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+---
+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.
+
+