Entangled photons and polarisation.

[Dadface]

I have been looking up the practical details of certain quantum eraser type experiments and got stuck on a couple of points. I refer to a particular experiment which can be found by googling:

"A DOUBLE SLIT QUANTUM ERASER EXPERIMENT" (Sorry I don't know how to include the address)

In the experiment which path information is achieved by placing quarter wave plates in front of the slits. It is claimed that the function of the plates is to change the linear polarisation of the incoming photon into circular polarisation, the direction of which depends on which of the two plates the photon passes through.
That seems to make sense but isn't elliptical polarisation the more likely outcome? I thought that circular polarisation was a special case when the plane of the electric vector of the incident photon was at 45 degrees to the principle axis of the plate. Furthermore I thought that the plane of polarisation of the photon wasn't known until it was detected.
It may be a trivial point but It's making me wonder if I'm overlooking something.
Thanks

 

[Cthugha]

That seems to make sense but isn't elliptical polarisation the more likely outcome? I thought that circular polarisation was a special case when the plane of the electric vector of the incident photon was at 45 degrees to the principle axis of the plate. Furthermore I thought that the plane of polarisation of the photon wasn't known until it was detected.

The output polarization of course depends strongly on what goes in. Circular polarization indeed is a special case, but in that kind of experiment you can always assure that the angle between the incoming polarization and the relevant axis of the waveplate is 45 degrees.

In this kind of entanglement the polarization you get out of the non-linear crystal for your photon pair depends on how your crystal is cut. You need to realize phase matching and there are just a few angles and polarizations which obey all the necessary conservation rules simultaneously. Indeed the exact polarization of any single photon "emitted" from the non-linear crystal is not know a priori, but you can tailor the crystal such that you get a state where either photon 1 is horizontally polarized and photon 2 is vertically polarized or vice versa. So although you do not know the exact polarization, you know that it will always be at 45 degrees with respect to the relevant axis of the waveplate.

 

[Dadface]

The output polarization of course depends strongly on what goes in. Circular polarization indeed is a special case, but in that kind of experiment you can always assure that the angle between the incoming polarization and the relevant axis of the waveplate is 45 degrees.

In this kind of entanglement the polarization you get out of the non-linear crystal for your photon pair depends on how your crystal is cut. You need to realize phase matching and there are just a few angles and polarizations which obey all the necessary conservation rules simultaneously. Indeed the exact polarization of any single photon "emitted" from the non-linear crystal is not know a priori, but you can tailor the crystal such that you get a state where either photon 1 is horizontally polarized and photon 2 is vertically polarized or vice versa. So although you do not know the exact polarization, you know that it will always be at 45 degrees with respect to the relevant axis of the waveplate.

I have spent a long time searching on this but couldn't find any mention that the crystal could be tailored in such a way. It was probably in the literature but I probably missed it or misunderstood it. Your post has clarified it all. Thank you so much.

 

[Cthugha]

Phase matching (and therefore crystal geometry) is of crucial importance for many effects in non-linear optics (including spontaneous parametric downconversion and second harmonic generation).

If you are interested in details, you might want to read some older PhD theses on entangled photons. Paul Kwiat's thesis is for example surprisingly readable (http://research.physics.illinois.edu/QI/Photonics/theses/kwiat-thesis.pdf). It does not explain all the details, but it gives the necessary references.

 

[Dadface]

Thank you again. I have looked at the link and it seems promising.It will take some time for me to go through it.
I will be grateful if you are able to clarify a point made in your previous post regarding the tailored 45 degree angle. Is the angle known to be 45 degrees but only on arrival at the waveplates? For any location between the crystal and waveplates is the angle still known to be 45 degrees or is it unknown?

 

[Cthugha]

I will be grateful if you are able to clarify a point made in your previous post regarding the tailored 45 degree angle. Is the angle known to be 45 degrees but only on arrival at the waveplates? For any location between the crystal and waveplates is the angle still known to be 45 degrees or is it unknown?

Assuming that we are discussing only the right emission angles (where you get entangled photons) and have chosen the correct geometry to create a Bell state, the state may look like something that: $$\frac{1}{\sqrt 2}(|H>_a \otimes|V>_b +|V>_a \otimes|H>_b) .$$

So you know that either photon will either have horizontal or vertical polarization when you perform a measurement. So if you place the waveplate at 45 degrees (assuming horizontal as 0 degrees and vertical as 90 degrees), you know that you will get a +/- 45 degrees angle to the polarization of the photon (unless you do something to the photons beforehand, of course).

 

[Dadface]

Does the photon have either vertical or horizontal polarisation regardless of where the measurement is made? Suppose,for example, that the experiment was correctly set up with the waveplates being at a distance of one metre from the source, but then this distance was reduced to say 0.5m with all other adjustments being kept unchanged. At this reduced, or any other distance, would the angle still be 45 degrees on detection?
Sorry if the answer is given in your equation. I don't even recognise the symbols. If I ever did this sort of maths it was donkeys years ago and I have forgotten it. Thank you .

 

[Cthugha]

Suppose,for example, that the experiment was correctly set up with the waveplates being at a distance of one metre from the source, but then this distance was reduced to say 0.5m with all other adjustments being kept unchanged. At this reduced, or any other distance, would the angle still be 45 degrees on detection?

For polarization entanglement, the distance between the crystal and the waveplate does not really matter. For momentum entanglement, the distance can become an issue (as a slit also acts as a momentum filter - placing it further away means a narrower range of momenta makes it through the slit).

 

[Dadface]

For polarization entanglement, the distance between the crystal and the waveplate does not really matter. For momentum entanglement, the distance can become an issue (as a slit also acts as a momentum filter - placing it further away means a narrower range of momenta makes it through the slit).

Thank you for confirming that. I had a reasonably good understanding of the experiment but until your posts the issue of polarisation angle was a sticking point for me. Interesting stuff.
Thank you again.