  The evolution of a quantum system towards thermal equilibrium is usually
studied by approximate methods, which have their limits of validity and should
be checked against analytically solvable models. In this paper, we propose an
analytically solvable model to investigate the energy transfer between a
bosonic bath and a fermionic chain which are nonlinearly-coupled to each other.
The bosonic bath consists of an infinite collection of non-interacting bosonic
modes, while the fermionic chain is represented by a chain of interacting
fermions with nearest-neighbor interactions. We compare behaviors of the
temperature-dependent energy current $J_{T}$ and temperature-independent energy
current $J_{TI}$ for different bath configurations. With respect to the bath
spectrum, $J_{T}$ decays exponentially for Lorentz-Drude type bath, which is
the same as the conventional approximations. On the other hand, the decay rate
is $1/t^{3}$ for Ohmic type and $1/t$ for white noise, which doesn't have
conventional counterparts. For the temperature-independent current $J_{TI}$,
the decay rate is divergent for the Lorentz-Drude type bath, $1/t^{4}$ for the
Ohmic bath, and $1/t$ for the white noise. When further considering the
dynamics of the fermionic chain, the current will be modulated based on the
envelope from the bath. As an example, for a bosonic bath with Ohmic spectrum,
when the fermionic chain is uniformly-coupled, we have $J_{T}\propto1/t^{6}$
and $J_{TI}\propto1/t^{3}$. Remarkably, for perfect state transfer (PST)
couplings, there always exists an oscillating quantum energy current $J_{TI}$.
Moreover, it is interesting that $J_{T}$ is proportional to $(N-1)^{1/2}$ at
certain times for PST couplings under Lorentz-Drude or Ohmic bath.
