Adiabatic Green's function technique and the transient behavior in time-dependent fermion-boson coupled models

Lang-Firsov Hamiltonian is a solvable model of interacting fermion-boson system with the fermion Green's function known to display sideband behavior reflecting the fermion-boson composite states. The exact solvability breaks down when time dependence is introduced in the fermion-boson coupling constant. Here we introduce a method, akin to the use of instantaneous basis states in solving the adiabatic evolution problem in quantum mechanics, by which the non-equilibrium two-time Green's function can be obtained in essentially exact form. With such "adiabatic Green's function" we analyze the transient behavior under the situation where the coupling is gradually tuned to zero. Another well-known model similar to the Lang-Firsov Hamiltonian is the spin-boson Hamiltonian. We analyze the non-equilibrium Green's function for the time-dependent version of this model as well. In both cases the sidebands arising from the fermion-boson coupling gradually lose their spectral weights in accord with expectations, but possibly with two distinct routes to recover the ground-state Green's function. We conclude with speculations on the transient dynamics in the time-resolved pump-probe experiments on the two-dimensional Dirac bands.