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Time-dependent perturbation theory in a harmonic oscillator with a time-dependent force

Lets assume the classic quantum harmonic oscillator (HO) with a Hamiltonian $$ H(t) = - \frac{\hbar^2}{2m} \frac{d^2}{d x^2} + \frac{m \omega^2 x^2}{2} + F(t) $$ where $F(t)$ is a time-dependent force defined via $$ F(t)=\frac{F_0 \tau / \omega}{\tau^2 + t^2} $$ At time $t \to -\infty$ the particle with mass $m$ is in the ground state $| 0 \rangle$ of the HO potential. Since this is a time-dependent problem, I try to use time-dependent perturbation theory to first order to obtain the probability that at $t \to \infty$ the particle is in the first excited state $|1\rangle$. From the lecture we have obtained the (slightly more general) formula $$ |c_f (t)|^2 = \frac{1}{\hbar^2} \left| \int_{t_0}^t V_{fi} (t') e^{i \omega_{fi} t'} dt' \right|^2 $$ which we already have applied to the kicked oscillator as shown here (I used the same notation to make life easier for the readers here). But, how can I (can I?) apply this approach to this problem? As I see it, I need to somehow use $F(t)$ to get an expression for $V_{fi}(t')$ that gives me a calculable expression.

In general, time-dependent perturbation theory poses a lot of problems for me, so I don't really have a starting point.