How to distinguish between diagonal vs horizontal/vertical polarization?

[JamesPaylow]

TL;DR

My photon has a polarization that either diagonally polarized or a superposition of horizontal/vertical polarization. How can I distinguish between the two conditions?

I have been working for some time on designing an experiment and have gotten stuck on one particular aspect. I would greatly appreciate any advice that can be offered. I'm using SPDC to produce two polarization-entangled photons. Through the course of the experiment I know that one of the photons will have a polarization that is diagonal (45 degrees) or a superposition of horizontal and vertical. I don't need to distinguish between horizontal and vertical, but need a way to be able to tell which condition is present with this photon. Is it diagonally polarized, or is it H/V polarized? I've been looking into interferometry, ellipsometry, and various other techniques, but haven't been able to nail down some specific viable technique. Of course, the less technically complex the better. Thanks in advance!

 

[Doc Al]

I know that one of the photons will have a polarization that is diagonal (45 degrees) or a superposition of horizontal and vertical.

A photon that is diagonally polarized is in a superposition of horizontal and vertical. You can always express diagonal polarization as a superposition of vertical and horizontal. (It's just a different basis.)

 

[JamesPaylow]

A photon that is diagonally polarized is in a superposition of horizontal and vertical. You can always express diagonal polarization as a superposition of vertical and horizontal. (It's just a different basis.)

I understand, but still I understand the two things to be distinct. Suppose I start with a set of two polarization-entangled photons. They are both in a H/V superposition and are othogonally polarized. One of them may or may not travel through a 45 degree linear polarizer. If it does, the entangled partner becomes -45 degree polarized. If it doesn't, the entangled partner remains in a H/V superposition. I want to evaluate the polarization basis of the partner to see if it's diagonal or still H/V.

 

[PeterDonis]

Through the course of the experiment I know that one of the photons will have a polarization that is diagonal

How do you know this?

Suppose I start with a set of two polarization-entangled photons. They are both in a H/V superposition and are othogonally polarized.

How do you know this?

 

[Nugatory]

TL;DR Summary: My photon has a polarization that either diagonally polarized or a superposition of horizontal/vertical polarization. How can I distinguish between the two conditions?

You cannot, unless you know how they were prepared and then you already know what you have even without a measurement.

This may be clearer if you are more careful with your use of natural language: There is no such thing as a diagonally (or vertically, or horizontally, or any other direction) polarized photon. Instead, you can have a photon whose state is such that it has a 100% probability that it will pass through a polarizing filter with a particular orientation.

The state that you are calling “diagonal” has a 100% probability of clearing a diagonally oriented polarizer and a 50% probability of clearing a vertically or horizontally oriented polarizer.

 

[JamesPaylow]

The ability to differentiate between circular (right-hand circular or left-hand circular) and linear (H/V superposition) polarization would also work.

 

[Doc Al]

I'm using SPDC to produce two polarization-entangled photons.

All this implies is that measurements of the photon's polarizations will be correlated with each other. The photons share a single, entangled state.

 

[Nugatory]

I want to evaluate the polarization basis of the partner to see if it's diagonal or still H/V.

But “diagonal” and “H/V” are the same state, just written differently because we’ve chosen to write it using a different basis. It’s as if we were trying to distinguish between an object’s velocity being $\vec{v}=v\vec{NW}$ and $\vec{v}=\frac{v}{\sqrt{2}}(\vec{N}+\vec{W})$ where $\vec{N}$, $\vec{NW}$, $\vec{W}$ are unit vectors in various compass directions.

The ability to differentiate between circular (right-hand circular or left-hand circular) and linear (H/V superposition) polarization would also work.

That is also impossible, for basically the same reason. What we call circular polarization can just as readily be written as a sum of linear polarizations with different phases.

 

[vanhees71]

Just start thinking about a classical plane em. wave in a given unknown elliptic polarization state. How would you measure in which state it is?

 

[gentzen]

The ability to differentiate between circular (right-hand circular or left-hand circular) and linear (H/V superposition) polarization would also work.

Sounds like you want to distinguish the (2D-)rotationally invariant states from the rest. One way to do that would be to measure (1) the total intensity, (2) the intensity through a horizontal polarizer, and (3) the intensity through a diagonal polarizer.

So you have to measure 3 of the 4 Stokes parameters. Why is that? Because the (2D-)rotationally invariant states are the one-parameter family of incoherent superpositions between circularly and anti-circularly polarized light. Except that this is actually a two parameter family, and you actually only need to determine 2 of the 4 Stokes parameters, $S_1$ and $S_2$, so at least in theory you don't need the total intensity $S_0$.

However, I strongly believe that you nevertheless need at least three different measurement configurations. But your question implicitly seems to ask for a single measurement configuration doing the same trick, or even stronger a measurement you could perform on a single photon (because you obviously cannot perform measurements corresponding to more than one configuration on a single photon). But of course, no Stokes parameter, not even the intensity $S_0$ can be determined for a single photon. (Still the question remains what is so special about having to change the measurement configuration that people can get so stuck trying to avoid it.)

 

[vanhees71]

A "single photon's" polarization state can only be measured completely on an "ensemble of equally prepared photons" as is the case with any quantum system and its observables.

 

[gentzen]

However, I strongly believe that you nevertheless need at least three different measurement configurations.

I was wrong, all I need is a polarizing beam splitter (in order to get away with just two different measurement configurations). Then I can determine the ratio of horizontal to vertical detection events, and the ratio of diagonal to anti-diagonal detection events. This measures just the "relevant part" of $S_1$ and $S_2$, without also determining the intensity $S_0$. Or maybe not, since of course I will also have learned the total number of detection events. But at least I don't need it, and don't need to worry about its accuracy. As long as both ratios are sufficiently close to 1, I can conclude that I have a (2D-)rotationally invariant state (within the limits of my measurement accuracy).

 

[JamesPaylow]

I was wrong, all I need is a beam splitter. Then I can determine the ratio of horizontal to vertical detection events, and the ratio of diagonal to anti-diagonal detection events. This measures just the "relevant part" of $S_1$ and $S_2$, without also determining the intensity $S_0$. Or maybe not, since of course I will also have learned the total number of detection events. But at least I don't need it, and don't need to worry about its accuracy. As long as both ratios are sufficiently close to 1, I can conclude that I have a (2D-)rotationally invariant state (within the limits of my measurement accuracy).

Thank you for your thoughtful replies. Do you mean a polarizing beam slitter or normal?

Edit: I think the challenge with using a BS or PBS is that I have either H/V or one specific something else (whether that is D, A, RHC, or LHC). So, I don't need to distinguish between H and V. Also, if I have a detector in one output of the BS that just detectors H or just detects V then presumably I'm also going to see about 50% hit rate with H/V superposition, which is the same thing I'd see for D, A, RHC, LHC since they are going to be randomly routed also roughly 50% of the time. Maybe I'm misunderstanding your suggestion.

 

[vanhees71]

You need a polarizing beam splitter, quarter- and half-wave plates, and two photodetectors. Then you can do "polarization-state tomography" (of course if you work with single electrons you need a single-photon source to prepare sufficiently large ensembles of "equally prepared photons"). Here's a corresponding lab instruction:

https://users.physics.ox.ac.uk/~lvovsky/443/2017/lab.pdf

 

[gentzen]

Do you mean a polarizing beam slitter or normal?

I meant a polarizing beam splitter. I edited my post, to clarify what I meant.

 

[JamesPaylow]

@vanhees71 , @gentzen super useful. This is what I needed. Thanks so much!