Categories: Optics, Physics, Quantum mechanics.

In quantum mechanics, from the derivation of time-dependent perturbation theory, we know that a time-dependent term \(\hat{H}_1\) in the Hamiltonian affects the state as follows, where \(c_n(t)\) are the coefficients of the linear combination of basis states \(\ket{n} \exp\!(-i E_n t / \hbar)\):

\[\begin{aligned} i \hbar \dv{c_m}{t} = \sum_{n} c_n(t) \matrixel{m}{\hat{H}_1}{n} \exp\!(i \omega_{mn} t) \end{aligned}\]

Where \(\omega_{mn} \equiv (E_m \!-\! E_n) / \hbar\) for energies \(E_m\) and \(E_n\). Note that this equation is exact, despite being used for deriving perturbation theory. Consider a two-level system where \(n \in \{a, b\}\), in which case the above equation can be expanded to the following:

\[\begin{aligned} \dv{c_a}{t} &= - \frac{i}{\hbar} \matrixel{a}{\hat{H}_1}{b} \exp\!(- i \omega_0 t) \: c_b - \frac{i}{\hbar} \matrixel{a}{\hat{H}_1}{a} \: c_a \\ \dv{c_b}{t} &= - \frac{i}{\hbar} \matrixel{b}{\hat{H}_1}{a} \exp\!(i \omega_0 t) \: c_a - \frac{i}{\hbar} \matrixel{b}{\hat{H}_1}{b} \: c_b \end{aligned}\]

Where \(\omega_0 \equiv \omega_{ba}\) is positive. We assume that \(\hat{H}_1\) has odd spatial parity, in which case Laporte’s selection rule states that the diagonal matrix elements vanish, leaving:

\[\begin{aligned} \dv{c_a}{t} &= - \frac{i}{\hbar} \matrixel{a}{\hat{H}_1}{b} \exp\!(- i \omega_0 t) \: c_b \\ \dv{c_b}{t} &= - \frac{i}{\hbar} \matrixel{b}{\hat{H}_1}{a} \exp\!(i \omega_0 t) \: c_a \end{aligned}\]

We now choose \(\hat{H}_1\) to be as follows, sinusoidally oscillating with a spatially odd \(V(\vec{r})\):

\[\begin{aligned} \hat{H}_1(t) = V \cos\!(\omega t) = \frac{V}{2} \Big( \exp\!(i \omega t) + \exp\!(-i \omega t) \Big) \end{aligned}\]

We insert this into the equations for \(c_a\) and \(c_b\), and define \(V_{ab} \equiv \matrixel{a}{V}{b}\), leading us to:

\[\begin{aligned} \dv{c_a}{t} &= - i \frac{V_{ab}}{2 \hbar} \Big( \exp\!\big(i (\omega \!-\! \omega_0) t\big) + \exp\!\big(\!-\! i (\omega \!+\! \omega_0) t\big) \Big) \: c_b \\ \dv{c_b}{t} &= - i \frac{V_{ab}}{2 \hbar} \Big( \exp\!\big(i (\omega \!+\! \omega_0) t\big) + \exp\!\big(\!-\! i (\omega \!-\! \omega_0) t\big) \Big) \: c_a \end{aligned}\]

Here, we make the *rotating wave approximation*: assuming we are close to resonance \(\omega \approx \omega_0\), we decide that \(\exp\!(i (\omega \!+\! \omega_0) t)\) oscillates so much faster than \(\exp\!(i (\omega \!-\! \omega_0) t)\), that its effect turns out negligible when the system is observed over a reasonable time interval.

In other words, over this reasonably-sized time interval, \(\exp\!(i (\omega \!+\! \omega_0) t)\) averages to zero, while \(\exp\!(i (\omega \!-\! \omega_0) t)\) does not. Dropping the respective terms thus leaves us with:

\[\begin{aligned} \boxed{ \begin{aligned} \dv{c_a}{t} &= - i \frac{V_{ab}}{2 \hbar} \exp\!\big(i (\omega \!-\! \omega_0) t \big) \: c_b \\ \dv{c_b}{t} &= - i \frac{V_{ba}}{2 \hbar} \exp\!\big(\!-\! i (\omega \!-\! \omega_0) t \big) \: c_a \end{aligned} } \end{aligned}\]

Now we can solve this system of coupled equations exactly. We differentiate the first equation with respect to \(t\), and then substitute \(\dv*{c_b}{t}\) for the second equation:

\[\begin{aligned} \dv[2]{c_a}{t} &= - i \frac{V_{ab}}{2 \hbar} \bigg( i (\omega - \omega_0) \: c_b + \dv{c_b}{t} \bigg) \exp\!\big(i (\omega \!-\! \omega_0) t \big) \\ &= - i \frac{V_{ab}}{2 \hbar} \bigg( i (\omega - \omega_0) \: c_b - i \frac{V_{ba}}{2 \hbar} \exp\!\big(\!-\! i (\omega \!-\! \omega_0) t \big) \: c_a \bigg) \exp\!\big(i (\omega \!-\! \omega_0) t \big) \\ &= \frac{V_{ab}}{2 \hbar} (\omega - \omega_0) \exp\!\big(i (\omega \!-\! \omega_0) t \big) \: c_b - \frac{|V_{ab}|^2}{(2 \hbar)^2} c_a \end{aligned}\]

In the first term, we recognize \(\dv*{c_a}{t}\), which we insert to arrive at an equation for \(c_a(t)\):

\[\begin{aligned} 0 = \dv[2]{c_a}{t} - i (\omega - \omega_0) \dv{c_a}{t} + \frac{|V_{ab}|^2}{(2 \hbar)^2} \: c_a \end{aligned}\]

To solve this, we make the ansatz \(c_a(t) = \exp\!(\lambda t)\), which, upon insertion, gives us:

\[\begin{aligned} 0 = \lambda^2 - i (\omega - \omega_0) \lambda + \frac{|V_{ab}|^2}{(2 \hbar)^2} \end{aligned}\]

This quadratic equation has two complex roots \(\lambda_1\) and \(\lambda_2\), which are found to be:

\[\begin{aligned} \lambda_1 = i \frac{\omega - \omega_0 + \tilde{\Omega}}{2} \qquad \quad \lambda_2 = i \frac{\omega - \omega_0 - \tilde{\Omega}}{2} \end{aligned}\]

Where we have defined the **generalized Rabi frequency** \(\tilde{\Omega}\) to be given by:

\[\begin{aligned} \boxed{ \tilde{\Omega} \equiv \sqrt{(\omega - \omega_0)^2 + \frac{|V_{ab}|^2}{\hbar^2}} } \end{aligned}\]

So that the general solution \(c_a(t)\) is as follows, where \(A\) and \(B\) are arbitrary constants, to be determined from initial conditions (and normalization):

\[\begin{aligned} \boxed{ c_a(t) = \Big( A \sin\!(\tilde{\Omega} t / 2) + B \cos\!(\tilde{\Omega} t / 2) \Big) \exp\!\big(i (\omega \!-\! \omega_0) t / 2 \big) } \end{aligned}\]

And then the corresponding \(c_b(t)\) can be found from the coupled equation we started at, or, if we only care about the probability density \(|c_a|^2\), we can use \(|c_b|^2 = 1 - |c_a|^2\). For example, if \(A = 0\) and \(B = 1\), we get the following probabilities

\[\begin{aligned} |c_a(t)|^2 &= \cos^2(\tilde{\Omega} t / 2) = \frac{1}{2} \Big( 1 + \cos\!(\tilde{\Omega} t) \Big) \\ |c_b(t)|^2 &= \sin^2(\tilde{\Omega} t / 2) = \frac{1}{2} \Big( 1 - \cos\!(\tilde{\Omega} t) \Big) \end{aligned}\]

Note that the period was halved by squaring. This periodic “flopping” of the particle between \(\ket{a}\) and \(\ket{b}\) is known as **Rabi oscillation**, **Rabi flopping** or the **Rabi cycle**. This is a more accurate treatment of the flopping found from first-order perturbation theory.

The name **generalized Rabi frequency** suggests that there is a non-general version. Indeed, the **Rabi frequency** \(\Omega\) is based on the special case of exact resonance \(\omega = \omega_0\):

\[\begin{aligned} \Omega \equiv \frac{V_{ba}}{\hbar} \end{aligned}\]

As an example, Rabi oscillation arises in the electric dipole approximation, where \(\hat{H}_1\) is:

\[\begin{aligned} \hat{H}_1(t) = - q \vec{r} \cdot \vec{E}_0 \cos\!(\omega t) \end{aligned}\]

After making the rotating wave approximation, the resulting Rabi frequency is given by:

\[\begin{aligned} \Omega = - \frac{\vec{d} \cdot \vec{E}_0}{\hbar} \end{aligned}\]

Where \(\vec{E}_0\) is the electric field amplitude, and \(\vec{d} \equiv q \matrixel{b}{\vec{r}}{a}\) is the transition dipole moment of the electron between orbitals \(\ket{a}\) and \(\ket{b}\). Apparently, some authors define \(\vec{d}\) with the opposite sign, thereby departing from its classical interpretation.

- D.J. Griffiths, D.F. Schroeter,
*Introduction to quantum mechanics*, 3rd edition, Cambridge.

© Marcus R.A. Newman, a.k.a. "Prefetch".
Available under CC BY-SA 4.0.