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

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

Contributors and Attributions

• Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)

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