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# 12.5: Harmonic Perturbation

Consider a (Hermitian) perturbation that oscillates sinusoidally in time. This is usually termed a harmonic perturbation. Such a perturbation takes the form

$\label{e13.51} H_1(t) = V\,\exp(\,{\rm i}\,\omega\,t) + V^\dagger\,\exp(-{\rm i}\,\omega\,t),$ where $$V$$ is, in general, a function of position, momentum, and spin operators.

It follows from Equations ([e13.48]) and ([e13.51]) that, to first-order, $c_f(t) = - \frac{\rm i}{\hbar}\int_0^t\left[V_{fi}\,\exp(\,{\rm i}\,\omega\,t') + V_{fi}^\dagger\,\exp(-{\rm i}\,\omega\,t')\right] \exp(\,{\rm i}\,\omega_{fi}\,t')\,dt',$ where

\begin{aligned} \label{e13.53} V_{fi}&= \langle f|V|i\rangle,\\[0.5ex] V_{fi}^\dagger &=\langle f|V^\dagger|i\rangle = \langle i|V|f\rangle^\ast.\end{aligned} Integration with respect to $$t'$$ yields \begin{aligned} c_f(t)&= - \frac{{\rm i}\,t}{\hbar}\left(V_{fi}\,\exp\left[\,{\rm i}\,(\omega+\omega_{fi})\,t/2\right]{\rm sinc}\left[(\omega+\omega_{fi})\,t/2\right]\right.\nonumber\\[0.5ex]& \left.\phantom{=}+V_{fi}^\dagger\,\exp\left[-{\rm i}\,(\omega-\omega_{fi})\,t/2\right]{\rm sinc}\left[(\omega-\omega_{fi})\,t/2\right]\right),\label{e13.55}\end{aligned} where ${\rm sinc}\, x\equiv \frac{\sin\,x}{x}.$ Figure 25: The functions $$\begin{equation}\operatorname{sinc}(x)\end{equation}$$ (dashed curve) and $$\begin{equation}\operatorname{sinc}^{2}(x)\end{equation}$$ (solid curve). The vertical dotted lines denote the region $$\begin{equation}|x| \leq \pi\end{equation}$$

Now, the function $${\rm sinc}(x)$$ takes its largest values when $$\begin{equation}|x| \lesssim \pi\end{equation}$$, and is fairly negligible when $$|x|\gg \pi$$. (See Figure [fsinc].) Thus, the first and second terms on the right-hand side of Equation ([e13.55]) are only non-negligible when

\begin{equation}\left|\omega+\omega_{f i}\right| \lesssim \frac{2 \pi}{t}\end{equation} and \begin{equation}\left|\omega-\omega_{f i}\right| \lesssim \frac{2 \pi}{t}\end{equation} respectively.

Clearly, as $$t$$ increases, the ranges in $$\omega$$ over which these two terms are non-negligible gradually shrink in size. Eventually, when $$t\gg 2\pi/|\omega_{fi}|$$, these two ranges become strongly non-overlapping. Hence, in this limit, $$P_{i\rightarrow f}=|c_f|^{\,2}$$ yields

$\label{e13.49} P_{i\rightarrow f}(t) = \frac{t^{\,2}}{\hbar^{\,2}}\left\{ |V_{fi}|^{\,2}\,{\rm sinc}^2\left[(\omega+\omega_{fi})\,t/2\right] + |V_{fi}^\dagger|^{\,2}\,{\rm sinc}^2\left[(\omega-\omega_{fi})\,t/2\right]\right\}.%\label{e13.59}$

Now, the function $${\rm sinc}^2(x)$$ is very strongly peaked at $$x=0$$, and is completely negligible for $$\begin{equation}|x| \gg \pi\end{equation}$$. (See Figure [fsinc].) It follows that the previous expression exhibits a resonant response to the applied perturbation at the frequencies $$\omega=\pm\omega_{fi}$$. Moreover, the widths of these resonances decease linearly as time increases. At each of the resonances (i.e., at $$\omega=\pm\omega_{fi}$$), the transition probability $$P_{i\rightarrow f}(t)$$ varies as $$t^{\,2}$$ [because $${\rm sinh} (0)=1$$]. This behavior is entirely consistent with our earlier result ([e13.28]), for the two-state system, in the limit $$\gamma\,t\ll 1$$ (recall that our perturbative solution is only valid as long as $$P_{i\rightarrow f}\ll 1$$).

The resonance at $$\omega=-\omega_{fi}$$ corresponds to $E_f - E_i = -\hbar\,\omega.$ This implies that the system loses energy $$\hbar\,\omega$$ to the perturbing field, while making a transition to a final state whose energy is less than the initial state by $$\hbar\,\omega$$. This process is known as stimulated emission. The resonance at $$\omega=\omega_{fi}$$ corresponds to $E_f - E_i = \hbar\,\omega.$ This implies that the system gains energy $$\hbar\,\omega$$ from the perturbing field, while making a transition to a final state whose energy is greater than that of the initial state by $$\hbar\,\omega$$. This process is known as absorption.

Stimulated emission and absorption are mutually exclusive processes, because the first requires $$\omega_{fi}<0$$, whereas the second requires $$\omega_{fi}>0$$. Hence, we can write the transition probabilities for both processes separately. Thus, from Equation ([e13.49]), the transition probability for stimulated emission is $P_{i\rightarrow f}^{stm}(t) = \frac{t^{\,2}}{\hbar^{\,2}}\, |V_{if}^\dagger|^{\,2}\,{\rm sinc}^2\left[(\omega-\omega_{if})\,t/2\right],$ where we have made use of the facts that $$\omega_{if}=-\omega_{fi}>0$$, and $$|V_{fi}|^{\,2}=|V_{if}^\dagger|^{\,2}$$. Likewise, the transition probability for absorption is

$\label{e13.63} P_{i\rightarrow f}^{abs}(t) = \frac{t^{\,2}}{\hbar^{\,2}}\, |V_{fi}^\dagger|^{\,2}\,{\rm sinc}^2\left[(\omega-\omega_{fi})\,t/2\right].$

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