  We established a theoretical framework in terms of the curl flux, population
landscape, and coherence for non-equilibrium quantum systems at steady state,
through exploring the energy and charge transport in molecular processes. The
curl quantum flux plays the key role in determining transport properties and
the system reaches equilibrium when flux vanishes. The novel curl quantum flux
reflects the degree of non-equilibriumness and the time-irreversibility. We
found an analytical expression for the quantum flux and its relationship to the
environmental pumping (non-equilibriumness quantified by the voltage away from
the equilibrium) and the quantum tunneling. Furthermore, we investigated
another quantum signature, the coherence, quantitatively measured by the
non-zero off diagonal element of the density matrix. Besides the
environment-assistance which can give dramatic enhancement of coherence and
quantum flux with high voltage at a fixed tunneling strength, the quantum flux
is promoted by the coherence in the regime of small tunneling while reduced by
the coherence in the regime of large tunneling, due to the non-monotonic
relationship between the coherence and tunneling. In view of the system as a
quantum heat engine, we studied the non-equilibrium thermodynamics and
established the analytical connections of curl quantum flux to the transport
quantities such as energy (charge) transfer efficiency, chemical reaction
efficiency, energy dissipation, heat and electric currents observed in the
experiments. We observed a perfect transfer efficiency in chemical reactions at
high voltage (chemical potential difference). Our theoretical predicted
behavior of the electric current with respect to the voltage is in good
agreements with the recent experiments on electron transfer in single
molecules.
