# Saturation, Allowed Transitions and Quantum Interference in Laser Cooling of Solids
## 2 Theory
### Quantum Interference for Self-Cooled Lasing without Inversion  
In this section, Fano resonance mediated by excited state mixing and electromagnetically induced transparency (without any state mixing) are investigated in the system depicted in Figure 7, modeled after Ce\({}^{3+}\) ions in the host crystal LiCaAlF\({}_{6}\). The Ce\({}^{3+}\) dopant ions are characterized by having two allowed transitions, from states 1 to 2 and from 2 to 3. These are located in the ultraviolet spectral region. The transition from 1 to 3 is assumed to be forbidden by the usual rule of Laporte, and the lifetime of state 3 is correspondingly long. Two types of quantum interference can manifest themselves in 3-level systems driven by a single pump wave. If there is _no mixing_ between the two excited states of the system, quantum interference is found to produce induced transparency on the 12 transition above a threshold corresponding to the optical saturation of the transition. If the two excited states are _mixed or coupled_ by a configuration interaction of strength \(V\) for example, transparency  
Figure 6: Energies of the sub-levels of the ground state of Ti:Al\({}_{2}\)O\({}_{3}\) (to scale). The three \({}^{2}\)Au electronic levels are labeled 0, 1, 2 in order of increasing energy, following Ref. [22]. The four vibrational sub-levels labeled a, b, c, d correspond to four allowed modes for \(\pi\)-polarization [21]. Only singly excited vibrational sub-levels are shown for each mode in association with each electronic state as only the first vibrational level is significantly populated at room temperature. Figure 7: 3-level system with dipole-allowed transitions between states 1 and 2 and between states 2 and 3. State 3 is assumed to be metastable. The frequency of incident light is \(\omega\). appears due to Fano resonance on the 13 transition. Interestingly, in each of these cases, gain appears even though the populations of states 1 and 3 always exceed that of state 2, and no special tuning requirement needs to be met [23]. Thus, lasing without inversion takes place at all frequencies close to either \(\omega_{12}\) or \(\omega_{13}\) depending on whether there is significant excited state mixing or not. In addition, the 12 transition in particular has gain over a range of red-detuned frequencies near the \(\omega_{12}\) resonance. Therefore, self-cooled lasing without inversion may be possible. To examine the behavior of this system, when irradiated by a single pump wave of frequency \(\omega\) tuned near the 12 transition frequency, the imaginary part of the susceptibility must be calculated. To do this, one can determine the density matrix elements \(\rho_{ij}\) between states \(i\) and \(j\) in the expression [23]  
\[\chi^{\prime\prime}\left(\omega\right)=\text{Im}\bigg{\{}\left(\frac{2N}{ \epsilon_{0}E_{0}}\right)\Big{(}\mu_{12}\widetilde{\rho}_{21}+\mu_{13} \widetilde{\rho}_{31}^{\left(-\right)}+\mu_{31}\widetilde{\rho}_{13}^{\left(- \right)}+\mu_{32}\widetilde{\rho}_{23}\Big{)}\bigg{\}}. \tag{12}\]  
Optical absorption is proportional to \(\chi^{\prime\prime}\left(\omega\right)\). Hence, when the expression in Equation (12) is evaluated, the absorption and gain spectra can be plotted. When the optical frequency is close to \(\omega_{12}\) or \(\omega_{13}\), the rotating wave approximation (RWA) can be made for the corresponding off-diagonal matrix elements, but not for the 13 transition. This accounts for the co- and counter-rotating terms (\(\widetilde{\rho}_{31}^{\left(-\right)}\) and \(\widetilde{\rho}_{13}^{\left(-\right)}\), respectively) in Equation (12). To plot the response near the 13 transition frequency, the RWA can be applied to the 13 transition but not to the other two. These requirements arise because of the relative placement of the levels assumed in Figure 7, which reflect those of Ce\({}^{3+}\). Steady-state expressions for the various matrix elements obtained by third-order perturbation theory are given in Appendix A.