# Saturation, Allowed Transitions and Quantum Interference in Laser Cooling of Solids
## 2 Theory
### Electric-Dipole-Allowed Transitions for Rapid Cooling  
The efficiency of laser cooling by the ASF method is independent of the transition rate of the coolant atoms. This is clear because the upper state lifetime of these atoms is absent from Equation (10). On the other hand, the number of transitions per unit time does determine the speed of cooling. Hence, the transition rate of coolant atoms has an important impact on how quickly bulk systems can be cooled with light. This is confirmed in the expression for heating power density \(H\), where the cooling term (second term on the right side of Equation (3) depends on the upper state radiative lifetime \(\tau_{r}\). From this equation, it  
Figure 3: Theoretical cooling efficiency plotted  
is evident that the net heating or cooling power varies with the transition rate of the coolant atoms and improves significantly when the radiative lifetime on allowed transitions is short. Another way to think about the cooling term is that \(N_{2}/\tau_{r}\) is proportional to the absorption cross-section. A medium 2, therefore, improves on the cooling power density of a medium 1 by the factor \(\tau_{r1}/\tau_{r2}=\rho_{2}/\sigma_{1}\), making it advantageous to use electric-dipole-allowed transitions for laser cooling, rather than forbidden transitions, because of their larger cross-sections. As most ED transitions have upper state decay times of less than 100 ns and forbidden transitions have lifetimes of \(\sim\)1 ms, the cooling power can theoretically be increased by a factor of roughly \(10^{4}\) by using allowed transitions. Thus, this could result in a substantial improvement of the cool-down times. Despite the high transition rate afforded by dipole-allowed transitions in solids, they normally incur heating due to non-radiative configuration relaxation. The mechanism is illustrated in Figure 4. The polarizability of excited state orbitals on allowed transitions in solids is susceptible to interactions with neighboring atoms. This shifts the minimum of the excited state potential to a greater distance from the nucleus than that of the ground state. According to the Franck-Condon principle, the absorption transition takes place rapidly enough so that the configuration coordinate \(Q\) does not change. Thus, the electron undergoes a vertical transition in Figure 4. which places it far from the equilibrium point at the bottom of the excited state well. This necessitates relaxation to the potential minimum, which is shifted to a larger value of the configuration coordinate \(Q\). This relaxation, and a similar one that follows the vertical radiative transition to the ground state potential, is mediated by the emission of the vibrational quanta of the atomic cluster, consisting of the atom itself together with its neighbors in its coordination sphere. Fortunately, the emission of phonons and consequent heating that takes place during configuration relaxation can be mitigated to achieve net cooling in the same way as for systems with forbidden transitions, such as Yb\({}^{3+}\). Theoretically, one simply tunes the pump wavelength to a value exceeding the average Stokes-shifted emission wavelength of the coolant species. In Section 3.2, this strategy is shown to work on an ED transition in Ti\({}^{3+}\):Al\({}_{2}\)O\({}_{3}\). Figure 4: Potential energy curves for the ground state (\(E_{1}\)) and the excited state (\(E_{2}\)) of an atom in a solid. The vertical blue and red arrows depict fast dipole-allowed transitions upward to \(E_{2}\) and downward to \(E_{1}\), respectively, followed by configuration relaxation (black arrows). Because optical transitions take place vertically according to the Franck–Condon Principle, the system is pushed far from equilibrium. The difference in energy between the absorption and emission transitions, known as the Stokes shift, is particularly large in systems that undergo configuration relaxation. The relaxation process is mediated by phonon emission, which causes heating.