Response functions for the quantum harmonic oscillator

I’m going through problems in Quantum Field Theory for the Gifted Amateur, and have been trying to understand a problem on the forced quantum oscillator [$L = frac{1}{2}dot{x}(t)^2-frac{1}{2}momega^2x(t)^2+f(t)x(t)$ ] and response functions.

The response function is

$$
langlepsi(t)|hat{x}(t)|psi(t)rangle = int_{-infty}^infty mathrm{d}t’chi(t-t’)f(t’)
$$

I want to show, using the interaction representation, that to first order in the force function $f_I(t)$

$$
|psi_I(t)rangle = |0rangle + iint_{-infty}^t mathrm{d}t’f_I(t’)hat{x}_I(t’)|0rangle
$$

Here is what I’ve done so far:

I started by taylor expanding the interacting ket:
$$ |psi_I(t)rangle = e^{i hat{H_0}(t)t}|psi(t)rangle = |psi(t)rangle + i hat{H_0}(t)t|psi(t)rangle+O(H_0^2)
$$

but I am confused about how to relate the wave function to the ground state, and how to use the information I have about the response function. When you have an expression for $|psirangle$ there is a procedure for finding the expectation value. I don’t know how to go the other way and around and extract a ket from the response function.

I also note that I can convert the response function to the interaction picture and it will have the same value, and that I can change f(t) to the interaction picture $f_I(t) = e^{i H_0 t}f(t)e^{-i H_0 t} =f(t) + O(H_0^2)$ since $H_0$ and f(t) commute.

Related: linear response for a simple harmonic oscillator