Coherent sources can have ultra short pulse durations provided there is at least one photon in frequency that span broad bandwidth (and are coherent)?
Does that mean that the pulse duration is infinitely long for beam of single photon (basically one frequency)?
Can we define precisely a photon energy or is it also uncertain?
Is the energy of single photon defined by uncertainty relation?
The energy of a photon is an unchanged value during the photons existence. Once emitted from a particle the photon is an indivisible unit and its energy content stays unchanged. But it is without doubt that we are not able to produce a serie of photons with equal energy content nor to measure it without uncertainty.
What is the pulse duration of sigle photon beam?
Interesting question. Do a photon has a shape? The answer depends from your choose of the physical theory you want to use. From the point of Quantum Mechanics a photon is an excitation of an overall existing electromagnetic field and by this has to be spreaded out to infinity.
From the point of experimental physics a photon has a cross section. Using a polarizer and experimenting with different wavelengths one get the result, that some wavelengths get deflected or absorbed, some range get polarised and in the best case 50% of the photons passes through the polarizer and some range of wavelengths passes through unchanged. From this it should be concluded that photons have a cross section.
The interesting point is what is the tend in the direction of movement. In a speculative way I would say that the "length" of a photon is in the range of the cross section dimension. Does this has any significance and for what I don't know.
There seems to be some confusion here.
The context of the question indicates that ``coherence'' here refers to temporal coherence. So I'll restrict the discussion to the notion of temporal coherence.
If we consider just one (laser) pulse, then the coherence length cannot be larger than the length of that pulse (measured in distance). The reason is simple. The temporal coherence is given by (is inversely proportional to) the width of the spectrum (measured in frequency). So, say the bandwidth of the light from that pulse is $\Delta\nu$, then the coherence length would be $$\ell_{\rm coh} = \frac{c}{\Delta\nu} . $$ The spectrum of the light is related to the time signal of the pulse by a Fourier transform, which means that the bandwidth is related to the duration of the pulse by $\Delta\nu=1/\Delta t$. Therefore $$\ell_{\rm coh} = c\Delta t . $$
Now, what if we have a periodic sequence of pulses? Well, it is possible to self-regulate such a pulsed laser so that the different pulses become mutually coherent. The resulting temporal coherence now becomes a very complicated function, depending on the separation distance between two points along the beam. The spectrum of such a laser looks like a comb (hence, the term comb laser). If we were to filter out just one such spectral line, we'll end up with a cw (continuous wave) laser with a very large coherence length, much larger than the length of the original pulses.
What does all this have to do with photons? Notice that we did not need to mention photons to discuss the coherence of the light. This kind of coherence is perfectly well defined in the classical context. There are other kinds of coherence, often referred to as quantum coherence where we need to consider the quantum properties of light. So from this point of view, the temporal coherence of light does not really have anything to do with photons.
Well, one can still ask how the temporal coherence of light would affect any quantum optical experiments where one detects single photons. How does the coherence length affects the interference seen for single photons? Well, it turns out that one can still use the coherence length as an indication even when we do the experiment with single photons.
So, imagine that I have some interferometer that divides the beam with a beam splitter and sends them along different paths before recombining them again to observe interference. (This is called a Mach-Zender interferometer.) I only send one photon through the interferometer at a time, but I do this many times so that I can build up an interferogram (an image to show me if there is interference). Now what if the path lengths of the two paths are different? Will I still see interference? It turns out that if the difference in path length is less than the coherence length of the light, then yes I will see interference and if the difference in path lengths is larger than the coherence length of the light, then no I will not see interference.
Coherent sources can have ultra short pulse durations
You are talking of the classical electromagnetic wave. The classical wave knows nothing about photons.
provided there is at least one photon in frequency that span broad bandwidth (and are coherent)?
One photon has no width in the framework of the current knowledge of physics. The photon is an elementary particle , a point particle , in the standard model of particle physics, i.e., has its values at an (x,y,z,t) point. Its values are energy=hnu, direction of motion, and spin + or - to its direction of motion . That is all. It is described by a quantum mechanical wavefunction, which is a complex function, Psi, whose Psi*Psi product gives the probability density for a photon to appear at an (x,y,z,t).
Does that mean that the pulse duration is infinitely long for beam of single photon (basically one frequency)?
There is no pulse duration for a photon, it is a disturbance in the four dimensional space carrying a specific energy and always traveling with velocity c.
Here is single photon at a time experiment of "photon scattering off double slit "
Note how the single hits accumulate to give a classical electromagnetic wave interference pattern. This pattern is the probability density distribution of the wavefunction of the system "photon scattering on double slits".
Can we define precisely a photon energy or is it also uncertain?
Its energy is defined by the frequency of the beam that will emerge from many such photons. It depends on the delta(energy) of the atomic transition or the compton scatter etc. A bunch of photons can have a spread in frequencies due to the energy width of the atomic or molecular transition that created them, (or the scattering setup) but a specific photon has an energy and it fixes its frequency by nu=E/h.
What happens with photons and the classical light beam where the terms "pulse" and "coherence" are clearly defined needs quantum field theory to be understood/modelled mathematically.
Hand waving :
Both the classical electromagnetic wave and the photon wavefunction depend on Maxwell's equations. The photon wavefunction obeys a form of maxwell's equations, and it is not surprising that the same frequencies appear, except that in the case of the photon, it is connected with its energy. The electric and magnetic fields of the classical em wave are built up by a superposition of the wave functions of innumerable same frequency photons. These complex wave functions have the electric and magnetic potentials in exponents, where also the frequency and the phases reside . When the functions are superposed and the Psi*Psi is taken, i.e. a measurement, an observation, the classical electromagnetic wave appears.
When one has a classical beam more complicated than a plane wave, the mathematical complexity of going from the photon quantum mechanical level to the classical light increases . It is not necessary though, because one has shown that the classical emerges from the quantum level, and one can trust classical Maxwell equation solutions to work perfectly, as long as one does not go to one photon at a time, where the quantum mechanical boundary conditions have to be considered. It is simple for the double slit , because it is still plane waves classically.
So the energy of a single photon depends on its production way, and that is subject to the quantum mechanical uncertainties and thus has a heisenberg uncertainty in the spread of the energy by its production. Onthe detection side, an equivalent quantum mechanical reaction will also obey the HUP.
There is only one photon , an elementry particle , and there is no pulse duration in the classical sense. Again the production and detection mechanism of a single photon will have a duration within the energy/time form of the heisenberg uncertainty, but is is a single photon, not a pulse.
