Transitions off resonance
Finally, we would like to find out how the states vary in the circumstance that the cavity frequency is nearly, but not exactly, equal to $\omega_0$. We could solve this problem exactly, but instead of trying to do that, we’ll take the important case that the electric field is small and also the period of time $T$ is small, so that $\mu\Efield_0T/\hbar$ is much less than one. Then, even in the case of perfect resonance which we have just worked out, the probability of making a transition is small. Suppose that we start again with $\gamma_{\slI}=1$ and $\gamma_{\slII}=0$. During the time $T$ we would expect $\gamma_{\slI}$ to remain nearly equal to one, and $\gamma_{\slII}$ to remain very small compared with unity. Then the problem is very easy. We can calculate $\gamma_{\slII}$ from the second equation in (9.45), taking $\gamma_{\slI}$ equal to one and integrating from $t=0$ to $t=T$. We get \begin{equation} \label{Eq:III:9:51} \gamma_{\slII}=\frac{\mu\Efield_0}{\hbar}\,\biggl[ \frac{1-e^{i(\omega-\omega_0)T}}{\omega-\omega_0} \biggr]. \end{equation}  This $\gamma_{\slII}$, used with Eq. (9.40), gives the amplitude to have made a transition from the state $\ketsl{\slI}$ to the state $\ketsl{\slII}$ during the time interval $T$. The probability $P(\slI\to\slII\,)$ to make the transition is $\abs{\gamma_{\slII}}^2$, or  \begin{equation} \label{Eq:III:9:52} P(\slI\to\slII\,)=\abs{\gamma_{\slII}}^2= \biggl[\frac{\mu\Efield_0T}{\hbar}\biggr]^2 \,\frac{\sin^2[(\omega-\omega_0)T/2]} {[(\omega-\omega_0)T/2]^2}. \end{equation} 
 \begin{gather} P(\slI\to\slII\,)=\abs{\gamma_{\slII}}^2\notag\\[1ex] \label{Eq:III:9:52} =\biggl[\frac{\mu\Efield_0T}{\hbar}\biggr]^2 \,\frac{\sin^2[(\omega-\omega_0)T/2]} {[(\omega-\omega_0)T/2]^2}. \end{gather} 
 It is interesting to plot this probability for a fixed length of time as a function of the frequency of the cavity in order to see how sensitive it is to frequencies near the resonant frequency $\omega_0$. We show such a plot of $P(\slI\to\slII\,)$ in Fig. 9–7. (The vertical scale has been adjusted to be $1$ at the peak by dividing by the value of the probability when $\omega=\omega_0$.) We have seen a curve like this in the diffraction theory, so you should already be familiar with it. The curve falls rather abruptly to zero for $(\omega-\omega_0)=2\pi/T$ and never regains significant size for large frequency deviations. In fact, by far the greatest part of the area under the curve lies within the range $\pm\pi/T$. It is possible to show5 that the area under the curve is just $2\pi/T$ and is equal to the area of the shaded rectangle drawn in the figure. Let’s examine the implication of our results for a real maser. Suppose that the ammonia molecule is in the cavity for a reasonable length of time, say for one millisecond. Then for $f_0=24{,}000$ megacycles, we can calculate that the probability for a transition falls to zero for a frequency deviation of $(f-f_0)/f_0=1/f_0T$, which is four parts in $10^8$. Evidently the frequency must be very close to $\omega_0$ to get a significant transition probability. Such an effect is the basis of the great precision that can be obtained with “atomic” clocks, which work on the maser principle. 