# Laser Cooling and Trapping of Rare-Earth-Doped Particles
## 2 Laser Cooling of Bulk Rare-Earth-Doped Solids  
In the majority of cases, a luminescent material interacting with light emits photons with a lower energy compared to the energy of the absorbed photons (Figure 1a). This effect is called the Stokes shift. The law of conservation of energy applied to these redshifted emitted photons explains heating of the optical material caused by phonon generation (Figure 1a). If the energy of the emitted photons exceeds the energy of absorbed ones, the anti-Stokes shift, accompanied by phonon absorption, takes place in the system (Figure 1b). As mentioned above, in 1929, [PERSON] proposed the use of anti-Stokes fluorescence for optical refrigeration [14]. In 1995, laser cooling with anti-Stokes fluorescence was reached for RE-doped solids [15]. In this first proof-of-principle experiment, an ytterbium (Yb\({}^{3+}\))-doped fluorozirconate ZrF\({}_{4}\)-BaF\({}_{2}\)-LaF\({}_{3}\)-AlF\({}_{3}\)-NaF-PbF\({}_{2}\) (ZBLANP) sample was cooled down to only 0.3 K  below the environmental temperature. Let us consider in detail a laser cooling cycle in a Yb\({}^{3+}\)-doped sample. As one can see in Figure 2, it includes photon absorption at the frequency \(\
u_{P}\), electron thermalization in the excited \({}^{2}F_{5/2}\) manifold accompanied by phonon absorption, and spontaneous photon emission from the excited \({}^{2}F_{5/2}\) manifold to the ground \({}^{2}F_{7/2}\) manifold. Figure 1: Energy levels of Yb\({}^{3+}\) ions doped in a solid. Arrows illustrate the excitation and decay processes for (**a**) Stokes and (**b**) anti-Stokes cycles. All photon absorption-emission cycles where the energy of the radiated photon exceeds the energy of the absorbed one (anti-Stokes fluorescence) result in cooling, as this energy difference has to be compensated by phonon absorption (Figure 2). All the photon absorption-emission cycles where the energy of the absorbed photons exceeds the energy of the radiated ones (Stokes fluorescence) causes heat generation in the sample, as the energy difference in these cycles has to be compensated by phonon generation (Figure 2). Spontaneous emission is characterized by the mean fluorescence frequency,  
\[\
u_{F}=\frac{\int\Phi(\
u)\
u d\
u}{\int\Phi(\
u)d\
u} \tag{1}\]  
where \(\Phi(\
u)\) is the fluorescence flux density, which can be obtained experimentally, and \(\
u_{F}=c/\lambda_{F}\). Here, \(\lambda_{F}\) is the mean fluorescence wavelength and \(c\) is the speed of light in vacuum. If one wants to cool a sample, the pump frequency, \(\
u_{P}\), is bound to be smaller than the mean fluorescence frequency, \(\
u_{F}\), and all radiated anti-Stokes photons must leave the sample (not to be reabsorbed by Yb\({}^{3+}\) ions). Indeed, these anti-Stokes reabsorbed photons serve as the source of heat generation in the sample, which is undesirable for cooling. The efficiency of the cooling cycle is the difference between the energy of the anti-Stokes emitted photon and the energy of the pump photon normalized by the energy of the pump photon that is  
\[\eta_{cool}=\frac{h\
u_{F}-h\
u_{P}}{h\
u_{P}}=\frac{\lambda_{P}}{\lambda_{F} }-1. \tag{2}\]  
The electrons excited to the \({}^{2}F_{5/2}\) manifold can also decay nonradiatively to the ground \({}^{2}F_{7/2}\) manifold, generating phonons. The \"competition\" between the radiative and nonradiative decays of the \({}^{2}F_{5/2}\) electrons can be quantified by the quantum efficiency,  
\[\eta_{q}=\frac{W_{r}}{W_{r}+W_{nr}} \tag{3}\]  
where \(W_{r}=1/\tau_{r}\) and \(W_{nr}=1/\tau_{nr}\) are the radiative and nonradiative decay rates, respectively, with \(\tau_{r}\) and \(\tau_{nr}\) being the radiative and nonradiative lifetimes of the excited manifold, respectively.