From 43c5b696aaf421dec7aee967002999d9145da35e Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Tue, 1 Feb 2022 11:23:35 +0100
Subject: Expand knowledge base

---
 content/know/concept/alfven-waves/index.pdc        | 249 +++++++++++++++++++++
 content/know/concept/bloch-sphere/index.pdc        |   1 +
 .../know/concept/einstein-coefficients/index.pdc   |   1 +
 content/know/concept/fermis-golden-rule/index.pdc  |   1 +
 content/know/concept/interaction-picture/index.pdc |   3 +-
 content/know/concept/ion-sound-wave/index.pdc      |   1 +
 content/know/concept/langmuir-waves/index.pdc      |   1 +
 .../know/concept/maxwell-bloch-equations/index.pdc |   5 +-
 .../know/concept/multi-photon-absorption/index.pdc |   3 +-
 content/know/concept/rabi-oscillation/index.pdc    |  15 +-
 .../concept/rotating-wave-approximation/index.pdc  | 126 +++++++++++
 11 files changed, 393 insertions(+), 13 deletions(-)
 create mode 100644 content/know/concept/alfven-waves/index.pdc
 create mode 100644 content/know/concept/rotating-wave-approximation/index.pdc

(limited to 'content/know/concept')

diff --git a/content/know/concept/alfven-waves/index.pdc b/content/know/concept/alfven-waves/index.pdc
new file mode 100644
index 0000000..ba87bee
--- /dev/null
+++ b/content/know/concept/alfven-waves/index.pdc
@@ -0,0 +1,249 @@
+---
+title: "Alfvén waves"
+firstLetter: "A"
+publishDate: 2022-01-31
+categories:
+- Physics
+- Plasma physics
+- Plasma waves
+
+date: 2022-01-30T19:26:33+01:00
+draft: false
+markup: pandoc
+---
+
+# Alfvén waves
+
+In the [magnetohydrodynamic](/know/concept/magnetohydrodynamics/) description of a plasma,
+we split the velocity $\vb{u}$, electric current $\vb{J}$,
+[magnetic field](/know/concept/magnetic-field/) $\vb{B}$
+and [electric field](/know/concept/electric-field/) $\vb{E}$ like so,
+into a constant uniform equilibrium (subscript $0$)
+and a small unknown perturbation (subscript $1$):
+
+$$\begin{aligned}
+    \vb{u}
+    = \vb{u}_0 + \vb{u}_1
+    \qquad
+    \vb{J}
+    = \vb{J}_0 + \vb{J}_1
+    \qquad
+    \vb{B}
+    = \vb{B}_0 + \vb{B}_1
+    \qquad
+    \vb{E}
+    = \vb{E}_0 + \vb{E}_1
+\end{aligned}$$
+
+Inserting this decomposition into the ideal form of the generalized Ohm's law
+and keeping only terms that are first-order in the perturbation, we get:
+
+$$\begin{aligned}
+    0
+    &= (\vb{E}_0 + \vb{E}_1) + (\vb{u}_0 + \vb{u}_1) \cross (\vb{B}_0 + \vb{B}_1)
+    \\
+    &= \vb{E}_1 + \vb{u}_1 \cross \vb{B}_0
+\end{aligned}$$
+
+We do this for the momentum equation too,
+assuming that $\vb{J}_0 \!=\! 0$ (to be justified later).
+Note that the temperature is set to zero, such that the pressure vanishes:
+
+$$\begin{aligned}
+    \rho \pdv{\vb{u}_1}{t}
+    = \vb{J}_1 \cross \vb{B}_0
+\end{aligned}$$
+
+Where $\rho$ is the uniform equilibrium density.
+We would like an equation for $\vb{J}_1$,
+which is provided by the magnetohydrodynamic form of Ampère's law:
+
+$$\begin{aligned}
+    \nabla \cross \vb{B}_1
+    = \mu_0 \vb{J}_1
+    \qquad \implies \quad
+    \vb{J}_1
+    = \frac{1}{\mu_0} \nabla \cross \vb{B}_1
+\end{aligned}$$
+
+Substituting this into the momentum equation,
+and differentiating with respect to $t$:
+
+$$\begin{aligned}
+    \rho \pdv[2]{\vb{u}_1}{t}
+    = \frac{1}{\mu_0} \bigg( \Big( \nabla \cross \pdv{\vb{B}_1}{t} \Big) \cross \vb{B}_0 \bigg)
+\end{aligned}$$
+
+For which we can use Faraday's law to rewrite $\pdv*{\vb{B}_1}{t}$,
+incorporating Ohm's law too:
+
+$$\begin{aligned}
+    \pdv{\vb{B}_1}{t}
+    = - \nabla \cross \vb{E}_1
+    = \nabla \cross (\vb{u}_1 \cross \vb{B}_0)
+\end{aligned}$$
+
+Inserting this into the momentum equation for $\vb{u}_1$
+thus yields its final form:
+
+$$\begin{aligned}
+    \rho \pdv[2]{\vb{u}_1}{t}
+    = \frac{1}{\mu_0} \bigg( \Big( \nabla \cross \big( \nabla \cross (\vb{u}_1 \cross \vb{B}_0) \big) \Big) \cross \vb{B}_0 \bigg)
+\end{aligned}$$
+
+Suppose the magnetic field is pointing in $z$-direction,
+i.e. $\vb{B}_0 = B_0 \vu{e}_z$.
+Then Faraday's law justifies our earlier assumption that $\vb{J}_0 = 0$,
+and the equation can be written as:
+
+$$\begin{aligned}
+    \pdv[2]{\vb{u}_1}{t}
+    = v_A^2 \bigg( \Big( \nabla \cross \big( \nabla \cross (\vb{u}_1 \cross \vu{e}_z) \big) \Big) \cross \vu{e}_z \bigg)
+\end{aligned}$$
+
+Where we have defined the so-called **Alfvén velocity** $v_A$ to be given by:
+
+$$\begin{aligned}
+    \boxed{
+        v_A
+        \equiv \sqrt{\frac{B_0^2}{\mu_0 \rho}}
+    }
+\end{aligned}$$
+
+Now, consider the following plane-wave ansatz for $\vb{u}_1$,
+with wavevector $\vb{k}$ and frequency $\omega$:
+
+$$\begin{aligned}
+    \vb{u}_1(\vb{r}, t)
+    &= \vb{u}_1 \exp\!(i \vb{k} \cdot \vb{r} - i \omega t)
+\end{aligned}$$
+
+Inserting this into the above differential equation for $\vb{u}_1$ leads to:
+
+$$\begin{aligned}
+    \omega^2 \vb{u}_1
+    = v_A^2 \bigg( \Big( \vb{k} \cross \big( \vb{k} \cross (\vb{u}_1 \cross \vu{e}_z) \big) \Big) \cross \vu{e}_z \bigg)
+\end{aligned}$$
+
+To evaluate this, we rotate our coordinate system around the $z$-axis
+such that $\vb{k} = (0, k_\perp, k_\parallel)$,
+i.e. the wavevector's $x$-component is zero.
+Calculating the cross products:
+
+$$\begin{aligned}
+    \omega^2 \vb{u}_1
+    &= v_A^2 \bigg( \Big( \begin{bmatrix} 0 \\ k_\perp \\ k_\parallel \end{bmatrix}
+    \cross \big( \begin{bmatrix} 0 \\ k_\perp \\ k_\parallel \end{bmatrix}
+    \cross ( \begin{bmatrix} u_{1x} \\ u_{1y} \\ u_{1z} \end{bmatrix}
+    \cross \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} ) \big) \Big)
+    \cross \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \bigg)
+    \\
+    &= v_A^2 \bigg( \Big( \begin{bmatrix} 0 \\ k_\perp \\ k_\parallel \end{bmatrix}
+    \cross \big( \begin{bmatrix} 0 \\ k_\perp \\ k_\parallel \end{bmatrix}
+    \cross \begin{bmatrix} u_{1y} \\ -u_{1x} \\ 0 \end{bmatrix} \big) \Big)
+    \cross \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \bigg)
+    \\
+    &= v_A^2 \bigg( \Big( \begin{bmatrix} 0 \\ k_\perp \\ k_\parallel \end{bmatrix}
+    \cross \begin{bmatrix} k_\parallel u_{1x} \\ k_\parallel u_{1y} \\ -k_\perp u_{1y} \end{bmatrix} \Big)
+    \cross \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \bigg)
+    \\
+    &= v_A^2 \bigg( \begin{bmatrix} -(k_\perp^2 \!+ k_\parallel^2) u_{1y} \\ k_\parallel^2 u_{1x} \\ -k_\perp k_\parallel u_{1x} \end{bmatrix}
+    \cross \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \bigg)
+    \\
+    &= v_A^2 \begin{bmatrix} k_\parallel^2 u_{1x} \\ (k_\perp^2 \!+ k_\parallel^2) u_{1y} \\ 0 \end{bmatrix}
+\end{aligned}$$
+
+We rewrite this equation in matrix form,
+using that $k_\perp^2 \!+ k_\parallel^2 = k^2 \equiv |\vb{k}|^2$:
+
+$$\begin{aligned}
+    \begin{bmatrix}
+        \omega^2 - v_A^2 k_\parallel^2 & 0 & 0 \\
+        0 & \omega^2 - v_A^2 k^2 & 0 \\
+        0 & 0 & \omega^2
+    \end{bmatrix}
+    \vb{u}_1
+    = 0
+\end{aligned}$$
+
+This has the form of an eigenvalue problem for $\omega^2$,
+meaning we must find non-trivial solutions,
+where we cannot simply choose the components of $\vb{u}_1$ to satisfy the equation.
+To achieve this, we demand that the matrix' determinant is zero:
+
+$$\begin{aligned}
+    \big(\omega^2 - v_A^2 k_\parallel^2\big) \: \big(\omega^2 - v_A^2 k^2\big) \: \omega^2
+    = 0
+\end{aligned}$$
+
+This equation has three solutions for $\omega^2$,
+one for each of its three factors being zero.
+The simplest case $\omega^2 = 0$ is of no interest to us,
+because we are looking for waves.
+
+The first interesting case is $\omega^2 = v_A^2 k_\parallel^2$,
+yielding the following dispersion relation:
+
+$$\begin{aligned}
+    \boxed{
+        \omega
+        = \pm v_A k_\parallel
+    }
+\end{aligned}$$
+
+The resulting waves are called **shear Alfvén waves**.
+From the eigenvalue problem, we see that in this case
+$\vb{u}_1 = (u_{1x}, 0, 0)$, meaning $\vb{u}_1 \cdot \vb{k} = 0$:
+these waves are **transverse**.
+The phase velocity $v_p$ and group velocity $v_g$ are as follows,
+where $\theta$ is the angle between $\vb{k}$ and $\vb{B}_0$:
+
+$$\begin{aligned}
+    v_p
+    = \frac{|\omega|}{k}
+    = v_A \frac{k_\parallel}{k}
+    = v_A \cos\!(\theta)
+    \qquad \qquad
+    v_g
+    = \pdv{|\omega|}{k}
+    = v_A
+\end{aligned}$$
+
+The other interesting case is $\omega^2 = v_A^2 k^2$,
+which leads to so-called **compressional Alfvén waves**,
+with the simple dispersion relation:
+
+$$\begin{aligned}
+    \boxed{
+        \omega
+        = \pm v_A k
+    }
+\end{aligned}$$
+
+Looking at the eigenvalue problem reveals that $\vb{u}_1 = (0, u_{1y}, 0)$,
+meaning $\vb{u}_1 \cdot \vb{k} = u_{1y} k_\perp$,
+so these waves are not necessarily transverse, nor longitudinal (since $k_\parallel$ is free).
+The phase velocity $v_p$ and group velocity $v_g$ are given by:
+
+$$\begin{aligned}
+    v_p
+    = \frac{|\omega|}{k}
+    = v_A
+    \qquad \qquad
+    v_g
+    = \pdv{|\omega|}{k}
+    = v_A
+\end{aligned}$$
+
+The mechanism behind both of these oscillations is magnetic tension:
+the waves are "ripples" in the field lines,
+which get straightened out by Faraday's law,
+but the ions' inertia causes them to overshoot and form ripples again.
+
+
+
+## References
+1.  M. Salewski, A.H. Nielsen,
+    *Plasma physics: lecture notes*,
+    2021, unpublished.
+
diff --git a/content/know/concept/bloch-sphere/index.pdc b/content/know/concept/bloch-sphere/index.pdc
index f0c48a9..27abb54 100644
--- a/content/know/concept/bloch-sphere/index.pdc
+++ b/content/know/concept/bloch-sphere/index.pdc
@@ -5,6 +5,7 @@ publishDate: 2021-03-09
 categories:
 - Quantum mechanics
 - Quantum information
+- Two-level system
 
 date: 2021-03-09T15:35:33+01:00
 draft: false
diff --git a/content/know/concept/einstein-coefficients/index.pdc b/content/know/concept/einstein-coefficients/index.pdc
index f0f0f96..b56af77 100644
--- a/content/know/concept/einstein-coefficients/index.pdc
+++ b/content/know/concept/einstein-coefficients/index.pdc
@@ -7,6 +7,7 @@ categories:
 - Optics
 - Electromagnetism
 - Quantum mechanics
+- Two-level system
 
 date: 2021-07-11T18:22:14+02:00
 draft: false
diff --git a/content/know/concept/fermis-golden-rule/index.pdc b/content/know/concept/fermis-golden-rule/index.pdc
index 5ed273e..6fcc482 100644
--- a/content/know/concept/fermis-golden-rule/index.pdc
+++ b/content/know/concept/fermis-golden-rule/index.pdc
@@ -5,6 +5,7 @@ publishDate: 2021-07-10
 categories:
 - Physics
 - Quantum mechanics
+- Two-level system
 - Optics
 
 date: 2021-07-03T14:41:11+02:00
diff --git a/content/know/concept/interaction-picture/index.pdc b/content/know/concept/interaction-picture/index.pdc
index 45950ff..89aff58 100644
--- a/content/know/concept/interaction-picture/index.pdc
+++ b/content/know/concept/interaction-picture/index.pdc
@@ -197,7 +197,8 @@ $$\begin{aligned}
     \mathcal{T} \bigg\{ \bigg( \int_{t_0}^{t} \hat{H}_{1,I}(t') \dd{t'} \bigg)^n \bigg\}
 \end{aligned}$$
 
-Here, we recognize the Taylor expansion of $\exp$,
+This construction is occasionally called the **Dyson series**.
+We recognize the well-known Taylor expansion of $\exp\!(x)$,
 leading us to a final expression for $\hat{K}_I$:
 
 $$\begin{aligned}
diff --git a/content/know/concept/ion-sound-wave/index.pdc b/content/know/concept/ion-sound-wave/index.pdc
index 657627d..38ab394 100644
--- a/content/know/concept/ion-sound-wave/index.pdc
+++ b/content/know/concept/ion-sound-wave/index.pdc
@@ -5,6 +5,7 @@ publishDate: 2021-10-31
 categories:
 - Physics
 - Plasma physics
+- Plasma waves
 - Perturbation
 
 date: 2021-10-31T09:38:14+01:00
diff --git a/content/know/concept/langmuir-waves/index.pdc b/content/know/concept/langmuir-waves/index.pdc
index c5cd23e..caf2294 100644
--- a/content/know/concept/langmuir-waves/index.pdc
+++ b/content/know/concept/langmuir-waves/index.pdc
@@ -5,6 +5,7 @@ publishDate: 2021-10-30
 categories:
 - Physics
 - Plasma physics
+- Plasma waves
 - Perturbation
 
 date: 2021-10-15T20:31:46+02:00
diff --git a/content/know/concept/maxwell-bloch-equations/index.pdc b/content/know/concept/maxwell-bloch-equations/index.pdc
index 3f090a2..e3a3680 100644
--- a/content/know/concept/maxwell-bloch-equations/index.pdc
+++ b/content/know/concept/maxwell-bloch-equations/index.pdc
@@ -5,6 +5,7 @@ publishDate: 2021-10-02
 categories:
 - Physics
 - Quantum mechanics
+- Two-level system
 - Electromagnetism
 
 date: 2021-09-09T21:17:52+02:00
@@ -78,8 +79,8 @@ $$\begin{aligned}
 \end{aligned}$$
 
 With these, the equations for $c_g$ and $c_e$ can be rewritten as shown below.
-Note that $\vb{E}^{-}$ and $\vb{E}^{+}$ include the driving plane wave,
-and the *rotating wave approximation* is still made:
+Note that $\vb{E}^{-}$ and $\vb{E}^{+}$ include the driving plane wave, and the
+[rotating wave approximation](/know/concept/rotating-wave-approximation/) is still made:
 
 $$\begin{aligned}
     \dv{c_g}{t}
diff --git a/content/know/concept/multi-photon-absorption/index.pdc b/content/know/concept/multi-photon-absorption/index.pdc
index cfdd234..b208cfe 100644
--- a/content/know/concept/multi-photon-absorption/index.pdc
+++ b/content/know/concept/multi-photon-absorption/index.pdc
@@ -29,7 +29,8 @@ $$\begin{aligned}
 
 Where $\vb{E}$ is the [electric field](/know/concept/electric-field/) amplitude,
 and $\vu{p} \equiv q \vu{x}$ is the transition dipole moment operator.
-Here, we have made the *rotating wave approximation*
+Here, we have made the
+[rotating wave approximation](/know/concept/rotating-wave-approximation/)
 to neglect the $e^{i \omega t}$ term,
 because it turns out to be irrelevant in this discussion.
 
diff --git a/content/know/concept/rabi-oscillation/index.pdc b/content/know/concept/rabi-oscillation/index.pdc
index a488de0..c6a1227 100644
--- a/content/know/concept/rabi-oscillation/index.pdc
+++ b/content/know/concept/rabi-oscillation/index.pdc
@@ -5,6 +5,7 @@ publishDate: 2021-09-22
 categories:
 - Physics
 - Quantum mechanics
+- Two-level system
 - Optics
 
 date: 2021-09-18T00:41:43+02:00
@@ -74,17 +75,13 @@ $$\begin{aligned}
     &= - i \frac{V_{ab}}{2 \hbar} \Big( \exp\!\big(i (\omega \!+\! \omega_0) t\big) + \exp\!\big(\!-\! i (\omega \!-\! \omega_0) t\big) \Big) \: c_a
 \end{aligned}$$
 
-Here, we make the *rotating wave approximation*:
+Here, we make the
+[rotating wave approximation](/know/concept/rotating-wave-approximation/):
 assuming we are close to resonance $\omega \approx \omega_0$,
-we decide that $\exp\!(i (\omega \!+\! \omega_0) t)$
-oscillates so much faster than $\exp\!(i (\omega \!-\! \omega_0) t)$,
-that its effect turns out negligible
+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.
-
-In other words, over this reasonably-sized time interval,
-$\exp\!(i (\omega \!+\! \omega_0) t)$ averages to zero,
-while $\exp\!(i (\omega \!-\! \omega_0) t)$ does not.
-Dropping the respective terms thus leaves us with:
+Dropping those terms leaves us with:
 
 $$\begin{aligned}
     \boxed{
diff --git a/content/know/concept/rotating-wave-approximation/index.pdc b/content/know/concept/rotating-wave-approximation/index.pdc
new file mode 100644
index 0000000..874dc96
--- /dev/null
+++ b/content/know/concept/rotating-wave-approximation/index.pdc
@@ -0,0 +1,126 @@
+---
+title: "Rotating wave approximation"
+firstLetter: "R"
+publishDate: 2022-02-01
+categories:
+- Physics
+- Quantum mechanics
+- Two-level system
+- Optics
+
+date: 2022-01-31T19:29:43+01:00
+draft: false
+markup: pandoc
+---
+
+# Rotating wave approximation
+
+Consider the following periodic perturbation $\hat{H}_1$ to a quantum system,
+which represents e.g. an [electromagnetic wave](/know/concept/electromagnetic-wave-equation/)
+in the [electric dipole approximation](/know/concept/electric-dipole-approximation/):
+
+$$\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.
+
+
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