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The evolution of a quantum system towards thermal equilibrium is usuallystudied by approximate methods, which have their limits of validity and shouldbe checked against analytically solvable models. In this paper, we propose ananalytically solvable model to investigate the energy transfer between abosonic bath and a fermionic chain which are nonlinearly-coupled to each other.The bosonic bath consists of an infinite collection of non-interacting bosonicmodes, while the fermionic chain is represented by a chain of interactingfermions with nearest-neighbor interactions. We compare behaviors of thetemperature-dependent energy current $J_{T}$ and temperature-independent energycurrent $J_{TI}$ for different bath configurations. With respect to the bathspectrum, $J_{T}$ decays exponentially for Lorentz-Drude type bath, which isthe same as the conventional approximations. On the other hand, the decay rateis $1/t^{3}$ for Ohmic type and $1/t$ for white noise, which doesn't haveconventional counterparts. For the temperature-independent current $J_{TI}$,the decay rate is divergent for the Lorentz-Drude type bath, $1/t^{4}$ for theOhmic bath, and $1/t$ for the white noise. When further considering thedynamics of the fermionic chain, the current will be modulated based on theenvelope 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 $(N-1)^{1/2}$ atcertain times for PST couplings under Lorentz-Drude or Ohmic bath.

Quantum energy transfer between nonlinearly-coupled bosonic bath and a fermionic chain: an exactly solvable model