Categories: Optics, Physics, Quantum mechanics, Two-level system.

# Rabi oscillation

In quantum mechanics, we know from the amplitude rate equations 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 unperturbed basis states $\ket{n} e^{-i E_n t / \hbar}$:

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

Where $\omega_{mn} \equiv (E_m \!-\! E_n) / \hbar$ for energies $E_m$ and $E_n$. Consider a two-level system $\{\ket{a}, \ket{b}\}$ with $E_a < E_b$, in which case the above equation can be written out as:

\begin{aligned} \dv{c_a}{t} &= - \frac{i}{\hbar} \matrixel{a}{\hat{H}_1}{b} e^{-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} e^{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} e^{-i \omega_0 t} \: c_b \\ \dv{c_b}{t} &= - \frac{i}{\hbar} \matrixel{b}{\hat{H}_1}{a} e^{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( e^{i \omega t} + e^{-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( e^{i (\omega - \omega_0) t} + e^{-i (\omega + \omega_0) t} \Big) \: c_b \\ \dv{c_b}{t} &= - i \frac{V_{ab}}{2 \hbar} \Big( e^{i (\omega + \omega_0) t} + e^{-i (\omega - \omega_0) t} \Big) \: c_a \end{aligned}

Here, we make the rotating wave approximation: assuming we are close to resonance $\omega \approx \omega_0$, we argue that $e^{i (\omega + \omega_0) t}$ oscillates so fast that its effect is negligible when the system is observed over a reasonable time interval. Dropping those terms leaves us with:

\begin{aligned} \boxed{ \begin{aligned} \dv{c_a}{t} &= - i \frac{V_{ab}}{2 \hbar} \: e^{i (\omega - \omega_0) t} \: c_b \\ \dv{c_b}{t} &= - i \frac{V_{ba}}{2 \hbar} \: e^{-i (\omega - \omega_0) t} \: 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 $\idv{c_b}{t}$ for the second equation:

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

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

\begin{aligned} 0 = \dvn{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 $c_a(t) = e^{\lambda t}$, 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) e^{i (\omega - \omega_0) t / 2} } \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 time-dependent 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}

Some authors use $|V_{ba}|$ instead, but not doing that lets us use $\Omega$ as a nice abbreviation. 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.

## References

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