  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, in spite of their potential to give
unphysical results.
