12.2: Two-State System

Consider a system in which the time-independent Hamiltonian possesses two eigenstates, denoted

\[\begin{align} H_0\,\psi_1 &= E_1\,\psi_1,\\[4pt] H_0\,\psi_2&=E_2\,\psi_2.\end{align} \nonumber \]

Suppose, for the sake of simplicity, that the diagonal elements of the interaction Hamiltonian, \(H_1\), are zero: that is,

\[\langle 1|H_1|1\rangle = \langle 2|H_1|2\rangle = 0. \nonumber \]

The off-diagonal elements are assumed to oscillate sinusoidally at some frequency \(\omega\): that is,

\[\langle 1|H_1|2\rangle = \langle 2|H_1|1\rangle^\ast = \gamma\,\hbar\,\exp(\,{\rm i}\,\omega\,t), \nonumber \]

where \(\gamma\) and \(\omega\) are real. Note that it is only the off-diagonal matrix elements which give rise to the effect which we are interested in: namely, transitions between states 1 and 2.

For a two-state system, Equation \ref{e13.12} reduces to

\[\begin{align} {\rm i}\,\frac{dc_1}{dt} &= \gamma\,\exp\left[+{\rm i}\,(\omega-\omega_{21})\,t\right]c_2,\\[4pt] {\rm i}\,\frac{dc_2}{dt} &=\gamma\,\exp\left[-{\rm i}\,(\omega-\omega_{21})\,t\right]c_1,\label{e13.20}\end{align} \]

where \(\omega_{21}=(E_2-E_1)/\hbar\). The previous two equations can be combined to give a second-order differential equation for the time-variation of the amplitude \(c_2\): that is,

\[ \frac{d^2 c_2}{dt^2} + {\rm i}\,(\omega-\omega_{21})\,\frac{dc_2}{dt}+\gamma^2\,c_2=0. \label{e13.21} \]

Once we have solved for \(c_2\), we can use Equation \ref{e13.20} to obtain the amplitude \(c_1\). Let us search for a solution in which the system is certain to be in state 1 (and, thus, has no chance of being in state 2) at time \(t=0\). Thus, our initial conditions are \(c_1(0)=1\) and \(c_2(0)=0\). It is easily demonstrated that the appropriate solutions to \ref{e13.21} and \ref{e13.20} are

\[\begin{align} c_2(t)&= \left(\frac{-{\rm i}\,\gamma}{\Omega}\right)\exp\!\left[\frac{-{\rm i}\,(\omega-\omega_{21})\,t}{2}\right]\sin(\Omega\,t),\\[4pt] c_1(t)&= \exp\!\left[\frac{{\rm i}\,(\omega-\omega_{21})\,t}{2}\right] \cos(\Omega\,t)- \left[\frac{{\rm i}\,(\omega-\omega_{21})}{2\,\Omega} \right]\exp\!\left[\frac{{\rm i}\,(\omega-\omega_{21})\,t}{2}\right]\sin(\Omega\,t),\end{align} \nonumber \]

where

\[\Omega = \sqrt{\gamma^2 + (\omega-\omega_{21})^2/4}. \nonumber \]

Now, the probability of finding the system in state 1 at time \(t\) is simply \(P_1(t)=|c_1(t)|^2\). Likewise, the probability of finding the system in state 2 at time \(t\) is \(P_2(t)= |c_2(t)|^2\). It follows that

\[\begin{align} P_1(t)&=1-P_2(t),\\[4pt] P_2(t)&= \left[\frac{\gamma^2}{\gamma^2 + (\omega-\omega_{21})^2/4}\right] \sin^2(\Omega\,t).\label{e13.25}\end{align} \]

This result is known as Rabi’s formula .

Equation \ref{e13.25} exhibits all the features of a classic resonance . At resonance, when the oscillation frequency of the perturbation, \(\omega\), matches the frequency \(\omega_{21}\), we find that

\[\begin{align} P_1(t)&=\cos^2(\gamma\,t),\\[4pt] P_2(t)&=\sin^2(\gamma\,t).\label{e13.28}\end{align} \]

According to the previous result, the system starts off in state 1 at \(t=0\). After a time interval \(\pi/(2\,\gamma)\) it is certain to be in state 2. After a further time interval \(\pi/(2\,\gamma)\) it is certain to be in state 1 again, and so on. Thus, the system periodically flip-flops between states 1 and 2 under the influence of the time-dependent perturbation. This implies that the system alternatively absorbs and emits energy from the source of the perturbation.

The absorption-emission cycle also takes place away from the resonance, when \(\omega\neq \omega_{21}\). However, the amplitude of the oscillation in the coefficient \(c_2\) is reduced. This means that the maximum value of \(P_2(t)\) is no longer unity, nor is the minimum of \(P_1(t)\) zero. In fact, if we plot the maximum value of \(P_2(t)\) as a function of the applied frequency, \(\omega\), then we obtain a resonance curve whose maximum (unity) lies at the resonance, and whose full-width half-maximum (in frequency) is \(4\,\gamma\). Thus, if the applied frequency differs from the resonant frequency by substantially more than \(2\,\gamma\) then the probability of the system jumping from state 1 to state 2 is always very small. In other words, the time-dependent perturbation is only effective at causing transitions between states 1 and 2 if its frequency of oscillation lies in the approximate range \(\omega_{21}\pm 2\,\gamma\). Clearly, the weaker the perturbation (i.e., the smaller \(\gamma\) becomes), the narrower the resonance.