12.2: Two-State System

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

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

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

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

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.

Contributors and Attributions

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

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