  Recently, Nathan and Rudner derived a Gorini-Kossakowski-Sudarshan-Lindblad
master equation from the Redfield equation. The claim is that the level of
approximation is equal to that of the Redfield equation. Here we benchmark the
Nathan-Rudner equation (NRE) against the exact solution of a damped harmonic
oscillator and compare its performance to that of the time-dependent Redfield
equation (RE). We find that which of the equations performs better depends on
the regime considered. It turns out that the short-time dynamics is generally
much better captured by the RE, whereas the NRE delivers results comparable to
those of the rotating-wave approximation. For the steady state, in the
high-temperature limit the RE again performs better and its solution approaches
the exact result for ultrahigh temperatures. Nevertheless, here also the NR
equation constitutes a good approximation. In the low-temperature limit, in
turn, the NRE provides a better approximation than the RE. For too strong
coupling, here the RE might even fail completely by predicting unphysical
behaviour.
