# Laser Cooling beyond Rate Equations: Approaches from Quantum Thermodynamics
## 2 Materials and Methods
### Master Equations for Open Quantum Systems  
Figure 1 depicts an open quantum system: one which interacts, explicitly or implicitly, with a wider environment. These interactions lead to an exchange of energy between system and environment, and dephasing and decoherence effects. Here, we have an environment comprising the phonons in the host crystal of the impurity and the photons associated with the radiative decay of the upper level. The dynamics of an open quantum system can, in certain circumstances, be described by a time-local master equation for its reduced density matrix [24]. Such equations can be obtained from microscopic models which consider the environment explicitly in making the weak-coupling and Markovian approximations. They are also often postulated phenomenologically, based on the observation that the most general equation of motion is one of Lindblad form. However, there are several different forms of equations that can result from a microscopic model, depending on the details of the approximations made. The predictions of these forms can, furthermore, differ from those based on phenomenological Lindblad forms. These issues have been discussed in previous works [26; 28] which suggest that the full Bloch-Redfield equation--obtained by using the weak-coupling and Markovian approximations, but without making the secular approximation--gives a good description of the dynamics. This is in spite of the fact that the Bloch-Redfield equation does not guarantee that the eigenvalues of the reduced density matrix remain positive [24; 34]. For a system where there are no degeneracies, or near-degeneracies, that issue can be cured by secularization [34; 39], which corresponds to eliminating oscillating terms in the dissipator that average to zero over time. This leads to a Lindblad form [40; 41] with positive rates. It is, however, a priori invalid for the laser-cooling protocol considered here, where weak  
Figure 1: Energy levels of an impurity in a model laser-cooling process. The two states of a ground-state manifold, \(|g_{u}\rangle\), and \(|g_{l}\rangle\), are coupled by the emission and absorption of lattice phonons (vertical solid lines). Laser driving occurs on the transition from the upper level of the ground-state manifold to an excited state \(|e\rangle\) (block arrow). This state decays radiatively to the ground state (wavy arrow). driving near resonance means we have \(\Omega\approx 0\) and \(\delta\approx 0\), so that two of the eigenstates of Equation (2), \(|g_{u}\rangle\) and \(|e\rangle\), are almost degenerate in the frame where the laser field is time independent. A fairly generic form for the Hamiltonian of an open quantum system is  
\[H =H_{S}+H_{B}+H_{SB} \tag{3}\] \[H_{SB} =\sum_{k}g_{k}O(b_{k}+b_{k}^{\ddagger}). \tag{4}\]  
Here \(H_{S}\) is the Hamiltonian for the system, \(H_{B}\) for its environment, or bath, and \(H_{SB}\) is the system-bath coupling. We consider the common situation in which the bath comprises a set of harmonic oscillators [24], which we index using a quantity or quantities labeled \(r\). Note that \(r\) denotes the full set of quantum numbers required to label the modes. The oscillators have frequencies \(\omega_{r}\), and ladder operators \(b_{r}\) and \(b_{r}^{\ddagger}\). The displacement of the \(r\)th bath mode is coupled to the system operator \(O\), with coupling strength \(g_{r}\). The dissipative effects of the bath depend on its spectral density, \(J(\omega)=\sum_{g}g_{r}^{2}\delta(\omega-\omega_{r})\). To fix notation we recall the standard procedure for deriving a Bloch-Redfield master equation [24; 42; 43]. We work in the interaction picture with respect to \(H_{S}+H_{B}\), so that \(O(t)=e^{iH_{S}}Oe^{-iH_{S}t}\). Note that where necessary we will distinguish operators in the interaction and Schrodinger pictures as, for example, \(O(t)\) and \(O\).