12.2: Two-State System

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Consider a system in which the time-independent Hamiltonian possesses two eigenstates, denoted \begin{aligned} H_0\,\psi_1 &= E_1\,\psi_1,\\[0.5ex] H_0\,\psi_2&=E_2\,\psi_2.\end{aligned} Suppose, for the sake of simplicity, that the diagonal elements of the interaction Hamiltonian, $$H_1$$, are zero: that is, $\langle 1|H_1|1\rangle = \langle 2|H_1|2\rangle = 0.$ The off-diagonal elements are assumed to oscillate sinusoidally at some frequency $$\omega$$: that is, $\langle 1|H_1|2\rangle = \langle 2|H_1|1\rangle^\ast = \gamma\,\hbar\,\exp(\,{\rm i}\,\omega\,t),$ 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 ([e13.12]) reduces to

\begin{aligned} {\rm i}\,\frac{dc_1}{dt} &= \gamma\,\exp\left[+{\rm i}\,(\omega-\omega_{21})\,t\right]c_2,\\[0.5ex] {\rm i}\,\frac{dc_2}{dt} &=\gamma\,\exp\left[-{\rm i}\,(\omega-\omega_{21})\,t\right]c_1,\label{e13.20}\end{aligned} 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,

$\label{e13.21} \frac{d^{\,2} c_2}{dt^{\,2}} + {\rm i}\,(\omega-\omega_{21})\,\frac{dc_2}{dt}+\gamma^{\,2}\,c_2=0.$ Once we have solved for $$c_2$$, we can use Equation ([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 ([e13.21]) and ([e13.20]) are

$$c_{2}(t)=\left(\frac{-\mathrm{i} \gamma}{\Omega}\right) \exp \left[\frac{-\mathrm{i}\left(\omega-\omega_{21}\right) t}{2}\right] \sin (\Omega t)$$

\begin{aligned}
c_{1}(t)=& \exp \left[\frac{\mathrm{i}\left(\omega-\omega_{21}\right) t}{2}\right] \cos (\Omega t) \\
&-\left[\frac{\mathrm{i}\left(\omega-\omega_{21}\right)}{2 \Omega}\right] \exp \left[\frac{\mathrm{i}\left(\omega-\omega_{21}\right) t}{2}\right] \sin (\Omega t)
\end{aligned}

where $$\Omega=\sqrt{\gamma^{2}+\left(\omega-\omega_{21}\right)^{2} / 4}$$

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{aligned} P_1(t)&=1-P_2(t),\\[0.5ex] P_2(t)&= \left[\frac{\gamma^{\,2}}{\gamma^{\,2} + (\omega-\omega_{21})^{\,2}/4}\right] \sin^2({\mit\Omega}\,t).\label{e13.25}\end{aligned} This result is known as Rabi’s formula .

Equation ([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{aligned} P_1(t)&=\cos^2(\gamma\,t),\\[0.5ex] P_2(t)&=\sin^2(\gamma\,t).\label{e13.28}\end{aligned} 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.

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