What's the interval between photons in stimulated emission?

[Enty]

What's the interval between photons in stimulated emission?

In stimulated emission one photon induces the emission of a second photon whose coherence length, energy, polarisation and direction of travel are all identical to its own. There must be a delay between the two photons, see below, so would anyone care to speculate on the magnitude of the time interval between the emission of the two photons?

Starting with an electron orbiting a molecule in its unexcited or ground state. If a photon is absorbed by a dipole between that electron and a nucleus and the photon does not posses sufficient energy to promote the electron to an electrically excited state the energy is rapidly re-emitted as a photon, Rayleigh scattering: this is the phenomenon that gives rise to transparent media exhibiting a refractive index. If a second photon arrives at the dipole and it is also absorbed before the first one has been re-emitted and their combined energies are sufficient to promote the electron to an electrically excited state, that is what occurs: second harmonic or two-photon excitation, also known as frequency doubling. This demonstrates that if the interval between two identical photons arriving at dipole is sufficiently short their effects are indistinguishable from the actions of a single photon that posses twice their energy: the relative polarisation of the photons can be ignored because with stimulated emission this is identical. Consequently if, in addition to identical directions of travel, the pair of photons produced by the process of stimulated emission also possessed instantaneously identical loci they would often, if not always, behave like a single photon with twice the energy. Now the amplification part of the acronym 'laser' is the product of stimulated emission, most laser emission is notoriously monochromatic and certainly does not behave as if it were frequency doubled. Thus the two photons generated by stimulated emission must occupy different loci and consequentially there must be delay between them.

I confess, I'm a biologist and light microscopist and don't posses the mathematical skills to even follow vector calculus, let alone QED, but nevertheless I would like to know the answer to the above question or at least where the flaw in the logic that suggests the question can be posed is.

 

[DrClaude]

That's too much classical thinking to be applicable to the situation at hand. For one, one can't define the position of a photon (there is no photon wave function or position operator, see https://www.physicsforums.com/threads/why-no-position-operator-for-photon.906932/).

This demonstrates that if the interval between two identical photons arriving at dipole is sufficiently short their effects are indistinguishable from the actions of a single photon that posses twice their energy

That's not correct. Two-photon processes are not equivalent to single-photon processes, even if they can lead to the same final state. For instance, selection rules for two-photons processes are different than the usual selection rules. The cross-section for absorption also differs a lot (two-photon absorption usually requires intense laser fields).

Stimulated emission comes from the fact that the probability of waiting a photon in a certain electromagnetic mode increases with the population of that same mode. Going from N photons in a mode to N+1 photons in the same mode after interaction with an excited atom is a coherent process.

 

[Enty]

DrClaude.

Thank you for your answer but unfortunately my mathematical abilities leave me stuck with classical conceptualism. In reply to the three, thought provoking, points that you make:-

Even if you can't define the position of a photon it must posses something akin to a probability density function so, where necessary, instead of position, the centroid of the probability density function could be considered, this makes it overt that the intervals are described by distribution and, initially I would like some idea of the mean and spread of that distribution in time. Alternatively, if it helps, you could regard my question as being, 'what is the order of magnitude of the life-time of the excited state of an absorbing dipole in Rayleigh scattering?'

As to your second point, perhaps it would have been less ambiguous if I had written,
'This demonstrates that if the interval between two identical photons arriving at dipole is sufficiently short the consequences are indistinguishable from the actions of a single photon that posses twice their energy.'

In any event I have been led to believe that two photon absorption by flurorophores does not require intense laser fields and it has been demonstrated that in two-photon confocal laser scanning microscopy the number of fluorescently emitted photons detected continues to be proportional to the square of the intensity of the illumination well into pulse lengths in the pico-second range (Brewersdorf, J. & Hell, S.W. (1998) Picosecond pulsed two-photon imaging with repetition rates of 200 and 400 MHz. J. Microscp. 192, 28 - 38). The only reason this research was not pursued further was because the yield of fluorescently emitted photons was too low to operate a commercially viable instrument. Moreover, in discussions with the theoreticians in that field (confocal microscopy) they appeared to think that the two photon phenomenon operated at any density of photons: this was a once a very hot topic in confocal microscopy which revolved around a patent dispute.

It appears that stimulated emission can be induced from small populations of, and even single, excited fluorophores, so it does not have to a population phenomenon per se., as examples, Rephaeli E. & Fan S. (2012) Stimulated emission from a single excited atom in a waveguide, Phys. Rev. Lett. 108, 143602. Bianchini et. al., (2012) Single-wavelength two-photon excitation–stimulated emission depletion (SW2PE-STED) superresolution imaging, PNAS 109, 6390 - 6393.

So my question, albeit ill-expressed, remains.