Waveform of Photons: Current Thinking & Analysis

[exmarine]

What is the current thinking about the waveform of photons? How can finite signals in the time domain produce a discrete frequency response in the frequency domain? As anyone who has worked with signal analysis remembers, the Fourier transforms of time signals in the real world always produce imperfect frequency plots. The transform of the product of time signals equals the convolution of the transforms of each signal - convolution theorem, hanning windows, etc. One can never get a clean impulse in the frequency domain.

Consider an AM RF transmitter and receiver. Presumably the fixed frequency and amplitude modulated signal is sent from the transmitter to the receiver antenna via photons. Each photon has a fixed amount of energy - E=hf - and some finite duration in the time domain. Yet it produces a single frequency response in the receiver antenna. I am no RF engineer, but I think that is correct. So how can such finite wave packets in the time domain produce clean single-frequency responses in the receiver?

 

[Cthugha]

Each photon has a fixed amount of energy - E=hf - and some finite duration in the time domain. Yet it produces a single frequency response in the receiver antenna.

This use of the term "photon" is purely a mathematical tool used in theory. You just perform a Fourier decomposition of the field and use these modes as a basis. The single photons created in a lab are never truly monochromatic and in fact can have pretty much any spectral and temporal shape. You get a fixed photon number of 1, but the energy is statistically distributed and will show some finite width when averaging over an ensemble of identically prepared systems.

In most other light sources like lasers or flashlights the photon number is not even fixed.

 

[exmarine]

Are you referring to the famous black-body radiation spectrum? Are you saying that even an RF transmitter has such a spectrum? My question is about the photons between radio transmitters and receivers.

 

[Cthugha]

Are you referring to the famous black-body radiation spectrum?

No, that was not what I had in mind.

Are you saying that even an RF transmitter has such a spectrum? My question is about the photons between radio transmitters and receivers.

Cavities (in optics) or antennas (as their RF equivalent) are similar to oscillating circuits. Their bandwidth is determined by the quality factor of the structure. If you do not lose much energy per round trip, you get a long oscillating decay and a narrow spectral line. If you lose a lot of energy per round trip, you get a broad spectral line and a pretty fast decay. This is a basic result resulting from Fourier transforms as I guess you know.

This is not different if you want to go to the single photon level. You just need to reinterpret things probabilistically. For RF stuff it is also absolutely not necessary to go to the single photon level. Classical waves explain pretty much everything in this regime.

So let me paraphrase:

So how can such finite wave packets in the time domain produce clean single-frequency responses in the receiver?

They simply do not. You will get a finite linewidth given by the typical Fourier transform limit between temporal and spectral width. If you really insisted on going to the single photon level, you would find this finite linewidth in averaging over many detection events.
edit: In case you are worrying about each detection event giving a discrete result: Yes, this is a result of being able to detect each photon only once. However, the quantity of interest is the mean energy and the width of the energy distribution averaged over many photon detection events under identical conditions. These are different animals.

 

[exmarine]

Yes, am quite familiar with damped frequency response curves for oscillating systems. But I regard those as POTENTIAL response frequencies. The off-resonant frequencies could only respond if subjected to broad-band excitation, or at least broad enough in that region. The REAL response frequencies in a particular situation COULD be narrower, could they not?

I suppose I should give some background to explain my motivation for this question. Optical wavelength experiments are totally out of reach for amateurs like me. And I’ve often wondered why we can’t study photons by doing experiments at much longer wavelengths. For example, could one do a double-slit experiment at infrared wave lengths?

I’ve also wondered why I’ve never seen any characterization of the wave-packet envelope shape, length, etc. It seems those might be important features. Are they uniform, consistent, or maybe random as you seem to suggest? I’d be surprised if they were random, given the fixed quanta of energy and momentum for each one.

Suppose one had very tight line width limits for an AM transmitter - I am guessing the FCC insists on that. Then suppose we varied the resistance of the receiver circuit, or just did frequency sweeps to determine if the arriving excitation caused responses wider than the transmitter line width. If the excitation photons are indeed arriving in wave packets, it seems that that would have to be the case.

I have no idea of what use this might be, but it seems like it would be interesting and could be important. All this might already be known in the RF engineering world. I should probably order another textbook.

 

[f95toli]

Yes, am quite familiar with damped frequency response curves for oscillating systems. But I regard those as POTENTIAL response frequencies. The off-resonant frequencies could only respond if subjected to broad-band excitation, or at least broad enough in that region. The REAL response frequencies in a particular situation COULD be narrower, could they not?

No, read what Cthugha wrote again. Even single photon sources will have linewidths.

I suppose I should give some background to explain my motivation for this question. Optical wavelength experiments are totally out of reach for amateurs like me. And I’ve often wondered why we can’t study photons by doing experiments at much longer wavelengths. For example, could one do a double-slit experiment at infrared wave lengths?

Sure, and this has been done lots of times. The problem is usually that as you go to to longer wavelengths you will find that that the available detectors are not as good. You also run into problem with black body radiation from the environment meaning you have to cool your experiment, this is doable but makes things more complicated.

I’ve also wondered why I’ve never seen any characterization of the wave-packet envelope shape, length, etc. It seems those might be important features. Are they uniform, consistent, or maybe random as you seem to suggest? I’d be surprised if they were random, given the fixed quanta of energy and momentum for each one.

This has also been done and you should be able to find papers if you use Google scholar. One the groups in Oxford recently published a few papers on this (I can't remember the name of the author).


Also, much work in cavity-QED is actully done in the GHz range and more recently you also have experiments on circuit-QED which does "quantum optics on a chip", usually using superconducting circuits. Hence, quantum optics in the RF range is a very well developed field. However, you always have to cool your experiment to low temperatures (mK) or otherwise the black body radiation would swamp your signal.