Why is two-photon absorption a third-order non-linear process? I've just started learning about non-linear optics where I could not understand why the two photon absorption is a $\chi^{(3)}$ process. The way I understand susceptibility is that the order of $\chi$ tells us the number of frequencies added/subtracted(not a precise defn.), while the real/imaginary part tell us if it's corresponding to the phase/absorption. So if in TPA only two photons are absorbed, why is it not described by imaginary part of $\chi^{(2)}$ ?
 A: This is actually rather confusing.  So, good question!
This describes single-photon interaction with a material:
$$
\mathbf{P}=\chi^{(1)}\mathbf{E}
$$
Or rewritten to assume time-harmonic fields,
$$
P_ie^{i \omega_1 t}=\chi^{(1)}_{ij}E_je^{i \omega_1 t}
$$
All of the subscripts refer to the spatial components of the field polarization, making $\chi^{(1)}_{ij}$ a tensor describing the material response.
So $\chi^{(1)}$ is taking in the frequency $\omega_1$ (or energy = $\hbar\omega_1$) and generating a material polarization oscillating with the same frequency.  This can be alternately expressed in terms of photon energy conservation as $\chi^{(1)}(\omega_1=\omega_1)$ or $\chi^{(1)}(0=\omega_1-\omega_1)$.  And, as you rightly point out, the imaginary component of $\chi$ corresponds to absorption.  In the absence of absorption, this process can represent a coherent promotion of an electron from the ground state to an excited state and back down again, continually.
But power flow for EM waves is determined by the Poynting vector, $\mathbf{S}=\mathbf{E}\times\mathbf{H}$.  Since power is proportional to the number of photons, you can think of a photon as needing two factors of field.  So to add another photon to the existing single-photon interaction, you need to multiply by two more fields and use $\chi^{(3)}$:
$$
P_ie^{i \omega_1 t}=\chi^{(3)}_{ijkl}E_je^{i \omega_1 t}E_ke^{i \omega_2 t}E_le^{-i \omega_2 t}
$$
The energy conservation of this process can be expressed as $\chi^{(3)}(\omega_1=\omega_1+\omega_2-\omega_2)$ or $\chi^{(3)}(0=\omega_1-\omega_1+\omega_2-\omega_2)$.  Again, $\chi^{(3)}$ is generally complex, to support absorption and phase shift.  The choice of two distinct frequencies reflects the two colors of the photons for two-photon absorption.
I hope this helps!
A: The relation between the number of photons - or, equivalently, order of the electronic transitions - involved in a TPA process (two) and the order of the corresponding nonlinear susceptibility (third) may be understood using the optical theorem. This theorem relates the imaginary part of an all-optical process of a given perturbation order $m$ with a process involving charge carriers with half the perturbation order, i.e. $m/2$. To apply this theorem it is important to consider that the order in perturbation theory to calculate the probability amplitude of an all-optical $\chi ^{(n)}$ process is $m=n+1$. Since in the case of TPA there are electronic transitions of the second order involved ($m/2=2$), it results from the optical theorem that the order of the nonlinear susceptibility is $n=m-1=3$, i.e. it is a $\chi ^{(3)}$ process.
Note also that in the case of processes involving only virtual electronic transitions the number of photons that take place in the process directly corresponds to the order of the nonlinear susceptibility. For instance, second harmonic generation (SHG) exploits a $\chi^{(2)}$ nonlinearity.
A: I think Gilbert explained it very well but to help you with the intuition,a very rough and short answer: since we are saying "two photons" we are referring to intensity (number of photons). However, for example in a second-harmonic generation, we refer to square of the electric field not the square of intensity.
A: As I learned from somewhere else, a non-linear process is of n-th order when n+1 waves are involved. In two photon absorption, 2 photons are absorbed and then 2 photons are emitted later, adding up to 4 light waves, which means it's a 3rd order non-linear process. By this criterion one can easily infer that three photon absorption is a fifth order non-linear process.
A: Two photon absorption phenomena is characteristic of materials namely semiconductor, insulators. When bandgap energy $E_\mathrm{g}=\frac{\hbar\omega}{2\pi}$ is larger then a photon being incident, there is simultaneous absorption of two photons. And these materials actually exhibit third order nonlinearity
