In this SE question, I asserted that mixed-frequency photons exist. Elsewhere I have asserted that a photon can have a finite coherence length. The two assertions are, of course, closely related: finite coherence length in a laser beam implies that the beam has finite frequency line width. My position is that frequency is a quantum state whose value lies on a continuum, and that the quantum state can be mixed. That is, the frequency of a photon is best described as a probability distribution of frequencies, just as the polarization of a photon is best described as a probability distribution of polarization values. There has been a fair amount of push-back when I've stated my position. In this SE question, there was more discussion of the idea that individual photons may have a coherence length.
Here I propose an experiment that might be able to settle the question. The experiment is analogous to the single-photon version of the double-slit interferometer (DSI). The DSI accumulates an interference pattern only if both slits are open, even when the beam passing through the DSI is attenuated so much that there is a very high probability only one photon can be in the DSI at a time; which only makes sense if the individual photons are somehow spread out in space until they are detected at single points. The proposed experiment is set up to produce a result that only makes sense if individual photons are somehow spread out in frequency space until they are detected at a single frequency.
The experiment: The setup shown below is a basic pulse shaper.
A train of femtosecond pulses is directed to a pair of diffraction gratings, The pulse is spread out into its frequency components by the gratings, then passes through a mask, is reflected back through the mask by a mirror, then passes back through the gratings in the reverse direction. The pulse is reconstituted as it exits the left-hand grating. The beamsplitter enables separation of the reconstituted pulse train from the oppositely propagating original pulse train.
If the mask is simply an open aperture, each reconstituted pulse is essentially identical to its original pulse. A femtosecond-duration pulse going in becomes a femtosecond-duration pulse going out. But if the mask blocks some of the frequency components of the pulse, it effectively acts as a temporal Fourier domain filter and thus alters the temporal profile of the reconstituted pulse relative to its original pulse. In essence, the pulse is reconstituted via coherent superposition of the various frequency components. That is, the pulse is reconstituted by interference of the frequency components. The presence of a mask that blocks any of the frequency components will necessarily spread out the reconstituted pulse temporally. Instead of having a duration of a femtosecond, it will have a longer duration.
The duration of the pulse can be measured any of several ways. In the proposed experiment, I suggest attenuating the incoming pulse until there is most likely only one photon at a time in the beam shaper.
If the photon exists in a mixed frequency state, its wavefunction will be separated into frequency components by the diffraction gratings, filtered (much as the double slit in the DSI filters the wavefunction), then reconstituted.
A photomultiplier is used to detect the time of arrival of each reconstituted photon at the photomultiplier relative to the time of injection of the original pulse.
We know, within the temporal resolution of the photomultipliers, precisely when the pulses enter the beam shaper and when they are detected after they leave the beam shaper. If the frequency bandpass of the pulse shaper is reduced enough, the reconstituted pulse duration can be made arbitrarily large (at the expense of throwing away most of the pulse energy). If the same thing happens for individual photons as happens for pulses consisting of large numbers of photons, then it will be hard to escape the conclusion that the photons in a pulse exist in a mixed frequency state that echoes the finite bandwidth of the pulses from which the photons are extracted.
I hope some of the QM experts in our community will comment on the likely outcomes of the experiment and how those outcomes should be interpreted.