  Recent experiments in hybrid-quantum systems facilitate the potential
realization of one of the most fundamental interacting Hamiltonian-Reservoir
system, namely, the single-site Bose-Hubbard model coupled to two reservoirs at
different temperatures. Using Redfield equations in Born-Markov approximation,
we compute non-equilibrium average particle number, energy and currents beyond
linear response regime, both time-dynamics and steady state and investigate its
dependence on various tunable parameters analytically. We find interesting
scaling laws in high temperature regimes that are independent of choice of bath
spectral functions. We also demonstrate that the system shows very interesting
particle and energy current rectification properties which can be controlled
via the relative strength of interaction and temperatures, as well as via the
degree of asymmetry in system-bath coupling. Specifically, we find inversion of
direction of thermal rectification as a function of the relative strength of
the interaction strength and the temperatures. We also show that, in the limit
of low-temperature and high interaction strength our results are consistent
with the non-equilibrium spin-Boson model. Our results are experimentally
relevant not only to hybrid quantum systems, but also in other areas such as
molecular junctions.
