In Fermi's Golden Rule, does the transition probability increase linearly with time or quadratically with time?

Question

When deriving Fermi's Golden rule, we get that the probability of a quantum system transitioning from an initial state $|i\rangle$ to a final state $|f\rangle$ is $$P_{i\rightarrow f}(t)=\frac{|V_{fi}|^2}{\hbar^2}\frac{\sin^2\left(\frac{\omega_{fi}-\omega}{2}\cdot t\right)}{\left(\frac{\omega_{fi}-\omega}{2}\right)^2}.\tag{1}$$ If we wish to examine the behaviour of this probability when the resonance condition $\omega_{fi}=\omega$ is met, we can do the following $$P_{i\rightarrow f}(t)=\frac{|V_{fi}|^2}{\hbar^2}\frac{\sin^2\left(\frac{\omega_{fi}-\omega}{2}\cdot t\right)}{\left(\frac{\omega_{fi}-\omega}{2}\cdot t\right)^2}\cdot t^2\\ \Rightarrow P_{i\rightarrow f}(t)=\frac{|V_{fi}|^2}{\hbar^2}\text{sinc}^2\left(\frac{\omega_{fi}-\omega}{2}\cdot t\right)\cdot t^2 $$ Now since $sinc(0)=1$, we have that when $\omega_{fi}=\omega$, the probability of transition increases quadratically with time: $$P_{i\rightarrow f}(t)=\frac{|V_{fi}|^2}{\hbar^2}t^2 \tag{2}$$ This quadratic increase in time is actually not unique to the resonance condition. It occurs for all the secondary peaks in the $P_{i\rightarrow f}(t)$ function. So seemingly, the probability for a transition should increase quadratically with time. However, all the books I have read (McIntyre, Shankar, Griffiths etc) as well as all the responses on this site pertaining to Fermi's Golden Rule that I have seen then go on to say that $$\lim_{t\rightarrow \infty}P_{i\rightarrow f}(t)=\lim_{t\rightarrow \infty}\frac{|V_{fi}|^2}{\hbar^2}\frac{\sin^2\left(\frac{\omega_{fi}-\omega}{2}\cdot t\right)}{\left(\frac{\omega_{fi}-\omega}{2}\right)^2}$$ $$\therefore \lim_{t\rightarrow \infty}P_{i\rightarrow f}(t)=\frac{2\pi}{\hbar^2}|V_{fi}|^2\delta(\omega_{fi}-\omega)\cdot t \tag{3}$$ This result now seems to indicate that the probability of transition increases linearly with respect to time provided we are looking at $P_{i\rightarrow f}(t)$ for large enought times. We no longer have the quadratic time dependence that equation (2) suggests we should. So what has happened here? Why do equation (1) and (2) imply that the probability should increase quadratically with time (regardless of how large $t$ is) but when we take the limit as $t\rightarrow \infty$, all of a sudden we now only have a linear increase with time?

I have seen the post over here (Transition probability derivation: How to prove $\lim_{\alpha\rightarrow\infty} \frac{\sin^2\alpha x}{\alpha x^2} ~=~\pi\delta(x)$?) which proves the limit but the answer does not explain how the transition probability $P_{i\rightarrow f}(t)$ somehow seems to be both increasing quadratically with time according to equations (1) and (2) but only linearly according to equation (3). Any help on this issue would be greatly appreciated!

Answer 1

1. OP is considering an argument $x$ for the ${\rm sinc}^2(x)$ function with $|x|\ll 1$, where the probability $P \propto t^2$ is indeed quadratic in $t$.

2. But Fermi's golden rule applies to the region $|x|\gtrsim 1$, where $P \propto t$ is linear in $t$, cf. e.g. this, this and this Phys.SE posts.