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Recent experiments in hybrid-quantum systems facilitate the potentialrealization of one of the most fundamental interacting Hamiltonian-Reservoirsystem, namely, the single-site Bose-Hubbard model coupled to two reservoirs atdifferent temperatures. Using Redfield equations in Born-Markov approximation,we compute non-equilibrium average particle number, energy and currents beyondlinear response regime, both time-dynamics and steady state and investigate itsdependence on various tunable parameters analytically. We find interestingscaling laws in high temperature regimes that are independent of choice of bathspectral functions. We also demonstrate that the system shows very interestingparticle and energy current rectification properties which can be controlledvia the relative strength of interaction and temperatures, as well as via thedegree of asymmetry in system-bath coupling. Specifically, we find inversion ofdirection of thermal rectification as a function of the relative strength ofthe interaction strength and the temperatures. We also show that, in the limitof low-temperature and high interaction strength our results are consistentwith the non-equilibrium spin-Boson model. Our results are experimentallyrelevant not only to hybrid quantum systems, but also in other areas such asmolecular junctions.

Non-linear transport in an out-of-equilibrium single-site Bose Hubbard model: scaling, rectification and time dynamics