# Laser Cooling beyond Rate Equations: Approaches from Quantum Thermodynamics
## 4 Discussion  
Figures 2 and 4 suggest that the full Bloch-Redfield equation gives an accurate account of the cooling profile, in both the weak- and strong-driving cases. In the weak-driving case, it agrees with the phenomenological theory, which is well-justified for weak-driving, while in the strong-driving case it agrees with the secular theory, which is well-justified  
Figure 4: Heat absorption rates, as a function of the detuning, computed using the full Bloch-Redfield equation (solid black curves) and a numerically exact method (red squares), for two different Rabi frequencies. The Rabi frequencies \(\Omega\) are (**a**) 0.5 and (**b**) 1.0. Figure 3: Rate of heat absorption from the phonon bath, as a function of temperature, for the different approaches. Results are shown for weak driving, \(\Omega=0.1\), at resonance, \(\delta=0\). The heat current is computed using the full Bloch-Redfield equation (solid black curve), the phenomenological Lindblad equation (dashed orange curve), and the Bloch-Redfield equation in the secular approximation without the principal value terms (dashed black curve). there. Furthermore, it agrees with exact numerical results, in the strong-driving regime where such simulations are possible. Thus the Bloch-Redfield equation allows a complete treatment of both regimes, using a single equation. This conclusion is similar to previous conclusions on the dynamics and thermodynamics of other open quantum systems, where the Bloch-Redfield equation has similarly been argued to provide the most accurate description [10; 26; 28; 29; 30]. This is in spite of the possibility that it produces unphysical density matrices with non-positive eigenvalues. That possibility does not occur in our results, except at very low temperatures where laser cooling would not, in any case, be expected to operate. In previous works, it has been noted that the secular approximation does not allow for the presence of bath-induced or noise-induced coherence [51] in multilevel systems where near-degenerate levels have different couplings to the bath [26]. This phenomenon can play an important role for the heat currents, as has been pointed out previously for quantum absorption refrigerators [10; 30]. Its significance in our case can be seen by comparing the secular result (where there is no bath-induced coherence) and the Bloch-Redfield result (where there is) in Figure 2. When \(\delta=0,\Omega=0\) our Hamiltonian has two degenerate eigenstates, but the form of those eigenstates depends on how the limit is taken: for \(\delta=0,\Omega\
eq 0\) they are \(|\pm\rangle=|g_{u}\rangle\pm|e\rangle\), but for \(\Omega=0,\delta\
eq 0\) they are \(|g_{u}\rangle\) and \(|e\rangle\). The naive form of secular approximation in Equation (10) produces a dissipator which populates the states \(|+\rangle\) and \(|-\rangle\), and destroys coherences between them. However, we observe that the phonon bath couples only to \(|g_{u}\rangle\propto|+\rangle+|-\rangle\), and not to \(|e\rangle\propto|+\rangle-|-\rangle\), so in the weak-driving case the correct dissipator should affect the population of the first combination, while leaving that of the second undamped. This means that there are undamped coherences in the \(|\pm\rangle\) basis, which survive in the steady-state [26], and produce corrections to the heat currents relative to the results of the secular approximation. Conceptualization, methodology, software, analysis, investigation, C.N.M., L.T.T. and P.R.E. Writing--original draft preparation, P.R.E. ; writing--review and editing, C.N.M. and L.T.T. ; visualization, supervision, P.R.E. All authors have read and agreed to the published version of the manuscript. This research was funded by the Irish Research Council grant number GOIPG/2017/1091. Some calculations were performed on the Boyle cluster maintained by the Trinity Centre for High-Performance Computing. This cluster was funded through grants from the European Research Council and Science Foundation Ireland. Not applicable.