# Laser Cooling beyond Rate Equations: Approaches from Quantum Thermodynamics
## 2 Materials and Methods
### Heat Flows from Master Equations  
The method of full counting statistics [25] allows one to extend the approaches above so as to compute the heat transferred to the bath. It has been used, often with the secular approximation [44], to obtain master equations and study heat statistics in various systems, including driven quantum-dot excitons [11; 15], a driven two-level system [45], a steady-state (absorption) refrigerator [10; 30; 36], and a two-bath spin-boson model [32; 33]. The absorption refrigerator and spin-boson model have been studied using the full Bloch-Redfield approach, without the secular approximation, which highlights the role of coherences [10; 30; 36]. Here we give an outline of the method and present a complete form for the full counting-field Bloch-Redfield equation, which we shall use to calculate laser cooling spectra. The heat transferred to a bath is, by definition, the change in its energy between two times. Thus we consider a process involving projective measurements of the bath energy at two times. We take the initial time to be \(t_{i}=0\), and suppose that at this time the system and bath are in a product state, \(\rho_{S}(0)\otimes\rho_{B}\). We can then consider the probability distribution of the heat, \(P(Q,t)\), which is the probability that the energy measurements of the bath at times \(t_{i}\) and \(t\) give results differing by \(Q\). It is convenient also to introduce the characteristic function of the heat distribution, \(\chi(u,t)=\int dQP(Q,t)e^{iuQ}\). The variable \(u\) is known as the counting field. (This term should not be taken to imply that heat is necessarily a discrete, countable quantity. It arises from other uses of the method, such as calculations of the number of electrons transferred across a tunnel junction [25].). One can evaluate \(\chi(u,t)\) by introducing an annotated density operator, \(\rho_{u}(t)\), such that \(\chi(u,t)=\operatorname{Tr}\rho_{u}(t)\). \(\rho_{u}(t)\) has a non-unitary time evolution given by  
\[\rho_{u}(t)=U_{u/2}\rho_{u}(0)U_{-u/2}^{t}, \tag{11}\]  
where \(U_{u}\) is related to the normal time-evolution operator, \(U=e^{-iHt}\), by  
\[U_{u}=e^{iuH_{B}}UE^{-iuH_{B}}. \tag{12}\]Note the similarity between these phase factors and the factor \(e^{i\mu Q}\) in the definition of the characteristic function; it is these factors that incorporate the results of the measurements of the bath energy, \(H_{B}\), into \(\rho_{u}(t)\). At the initial time the annotated density matrix is given by \(\rho_{u}(0)=\rho(0)\). For a general operator \(P\) we define the annotated version \(P_{u}=e^{i\mu H_{B}}Pe^{-i\mu H_{B}}\), which obeys the Heisenberg-like equation  
\[i\frac{dP_{u}}{du}=[P_{u},H_{B}].\]  
For the lowering operator appearing in Equation (4) we have \(b_{u,k}=e^{-i\omega_{k}u}b_{0,k}\). Thus, the time-evolution operators, \(U_{\pm u/2}\), can be obtained from the standard form, \(e^{-i\mu H}\), by replacing the coupling Hamiltonian, Equation (4), with \(H_{SB}^{\pm}=\sum g_{k}C(b_{k}e^{\mp i\omega u/2}+b_{k}^{\dagger}e^{\pm i\omega u /2})\). A master equation for the reduced annotated density matrix, \(\rho_{u,S}(t)\) can now be obtained, following the steps above. The essential difference is that the von Neumann equation for \(\rho(t)\), in the interaction picture, must be replaced by  
\[\frac{d\rho_{u}(t)}{dt}=-i(H_{SB}^{+}\rho_{u}(t)-\rho_{u}(t)H_{SB}^{-}). \tag{13}\]  
The result is  
\[ \begin{split}\frac{d\rho_{u,S}(t)}{dt}=\sum_{ij}\big{\{}A_{ij}[e^ {iu(E_{i}-E_{j})}(\hat{O}_{ji}(t)\rho_{u,S}(t)O(t)+O(t)\rho_{u,S}(t)\hat{O}_{ij }(t))\] \[-\rho_{u,S}(t)\hat{O}_{ij}(t)O(t)-O(t)\hat{O}_{ji}(t)\rho(t)] \\ -iB_{ij}[e^{iu(E_{i}-E_{j})}(\hat{O}_{ji}(t)\rho_{u,S}(t)O(t)-O(t) \rho_{u,S}(t)\hat{O}_{ij}(t))\] \[+\rho_{u,S}(t)\hat{O}_{ij}(t)O(t)-O(t)\hat{O}_{ji}(t)\rho(t)]\big{\}}. \end{split}  \tag{14}\]  
This form differs from Equation (7) by the addition of phase factors in the four terms that cause transitions between the system eigenstates.