path: root/source/know/concept/rabi-oscillation/index.md

---
title: "Rabi oscillation"
sort_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.