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Date: Sun, 30 Jan 2022 11:27:00 +0100
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+---
+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.
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