# Laser Cooling beyond Rate Equations: Approaches from Quantum Thermodynamics
## 2 Materials and Methods
### Master Equations for Open Quantum Systems
From the von Neumann equation we obtain the form  
\[\frac{d\rho(t)}{dt}=-i[H_{SB}(t),\rho(0)]-\int_{0}^{t}dt^{\prime}[H_{SB}(t),[H_ {SB}(t^{\prime}),\rho(t^{\prime})]] \tag{5}\]  
where \(\rho(t)\) is the full density operator of the system and environment. For weak coupling to a bath one can replace \(\rho(t^{\prime})\approx\rho_{S}(t^{\prime})\otimes\rho_{B}(t^{\prime})\) on the right-hand side, where \(\rho_{S}\) is the reduced density matrix of the system, and \(\rho_{B}\) that of the bath. Since the bath is macroscopic it can be assumed to be unperturbed by the system, and \(\rho_{B}\) taken to be a thermal state at inverse temperature \(\beta\). For a Markovian system one may, furthermore, approximate \(\rho_{S}(t^{\prime})\approx\rho_{S}(t)\). We can write the coupling operator in the eigenbasis of \(H_{S}\) as  
\[O(t)=\sum_{ij}e^{i(E_{i}-E_{j})t}\langle i|O|j\rangle|i\rangle\langle j|\equiv \sum_{ij}\hat{O}_{ij}(t). \tag{6}\]  
Taking the trace of Equation (5) over the environment's degrees-of-freedom we find  
\[ \begin{split}\frac{d\rho_{S}(t)}{dt}=\sum_{ij}\big{\{}A_{ij}[ \hat{O}_{ji}(t)\rho_{S}(t)O(t)+O(t)\rho_{S}(t)\hat{O}_{ij}(t)\\ -\rho_{S}(t)\hat{O}_{ij}(t)O(t)-O(t)\hat{O}_{ji}(t)\rho_{S}(t)] \\ -iB_{ij}[\hat{O}_{ji}(t)\rho_{S}(t)O(t)-O(t)\rho_{S}(t)\hat{O}_{ ij}(t)\\ +\rho_{S}(t)\hat{O}_{ij}(t)O(t)-O(t)\hat{O}_{ji}(t)\rho_{S}(t)] \big{\}}.\end{split}  \tag{7}\]  
The quantities \(A_{ij}\) and \(B_{ij}\) are related to the the real-time Green's functions of the environment at the transition frequency \(\
u_{ij}=E_{i}-E_{j}\) connecting levels \(i\) and \(j\). The quantities \(A_{ij}\) are associated with dissipation, and are  
\[A_{ij}=\pi\big{\{}[n(\
u_{ij})+1]J(\
u_{ij})+n(\
u_{ji})J(\
u_{ji})\big{\}}. \tag{8}\]  
Here \(n(\
u>0)=1/(\exp(\beta\
u)-1)\) is the Bose function describing the bath occupation, and \(J(\
u)=0\) for \(\
u<0\). The first term in \(A_{ij}\) corresponds to the creation of a bath quantum as the system transitions from a state \(i\) to \(j\) with \(E_{i}-E_{j}>0\), whereas the second corresponds to the absorption of a bath quantum in the opposite case, \(E_{i}-E_{j}<0\). The quantities \(B_{ij}\) are associated with energy shifts, and are given by the principal value integral \[B_{ij}=\mathcal{P}\int J(\omega)\frac{\omega+(2n(\omega)+1)(E_{i}-E_{j})}{\omega^ {2}-(E_{i}-E_{j})^{2}}d\omega. \tag{9}\]  
Equation (7) can be used directly, but is often further approximated, leading to other forms of equation-of-motion for an open quantum system. One very common approximation is to drop the principal value terms proportional to \(B_{ij}\). Another common approximation is to _secularize_ the equation-of-motion. This is done by decomposing the remaining coupling operators, \(O(t)\), into the energy eigenbasis: \(O(t)=\sum_{kl}\hat{O}_{kl}(t)\). Every term in Equation (7) then involves a product of operators corresponding to two transitions, one involving the pair of levels \(i\) and \(j\), and one involving the pair \(k\), \(l\). If the levels are non-degenerate these products of operators are, in general, time-dependent in the interaction picture, and average to zero. The exception is where a transition in one direction is paired with the same transition in the opposite direction, so that the time-dependence cancels out. Retaining only those terms the dissipative part of Equation (7) becomes  
\[\frac{d\rho_{S}(t)}{dt}=\sum_{ij}2A_{ij}\bigg{(}\hat{O}_{ji}(t)\rho_{S}(t)\hat {O}_{ij}(t)-\frac{1}{2}[\rho_{S}(t),\hat{O}_{ij}(t)\hat{O}_{ji}(t)]_{+}\bigg{)}, \tag{10}\]  
where \([A,B]_{+}=AB+BA\) is an anticommutator. This is of Lindblad form, and therefore guarantees the positivity of the density operator. It has a straightforward physical interpretation: the environment causes transitions from the system state \(i\) to the system state \(j\) at rate \(2A_{ij}\).