# Saturation, Allowed Transitions and Quantum Interference in Laser Cooling of Solids
## 3 Methods and Results
### Fano Resonance and Induced Transparency in Laser Cooling  
The 3-level system of Ce\({}^{3+}\) can be viewed either as a \(\Lambda\)-system or a V-system with a large detuning from at least one of the three transitions. The validity of these two perspectives explains why two forms of quantum interference are possible in this system, namely Fano resonance [35] and induced transparency [36,37]. The former requires mixing of the two distinct excited states by some interaction (V-picture), and the latter does not (\(\Lambda\)-picture). The two features that result from interference are the negative peaks displayed in the plots of Figure 16a,b respectively, which display results calculated from Equation (12) using input parameters [38] that specialize the problem to Ce\({}^{3+}\): LiCaAlF6.  
In Figure 16a, the coupling frequency was assumed to have a value of \(V/\hbar=10^{8}\) rad/s. This is considerably less than known optical dephasing and decay parameters of trivalent Ce ions in LiCaAlF6 [37]: \(\Gamma_{12}=10^{12}\) rad/s, \(\Gamma_{23}=10^{12}\) rad/s, \(\gamma_{12}=3.3\times 10^{7}\) s\({}^{-1}\), \(\gamma_{13}=1.0\times 10^{4}\) s\({}^{-1}\), and \(\gamma_{23}=1.1\times 10^{7}\) s\({}^{-1}\). In Figure 16b, its value was set to \(V/\hbar=0\) rad/s. The negative absorption (gain) peak in Figure 16a disappeared when the coupling frequency was set equal to zero. In addition, this feature was found to appear at arbitrary intensity of the pump wave. This identifies it as a Fano resonance that results in gain when the system is pumped at frequency \(\omega_{13}\). At the low pump intensity assumed in the plot (\(I\)\(\sim\)\(10^{8}\) W/m\({}^{2}\)) it may also readily be confirmed that the upper state 3 of the transition is less populated than the ground state 1, showing that lasing without inversion can take place under these circumstances. For example, the steady-state solution for the ratio of populations in states 2 and 3 is \(\rho_{22}/\rho_{33}=\gamma_{13}/\gamma_{12}\), which is always small when state 3 is metastable. Near the \(\omega_{12}\) resonance in Figure 16b, negative absorption (gain) is also induced, but only above a threshold value of pump intensity. The apparent need to meet a threshold requirement for the appearance of gain, in this case, indicates an induced transparency origin of the interference.  
Figure 17 presents the theoretical cooling efficiency of Ce\({}^{3+}\) ions in the wavelength range where the absorption and emission curves overlap. Net cooling is possible from 293-310 nm on the red side of the \(\omega_{12}\) resonance when the quantum efficiency is one. This  
Figure 16: Plots of optical absorption versus frequency in a 3-level system modeled after Ce\({}^{3+}\):LiCaAlF\({}_{6}\) showing **(a)** a gain feature on the low frequency transition at \(\omega_{13}\) for an intensity of \(I=0.5\times 10^{8}\) W/m\({}^{2}\) and coupling parameter \(V/\hbar=10^{8}\) rad/s and **(b)** the two short wavelength transitions of Ce\({}^{3+}\) at low-, medium-, and high-intensity, with \(V=0\). Negative absorption indicates gain without population inversion as the populations of states 1 and 3 exceed that of state 2 at all intensities. The shaded spectral region at red detunings within a linewidth of the \(\omega_{12}\) transition illustrates the range over which self-cooled LWI may be possible.  
range encroaches on the shaded region of Figure 16b, where lasing without inversion can take place. Hence, self-cooled lasing without inversion may be possible in this system. While no experiments have been performed to date to test this prediction, this intriguing possibility indicates that quantum interference could play a role in future schemes for the laser cooling of solids.