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---
title: "Rotating wave approximation"
sort_title: "Rotating wave approximation"
date: 2022-02-01
categories:
- Physics
- Quantum mechanics
- Two-level system
- Optics
layout: "concept"
---
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 [amplitude rate equations](/know/concept/amplitude-rate-equations/),
we know that the general superposition 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.
|
