Expand knowledge base, remove comments

26 files changed, 370 insertions, 50 deletions

diff --git a/content/know/category/nonlinear-dynamics.md b/content/know/category/nonlinear-optics.md
index e0305f4..9432bee 100644
--- a/content/know/category/nonlinear-dynamics.md
+++ b/content/know/category/nonlinear-optics.md
@@ -1,5 +1,5 @@
 ---
-title: "Nonlinear dynamics"
+title: "Nonlinear optics"
 firstLetter: "N"
 date: 2021-02-26T20:30:17+01:00
 draft: false
diff --git a/content/know/concept/cauchy-strain-tensor/index.pdc b/content/know/concept/cauchy-strain-tensor/index.pdc
index cb48377..bfbe283 100644
--- a/content/know/concept/cauchy-strain-tensor/index.pdc
+++ b/content/know/concept/cauchy-strain-tensor/index.pdc
@@ -246,7 +246,6 @@ $$\begin{aligned}
     \boxed{
         \delta(\dd{\va{l}})
         = (\dd{\va{l}} \cdot \nabla) \va{u}
-        %= (\nabla \vec{u})^\top \cdot \dd{\va{l}}
     }
 \end{aligned}$$
 
diff --git a/content/know/concept/drude-model/index.pdc b/content/know/concept/drude-model/index.pdc
index a738dff..703ddbb 100644
--- a/content/know/concept/drude-model/index.pdc
+++ b/content/know/concept/drude-model/index.pdc
@@ -74,7 +74,6 @@ which depends on $\omega$:
 $$\begin{aligned}
     \boxed{
         \varepsilon_r(\omega)
-        %= 1 - \frac{N q^2}{\varepsilon_0 m} \frac{1}{\omega^2 + i \gamma \omega}
         = 1 - \frac{\omega_p^2}{\omega^2 + i \gamma \omega}
     }
 \end{aligned}$$
@@ -190,7 +189,6 @@ The dielectric function $\varepsilon_r(\omega)$ is therefore given by:
 $$\begin{aligned}
     \boxed{
         \varepsilon_r(\omega)
-        %= \varepsilon_{\mathrm{int}}(\omega) - \frac{N q^2}{\varepsilon_0 m^*} \frac{1}{\omega^2 + i \gamma \omega}
         = \varepsilon_{\mathrm{int}} \Big( 1 - \frac{\omega_p^2}{\omega^2 + i \gamma \omega} \Big)
     }
 \end{aligned}$$
diff --git a/content/know/concept/electric-dipole-approximation/index.pdc b/content/know/concept/electric-dipole-approximation/index.pdc
index 67c73ee..34427bf 100644
--- a/content/know/concept/electric-dipole-approximation/index.pdc
+++ b/content/know/concept/electric-dipole-approximation/index.pdc
@@ -6,6 +6,8 @@ categories:
 - Physics
 - Quantum mechanics
 - Optics
+- Electromagnetism
+- Perturbation
 
 date: 2021-09-14T13:11:54+02:00
 draft: false
diff --git a/content/know/concept/electromagnetic-wave-equation/index.pdc b/content/know/concept/electromagnetic-wave-equation/index.pdc
index 84946bb..59e0125 100644
--- a/content/know/concept/electromagnetic-wave-equation/index.pdc
+++ b/content/know/concept/electromagnetic-wave-equation/index.pdc
@@ -58,9 +58,7 @@ and substitute $\nabla \cross \vb{E}$ according to Faraday's law:
 
 $$\begin{aligned}
     \nabla \cross (\nabla \cross \vb{B})
-    %= \nabla \cross \Big( \mu_0 \mu_r \varepsilon_0 \varepsilon_r \pdv{\vb{E}}{t} \Big)
     = \mu_0 \mu_r \varepsilon_0 \varepsilon_r \pdv{t} (\nabla \cross \vb{E})
-    %= \mu_0 \mu_r \varepsilon_0 \varepsilon_r \pdv{t} \Big( \!-\! \pdv{\vb{B}}{t} \Big)
     = - \mu_0 \mu_r \varepsilon_0 \varepsilon_r \pdv[2]{\vb{B}}{t}
 \end{aligned}$$
 
@@ -79,9 +77,7 @@ taking the curl of Faraday's law yields:
 
 $$\begin{aligned}
     \nabla \cross (\nabla \cross \vb{E})
-    %= - \nabla \cross \pdv{\vb{B}}{t}
     = - \pdv{t} (\nabla \cross \vb{B})
-    %= - \pdv{t} \Big( \mu_0 \mu_r \varepsilon_0 \varepsilon_r \pdv{\vb{E}}{t} \Big)
     = - \mu_0 \mu_r \varepsilon_0 \varepsilon_r \pdv[2]{\vb{E}}{t}
 \end{aligned}$$
 
diff --git a/content/know/concept/ion-sound-wave/index.pdc b/content/know/concept/ion-sound-wave/index.pdc
index 5cba1d0..657627d 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
+- Perturbation
 
 date: 2021-10-31T09:38:14+01:00
 draft: false
diff --git a/content/know/concept/jellium/index.pdc b/content/know/concept/jellium/index.pdc
index 15c3308..9743514 100644
--- a/content/know/concept/jellium/index.pdc
+++ b/content/know/concept/jellium/index.pdc
@@ -5,6 +5,7 @@ publishDate: 2021-11-23
 categories:
 - Physics
 - Quantum mechanics
+- Perturbation
 
 date: 2021-10-15T20:31:12+02:00
 draft: false
@@ -363,17 +364,11 @@ $$\begin{aligned}
     &= \frac{- e^2 V}{4 \pi^4 \varepsilon_0} \int_0^{2 k_F}
     \bigg[ \frac{k_F^3}{3} x + \frac{|\vb{q}|^3}{48 \xi^2} \bigg]_{|\vb{q}| / (2 k_F)}^1 \dd{|\vb{q}|}
     \\
-    %&= \frac{- e^2 V}{4 \pi^4 \varepsilon_0} \int_0^{2 k_F}
-    %\bigg( \frac{k_F^3}{3} + \frac{|\vb{q}|^3}{48} - \frac{k_F^3}{3} \frac{|\vb{q}|}{2 k_F} - \frac{4 k_F^2 |\vb{q}|^3}{48 |\vb{q}|^2} \bigg)
-    %\dd{|\vb{q}|}
-    %\\
     &= \frac{- e^2 V}{4 \pi^4 \varepsilon_0} \int_0^{2 k_F}
     \bigg( \frac{k_F^3}{3} + \frac{|\vb{q}|^3}{48} - \frac{k_F^2 |\vb{q}|}{4} \bigg) \dd{|\vb{q}|}
     \\
     &= \frac{- e^2 V}{4 \pi^4 \varepsilon_0} \bigg[ \frac{k_F^3 |\vb{q}|}{3} + \frac{|\vb{q}|^4}{192} - \frac{k_F^2 |\vb{q}|^2}{8} \bigg]_0^{2 k_F}
     \\
-    %&= \frac{- e^2 V}{4 \pi^4 \varepsilon_0} \bigg( \frac{k_F^3}{3} 2 k_F + \frac{16 k_F^4}{192} - \frac{k_F^2 4 k_F^2}{8} \bigg)
-    %\\
     &= \frac{- e^2 V}{16 \pi^4 \varepsilon_0} k_F^4
     = \frac{- e^2 N}{16 \pi^4 \varepsilon_0 n} k_F^4
     = -\frac{3 e^2 N}{16 \pi^2 \varepsilon_0} k_F
diff --git a/content/know/concept/kramers-kronig-relations/index.pdc b/content/know/concept/kramers-kronig-relations/index.pdc
index 01e5a3a..7a9093d 100644
--- a/content/know/concept/kramers-kronig-relations/index.pdc
+++ b/content/know/concept/kramers-kronig-relations/index.pdc
@@ -43,7 +43,6 @@ where $A$, $B$ and $s$ are constants from the FT definition:
 
 $$\begin{aligned}
     \tilde{\chi}(\omega)
-    %= \hat{\mathcal{F}}\{\chi_c(t) \: \Theta(t)\}
     = (\tilde{\chi} * \tilde{\Theta})(\omega)
     = B \int_{-\infty}^\infty \tilde{\chi}(\omega') \: \tilde{\Theta}(\omega - \omega') \dd{\omega'}
 \end{aligned}$$
diff --git a/content/know/concept/kubo-formula/index.pdc b/content/know/concept/kubo-formula/index.pdc
index f5430da..1df4780 100644
--- a/content/know/concept/kubo-formula/index.pdc
+++ b/content/know/concept/kubo-formula/index.pdc
@@ -5,6 +5,7 @@ publishDate: 2021-09-23
 categories:
 - Physics
 - Quantum mechanics
+- Perturbation
 
 date: 2021-09-23T16:21:51+02:00
 draft: false
@@ -54,9 +55,6 @@ $$\begin{aligned}
     &\approx \bigg( 1 + \frac{i}{\hbar} \int_{t_0}^t \hat{H}_{1,I}(t') \dd{t'} \bigg) \hat{A}_I(t)
     \bigg( 1 - \frac{i}{\hbar} \int_{t_0}^t \hat{H}_{1,I}(t') \dd{t'} \bigg)
     \\
-    %&= \bigg( \hat{A}_I + \frac{i}{\hbar} \int_{t_0}^t \hat{H}_{1,I}(t') \hat{A}_I \dd{t'} \bigg)
-    %\bigg( 1 - \frac{i}{\hbar} \int_{t_0}^t \hat{H}_{1,I}(t') \dd{t'} \bigg)
-    %\\
     &\approx \hat{A}_I(t)
     - \frac{i}{\hbar} \int_{t_0}^t \hat{A}_I(t) \hat{H}_{1,I}(t') \dd{t'}
     + \frac{i}{\hbar} \int_{t_0}^t \hat{H}_{1,I}(t') \hat{A}_I(t) \dd{t'}
@@ -102,7 +100,6 @@ describing the response of $\expval*{\hat{A}}$ to first order in $\hat{H}_1$:
 $$\begin{aligned}
     \boxed{
         \delta\expval*{\hat{A}}(t)
-        %= - \frac{i}{\hbar} \int_{t_0}^\infty \Theta(t \!-\! t') \expval{\comm{\hat{A}_I(t)}{\hat{H}_{1,I}(t')}}_0 \dd{t'}
         = \int_{t_0}^\infty C^R_{A H_1}(t, t') \dd{t'}
     }
 \end{aligned}$$
diff --git a/content/know/concept/langmuir-waves/index.pdc b/content/know/concept/langmuir-waves/index.pdc
index 3c3f1d2..c5cd23e 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
+- Perturbation
 
 date: 2021-10-15T20:31:46+02:00
 draft: false
diff --git a/content/know/concept/lindhard-function/index.pdc b/content/know/concept/lindhard-function/index.pdc
index aedf0f5..8af7416 100644
--- a/content/know/concept/lindhard-function/index.pdc
+++ b/content/know/concept/lindhard-function/index.pdc
@@ -35,7 +35,7 @@ Now consider a harmonic $\hat{H}_1$:
 
 $$\begin{aligned}
     \hat{H}_{1,S}(t)
-    = e^{i (\omega + i \eta) t} \int_{-\infty}^\infty U(\vb{r}) \: \hat{n}(\vb{r}) \dd{\vb{r}}
+    = e^{i (\omega + i \eta) t} \int_{-\infty}^\infty U(\vb{r}) \: \hat{n}_S(\vb{r}) \dd{\vb{r}}
 \end{aligned}$$
 
 Where $S$ is the Schrödinger picture,
diff --git a/content/know/concept/lubrication-theory/index.pdc b/content/know/concept/lubrication-theory/index.pdc
index 7641440..a883023 100644
--- a/content/know/concept/lubrication-theory/index.pdc
+++ b/content/know/concept/lubrication-theory/index.pdc
@@ -124,8 +124,6 @@ Then we insert this into our earlier expression for $v_x$, yielding:
 $$\begin{aligned}
     v_x
     &= 3 y (y - h) \Big( \frac{U}{h^2} - \frac{2 Q}{h^3} \Big) - \frac{U h}{h^2} y + \frac{U h^2}{h^2}
-    %\\
-    %&= U \frac{3 y (y - h) - h y + h^2}{h^2} - Q \frac{6 y (y - h)}{h^3}
 \end{aligned}$$
 
 Which, after some rearranging, can be written in the following form:
diff --git a/content/know/concept/modulational-instability/index.pdc b/content/know/concept/modulational-instability/index.pdc
index 6856941..993dec9 100644
--- a/content/know/concept/modulational-instability/index.pdc
+++ b/content/know/concept/modulational-instability/index.pdc
@@ -7,7 +7,7 @@ categories:
 - Fiber optics
 - Optics
 - Perturbation
-- Nonlinear dynamics
+- Nonlinear optics
 
 date: 2021-02-26T20:36:22+01:00
 draft: false
diff --git a/content/know/concept/multi-photon-absorption/index.pdc b/content/know/concept/multi-photon-absorption/index.pdc
new file mode 100644
index 0000000..cfdd234
--- /dev/null
+++ b/content/know/concept/multi-photon-absorption/index.pdc
@@ -0,0 +1,357 @@
+---
+title: "Multi-photon absorption"
+firstLetter: "M"
+publishDate: 2022-01-30
+categories:
+- Physics
+- Quantum mechanics
+- Nonlinear optics
+- Perturbation
+
+date: 2022-01-08T14:22:15+01:00
+draft: false
+markup: pandoc
+---
+
+# Multi-photon absorption
+
+Consider a quantum system where there are many eigenstates $\ket{n}$,
+e.g. atomic orbitals, for an electron to occupy.
+Suppose an [electromagnetic wave](/know/concept/electromagnetic-wave-equation/)
+passes by, such that its Hamiltonian gets perturbed by $\hat{H}_1$, given in the
+[electric dipole approximation](/know/concept/electric-dipole-approximation/) by:
+
+$$\begin{aligned}
+    \hat{H}_1(t)
+    = -\vu{p} \cdot \vb{E} \cos\!(\omega t)
+    \approx -\vu{p} \cdot \vb{E} e^{-i \omega t}
+\end{aligned}$$
+
+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*
+to neglect the $e^{i \omega t}$ term,
+because it turns out to be irrelevant in this discussion.
+
+
+We call the ground state $\ket{0}$,
+but other than that, the other states need *not* be sorted by energy.
+However, we demand that the following holds
+for all even-numbered states $\ket{e}$ and $\ket{e'}$,
+and for all odd-numbered ($u$neven) states $\ket{u}$ and $\ket{u'}$:
+
+$$\begin{aligned}
+    \matrixel{e}{\hat{H}_1}{e'} = \matrixel{u}{\hat{H}_1}{u'} = 0
+    \qquad \quad
+    \matrixel{e}{\hat{H}_1}{u} \neq 0
+\end{aligned}$$
+
+This is justified for atomic orbitals thanks to
+[Laporte's selection rule](/know/concept/selection-rules/).
+Therefore, [time-dependent perturbation theory](/know/concept/time-dependent-perturbation-theory/)
+says that the $N$th-order coefficient corrections are:
+
+$$\begin{aligned}
+    c_e^{(N)}(t)
+    &= -\frac{i}{\hbar} \sum_{u}^{\mathrm{odd}} \int_0^t \matrixel{e}{\hat{H}_1(\tau)}{u} \: c_u^{(N-1)}(\tau) \: e^{i \omega_{eu} \tau} \dd{\tau}
+    \\
+    c_u^{(N)}(t)
+    &= -\frac{i}{\hbar} \sum_{e}^{\mathrm{even}} \int_0^t \matrixel{u}{\hat{H}_1(\tau)}{e} \: c_e^{(N-1)}(\tau) \: e^{i \omega_{ue} \tau} \dd{\tau}
+\end{aligned}$$
+
+Where $\omega_{eu} = (E_e \!-\! E_u) / \hbar$.
+For simplicity, the electron starts in the lowest-energy state $\ket{0}$:
+
+$$\begin{aligned}
+    c_0^{(0)} = 1
+    \qquad \qquad
+    c_u^{(0)} = c_{e \neq 0}^{(0)} = 0
+\end{aligned}$$
+
+Finally, we prove the following useful relation for large $t$,
+involving a [Dirac delta function](/know/concept/dirac-delta-function/) $\delta$:
+
+$$\begin{aligned}
+    \lim_{t \to \infty} \bigg| \frac{e^{i x t} - 1}{x} \bigg|^2
+    = 2 \pi \: \delta(x) \: t
+\end{aligned}$$
+
+<div class="accordion">
+<input type="checkbox" id="proof-sinc"/>
+<label for="proof-sinc">Proof</label>
+<div class="hidden">
+<label for="proof-sinc">Proof.</label>
+First, observe that we can rewrite the fraction using an integral:
+
+$$\begin{aligned}
+    \frac{e^{i x t} - 1}{x}
+    = e^{i x t / 2} \frac{e^{i x t / 2} - e^{-i x t / 2}}{x}
+    = i e^{i x t / 2} \int_{-t/2}^{t/2} e^{i x \tau} \dd{\tau}
+\end{aligned}$$
+
+By taking the limit $t \to \infty$,
+it can be turned into a nascent Dirac delta function:
+
+$$\begin{aligned}
+    \lim_{t \to \infty} \frac{e^{i x t} - 1}{x}
+    = \lim_{t \to \infty} i e^{i x t / 2} \frac{2 \pi}{2 \pi} \int_{-\infty}^{\infty} e^{i x \tau} \dd{\tau}
+    = \lim_{t \to \infty} i 2 \pi e^{i x t / 2} \: \delta(x)
+\end{aligned}$$
+
+Consequently, the absolute value squared is as follows:
+
+$$\begin{aligned}
+    \lim_{t \to \infty} \bigg| \frac{e^{i x t} - 1}{x} \bigg|^2
+    = 4 \pi^2 \delta^2(x)
+\end{aligned}$$
+
+However, a squared delta function $\delta^2$ is not ideal,
+so we take a step back:
+
+$$\begin{aligned}
+    \delta^2(x)
+    = \delta(x) \lim_{t \to \infty} \frac{1}{2 \pi} \int_{-t/2}^{t/2} e^{i x \tau} \dd{\tau}
+    = \delta(x) \lim_{t \to \infty} \frac{t}{2 \pi}
+\end{aligned}$$
+
+Where we have set $x = 0$ according to the first delta function.
+This gives the target:
+
+$$\begin{aligned}
+    \lim_{t \to \infty} \bigg| \frac{e^{i x t} - 1}{x} \bigg|^2
+    = 4 \pi^2 \delta^2(x)
+    = 2 \pi \: \delta(x) \: t
+\end{aligned}$$
+</div>
+</div>
+
+
+## One-photon absorption
+
+To warm up, we start at first-order perturbation theory.
+Thanks to our choice of initial condition,
+nothing at all happens to any of the even-numbered states $\ket{e}$:
+
+$$\begin{aligned}
+    c_e^{(1)}(t)
+    &= -\frac{i}{\hbar} \sum_{u}^{\mathrm{odd}} \int_0^t \matrixel{e}{\hat{H}_1(\tau)}{u} \: c_u^{(0)} \: e^{i \omega_{eu} \tau} \dd{\tau}
+    = 0
+\end{aligned}$$
+
+While the odd-numbered states $\ket{u}$ have a nonzero correction $c_u^{(1)}$,
+where $\vb{p}_{u0} = \matrixel{u}{\vu{p}}{0}$:
+
+$$\begin{aligned}
+    c_u^{(1)}(t)
+    &= -\frac{i}{\hbar} \int_0^t \matrixel{u}{\hat{H}_1(\tau)}{0} \: c_0^{(0)} \: e^{i \omega_{u0} \tau} \dd{\tau}
+    \\
+    &= i \frac{\vb{p}_{u0} \cdot \vb{E}}{\hbar} \int_0^t e^{i (\omega_{u0} - \omega) \tau} \dd{\tau}
+    \\
+    &= i \frac{\vb{p}_{u0} \cdot \vb{E}}{\hbar} \bigg[ \frac{e^{i (\omega_{u0} - \omega) \tau}}{i (\omega_{u0} - \omega)} \bigg]_0^t
+\end{aligned}$$
+
+Consequently, the first-order correction
+(in the rotating wave approximation) is given by:
+
+$$\begin{aligned}
+    \boxed{
+        c_u^{(1)}(t)
+        \approx \frac{\vb{p}_{u0} \cdot \vb{E}}{\hbar} \frac{e^{i (\omega_{u0} - \omega) t} - 1}{\omega_{u0} - \omega}
+    }
+\end{aligned}$$
+
+Since $\big| c_u^{(1)}(t) \big|^2$ is the probability
+of finding the electron in $\ket{u}$,
+its transition rate $R_u^{(1)}(t)$ is as follows,
+averaged since the beginning $t = 0$:
+
+$$\begin{aligned}
+    R_u^{(1)}(t)
+    = \frac{\big| c_u^{(1)}(t) \big|^2}{t}
+    = \frac{1}{t} \bigg| \frac{\vb{p}_{u0} \cdot \vb{E}}{\hbar} \bigg|^2
+    \cdot \bigg| \frac{e^{i (\omega_{u0} - \omega) t} - 1}{\omega_{u0} - \omega} \bigg|^2
+\end{aligned}$$
+
+For large $t \to \infty$, we can use the formula we proved earlier
+to get [Fermi's golden rule](/know/concept/fermis-golden-rule/):
+
+$$\begin{aligned}
+    \boxed{
+        R_u^{(1)}
+        = 2 \pi \bigg| \frac{\vb{p}_{u0} \cdot \vb{E}}{\hbar} \bigg|^2 \delta(\omega_{u0} - \omega)
+    }
+\end{aligned}$$
+
+This well-known formula represents **one-photon absorption**:
+it peaks at $\omega_{u0} = \omega$, i.e. when one photon $\hbar \omega$
+has the exact energy of the transition $\hbar \omega_{u0}$.
+Note that this transition is only possible when $\matrixel{u}{\vu{p}}{0} \neq 0$,
+i.e. for any odd-numbered final state $\ket{u}$.
+
+
+## Two-photon absorption
+
+Next, we go to second-order perturbation theory.
+Based on the previous result, this time
+all odd-numbered states $\ket{u}$ are unaffected:
+
+$$\begin{aligned}
+    c_u^{(2)}(t)
+    &= -\frac{i}{\hbar} \sum_{e}^{\mathrm{even}} \int_0^t \matrixel{u}{\hat{H}_1(\tau)}{e} \: c_e^{(1)}(\tau) \: e^{i \omega_{ue} \tau} \dd{\tau}
+    = 0
+\end{aligned}$$
+
+While the even-numbered states $\ket{e}$ have the following correction,
+using $\omega_{eu} \!+\! \omega_{u0} = \omega_{e0}$:
+
+$$\begin{aligned}
+    c_e^{(2)}(t)
+    &= -\frac{i}{\hbar} \sum_{u}^{\mathrm{odd}} \int_0^t \matrixel{e}{\hat{H}_1(\tau)}{u} \: c_u^{(1)}(\tau) \: e^{i \omega_{eu} \tau} \dd{\tau}
+    \\
+    &= i \sum_{u}^{\mathrm{odd}} \frac{(\vb{p}_{eu} \cdot \vb{E}) (\vb{p}_{u0} \cdot \vb{E})}{\hbar^2 (\omega_{u0} - \omega)}
+    \int_0^t e^{i (\omega_{eu} + \omega_{u0} - 2 \omega) \tau} - e^{i (\omega_{eu} - \omega) \tau} \dd{\tau}
+    \\
+    &= i \sum_{u}^{\mathrm{odd}} \frac{(\vb{p}_{eu} \cdot \vb{E}) (\vb{p}_{u0} \cdot \vb{E})}{\hbar^2 (\omega_{u0} - \omega)}
+    \bigg[ \frac{e^{i (\omega_{e0} - 2 \omega) \tau}}{i (\omega_{e0} - 2 \omega)}
+    - \frac{e^{i (\omega_{eu} - \omega) \tau}}{i (\omega_{eu} - \omega)} \bigg]_0^t
+\end{aligned}$$
+
+The second term represents one-photon absorption between $\ket{u}$ and $\ket{e}$.
+We do not care about that, so we drop it, leaving only the first term:
+
+$$\begin{aligned}
+    \boxed{
+        c_e^{(2)}(t)
+        \approx \sum_{u}^{\mathrm{odd}} \frac{(\vb{p}_{eu} \cdot \vb{E}) (\vb{p}_{u0} \cdot \vb{E})}{\hbar^2 (\omega_{u0} - \omega)}
+        \frac{e^{i (\omega_{e0} - 2 \omega) t} - 1}{\omega_{e0} - 2 \omega}
+    }
+\end{aligned}$$
+
+As before, we can define a rate $R_e^{(2)}(t)$
+for all transitions represented by this term:
+
+$$\begin{aligned}
+    R_e^{(2)}(t)
+    = \frac{\big| c_e^{(2)}(t) \big|^2}{t}
+    = \frac{1}{t} \bigg| \sum_{u}^{\mathrm{odd}} \frac{(\vb{p}_{eu} \cdot \vb{E}) (\vb{p}_{u0} \cdot \vb{E})}{\hbar^2 (\omega_{u0} - \omega)} \bigg|^2
+    \cdot \bigg| \frac{e^{i (\omega_{e0} - 2 \omega) t} - 1}{\omega_{e0} - 2 \omega} \bigg|^2
+\end{aligned}$$
+
+Which for $t \to \infty$ takes a similar form to Fermi's golden rule,
+using the formula we proved:
+
+$$\begin{aligned}
+    \boxed{
+        R_e^{(2)}
+        = 2 \pi \bigg| \sum_{u}^{\mathrm{odd}} \frac{(\vb{p}_{eu} \cdot \vb{E}) (\vb{p}_{u0} \cdot \vb{E})}{\hbar^2 (\omega_{u0} - \omega)} \bigg|^2
+        \delta(\omega_{e0} - 2 \omega)
+    }
+\end{aligned}$$
+
+This represents **two-photon absorption**, since it peaks at $\omega_{e0} = 2 \omega$:
+two identical photons $\hbar \omega$ are absorbed simultaneously
+to bridge the energy gap $\hbar \omega_{e0}$.
+Suprisingly, such a transition can only occur when $\matrixel{e}{\vu{p}}{0} = 0$,
+i.e. for any even-numbered final state $\ket{e}$.
+Notice that the rate is proportional to $|\vb{E}|^4$,
+so this effect is only noticeable at high light intensities.
+
+
+## Three-photon absorption
+
+For third-order perturbation theory,
+all even-numbered states $\ket{e}$ are unchanged:
+
+$$\begin{aligned}
+    c_e^{(3)}(t)
+    &= -\frac{i}{\hbar} \sum_{u}^{\mathrm{odd}} \int_0^t \matrixel{e}{\hat{H}_1(\tau)}{u} \: c_u^{(2)}(\tau) \: e^{i \omega_{eu} \tau} \dd{\tau}
+    = 0
+\end{aligned}$$
+
+And the odd-numbered states $\ket{u}$ get the following third-order corrections:
+
+$$\begin{aligned}
+    c_u^{(3)}(t)
+    &= -\frac{i}{\hbar} \sum_{e}^{\mathrm{even}} \int_0^t \matrixel{u}{\hat{H}_1(\tau)}{e} \: c_e^{(2)}(\tau) \: e^{i \omega_{ue} \tau} \dd{\tau}
+    \\
+    &= i \sum_{e}^{\mathrm{even}} \sum_{u'}^{\mathrm{odd}}
+    \frac{(\vb{p}_{ue} \cdot \vb{E}) (\vb{p}_{eu'} \cdot \vb{E}) (\vb{p}_{u'0} \cdot \vb{E})}{\hbar^3 (\omega_{u'0} - \omega) (\omega_{e0} - 2 \omega)}
+    \int_0^t e^{i (\omega_{ue} + \omega_{e0} - 3 \omega) \tau} - e^{i (\omega_{ue} - \omega) \tau} \dd{\tau}
+    \\
+    &= i \sum_{e}^{\mathrm{even}} \sum_{u'}^{\mathrm{odd}}
+    \frac{(\vb{p}_{ue} \cdot \vb{E}) (\vb{p}_{eu'} \cdot \vb{E}) (\vb{p}_{u'0} \cdot \vb{E})}{\hbar^3 (\omega_{u'0} - \omega) (\omega_{e0} - 2 \omega)}
+    \bigg[ \frac{e^{i (\omega_{u0} - 3 \omega) \tau}}{i (\omega_{u0} - 3 \omega)}
+    - \frac{e^{i (\omega_{ue} - \omega) \tau}}{i (\omega_{ue} - \omega)} \bigg]_0^t
+\end{aligned}$$
+
+Once again, the second term is uninteresting,
+so we drop it and look at the first term only:
+
+$$\begin{aligned}
+    \boxed{
+        c_u^{(3)}(t)
+        \approx \sum_{e}^{\mathrm{even}} \sum_{u'}^{\mathrm{odd}}
+        \frac{(\vb{p}_{ue} \cdot \vb{E}) (\vb{p}_{eu'} \cdot \vb{E}) (\vb{p}_{u'0} \cdot \vb{E})}
+        {\hbar^3 (\omega_{u'0} - \omega) (\omega_{e0} - 2 \omega)}
+        \frac{e^{i (\omega_{u0} - 3 \omega) t} - 1}{\omega_{u0} - 3 \omega}
+    }
+\end{aligned}$$
+
+The resulting transition rate $R_u^{(3)}(t)$
+is found to have the following familiar form:
+
+$$\begin{aligned}
+    R_u^{(3)}(t)
+    = \frac{\big| c_u^{(3)}(t) \big|^2}{t}
+    = \frac{1}{t} \bigg| \sum_{e}^{\mathrm{even}} \sum_{u'}^{\mathrm{odd}}
+    \frac{(\vb{p}_{ue} \cdot \vb{E}) (\vb{p}_{eu'} \cdot \vb{E}) (\vb{p}_{u'0} \cdot \vb{E})}
+    {\hbar^3 (\omega_{u'0} - \omega) (\omega_{e0} - 2 \omega)} \bigg|^2
+    \cdot \bigg| \frac{e^{i (\omega_{u0} - 3 \omega) t} - 1}{\omega_{u0} - 3 \omega} \bigg|^2
+\end{aligned}$$
+
+Applying our formula to this yields the following analogue of Fermi's golden rule:
+
+$$\begin{aligned}
+    \boxed{
+        R_u^{(3)}
+        = 2 \pi \bigg| \sum_{e}^{\mathrm{even}} \sum_{u'}^{\mathrm{odd}}
+        \frac{(\vb{p}_{ue} \cdot \vb{E}) (\vb{p}_{eu'} \cdot \vb{E}) (\vb{p}_{u'0} \cdot \vb{E})}
+        {\hbar^3 (\omega_{u'0} - \omega) (\omega_{e0} - 2 \omega)} \bigg|^2 \delta(\omega_{u0} - 3 \omega)
+    }
+\end{aligned}$$
+
+This represents **three-photon absorption**, since it peaks at $\omega_{u0} = 3 \omega$:
+three identical photons $\hbar \omega$ are absorbed simultaneously
+to bridge the energy gap $\hbar \omega_{u0}$.
+This process is similar to one-photon absorption,
+in the sense that it can only occur if $\matrixel{u}{\vu{p}}{0} \neq 0$.
+The rate is proportional to $|\vb{E}|^6$,
+so this effect only appears at extremely high light intensities.
+
+
+## N-photon absorption
+
+A pattern has appeared in these calculcations:
+in $N$th-order perturbation theory,
+we get a term representing $N$-photon absorption,
+with a transition rate proportional to $|\vb{E}|^{2N}$.
+Indeed, we can derive infinitely many formulas in this way,
+although the results become increasingly unrealistic
+due to the dependence on $\vb{E}$.
+
+If $N$ is odd, only odd-numbered destinations $\ket{u}$ are allowed
+(assuming the electron starts in the ground state $\ket{0}$),
+and if $N$ is even, only even-numbered destinations $\ket{e}$.
+Note that nothing has been said about the energies of these states
+(other than $\ket{0}$ being the minimum);
+everything is determined by the matrix elements $\matrixel{f}{\vu{p}}{i}$.
+
+
+
+## References
+1.  R.W. Boyd,
+    *Nonlinear optics*, 4th edition,
+    Academic Press.
+2.  R. Shankar,
+    *Principles of quantum mechanics*, 2nd edition,
+    Springer.
diff --git a/content/know/concept/navier-cauchy-equation/index.pdc b/content/know/concept/navier-cauchy-equation/index.pdc
index d3802d3..a17d69d 100644
--- a/content/know/concept/navier-cauchy-equation/index.pdc
+++ b/content/know/concept/navier-cauchy-equation/index.pdc
@@ -72,7 +72,6 @@ we start by inserting Hooke's law into Newton's law:
 
 $$\begin{aligned}
     \rho \pdv[2]{u_i}{t}
-    %= f_i + \sum_{j} \nabla_j \sigma_{ij}
     &= f_i + 2 \mu \sum_{j} \nabla_j u_{ij} + \lambda \sum_{j} \nabla_j \bigg( \delta_{ij} \sum_{k} u_{kk} \bigg)
     \\
     &= f_i + 2 \mu \sum_{j} \nabla_j u_{ij} + \lambda \nabla_i \sum_{j} u_{jj}
diff --git a/content/know/concept/optical-wave-breaking/index.pdc b/content/know/concept/optical-wave-breaking/index.pdc
index 2ab3ff1..30305f5 100644
--- a/content/know/concept/optical-wave-breaking/index.pdc
+++ b/content/know/concept/optical-wave-breaking/index.pdc
@@ -6,7 +6,7 @@ categories:
 - Physics
 - Optics
 - Fiber optics
-- Nonlinear dynamics
+- Nonlinear optics
 
 date: 2021-02-27T10:09:46+01:00
 draft: false
@@ -75,9 +75,6 @@ the instantaneous frequencies for these separate effects:
 $$\begin{aligned}
     \omega_i(z,t)
     &\approx \omega_\mathrm{GVD}(z,t) + \omega_\mathrm{SPM}(z,t)
-%    &= \frac{\beta_2 z / T_0^2}{1 + \beta_2^2 z^2 / T_0^4} \frac{t}{T_0^2}
-%    + \frac{2\gamma P_0 z}{T_0^2} t \exp\!\Big(\!-\frac{t^2}{T_0^2}\Big)
-%    \\
     = \frac{tz}{T_0^2} \bigg( \frac{\beta_2 / T_0^2}{1 + \beta_2^2 z^2 / T_0^4}
     + 2\gamma P_0 \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big) \bigg)
 \end{aligned}$$
@@ -210,7 +207,6 @@ be approximately reduced to:
 
 $$\begin{aligned}
     \omega_\mathrm{SPM}(L_\mathrm{WB}, t)
-%   = \frac{2 \gamma P_0 t \exp(-t^2 / T_0^2)}{\beta_2 \sqrt{N_\mathrm{sol}^2 / N_\mathrm{min}^2 - 1}}
     \approx \frac{2 \gamma P_0 t}{\beta_2 N_\mathrm{sol}} \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big)
     = 2 \sqrt{\frac{\gamma P_0}{\beta_2}} \frac{t}{T_0} \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big)
 \end{aligned}$$
diff --git a/content/know/concept/quantum-fourier-transform/index.pdc b/content/know/concept/quantum-fourier-transform/index.pdc
index 627f2de..5a3de7b 100644
--- a/content/know/concept/quantum-fourier-transform/index.pdc
+++ b/content/know/concept/quantum-fourier-transform/index.pdc
@@ -102,8 +102,6 @@ decompose $x$ as $x_1 2^{n-1} + x_2 2^{n-2} + ... + x_n 2^{0}$:
 
 $$\begin{aligned}
     \ket{\tilde{x}}
-%    &= \bigotimes_{j = 1}^{n} \frac{1}{\sqrt{2}} \bigg( \ket{0} + \exp\!\Big( \frac{i 2 \pi}{2^j} \sum_{r = 1}^{n} x_r 2^{r-1} \Big) \ket{1} \bigg)
-%    \\
     &= \bigotimes_{j = 1}^{n} \frac{1}{\sqrt{2}} \bigg( \ket{0} + \exp\!\Big( i 2 \pi \sum_{r = 1}^{n} x_r 2^{n-r-j} \Big) \ket{1} \bigg)
 \end{aligned}$$
 
diff --git a/content/know/concept/rayleigh-plateau-instability/index.pdc b/content/know/concept/rayleigh-plateau-instability/index.pdc
index 59407d6..ccb2e7c 100644
--- a/content/know/concept/rayleigh-plateau-instability/index.pdc
+++ b/content/know/concept/rayleigh-plateau-instability/index.pdc
@@ -146,7 +146,6 @@ so, by inserting the definition of $p_i = p_o + \alpha / R_0$:
 
 $$\begin{aligned}
     p_o + \alpha \Big( \frac{1}{R_1} + \frac{1}{R_2} \Big)
-    %= p_i + p_\epsilon(r\!=\!R)
     = p_o + \frac{\alpha}{R_0} + p_\epsilon(r\!=\!R)
 \end{aligned}$$
 
diff --git a/content/know/concept/selection-rules/index.pdc b/content/know/concept/selection-rules/index.pdc
index 22dfd64..9fda878 100644
--- a/content/know/concept/selection-rules/index.pdc
+++ b/content/know/concept/selection-rules/index.pdc
@@ -304,10 +304,6 @@ By inserting the expressions we found earlier for these commutators, we get:
 
 $$\begin{aligned}
     \comm*{\hat{L}^2}{\comm*{\hat{L}^2}{\hat{x}}}
-    %&= - 2 \hbar^2 \big( 2 (\hat{z} \hat{L}_x - \hat{x} \hat{L}_z - i \hbar \hat{y}) \hat{L}_z
-    %- 2 (\hat{x} \hat{L}_y - \hat{y} \hat{L}_x - i \hbar \hat{z}) \hat{L}_y
-    %- (\hat{L}^2 \hat{x} - \hat{x} \hat{L}^2) \big)
-    %\\
     &= - 4 \hbar^2 \big( \hat{z} \hat{L}_x \hat{L}_z - \hat{x} \hat{L}_z^2  - i \hbar \hat{y} \hat{L}_z
     + \hat{y} \hat{L}_x \hat{L}_y - \hat{x} \hat{L}_y^2 + i \hbar \hat{z} \hat{L}_y \big) \\
     &\qquad\qquad + 2 \hbar^2 \big( \hat{L}^2 \hat{x} - \hat{x} \hat{L}^2 \big)
@@ -334,10 +330,6 @@ which we use to arrive at:
 
 $$\begin{aligned}
     \comm*{\hat{L}^2}{\comm*{\hat{L}^2}{\hat{x}}}
-    %&= - 4 \hbar^2 \big( \hat{z} \hat{L}_z \hat{L}_x + \hat{y} \hat{L}_y \hat{L}_x + \hat{x} \hat{L}_x^2
-    %- \hat{x} \hat{L}_x^2 - \hat{x} \hat{L}_y^2 - \hat{x} \hat{L}_z^2 \big)
-    %+ 2 \hbar^2 \big( \hat{L}^2 \hat{x} - \hat{x} \hat{L}^2 \big)
-    %\\
     &= - 4 \hbar^2 \big( \hat{z} \hat{L}_z \hat{L}_x + \hat{y} \hat{L}_y \hat{L}_x + \hat{x} \hat{L}_x^2 - \hat{x} \hat{L}^2 \big)
     + 2 \hbar^2 \big( \hat{L}^2 \hat{x} - \hat{x} \hat{L}^2 \big)
     \\