temperature of the sample, \(T_{S}\), can be estimated by the Stefan-Boltzmann law,  
\[P_{cool}^{V}=\varepsilon\sigma_{B}S\Big{(}T_{r}^{4}-T_{S}^{4}\Big{)} \tag{7}\]  
where \(S\) is the total surface area of the sample, \(T_{r}\) is room temperature, and \(\varepsilon\) is the sample emissivity. Since the first experimental observation in 1995, laser cooling of solids has been realized with Yb\({}^{+3}\), Er\({}^{+3}\), Tm\({}^{3+}\), and Ho\({}^{3+}\) ions doped in a wide variety of low-phonon glasses and crystals including ZBLAN (maximum phonon energy of 580 cm\({}^{-1}\)), ZBLANP (505 cm\({}^{-1}\)), CBNZn (370 cm\({}^{-1}\)), YAG (630 cm\({}^{-1}\)), BYF (450 cm\({}^{-1}\)), YLF (350 cm\({}^{-1}\)), KPC (203 cm\({}^{-1}\)), and LLF (420 cm\({}^{-1}\)) [16; 19; 20; 21; 22; 23; 24; 25; 26]. Yb\({}^{+3}\) ions are the most suitable RE ions for laser cooling application. Yb\({}^{+3}\) ions have only one excited manifold. They are free from excited-state absorption, a source of undesirable nonradiative decay resulting in heat generation. The resonant absorption coefficient, \(\alpha_{r}(\
u,T)\), can be estimated as a product of the absorption cross-section \(\sigma_{a}(\
u,T)\) and the thermal population density, \(N_{0}(E_{0i},T)\), of the initial state with the energy \(E_{0 ii}\) that is \(\alpha_{r}(\
u,T)=\sigma_{a}(\
u,T)\cdot N_{0}(E_{0i},T)\). Following the Boltzmann distribution, the thermal population density of the initial state is \(N_{0}(E_{0i},T)\sim exp(-E_{0i}/\kappa_{B}T)\).