# Laser Cooling beyond Rate Equations: Approaches from Quantum Thermodynamics
## 3 Results  
The parameters in our model are the energy splitting of the ground-state manifold, the detuning and Rabi frequency of the driving, the radiative decay rate, \(\gamma\), the cut-off frequency, \(\omega_{c}\), the dimensionless coupling, \(\alpha\), and the temperature \(T\). We choose the energy and time units such that \(E_{\text{man}}=2\). For the remaining parameters we take \(\gamma=0.5\), \(\omega_{c}=1\), \(\alpha=0.01\), and \(T=3\). These parameters are not intended to be realistic but are chosen to allow us to compute the exact solutions with a reasonable effort, and compare the results of the different master equations. In particular, we choose a large value for the radiative decay rate, \(\gamma\), to increase the magnitude of the heat current. It may be noted that for these parameters the phonon absorption rate, \(\gamma_{+}\approx 0.14\), is comparable to, but smaller than, the radiative decay rate. This differs from the situation for conventional laser cooling, appropriate in systems such as rare-earth ions, where the phonon rates are much larger than those for radiative decay [1], and the electronic populations are very close to equilibrium. Itimplies that, in our case, the heat current will be limited by the driving strength (for weak driving) or the phonon rate (for strong driving), and not the radiative lifetime. Figure 2 shows the calculating cooling power as a function of the detuning, \(\delta\), for four different strengths of the driving field. The different curves are computed using the full Bloch-Redfield equation, (7), the phenomenological Lindblad form, Equation (17), and the secular Bloch-Redfield equation, (10). Considering first weak driving, in Figure 2a, we see that the Bloch-Redfield and phenomenological theories agree well, and give a cooling profile that appears to be Lorentzian, as one would expect. While the secular Bloch-Redfield equation agrees away from the resonance, we see that it fails close to it, massively overestimating the cooling power. The secular approximation is, of course, not justified here, because there are near degeneracies in the Hamiltonian. Nonetheless, the level of disagreement seems surprising, given the agreement away from resonance. In the converse, strong-driving region, Figure 2d, all three methods give similar results. However, there is a noticeable difference on the high-energy side of the transition, with the phenomenological theory giving, as before, a Lorentzian profile, while the other theories predict the heat current drops off more rapidly, and indeed switches direction, from cooling to heating, in the range of detunings shown. Figure 3 shows the temperature dependence of the cooling power for weak resonant driving. As noted above, the secular approximation is inappropriate in this regime and massively overestimates the cooling power. The other theories agree closely and appear physically reasonable, predicting that the cooling power drops rapidly once the temperature is lowered below the splitting \(E_{\text{man}}\). This is the expected physical behavior for laser cooling in a discrete level structure, caused by the vanishing of the phonon occupation at temperatures much less than \(E_{\text{man}}\). The precise behavior at very low temperatures is not relevant to laser cooling since it is not expected to operate there. Nonetheless, it may be noted that, for these parameters, the Bloch-Redfield theory predicts a very small but negative cooling power, i.e., net heating, at very low temperatures (\(T<0.38\)). However, this is an approximate theory whose accuracy is not sufficient to discern the true behavior of the heat current in this regime. Indeed, we find that at these very low temperatures the theory does not predict a physical density matrix, giving one which has a negative, albeit very small, eigenvalue. Figure 2: Rates of heat absorption from the phonon bath, as a function of the detuning \(\delta=\omega_{l}-E_{0}\) of the driving laser from resonance, for four different Rabi frequencies.