The two-state system

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

$$H_0 \vert 1\rangle = E_1 \vert 1\rangle,$$ (743)

$$H_0 \vert 2\rangle = E_2 \vert 2\rangle.$$ (744)

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

$$\langle 1\vert H_1\vert 1\rangle = \langle 2\vert H_1\vert 2\rangle = 0.$$ (745)

The off-diagonal matrix elements are assumed to oscillate sinusoidally at some frequency $\omega$:

$$\langle 1\vert H_1\vert 2\rangle = \langle 2\vert H_1\vert 1\rangle^\ast = \gamma \exp({\rm i} \omega t),$$ (746)

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. (740) reduces to

$${\rm i} \hbar \frac{d c_1}{dt} = \gamma \exp[+{\rm i} (\omega-\omega_{21}) t ] c_2,$$ (747)

$${\rm i} \hbar \frac{d c_2}{dt} = \gamma \exp[-{\rm i} (\omega-\omega_{21}) t ] c_1,$$ (748)

where $\omega_{21} = (E_2 - E_1)/\hbar$, and assuming that $t_0=0$. Equations (747) and (748) can be combined to give a second-order differential equation for the time variation of the amplitude $c_2$:

$$\frac{d^2 c_2}{dt^2} + {\rm i} (\omega-\omega_{21})\frac{d c_2}{dt} + \frac{\gamma^2}{\hbar^2} c_2 = 0.$$ (749)

Once we have solved for $c_2$, we can use Eq. (748) to obtain the amplitude $c_1$. Let us look for a solution in which the system is certain to be in state 1 at time $t=0$. Thus, our boundary conditions are $c_1(0) = 1$ and $c_2(0) = 0$. It is easily demonstrated that the appropriate solutions are

$$c_2(t) = \frac{-{\rm i} \gamma/\hbar} {\sqrt{\gamma^2/\hbar^2 + (\omega-\omega_{21})^2/4}} \exp[-{\rm i} (\omega-\omega_{21}) t/2]$$

$$\times\sin\!\left(\sqrt{\gamma^2/\hbar^2+(\omega-\omega_{21})^2/4} t\right),$$ (750)

$$c_1(t) = \exp[ {\rm i} (\omega-\omega_{21}) t/2] \cos\!\left( \sqrt{\gamma^2/\hbar^2+(\omega-\omega_{21})^2/4} t\right)$$

$$- \frac{{\rm i} (\omega-\omega_{21})/2 }{\sqrt{\gamma^2/\hbar^2 + (\omega-\omega_{21})^2/4}} \exp[ {\rm i} (\omega-\omega_{21}) t/2]$$

$$\times\sin\!\left( \sqrt{\gamma^2/\hbar^2+(\omega-\omega_{21})^2/4} t\right).$$ (751)

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

$$P_2(t) = \frac{\gamma^2/\hbar^2}{ \gamma^2/\hbar^2 + (\omega-\omega_{21})^2/4}$$

$$\times \sin^2\!\left(\sqrt{\gamma^2/\hbar^2+ (\omega-\omega_{21})^2/4} t\right),$$ (752)

$$P_1(t) = 1 - P_2(t).$$ (753)

This result is known as Rabi's formula.

Equation (752) 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

$$P_1(t) = \cos^2 (\gamma t / \hbar),$$ (754)

$$P_2(t) = \sin^2 (\gamma t/\hbar ).$$ (755)

According to the above result, the system starts off at $t=0$ in state $1$. After a time interval $\pi \hbar/2 \gamma$ it is certain to be in state 2. After a further time interval $\pi \hbar/2 \gamma$ it is certain to be in state 1, 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 take place away from the resonance, when $\omega\neq \omega_{21}$. However, the amplitude of oscillation of the coefficient $c_2$ is reduced. This means that the maximum value of $P_2(t)$ is no longer unity, nor is the minimum value 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 $4 \gamma/\hbar$. Thus, if the applied frequency differs from the resonant frequency by substantially more than $2 \gamma/\hbar$ then the probability of the system jumping from state 1 to state 2 is 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 $\omega_{21} \pm 2 \gamma/\hbar$. Clearly, the weaker the perturbation (i.e., the smaller $\gamma$ becomes), the narrower the resonance.