Can photons move in parallel? Is it possible that two photons move in parallel, on the same trajectory - having the same wavelength, but differ in phase?


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*Is it fundamentally possible on the level of quantum mechanics?

*Is it potentially possible to build an experiment demonstrating it?

*Was such an experiment actually done?

 A: 
Is it possible that two photons move in parallel, on the same trajectory - having the same wavelength, but differ in phase?

Photons are quantum mechanical entities/particles listed in the elementary particle table which is the basis of the standard model of particle physics.
When measured individually, it just gives an (x,y,z,t) , by interacting with some other particles and an energy can be measured equal to h*nu , where  nu is the frequency of the macroscopic classical electromagnetic wave that can be built up out of an enormous number of same energy photons.
As a quantum mechanical entity it has a wave function, although it is not often thought about, since classical electromagnetism describes so well the behavior of light.  It is the wavefunction which will answer whether two photons can be on the same trajectory . The wavefunction of a single photon is a complex function and the electric and magnetic fields appear in the phases of this complex function. In an ensemble of photons, a light beam, they build up the E nd B fields of the classical light wave. The quantum field theoretical framework is used for the calculations,  where creation and annihilation operators operate on the free photon wavefunction.
For two photons:
Photon photon interactions are so small they can be ignored for frequencies less than gamma ray frequencies. Therefore it is only the superposition of the two photon wavefunctions that will have an effect. The square of this superposition will give the probability density for finding the two photons in (x1,y1,z1,t1) and (x2,y2,z2,t2)


Standing wave interference pattern (showing the optical intensity) from the superposition of two elliptical Gaussian beams under some angle.  

italics mine
This by definition is  the probability distribution of adding two photons with the spatial and temporal phases of the laser beams. 

Is it fundamentally possible on the level of quantum mechanics?

Yes.

Is it potentially possible to build an experiment demonstrating it?

There have been a number of experiments with single photons hitting a double slit. Conceptually one might design an experiment on the lines of the figure above where the beams are controlled two photons at the time.

Was such an experiment actually done?

It seems that people have been looking at two single photon interference patterns 
