  The use of two-site Lindblad dissipator to generate thermal states and study
heat transport raised to prominence since [J. Stat. Mech. (2009) P02035] by
Prosen and \v{Z}nidari\v{c}. Here we propose a variant of this method based on
detailed balance of internal levels of the two site Hamiltonian and
characterize its performance. We study the thermalization profile in the chain,
the effective temperatures achieved by different single and two-site
observables, and we also investigate the decay of two-time correlations. We
find that at a large enough temperature the steady state approaches closely a
thermal state, with a relative error below 1% for the inverse temperature
estimated from different observables.
