  We present a comprehensive and systematic study of thermal rectification in a
prototypical low-dimensional quantum system -- a non-linear resonator: we
identify necessary conditions to observe thermal rectification and we discuss
strategies to maximize it. We focus, in particular, on the case where
anharmonicity is very strong and the system reduces to a qubit. In the latter
case, we derive general upper bounds on rectification which hold in the weak
system-bath coupling regime, and we show how the Lamb shift can be exploited to
enhance rectification. We then go beyond the weak-coupling regime by employing
different methods: i) including co-tunneling processes, ii) using the
non-equilibrium Green's function formalism and iii) using the Feynman-Vernon
path integral approach. We find that the strong coupling regime allows us to
violate the bounds derived in the weak-coupling regime, providing us with clear
signatures of high order coherent processes visible in the thermal
rectification. In the general case, where many levels participate to the system
dynamics, we compare the heat rectification calculated with the equation of
motion method and with a mean-field approximation. We find that the former
method predicts, for a small or intermediate anharmonicity, a larger
rectification coefficient.
