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fxdung

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DrChinese

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A single photon has a single frequency (what you might call "color"). However, it is possible to split a photon into 2 photons using "parametric down conversion" (PDC) using a nonlinear crystal (although the proper description of that process is complicated). When PDC occurs, there is conservation of momentum, energy, etc. as would be expected. I.e. total energy of 1 photon in equals total energy of 2 photons out.

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PeterDonis

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Actually, I think even this is not always true. AFAIK the particle number operator and the frequency operator do not commute, so they have no joint eigenstates; so a "single photon" that is a number eigenstate will not have a well-defined frequency. A "single photon" that is actually a coherent state will, but then it is not an eigenstate of the number operator and the term "single photon" has to be interpreted differently when applied to it (in terms of the expectation value of photon number).A single photon has a single frequency

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PeterDonis

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No, we don't know that, because the term "a single photon" in quantum field theory does not mean what you are thinking it means. In fact, as my post #3 just now notes, the term does not even have a single meaning; it can mean different things for different types of quantum field states.We know that we can not cut a single photon into many photons.

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fxdung

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What is mathematical formula of frequency operator?

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What do you mean by "frequency operator"? I guess you mean the Hamiltonian. For a single-photon mode with momentum ##\vec{k}## and helicity ##h## the energy eigenvalues are ##E=|\vec{k}|## (using natural units with ##\hbar=c=1##), and ##E=\omega##.Actually, I think even this is not always true. AFAIK the particle number operator and the frequency operator do not commute, so they have no joint eigenstates; so a "single photon" that is a number eigenstate will not have a well-defined frequency. A "single photon" that is actually a coherent state will, but then it is not an eigenstate of the number operator and the term "single photon" has to be interpreted differently when applied to it (in terms of the expectation value of photon number).

This single-photon Fock state of course is not a proper state vector representing a photon, because as any plane wave, it's not normalizable to one. They are generalized eigenstates "normalizable to a ##\delta##-distribution". A true single-photon state is always a "wave packet", i.e., something like

$$|\psi \rangle \propto \int_{\mathbb{R}^3} \mathrm{d}^3 k A(\vec{k}) \hat{a}^{\dagger}(\vec{k},h) |\Omega \rangle,$$

where ##A(\vec{k})## is square integrable. I've written down a one-photon state with helicity ##h##. You can of course have arbitrary superpositions (i.e., arbitrary polarization states). Since ##A(\vec{k})## has a finite width, indeed such a state is not a monochromatic one (although you can make it arbitrarily close to one; you only have to make ##A(\vec{k})## sharply peaked around some momentum ##\vec{k}_0##); correctly normalized ##|A(\vec{k})|^2## gives the probability distribution for detecting a photon with momentum ##\vec{k}## (and thus energy/frequency ##|\vec{k}|##). It's an eigenstate of the photon-number operator with eigenvalue 1, i.e., you have prepared precisely one photon.

A coherent state is not a single-photon state. It is not even an eigen state of the photon-number operator but an eigen state of an annihilation operator. The photon number is Poisson distributed in such a state, which represents (if of not too low intensity) rather a classical em. wave than "photons". Sometimes popular-science sources mix up single-photon states with low-intensity coherent states. You can make the expectation value of the photon number arbitrarily small for such a coherent state, even smaller than 1. Then this coherent state is pretty close to the vacuum state, and if you detect something it's very likely to be the response to a single photon, but the probability to detect 2 or more photons is also not exactly 0.

- #7

A. Neumaier

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Not necessarily!A single photon has a single frequency (what you might call "color").

The most general photon is an arbitrary solution of the free Maxwell equations. For experimental reasons one usually prepares the photons to have a significant energy density only along a narrow beam.

A single photon in a beam is generally in a superposition of infinitely many frequencies, which determine its color. When the photon is produced by a monochromatic laser, all these frequencies are very close to the nominal frequency of the laser (within the width of the corresponding spectral line), and one speaks of a monochromatic photon.

A white photon will go through the prism and split into a superposition of coherent photons with different wave vectors corresponding to the different directions the contributions of different colors take. Thus it will no longer propagate along a beam but along a fan.

If the white beam (or the fan) passes through a color filter, the photon will pass with a probability corresponding to the contribution in its wave function of the color(s) that can pass. If it passes, its state will have been changed to that of a (nearly) monochromatic photon. This is standard state reduction (collapse of the wave function). Thus the photon will change its color rather than split into many.

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