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

Question

Lets assume the classic quantum harmonic oscillator (HO) with a Hamiltonian $$ \hat{H}(t) = - \frac{\hbar^2}{2m} \frac{d^2}{d x^2} + \frac{m \omega^2 \hat{x}^2}{2} + F(t) \hat{x} $$ 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.

Answer 1

Your hamiltonian looks pretty wonky, and the notation should give itself away. $F(t)$ is a force, so what is it doing inside the hamiltonian without an $x$?

Instead, you might want to consider the hamiltonian $$ H(t) = - \frac{\hbar^2}{2m} \frac{d^2}{d x^2} + \frac{1}{2}m \omega^2 x^2 + F(t)x, $$ where $V(t) = F(t)x$ is a bona fide potential. This will save you from some embarrassing features of your original hamiltonian, like the fact that if you just set $V(t) = F(t)$ then its matrix elements will just be $$ V_{fi}(t) = \langle f|V(t)|i\rangle = \langle f|F(t)|i\rangle = F(t)\langle f|i\rangle = \delta_{fi}F(t), $$ and you will not get any transitions at all. If, instead, you use the actual potential $V(t)=F(t)x$, the matrix elements are $$ V_{fi}(t) = \langle f|V(t)|i\rangle = \langle f|F(t)x|i\rangle = F(t)\langle f|x|i\rangle, $$ and they can be calculated relatively easily by expressing $x$ as a sum of $a$ and $a^\dagger$, giving you an expression proportional to $\delta_{f,i-1}+\delta_{f,i+1}$. In the end, this will reduce rather easily to some prefactor multiplying the integral $$ \int_{-\infty}^\infty F(t) e^{i\omega t}\mathrm dt, $$ which requires some handling of the exponential integral Ei function (see also here) (probably after some partial-fractions handling of the denominator? e.g. you can split $$\frac{2}{\tau^2+t^2}=\frac{1}{t+i\tau}+\frac{1}{t-i\tau}$$ or something) but which should not be particularly challenging.