Do you have in mind a Mach-Zehnder inerferometer where in one of the arms a 1/4 wave plate has been put to rotate the plane of polarization at PI/2 to the original one?

I thought about such a scheme but would like to know is it realized?

Your answer is in the attached screenshot. Abbreviations used are: QWP: Quarter wave plate, HWP: Half wave plate, LCP: Left circularly polarized and RCP: Right circularly polarized.

Thank you!

Do you know what happens to the spin. The linear polarized photon has s=0 but the LCP s=1 or -1 (nevermind).

LCP is -1, RCP is 1. See link.

https://en.wikipedia.org/wiki/Photon_polarization#Spin_operator

This is in fact matter of conevention. What I ask is how about spin conservation. Angular momentum h/2pi must stay in the plate. If so, this not what I  am looking for.

Up to my knowledge,one photon can be split in to two photons by only nonlinear process (ex: down conversion process). where one linear polarization photon either horizontal or vertical photon can be split in two two orthogonal linear popularization photons. I don't know, it may be possible that split a linear polarized photon into left and right polarized photons by nonlinear process

It is advisable not to talk about "splitting photons" in any linear system. The photonic nature of light is not evident in propagation, there is just the propagating field which can be split or manipulated using a polarisation beam splitter or a birefringent element. Subsequent detection of each beam exhibits quantum like behaviour.

As A S V Rao and Richard  Epworth point out, you cannot split a single photon by a linear process.

I don't know the context, but I suspect Dirac described a linearly polarised photon as the superposition of two orthogonal, circularly polarised, quantum states.  If you detect the photon using a sensor which discriminates between circular polarisations, then you will record either a single right polarised photon or a single left polarised photon, but never two photons.  Detecting more than one photon of the same wavelength would violate conservation of energy.

Non-linear down-conversion is used to create entangled pairs of orthogonally polarised photons, each with half the energy (and twice the wavelength) of the original photon.

Yes I know. Its splitting of the wave function. But talking of photon splitting is more concise.

Yes I know. Its splitting of the wave function. But talking of photon splitting is more concise.

I can reformulate the question in the form: How is the Dirac statement proved experimentally?

Yes, of course there is: cyclically birefringent crystals for example.

When the axis of polarization is changing while passing through the crystal it is precisely this what is happening.

Polarisation is an internal degree of freedom, like the z-component of a spin for example. An electron may also have different components in its wave funtion, but it cannot be split (at least not following nowadays knowledge).

In the same way one must see the photon: it may have different internal degrees of freedom, but you cannot split it into two "halves". I think, QED is old enough for being checked to the details, and polarisation is NOT a detail; it's essential.

You could of course use this internal degree of freedom to create entangled states in two photons...(hyper-entanglement)

Is this some kind of convincing?

Yes, this superposition state can be experimentally measured using a refraction of an optically active medium such as quartz or a sugar solution in the form of a prism. Left- or right polarized photons to be divided by the angle refraction, and their statistics will indicate the values of the two coefficients of the superposition state. But I not recommend naming this superposition state as the "linear polarized photon".

Andreas thank you!

It was very hard to find something about cyclically birefringent crystals. In fact I found some material  stating that the effect of splitting left from right rotation is 1000 to 100 000 times weaker than horizontal/vertical plane splitting. I lost this and can not find it again. I would be indebted if you can point out on some source for cyclically birefringent crystals. Nevertheless I am convinced (partially). Look further in the answer to Vahram.

Vahram,

I think that sugar and quartz only rotate the polarization plane. If it is a consequence of lag of left to right state, than their ways are not split (which was my problem).

I came recently about another puzzling aspect of this superposition of left and right polarized states in the 'linear' polarized photon. There are dichroic materials which can absorb only left photons. What is their effect on this superposition? They would absorb the left state and let the right. It turns out that there are two photons in a linear photon! The mechanical momentum of the dichroic material must be measurable in principle. It would be expected that there is some energy hv absorbed also but it must be in the right photon too. Conservation of energy ..? If there is no momentum and not energy absorbed in the dichroic the conservation laws are also severely harmed. At least something happens to that photon (linear to right which is easy to show) but nothing happens to the causer of this event (namely to the dichroic). I see that there must be wavefunction collapse for the photon and in fact the question boils out to that is something physical happening to the machinery causing the collapse?

Answer to Ilian Peruhov:

Well in turning the direction of polarization you are there. The right polarized electron travels with a different speed than the left oriented one. With this I mean cyclically birefringent.

An absorber: yes, in this way you can test whether a linear polarized beam is made of pairs of photons or whether it is only a superposition, as I think it is.

You have to make "single" photon experiments and be sure that "it is" linearly polarized, and with a right- or left-absorber in the middle. If after an absorbtion nothing is left, then polarization is an inner degree of freedom, as I think it should be. If after absorbtion there is the corresponding left or right polarization left over, then it is not an inner degree of freedom but a particle property. Then two different photons must exist, as you say: one with left polarization and one with right polarization.

Then however, you would have to explain, why this two photon process (creating a linearly polarized "photon") is not supressed!?

So either lots of work to be done, or it really is an inner degree of freedom.

Conservation laws: of course energy conservation and also momentum conservation must also be satisfied. The momentum transfer of a photon (or torque transfer, which has been measured, but I don't remember where) can be easily satisfied by each massive particle which carries the interaction: it is so tiny.

The absorber also feels the momentum kick; it is not unharmed by the interaction.

Of course, the interaction of the beam with the medium, not a trivial task.

To have a definite answer, you should find out what kind of a beam in that case, from which kind of photons it is made. Namely:

1) only from the same left- or right-polarized photons

2) only biphotons - pairs of left- and right-polarized photons

3) an electromagnetic wave in the classic sense with a particular polarization (linear, circular, elliptical) or not having.

The physical processes of interaction of these beams with the medium significantly different, and the corresponding beam should be prepared for this experiment.

On the other hand, in the case of single photons, which have only two states of polarization and do not overlap each other, we have unambiguous character of interaction. No imagination could not circumvent the laws of conservation of energy and momentum.

Similarly, chiral mediums (such as quartz, sugar solution) have different refractive index for single photons left- and right-polarized and can be spatially separated when using a prism of such media (see attached image).

Hello Ilian,

since you are asking about absorption: in an absorption process the photon is always eintirely destroyed. 

Now, a situation, in which such a distinction between absorption of left and right polarized photons exists is given is in magnetically polarized materials (e.g. core level x-ray absorption from strongly magnetically polarized atoms). That would be called circular magnetic (x-ray) dichroism. Alternatively check the physics of the optical absorption and emission in the Zeeman effect on hydrogen. There is a very mundane (i.e. down to earth) way to describe what will happen.

The (probability of) optical absorption is essentially given by the dipole matrix element (squared) and energy conservation. For simplicity, think of a nonmagnetic initial state |i> (e.g. a rare gas atom) and two (orthogonal) final states |f+> and |f->, which are degenerate in absence but do energy split in a magnetic field. Let these be so that the dipole matrix elements are

||2 = M = ||2

and 

=  = 0

where T+ and T- are the interaction hamiltonians for "+" and "-" circular polarized radiation.

(This is nothing very special, just check the dipole selection rules.) Now let us consider the mutually orthogonal three photon polarization states |+>, |-> and |z> (analogous to the magnetic quantum numbers of angular momentum L=1. Note though, that since light polarization is always transverse, the z polarization requires a different propagation direction. Exactly what is observed in the Zeeman effect.)

What you will see is that left and right absorption strength is the same, and - in absence of a magnetic field - they will occur at the same photon energy. As a magnetic field is turned on (along the direction of photon propagation!!), both absorptions will continue to be of same strength but split in energy.

Now let's prepare photons in superimposed (again mutually orthogonal) polarization states, e.g.

|A> = (1/sqrt(2))*(|+> + |->)

|B> = (1/sqrt(2))*(|+> - |->)

Since we are dealing with linear physics, the corresponding dipole interaction Hamiltonians are given by the analogous superpositions:

TA =  (1/sqrt(2))*(T+ + T-)

TB = (1/sqrt(2))*(T+ - T-)

You can find out that one corresponds to linear polarization along x, the other along y (but for a factor of i). All this is very similar to your more recent question on spins, actually.

now you will easily find that

||2 = ||2 = 0.5*M

||2 = ||2 = 0.5*M

I.e. with linear polarization both final states can be reached, and the absorption probability for each is halved (This is the "true" difference between the situations you asked. The photon is always in one state. Depending on what the states are, absorption probabilities vary. When absorption happens, the entire photon "is gone").

In absence of a magnetic field the spectrum for all four polarization states will look the same. When the magnetic field is turned on, the linear polarized absorption spectra will both look the same and exhibit two absorption peaks each at exactly those non-degenerate energies where each of the circular spectra gives only one absorption line.

So, coming back to the formulation of your question: within linear physics (thus excluding 1photon -> 2photon conversions discussed by others) in a setup which forces initially circularly polarized photons to tell whether they-'re linearly 'x' or 'y' polarized you will always only get one single answer per photon (ad you can't tell in advance whether that answer shall be 'x' or 'y'). But in the limit of large numbers of asking the question to identically prepared photons, you will get both answers with the same abundance.

this is true in both directions, i.e. by taking initially "+" or "-" photons and asking them whether they're "x" or "y" like. 

|+>, |->, |z>

and

|x>, |y> and |z>

are two equivalent sets of polarization basis states to describe the polarization of e/m radiation.

From how we usually teach atomic and quantum physics (with atomic eigenstates as eigenstates of total angular momentum and its z component [J2, JZ]), and since their magnetic splitting is most directly linked to the spectra with circular and z polarization, the {|+>, |->, |z>} is - as far as I see - given some preference, at least in the contexts I work in.

I can not concentrate on your remarks. The idea here is as far as I remember:

A linear photon which is linear combination of left and right circulation photon (Dirac), in the framework of the wavefunction, passes thru a medium (crystal) and thereafter is reduced to right circulating photon. The photon has passed thru so its energy has not changed. Its just QM reduction of the wavefunction. But it is already right photon. So the angular momentum of the crystal must have changed by 1h. The crystal must physically rotate. But there is not energy for this, as it is in the right photon totally.

I write this to remember what I mean later.

Ine aspect in the formulation: it is not a superposition of a "left" and a "right" photon, but the photon is in a superposition of left and right polarization states.

And if the absorption is a transition owing to the part of the dipole transition element involving the T- operator, then the final state of absorption does possess a different angular momentum compared to the initial state. That is not necessarily a state of rotation of the crystal.