Time correlation function
In this calculation, the bath spectrum used include Ohmic bath with exponential cutoff funciton and the Debye bath.
high temperature, $\beta=0.25$ low temperature,$\beta=10,50$ $\beta=1/k_BT=0.25,1,5,10,50$
The imaginary part of C(t) can be replaced by a single exponential function
$$C_I(t)=-\lambda\omega_c \exp(-\omega_c t).$$
Meanwhile, this need us to fit the real part more carefully.