I Polarization and Stimulated Emission

[Erik Ayer]

TL;DR Summary

If a photon in a polarization superposition with respect to a given axis causes stimulated emission, what is the polarization of the emitted photon?

If a laser beam has a definite diagonal polarization, it is in a superposition of horizontal and vertical polarization. If that beam were then sent through another lasing medium and caused stimulated emission of another photon (or possibly several photons), would those be in superposition between horizontal and vertical polarization?

What I would think is that they would be in superposition, but if the polarization were measured, they would have the same polarization - both vertical or both horizontal.

 

[Dale]

Lasers are coherent states. Coherent states are eigenstates of the creation operator so adding or removing a photon does not change the state.

EDIT: this post is incorrect. See below

 

[Erik Ayer]

Thank you for the reply.

So the original photon would still be in superposition between horizontal and vertical polarization ("does not change the state"). Is the newly emitted photon also in a superposition, and if measured will it match the original photon?

 

[Dale]

Is the newly emitted photon also in a superposition, and if measured will it match the original photon?

There isn’t a separate new photon. There is just the coherent state with an indefinite number of photons.

 

[Erik Ayer]

If the beam of an indefinite number of photons goes through a beam splitter, a definite number will transmit and a definite number will reflect. But that's beside the point, actually. If polarizations are measured, I'm wondering whether they would all be the same (all vertical or all horizontal).

I think the answer is "no". If diagonal photons cause stimulated emissions of other photons, all of those will be diagonally polarized and measuring horizontal or vertical polarization of them later will give random results.

What about the following: the original beam is split with a regular, old beamsplitter and the two sub-beams reflected with mirrors so they are right next to each other and parallel. If those both go through a lasing medium, they will cause an indeterminite number of stimulated emissions. Then a single photon would start out in superposition between the two beams and cause more photons to joing those beams, but would they all still be in superposition between which beam they were in? If measured, would they all be in one beam or the other?

The question comes down to asking whether the stimulated emission causes the original superposition to collapse.

 

[Dale]

a definite number will transmit and a definite number will reflect

I don’t think that is correct. I think that you wind up with two coherent states, each with a given fraction of the expected number of photons but neither with a definite number.

Regarding your question about the state of the lasing material with two beams exciting it. Unfortunately that is beyond my knowledge. I don’t know what the resulting state would be.

 

[PeterDonis]

Coherent states are eigenstates of the creation operator

No, they're not. They're eigenstates of the annihilation operator. So removing a photon doesn't change the state, but that is not true of adding a photon.

(The creation operator actually cannot have any eigenstates. If you look at how that operator acts on the photon number eigenstates, this should be obvious.)

 

[Dale]

No, they're not. They're eigenstates of the annihilation operator. So removing a photon doesn't change the state, but that is not true of adding a photon.

Oh, that is a big mistake in the context of this thread. I had thought that creation was just annhilation of a negative number of photons.

 

[PeterDonis]

If the beam of an indefinite number of photons goes through a beam splitter, a definite number will transmit and a definite number will reflect.

No, this is not correct. The beam splitter splits the entire state; it turns into a superposition of states that go each way.

I think that you wind up with two coherent states, each with a given fraction of the expected number of photons

More precisely, the beam splitter splits the state into two "copies" of the original state, each with a different complex coefficient, and each located in one of the two possible output paths of the splitter.

If you put a detector in each output path of the splitter, the measured intensity of photons at each detector will be roughly half the original intensity of photons before the splitter. But that is because of the measurement being made, which, heuristically, forces each individual "photon" to choose one arm or the other instead of being in a superposition of being in both.

 

[PeterDonis]

If a laser beam has a definite diagonal polarization, it is in a superposition of horizontal and vertical polarization. If that beam were then sent through another lasing medium and caused stimulated emission of another photon (or possibly several photons), would those be in superposition between horizontal and vertical polarization?

As far as I know, stimulated emission photons have the same polarization as the original "trigger" photon. So if the original photon had diagonal polarization, so would the stimulated emission photons.

 

[Erik Ayer]

Unfortunately it's beyond my knowledge too :) It seems like an entangled state could be produced, as long as the stimulated emission could be done in such a way that which beam the seed photon was in wasn't distinguishable or didn't create an effect that would carry information as to which beam it was in.

 

[Erik Ayer]

As far as I know, stimulated emission photons have the same polarization as the original "trigger" photon. So if the original photon had diagonal polarization, so would the stimulated emission photons.

Yeah, I came to that conclusion after burning up a few processor cycles. Initially I was thinking a beam could be split via polarization, then the two sub-beams put on top of each other before going through the lasing medium. The idea was that a photon would be in superposition of horizontal and vertical, it would cause stimulated emission that would match the polarization and be in a superposition of horizontal and vertical, then the two (or multiple) photons could be split apart with a polarizing beam splitter and the "clump" would go off in one direction or the other.

Unfortunately, recombining the beams gives the same polarization before they were split, so the emitted photons would all have that polarization and splitting them later would be random per photon. They would not all go one way or the other - they would not be entangled via polarization.

What about splitting the beam, then putting each sub-beam through a quarter wave plate to give them clockwise and counterclockwise polarization, recombining them and sending that through the lasing medium? I suspect that upon recombining, the polarization would again mix into something which the stimulated photons would match and splitting them apart would result in random, per-photon direction.

The other possibility would be to split the beam, reflect the sub beams so that they are next to each other and parallel, then send that through the lasing medium. A photon would be in superposition between the two sub-beams, and stimulated photons would also be in superposition between sub-beams but upon detection would have to all be in the same beam as the original photon. My suspicion is that the emission process would collapse the superposition to a mixed state so again, no entanglement.

 

[PeterDonis]

recombining the beams gives the same polarization before they were split

Yes.

What about splitting the beam, then putting each sub-beam through a quarter wave plate to give them clockwise and counterclockwise polarization, recombining them and sending that through the lasing medium? I suspect that upon recombining, the polarization would again mix into something which the stimulated photons would match

The polarization after recombination would be a linear combination of clockwise and counterclockwise, which is either horizontal or vertical depending on which combination you pick.

The other possibility would be to split the beam, reflect the sub beams so that they are next to each other and parallel, then send that through the lasing medium. A photon would be in superposition between the two sub-beams

More or less, yes. Describing this in ordinary language is difficult.

and stimulated photons would also be in superposition between sub-beams

No, they wouldn't, because only one of the sub-beams will match the atomic energy level transition that the stimulated emission photons are produced by. That's a key piece of the puzzle that you are missing: for stimulated emission to happen, the state that the trigger photon is in has to be a state that can be produced by an available energy level transition in the atoms of the lasing medium. Any such transition will give rise to a photon with a definite energy and polarization, which the trigger photon has to match. Only one of the sub-beams, as you've set things up, will be able to match it.

(Note that in the previous examples, where you recombined the sub-beams, you were implicitly assuming that the recombined state matched an available transition in the lasing medium, which wasn't an issue since there was only one state. Now you have two so that extra condition of matching an available transition has to be made explicit because it affects the analysis.)

 

[Erik Ayer]

Why would only one sub-beam match the atomic energy level transition? Photons initially split with a regular, old beamsplitter would not have different energy, just that the energy is in a superposition of being in one beam or the other. Now, if the initial photons were downconverted into to photons of different wavelength, it would make sense that each may or may not match the atomic transition energy.

Note that my intention is not to disagree but rather to illustrate my thought process in order to make it easier to point out where it's wrong.

 

[PeterDonis]

Why would only one sub-beam match the atomic energy level transition?

Because only one beam will have the right polarization. The two beams have different polarizations because of the splitter.

 

[Erik Ayer]

If the beam splitter was a regular, old beam splitter and *NOT* a polarizing beam splitter, there would not be a difference in the polarizations of the beams, correct?

I think I've mentioned variations that would use a PBS, but in this case I'm not. A beam would be split, then reflected close together and parallel. The photons should then be in superposition between the two sub-beams with neither having a definite polarization. Then, if stimulated emission occurred, would extra energy also be in superposition between the two sub-beams?

 

[PeterDonis]

If the beam splitter was a regular, old beam splitter and *NOT* a polarizing beam splitter, there would not be a difference in the polarizations of the beams, correct?

I'm not sure. The splitter changes the phase in one of the output beams. I would have to look at the math.

 

[Erik Ayer]

I guess what I'm getting at is the possibility that bunches of photons could be entangled. If they were all in superposition between two beams but when measured were all in the same beam, then the two beams could be set to interfere. If, however, the bunches were split roughly in half with a (regular old) beamsplitter, then some could interfere while the others would either erase or preserve the which-way information.

The advantage of using a lasing medium to "amplify" the light would be that it would all stay in phase. Assuming that signalling is impossible, which is a fairly good assumption, there has to be something critically wrong with the idea of duplicating the photons. I'm pretty sure the problems include that either the photons are re-merged together and the polarization superposition is lost, or with spatially-separated beams, the superposition will decohere.

 

[PeterDonis]

I guess what I'm getting at is the possibility that bunches of photons could be entangled.

Photon sources can certainly emit photons with various forms of correlations, two of which are called "bunching" and "antibunching":

https://en.wikipedia.org/wiki/Photon_antibunching

Pairs of entangled photons can also be emitted by parametric down conversion:

https://en.wikipedia.org/wiki/Spontaneous_parametric_down-conversion

These kinds of phenomena can't really be talked about properly using vague ordinary language. You need to use precise math to describe the states and processes involved.

 

[Erik Ayer]

Parametric down conversion has its good points, but the phases of the downconverted photons get all messed up. I haven't heard of bunching or antibunching but I'll read through the link you posted.

The initial idea I has was to have a surface that could be either transmitting or reflecting, but could also be a superposition of both. Then the light that hit it would become entangled with the state f the surface. Such a surface doesn't exist as far as I know, but are there other materials that can, say, deflect light at different angles and also be set to a superposition of states or diffraction angles? Again I bet it isn't possible since it would make certain things possible that shouldn't be possible :)

 

[A. Neumaier]

The creation operator actually cannot have any eigenstates.

Indeed, assuming that $\psi=\sum_k \psi_k|k\rangle$, the condition $a^*\psi=\lambda\psi$ gives a recurrence for the coefficients $\psi_k$ that force in turn $\psi_0,\psi_1,\ldots$ to be zero. Hence $\psi=0$.

Thus the behavior expected from matrices in place of operators is violated. Indeed, the relation between eigenvalues of non-selfadjoint operators and their adjoints is complicated in general.