Two photon absorption spectroscopy is well known non-linear optic phenomena [1]. In this phenomena the virtual level is present which allows to absorb the second photon and thus excite the state with energy equal to two times the energy of the original quantum. The corresponding two-photon excited fluorescence is well known (and now three-photon excited fluorescence is known well). The most important observation connected with multiple quanta absorption is the non-linear dependence on power: this allows easily distinguish it from other phenomena.
The fundamental hypothesis outlined in the previous post [2] concerning the harmonics of de Broglie waves may be also stated for the photon itself. Indeed, the initial hypothesis of Plank concerning the photons [3] was the energy of photons itself is: E=n*h*ν [3]. From observation of photoelectric effect Einstein deduced the more commonly known rule: E=h*ν, which eventually lead to the development of the quantum electrodynamics and numerous discoveries.
However, any mathematical expression is only the approximation to the natural law, and the idea of the photons having energy of only E=h*ν may be very successful but not finally correct. Indeed, the double energy photon (with E=2*h*ν) may be so rare that virtually non-observable and thus making the Einstein idea so exceptionally great fit to the natural law that it looks absolute. The double energy photons may easily decay into two ordinary photons or mutate into the photon with double frequency.
Even the probability of the existence of such photons would be governed by the usual Boltzman rule (from [2]):
the population of double energy photons would be exp[-hν/(kT)] less compare to the usual photon. That value for relatively small energy infrared photons (1064 nm wavelength, 1.17 eV ordinary photon energy) would be at room temperature only 2.35*10exp(-20). This means that even such photons exist, they are so rare that virtually non-observable.
However, the two photon excited fluorescence is a convenient way to check theirs presence. Indeed, in addition to the quadratic in power term of such fluorescence (due to virtual levels creation [1]) an extremely small linear in power fluorescence is predicted. This fluorescence is so weak that it is necessary to consider the background created by usual thermal excitation (with some non-zero probability the same excited level may be reached by the usual thermal excitation according to the Boltzman formula P/Po=exp[-E/(kT)].
The trick to subtract background is that the two photon fluorescence is a resonant phenomena. It means that for the deviation of the wavelength from the resonant value it will quickly disappear. Since for the linear phenomena search the laser should not be powerful (to prevent observation of the more common quadratic in power two photon fluorescence) it may be with tuned frequency and thus allowing to observe the linear in power resonant phenomenon. When the frequency of the laser deviates from the frequency necessary for two quanta fluorescence (this frequency may be obtained from the quadratic term of the induced fluorescence) the observed linear term should quickly disappear (and the thermal background stay the same).
In summary: the observation of the linear term in the two photon fluorescence is predicted due to hypothetical existence of double energy photons (from Planks rule E=n*h*ν [3]), the phenomenon would be really weak (20 orders of magnitude weaker compare to one-photon fluorescence at least for infrared photons) but resonant in photon frequency.
References.
1. https://en.wikipedia.org/wiki/Two-photon_absorption
2. https://tipikin.blogspot.com/2020/05/quantization-rule-and-harmonics-of.html
3. http://web.phys.ntnu.no/~stovneng/TFY4165_2013/BlackbodyRadiation.pdf
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