Categories: Nonlinear optics, Optics, Perturbation, Physics, Quantum mechanics.

Multi-photon absorption

Consider a quantum system where there are many eigenstates n\Ket{n}, e.g. atomic orbitals, for an electron to occupy. Suppose an electromagnetic wave passes by, such that its Hamiltonian gets perturbed by H^1\hat{H}_1, given in the electric dipole approximation by:

H^1(t)=p^Ecos(ωt)p^Eeiωt\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 E\vb{E} is the electric field amplitude, and p^qx^\vu{p} \equiv q \vu{x} is the transition dipole moment operator. Here, we have made the rotating wave approximation to neglect the eiωte^{i \omega t} term, because it turns out to be irrelevant in this discussion.

We call the ground state 0\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 e\Ket{e} and e\Ket{e'}, and for all odd-numbered (uuneven) states u\Ket{u} and u\Ket{u'}:

eH^1e=uH^1u=0eH^1u0\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. Therefore, time-dependent perturbation theory says that the NNth-order coefficient corrections are:

ce(N)(t)=iuodd0teH^1(τ)ucu(N1)(τ)eiωeuτdτcu(N)(t)=ieeven0tuH^1(τ)ece(N1)(τ)eiωueτdτ\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 ωeu=(Ee ⁣ ⁣Eu)/\omega_{eu} = (E_e \!-\! E_u) / \hbar. For simplicity, the electron starts in the lowest-energy state 0\Ket{0}:

c0(0)=1cu(0)=ce0(0)=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 tt, involving a Dirac delta function δ\delta:

limteixt1x2=2πδ(x)t\begin{aligned} \lim_{t \to \infty} \bigg| \frac{e^{i x t} - 1}{x} \bigg|^2 = 2 \pi \: \delta(x) \: t \end{aligned}

First, observe that we can rewrite the fraction using an integral:

eixt1x=eixt/2eixt/2eixt/2x=ieixt/2t/2t/2eixτdτ\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 tt \to \infty, it can be turned into a nascent Dirac delta function:

limteixt1x=limtieixt/22π2πeixτdτ=limti2πeixt/2δ(x)\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:

limteixt1x2=4π2δ2(x)\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 δ2\delta^2 is not ideal, so we take a step back:

δ2(x)=δ(x)limt12πt/2t/2eixτdτ=δ(x)limtt2π\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=0x = 0 according to the first delta function. This gives the target:

limteixt1x2=4π2δ2(x)=2πδ(x)t\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}

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 e\Ket{e}:

ce(1)(t)=iuodd0teH^1(τ)ucu(0)eiωeuτdτ=0\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 u\Ket{u} have a nonzero correction cu(1)c_u^{(1)}, where pu0=up^0\vb{p}_{u0} = \matrixel{u}{\vu{p}}{0}:

cu(1)(t)=i0tuH^1(τ)0c0(0)eiωu0τdτ=ipu0E0tei(ωu0ω)τdτ=ipu0E[ei(ωu0ω)τi(ωu0ω)]0t\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:

cu(1)(t)pu0Eei(ωu0ω)t1ωu0ω\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 cu(1)(t)2\big| c_u^{(1)}(t) \big|^2 is the probability of finding the electron in u\Ket{u}, its transition rate Ru(1)(t)R_u^{(1)}(t) is as follows, averaged since the beginning t=0t = 0:

Ru(1)(t)=cu(1)(t)2t=1tpu0E2ei(ωu0ω)t1ωu0ω2\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 tt \to \infty, we can use the formula we proved earlier to get Fermi’s golden rule:

Ru(1)=2πpu0E2δ(ωu0ω)\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 ωu0=ω\omega_{u0} = \omega, i.e. when one photon ω\hbar \omega has the exact energy of the transition ωu0\hbar \omega_{u0}. Note that this transition is only possible when up^00\matrixel{u}{\vu{p}}{0} \neq 0, i.e. for any odd-numbered final state u\Ket{u}.

Two-photon absorption

Next, we go to second-order perturbation theory. Based on the previous result, this time all odd-numbered states u\Ket{u} are unaffected:

cu(2)(t)=ieeven0tuH^1(τ)ece(1)(τ)eiωueτdτ=0\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 e\Ket{e} have the following correction, using ωeu ⁣+ ⁣ωu0=ωe0\omega_{eu} \!+\! \omega_{u0} = \omega_{e0}:

ce(2)(t)=iuodd0teH^1(τ)ucu(1)(τ)eiωeuτdτ=iuodd(peuE)(pu0E)2(ωu0ω)0tei(ωeu+ωu02ω)τei(ωeuω)τdτ=iuodd(peuE)(pu0E)2(ωu0ω)[ei(ωe02ω)τi(ωe02ω)ei(ωeuω)τi(ωeuω)]0t\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 u\Ket{u} and e\Ket{e}. We do not care about that, so we drop it, leaving only the first term:

ce(2)(t)uodd(peuE)(pu0E)2(ωu0ω)ei(ωe02ω)t1ωe02ω\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 Re(2)(t)R_e^{(2)}(t) for all transitions represented by this term:

Re(2)(t)=ce(2)(t)2t=1tuodd(peuE)(pu0E)2(ωu0ω)2ei(ωe02ω)t1ωe02ω2\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 tt \to \infty takes a similar form to Fermi’s golden rule, using the formula we proved:

Re(2)=2πuodd(peuE)(pu0E)2(ωu0ω)2δ(ωe02ω)\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 ωe0=2ω\omega_{e0} = 2 \omega: two identical photons ω\hbar \omega are absorbed simultaneously to bridge the energy gap ωe0\hbar \omega_{e0}. Surprisingly, such a transition can only occur when ep^0=0\matrixel{e}{\vu{p}}{0} = 0, i.e. for any even-numbered final state e\Ket{e}. Notice that the rate is proportional to E4|\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 e\Ket{e} are unchanged:

ce(3)(t)=iuodd0teH^1(τ)ucu(2)(τ)eiωeuτdτ=0\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 u\Ket{u} get the following third-order corrections:

cu(3)(t)=ieeven0tuH^1(τ)ece(2)(τ)eiωueτdτ=ieevenuodd(pueE)(peuE)(pu0E)3(ωu0ω)(ωe02ω)0tei(ωue+ωe03ω)τei(ωueω)τdτ=ieevenuodd(pueE)(peuE)(pu0E)3(ωu0ω)(ωe02ω)[ei(ωu03ω)τi(ωu03ω)ei(ωueω)τi(ωueω)]0t\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:

cu(3)(t)eevenuodd(pueE)(peuE)(pu0E)3(ωu0ω)(ωe02ω)ei(ωu03ω)t1ωu03ω\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 Ru(3)(t)R_u^{(3)}(t) is found to have the following familiar form:

Ru(3)(t)=cu(3)(t)2t=1teevenuodd(pueE)(peuE)(pu0E)3(ωu0ω)(ωe02ω)2ei(ωu03ω)t1ωu03ω2\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:

Ru(3)=2πeevenuodd(pueE)(peuE)(pu0E)3(ωu0ω)(ωe02ω)2δ(ωu03ω)\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 ωu0=3ω\omega_{u0} = 3 \omega: three identical photons ω\hbar \omega are absorbed simultaneously to bridge the energy gap ωu0\hbar \omega_{u0}. This process is similar to one-photon absorption, in the sense that it can only occur if up^00\matrixel{u}{\vu{p}}{0} \neq 0. The rate is proportional to E6|\vb{E}|^6, so this effect only appears at extremely high light intensities.

N-photon absorption

A pattern has appeared in these calculations: in NNth-order perturbation theory, we get a term representing NN-photon absorption, with a transition rate proportional to E2N|\vb{E}|^{2N}. Indeed, we can derive infinitely many formulas in this way, although the results become increasingly unrealistic due to the dependence on E\vb{E}.

If NN is odd, only odd-numbered destinations u\Ket{u} are allowed (assuming the electron starts in the ground state 0\Ket{0}), and if NN is even, only even-numbered destinations e\Ket{e}. Note that nothing has been said about the energies of these states (other than 0\Ket{0} being the minimum); everything is determined by the matrix elements fp^i\matrixel{f}{\vu{p}}{i}.


  1. R.W. Boyd, Nonlinear optics, 4th edition, Academic Press.
  2. R. Shankar, Principles of quantum mechanics, 2nd edition, Springer.