# Laser Cooling beyond Rate Equations: Approaches from Quantum Thermodynamics
###### Abstract  
Solids can be cooled by driving impurity ions with lasers, allowing them to transfer heat from the lattice phonons to the electromagnetic surroundings. This exemplifies a quantum thermal machine, which uses a quantum system as a working medium to transfer heat between reservoirs. We review the derivation of the Bloch-Redfield equation for a quantum system coupled to a reservoir, and its extension, using counting fields, to calculate heat currents. We use the full form of this equation, which makes only the weak-coupling and Markovian approximations, to calculate the cooling power for a simple model of laser cooling. We compare its predictions with two other time-local master equations: the secular approximation to the full Bloch-Redfield equation, and the Lindblad form expected for phonon transitions in the absence of driving. We conclude that the full Bloch-Redfield equation provides accurate results for the heat current in both the weak- and strong- driving regimes, whereas the other forms have more limited applicability. Our results support the use of Bloch-Redfield equations in quantum thermal machines, despite their potential to give unphysical results.  
quantum thermodynamics; open quantum systems; laser cooling; Bloch-Redfield theory +
Footnote †: journal: _Sci._  
occurs include quantum-dot excitons coupled to acoustic phonons [20], colour centres in diamond [21, 22], and superconducting circuits [23].  
These issues can be treated theoretically by studying models of an open quantum system in which the working medium interacts with its surrounding heat baths. Such models are tractable in the weak-coupling, Markovian regime, where they lead to time-local equations of motion such as the Bloch-Redfield equation [24]. Those approaches can be extended to allow calculations of heat and work in the quantum regime [25]. However, there are several time-local equations which can be obtained, using reasonable approximations, from a given model, and these can make differing predictions for the dynamics [26, 27]. This problem has been addressed by several groups, who argue that the Bloch-Redfield equation [10, 28, 29, 30, 31, 32, 33] is useful and indeed accurate, despite its potential pathologies [34]. In this paper, we extend such studies to explore the heat flows in a simple laser cooling process, with the aim of identifying an approximate time-local equation that can accurately model them.  
In the following, we first review the derivation of the Bloch-Redfield equation for an open quantum system, and outline its extension to calculate heat flows. We also discuss two other time-local equations which can be obtained by making further approximations: a Lindblad form in the energy eigenbasis, obtained by making the secular approximation, and a Lindblad form in the eigenbasis of the undriven system. We use these forms to calculate the cooling spectrum, i.e., the cooling power as a function of driving frequency, in a model of laser cooling. The model allows for strong driving and includes a spectral structure for the environment. We find that a complete description of the cooling spectrum, which covers both the weak-driving and strong-driving regimes, can be achieved using the full Bloch-Redfield equation. We provide further support for the correctness of the Bloch-Redfield master equation--whose use has been controversial because it does not guarantee positivity [34], and can lead to behavior inconsistent with thermodynamic principles [35]--by comparing its predictions to those of an exact numerical method. Our conclusions support the use of Bloch-Redfield equations to model laser cooling and other thermodynamic processes [10, 30, 32, 33, 36, 37].