---
title: "Rabi oscillation"
date: 2021-09-22
categories:
- Physics
- Quantum mechanics
- Two-level system
- Optics
layout: "concept"
---

In quantum mechanics, from the derivation of
[time-dependent perturbation theory](/know/concept/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](/know/concept/selection-rules/)
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](/know/concept/rotating-wave-approximation/):
assuming we are close to resonance $\omega \approx \omega_0$,
we argue that $\exp(i (\omega \!+\! \omega_0) t)$
oscillates so fast that its effect is negligible
when the system is observed over a reasonable time interval.
Dropping those terms leaves us with:

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

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

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

In the first term, we recognize $\idv{c_a}{t}$,
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 $c_a(t) = \exp(\lambda t)$,
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](/know/concept/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](/know/concept/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.