---
title: "Multi-photon absorption"
sort_title: "Multi-photon absorption"
date: 2022-01-30
categories:
- Physics
- Optics
- Quantum mechanics
- Nonlinear optics
- Perturbation
layout: "concept"
---
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](/know/concept/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/) $$\delta$$:
$$\begin{aligned}
\lim_{t \to \infty} \bigg| \frac{e^{i x t} - 1}{x} \bigg|^2
= 2 \pi \: \delta(x) \: t
\end{aligned}$$
## 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 $$\hbar \omega_{e0}$$.
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:
$$\begin{aligned}
c_e^{(3)}(t)
&= -\frac{i}{\hbar} \sum_{u}^{\mathrm{odd}} \int_0^t \matrixel{e}{\hat{H}_1(\tau)}{u} \: c_u^{(2)}(\tau) \: e^{i \omega_{eu} \tau} \dd{\tau}
= 0
\end{aligned}$$
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 calculations:
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.