Two-State System

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

$$H_0 \psi_1 = E_1 \psi_1,$$ (1031)

$$H_0 \psi_2 = E_2 \psi_2.$$ (1032)

Suppose, for the sake of simplicity, that the diagonal elements of the interaction Hamiltonian, $H_1$, are zero: i.e.,

$$\langle 1\vert H_1\vert 1\rangle = \langle 2\vert H_1\vert 2\rangle = 0.$$ (1033)

The off-diagonal elements are assumed to oscillate sinusoidally at some frequency $\omega$: i.e.,

$$\langle 1\vert H_1\vert 2\rangle = \langle 2\vert H_1\vert 1\rangle^\ast = \gamma \hbar \exp({\rm i} \omega t),$$ (1034)

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, Eq. (1028) reduces to

$${\rm i} \frac{dc_1}{dt} = \gamma \exp\left[+{\rm i} (\omega-\omega_{21}) t\right] c_2,$$ (1035)

$${\rm i} \frac{dc_2}{dt} = \gamma \exp\left[-{\rm i} (\omega-\omega_{21}) t\right] c_1,$$ (1036)

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

$$\frac{d^2 c_2}{dt^2} + {\rm i} (\omega-\omega_{21}) \frac{dc_2}{dt}+\gamma^2 c_2=0.$$ (1037)

Once we have solved for $c_2$, we can use Eq. (1036) 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 (1037) and (1036) are

$$c_2(t) = \left(\frac{-{\rm i} \gamma}{\Omega}\right)\exp\!\left[\frac{-{\rm i} (\omega-\omega_{21}) t}{2}\right]\sin(\Omega t),$$ (1038)

$$c_1(t) = \exp\!\left[\frac{{\rm i} (\omega-\omega_{21}) t}{2}\right] \cos(\Omega t)$$

$$- \left[\frac{{\rm i} (\omega-\omega_{21})}{2 \Omega} \right]\exp\!\left[\frac{{\rm i} (\omega-\omega_{21}) t}{2}\right]\sin(\Omega t),$$ (1039)

where

$$\Omega = \sqrt{\gamma^2 + (\omega-\omega_{21})^2/4}.$$ (1040)

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

$$P_1(t) = 1-P_2(t),$$ (1041)

$$P_2(t) = \left[\frac{\gamma^2}{\gamma^2 + (\omega-\omega_{21})^2/4}\right] \sin^2(\Omega t).$$ (1042)

This result is known as Rabi's formula.

Equation (1042) 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

$$P_1(t) = \cos^2(\gamma t),$$ (1043)

$$P_2(t) = \sin^2(\gamma t).$$ (1044)

According to the above 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, $\omega$, 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.