From 16555851b6514a736c5c9d8e73de7da7fc9b6288 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Thu, 20 Oct 2022 18:25:31 +0200
Subject: Migrate from 'jekyll-katex' to 'kramdown-math-sskatex'
---
.../know/concept/multi-photon-absorption/index.md | 115 +++++++++++----------
1 file changed, 58 insertions(+), 57 deletions(-)
(limited to 'source/know/concept/multi-photon-absorption')
diff --git a/source/know/concept/multi-photon-absorption/index.md b/source/know/concept/multi-photon-absorption/index.md
index af433ac..5dd9887 100644
--- a/source/know/concept/multi-photon-absorption/index.md
+++ b/source/know/concept/multi-photon-absorption/index.md
@@ -11,10 +11,10 @@ categories:
layout: "concept"
---
-Consider a quantum system where there are many eigenstates $\Ket{n}$,
+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
+passes by, such that its Hamiltonian gets perturbed by $$\hat{H}_1$$, given in the
[electric dipole approximation](/know/concept/electric-dipole-approximation/) by:
$$\begin{aligned}
@@ -23,19 +23,19 @@ $$\begin{aligned}
\approx -\vu{p} \cdot \vb{E} e^{-i \omega t}
\end{aligned}$$
-Where $\vb{E}$ is the [electric field](/know/concept/electric-field/) amplitude,
-and $\vu{p} \equiv q \vu{x}$ is the transition dipole moment operator.
+Where $$\vb{E}$$ is the [electric field](/know/concept/electric-field/) amplitude,
+and $$\vu{p} \equiv q \vu{x}$$ is the transition dipole moment operator.
Here, we have made the
[rotating wave approximation](/know/concept/rotating-wave-approximation/)
-to neglect the $e^{i \omega t}$ term,
+to neglect the $$e^{i \omega t}$$ term,
because it turns out to be irrelevant in this discussion.
-We call the ground state $\Ket{0}$,
+We call the ground state $$\Ket{0}$$,
but other than that, the other states need *not* be sorted by energy.
However, we demand that the following holds
-for all even-numbered states $\Ket{e}$ and $\Ket{e'}$,
-and for all odd-numbered ($u$neven) states $\Ket{u}$ and $\Ket{u'}$:
+for all even-numbered states $$\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
@@ -46,7 +46,7 @@ $$\begin{aligned}
This is justified for atomic orbitals thanks to
[Laporte's selection rule](/know/concept/selection-rules/).
Therefore, [time-dependent perturbation theory](/know/concept/time-dependent-perturbation-theory/)
-says that the $N$th-order coefficient corrections are:
+says that the $$N$$th-order coefficient corrections are:
$$\begin{aligned}
c_e^{(N)}(t)
@@ -56,8 +56,8 @@ $$\begin{aligned}
&= -\frac{i}{\hbar} \sum_{e}^{\mathrm{even}} \int_0^t \matrixel{u}{\hat{H}_1(\tau)}{e} \: c_e^{(N-1)}(\tau) \: e^{i \omega_{ue} \tau} \dd{\tau}
\end{aligned}$$
-Where $\omega_{eu} = (E_e \!-\! E_u) / \hbar$.
-For simplicity, the electron starts in the lowest-energy state $\Ket{0}$:
+Where $$\omega_{eu} = (E_e \!-\! E_u) / \hbar$$.
+For simplicity, the electron starts in the lowest-energy state $$\Ket{0}$$:
$$\begin{aligned}
c_0^{(0)} = 1
@@ -65,8 +65,8 @@ $$\begin{aligned}
c_u^{(0)} = c_{e \neq 0}^{(0)} = 0
\end{aligned}$$
-Finally, we prove the following useful relation for large $t$,
-involving a [Dirac delta function](/know/concept/dirac-delta-function/) $\delta$:
+Finally, we prove the following useful relation for large $$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
@@ -86,7 +86,7 @@ $$\begin{aligned}
= i e^{i x t / 2} \int_{-t/2}^{t/2} e^{i x \tau} \dd{\tau}
\end{aligned}$$
-By taking the limit $t \to \infty$,
+By taking the limit $$t \to \infty$$,
it can be turned into a nascent Dirac delta function:
$$\begin{aligned}
@@ -102,7 +102,7 @@ $$\begin{aligned}
= 4 \pi^2 \delta^2(x)
\end{aligned}$$
-However, a squared delta function $\delta^2$ is not ideal,
+However, a squared delta function $$\delta^2$$ is not ideal,
so we take a step back:
$$\begin{aligned}
@@ -111,7 +111,7 @@ $$\begin{aligned}
= \delta(x) \lim_{t \to \infty} \frac{t}{2 \pi}
\end{aligned}$$
-Where we have set $x = 0$ according to the first delta function.
+Where we have set $$x = 0$$ according to the first delta function.
This gives the target:
$$\begin{aligned}
@@ -119,6 +119,7 @@ $$\begin{aligned}
= 4 \pi^2 \delta^2(x)
= 2 \pi \: \delta(x) \: t
\end{aligned}$$
+
</div>
</div>
@@ -127,7 +128,7 @@ $$\begin{aligned}
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 $\Ket{e}$:
+nothing at all happens to any of the even-numbered states $$\Ket{e}$$:
$$\begin{aligned}
c_e^{(1)}(t)
@@ -135,8 +136,8 @@ $$\begin{aligned}
= 0
\end{aligned}$$
-While the odd-numbered states $\Ket{u}$ have a nonzero correction $c_u^{(1)}$,
-where $\vb{p}_{u0} = \matrixel{u}{\vu{p}}{0}$:
+While the odd-numbered states $$\Ket{u}$$ have a nonzero correction $$c_u^{(1)}$$,
+where $$\vb{p}_{u0} = \matrixel{u}{\vu{p}}{0}$$:
$$\begin{aligned}
c_u^{(1)}(t)
@@ -157,10 +158,10 @@ $$\begin{aligned}
}
\end{aligned}$$
-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 $t = 0$:
+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 $$t = 0$$:
$$\begin{aligned}
R_u^{(1)}(t)
@@ -169,7 +170,7 @@ $$\begin{aligned}
\cdot \bigg| \frac{e^{i (\omega_{u0} - \omega) t} - 1}{\omega_{u0} - \omega} \bigg|^2
\end{aligned}$$
-For large $t \to \infty$, we can use the formula we proved earlier
+For large $$t \to \infty$$, we can use the formula we proved earlier
to get [Fermi's golden rule](/know/concept/fermis-golden-rule/):
$$\begin{aligned}
@@ -180,17 +181,17 @@ $$\begin{aligned}
\end{aligned}$$
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 $\Ket{u}$.
+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:
+all odd-numbered states $$\Ket{u}$$ are unaffected:
$$\begin{aligned}
c_u^{(2)}(t)
@@ -198,8 +199,8 @@ $$\begin{aligned}
= 0
\end{aligned}$$
-While the even-numbered states $\Ket{e}$ have the following correction,
-using $\omega_{eu} \!+\! \omega_{u0} = \omega_{e0}$:
+While the even-numbered states $$\Ket{e}$$ have the following correction,
+using $$\omega_{eu} \!+\! \omega_{u0} = \omega_{e0}$$:
$$\begin{aligned}
c_e^{(2)}(t)
@@ -213,7 +214,7 @@ $$\begin{aligned}
- \frac{e^{i (\omega_{eu} - \omega) \tau}}{i (\omega_{eu} - \omega)} \bigg]_0^t
\end{aligned}$$
-The second term represents one-photon absorption between $\Ket{u}$ and $\Ket{e}$.
+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:
$$\begin{aligned}
@@ -224,7 +225,7 @@ $$\begin{aligned}
}
\end{aligned}$$
-As before, we can define a rate $R_e^{(2)}(t)$
+As before, we can define a rate $$R_e^{(2)}(t)$$
for all transitions represented by this term:
$$\begin{aligned}
@@ -234,7 +235,7 @@ $$\begin{aligned}
\cdot \bigg| \frac{e^{i (\omega_{e0} - 2 \omega) t} - 1}{\omega_{e0} - 2 \omega} \bigg|^2
\end{aligned}$$
-Which for $t \to \infty$ takes a similar form to Fermi's golden rule,
+Which for $$t \to \infty$$ takes a similar form to Fermi's golden rule,
using the formula we proved:
$$\begin{aligned}
@@ -245,19 +246,19 @@ $$\begin{aligned}
}
\end{aligned}$$
-This represents **two-photon absorption**, since it peaks at $\omega_{e0} = 2 \omega$:
-two identical photons $\hbar \omega$ are absorbed simultaneously
-to bridge the energy gap $\hbar \omega_{e0}$.
-Surprisingly, such a transition can only occur when $\matrixel{e}{\vu{p}}{0} = 0$,
-i.e. for any even-numbered final state $\Ket{e}$.
-Notice that the rate is proportional to $|\vb{E}|^4$,
+This represents **two-photon absorption**, since it peaks at $$\omega_{e0} = 2 \omega$$:
+two identical photons $$\hbar \omega$$ are absorbed simultaneously
+to bridge the energy gap $$\hbar \omega_{e0}$$.
+Surprisingly, such a transition can only occur when $$\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$$,
so this effect is only noticeable at high light intensities.
## Three-photon absorption
For third-order perturbation theory,
-all even-numbered states $\Ket{e}$ are unchanged:
+all even-numbered states $$\Ket{e}$$ are unchanged:
$$\begin{aligned}
c_e^{(3)}(t)
@@ -265,7 +266,7 @@ $$\begin{aligned}
= 0
\end{aligned}$$
-And the odd-numbered states $\Ket{u}$ get the following third-order corrections:
+And the odd-numbered states $$\Ket{u}$$ get the following third-order corrections:
$$\begin{aligned}
c_u^{(3)}(t)
@@ -294,7 +295,7 @@ $$\begin{aligned}
}
\end{aligned}$$
-The resulting transition rate $R_u^{(3)}(t)$
+The resulting transition rate $$R_u^{(3)}(t)$$
is found to have the following familiar form:
$$\begin{aligned}
@@ -317,31 +318,31 @@ $$\begin{aligned}
}
\end{aligned}$$
-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 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 $|\vb{E}|^6$,
+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 calculations:
-in $N$th-order perturbation theory,
-we get a term representing $N$-photon absorption,
-with a transition rate proportional to $|\vb{E}|^{2N}$.
+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 $\vb{E}$.
+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 $\Ket{e}$.
+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 $\matrixel{f}{\vu{p}}{i}$.
+(other than $$\Ket{0}$$ being the minimum);
+everything is determined by the matrix elements $$\matrixel{f}{\vu{p}}{i}$$.
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