---
title: "Rotating wave approximation"
firstLetter: "R"
publishDate: 2022-02-01
categories:
- Physics
- Quantum mechanics
- Two-level system
- Optics

date: 2022-01-31T19:29:43+01:00
draft: false
markup: pandoc
---

# Rotating wave approximation

Consider the following periodic perturbation $\hat{H}_1$ to a quantum system,
which represents e.g. an [electromagnetic wave](/know/concept/electromagnetic-wave-equation/)
in the [electric dipole approximation](/know/concept/electric-dipole-approximation/):

$$\begin{aligned}
    \hat{H}_1(t)
    = \hat{V} \cos\!(\omega t)
    = \frac{\hat{V}}{2} \Big( e^{i \omega t} + e^{-i \omega t} \Big)
\end{aligned}$$

Where $\hat{V}$ is some operator, and we assume that $\omega$
is fairly close to a resonance frequency $\omega_0$
of the system that is getting perturbed by $\hat{H}_1$.

As an example, consider a two-level system
consisting of states $\ket{g}$ and $\ket{e}$,
with a resonance frequency $\omega_0 = (E_e \!-\! E_g) / \hbar$.
From the derivation of
[time-dependent perturbation theory](/know/concept/time-dependent-perturbation-theory/),
we know that the state $\ket{\Psi} = c_g \ket{g} + c_e \ket{e}$ evolves as:

$$\begin{aligned}
    i \hbar \dv{c_g}{t}
    &= \matrixel{g}{\hat{H}_1(t)}{g} \: c_g(t) + \matrixel{g}{\hat{H}_1(t)}{e} \: c_e(t) \: e^{- i \omega_0 t}
    \\
    i \hbar \dv{c_e}{t}
    &= \matrixel{e}{\hat{H}_1(t)}{g} \: c_g(t) \: e^{i \omega_0 t} + \matrixel{e}{\hat{H}_1(t)}{e} \: c_e(t)
\end{aligned}$$

Typically, $\hat{V}$ has odd spatial parity, in which case
[Laporte's selection rule](/know/concept/selection-rules/)
reduces this to:

$$\begin{aligned}
    \dv{c_g}{t}
    &= \frac{1}{i \hbar} \matrixel{g}{\hat{H}_1}{e} \: c_e \: e^{- i \omega_0 t}
    \\
    \dv{c_e}{t}
    &= \frac{1}{i \hbar} \matrixel{e}{\hat{H}_1}{g} \: c_g \: e^{i \omega_0 t}
\end{aligned}$$

We now insert the general $\hat{H}_1$ defined above,
and define $V_{eg} \equiv \matrixel{e}{\hat{V}}{g}$ to get:

$$\begin{aligned}
    \dv{c_g}{t}
    &= \frac{V_{eg}^*}{i 2 \hbar}
    \Big( e^{i (\omega - \omega_0) t} + e^{- i (\omega + \omega_0) t} \Big) \: c_e
    \\
    \dv{c_e}{t}
    &= \frac{V_{eg}}{i 2 \hbar}
    \Big( e^{i (\omega + \omega_0) t} + e^{- i (\omega - \omega_0) t} \Big) \: c_g
\end{aligned}$$

At last, here we make the **rotating wave approximation**:
since $\omega$ is assumed to be close to $\omega_0$,
we argue that $\omega \!+\! \omega_0$ is so much larger than $\omega \!-\! \omega_0$
that those oscillations turn out negligible
if the system is observed over a reasonable time interval.

Specifically, since both exponentials have the same weight,
the fast ($\omega \!+\! \omega_0$) oscillations
have a tiny amplitude compared to the slow ($\omega \!-\! \omega_0$) ones.
Furthermore, since they average out to zero over most realistic time intervals,
the fast terms can be dropped, leaving:

$$\begin{aligned}
    \boxed{
        \begin{aligned}
            e^{i (\omega - \omega_0) t} + e^{- i (\omega + \omega_0) t}
            &\approx e^{i (\omega - \omega_0) t}
            \\
            e^{i (\omega + \omega_0) t} + e^{- i (\omega - \omega_0) t}
            &\approx e^{- i (\omega - \omega_0) t}
        \end{aligned}
    }
\end{aligned}$$

Such that our example set of equations can be approximated as shown below,
and its analysis can continue;
see [Rabi oscillation](/know/concept/rabi-oscillation/) for more:

$$\begin{aligned}
    \dv{c_g}{t}
    &= \frac{V_{eg}^*}{i 2 \hbar} c_e \: e^{i (\omega - \omega_0) t}
    \\
    \dv{c_e}{t}
    &= \frac{V_{eg}}{i 2 \hbar} c_g \: e^{- i (\omega - \omega_0) t}
\end{aligned}$$

This approximation's name is a bit confusing:
the idea is that going from the Schrödinger to
the [interaction picture](/know/concept/interaction-picture/)
has the effect of removing the exponentials of $\omega_0$ from the above equations,
i.e. multiplying them by $e^{i \omega_0 t}$ and $e^{- i \omega_0 t}$
respectively, which can be regarded as a rotation.

Relative to this rotation, when we split the wave $\cos\!(\omega t)$
into two exponentials, one co-rotates, and the other counter-rotates.
We keep only the co-rotating waves, hence the name.

The rotating wave approximation is usually used in the context
of the two-level quantum system for light-matter interactions,
as in the above example.
However, it is not specific to that case,
and it more generally refers to any approximation
where fast-oscillating terms are neglected.