I The HOM Effect, Path Entanglement, and Generation

[Erik Ayer]

TL;DR Summary

HOM seems to entangle the paths of pairs of photons, but what is a good way to generate them?

Reading about the HOM effect, two photons come to a half-silvered mirror from opposite sides but they both end up going the same way - one always reflects while the other transmits. Which path they take is random but they both take the same path. Is the path they end up on initially a superposition of both, until they run into something and pick a path? If so, this sounds very much like an entangled state.

Also as I have read, the HOM effect depends on the two initial photons being indistinguishable. If they have definite and opposite polarizations, that would be distinguishable but having different phases would not. How close to simultaneous do the two photons have to hit the half-mirror? One experiment used photons downconverted with a non-linear optic and equal path lengths, but "equal" really means very close - getting them exact with infinite precision isn't possible.

Another experiment I read about used quantum dots to emit photons very regularly, and shifted one path to the half-mirror such that a photon and the next would be the ones to meet at the half-mirror (there must have been two quantum dots each emitting a photon). I would think that the phases of those two photons would not match, therefore the HOM effect doesn't require it and different phases don't make photons distinguishable. This would also mean the two photons, while ending up on the same path, don't have the same phase necessarily.

If two identical lasers were used as the sources for HOM, what percentage of the outgoing light would be in HOM pairs (in HOMiny)? It seems like it would be very low, since the probability of two photons to hit the mirror simultaneously would be very low. Of course, this depends on just how simultaneous they have to be.

What about using a lasing medium such that photons can be sent through it, and the peak probability is that one stimulated emission will occur so that most of the time, two photons come out for each one sent it (the lasing medium would not have mirrors on the side, but just be used for amplification)? Then the resulting beam could be put through a beam splitter that would break roughly half the pairs apart. Those two paths could be the input to the HOM half-mirror and, on average, half would then be path-entangled pairs. That should be a lot better than sending in beams from two different lasers.

I think I know just enough to spout a bunch of BS. What I'm hoping is that someone will be kind enough to point out said BS and the next time I ask questions, there will be slightly less BS :)

 

[PeterDonis]

Reading about the HOM effect

Also as I have read

Another experiment I read about

Please give references.

I think I know just enough to spout a bunch of BS. What I'm hoping is that someone will be kind enough to point out said BS and the next time I ask questions, there will be slightly less BS :)

If we don't know where you're getting your current BS from, we can't help you improve it.

 

[Erik Ayer]

Fair enough, the first thing is the Wikipedia page:

https://en.wikipedia.org/wiki/Hong–Ou–Mandel_effectHere is the example using SPDC to create photon pairs, split apart by polarization, then subjected to HOM interference:

https://advances.sciencemag.org/content/4/5/eaap9416.fullHere is the experiment where two photons are emitted but one is delayed before reaching the beam-splitter:

https://singlequantum.com/technology/applications/One thing I realized recently is that, since HOM depends on a beam splitter where one of the reflections will not cause a pi phase shift but the other reflection will, is that if the photons exit one way, they will have an additional phase shift of pi. If the two beams are overlapped and interfered, that would cause interference and anti-interference to be present, as did the quantum eraser experiments.

Hopefully this removes one piece of BS :)

 

[Cthugha]

Reading about the HOM effect, two photons come to a half-silvered mirror from opposite sides but they both end up going the same way - one always reflects while the other transmits. Which path they take is random but they both take the same path. Is the path they end up on initially a superposition of both, until they run into something and pick a path? If so, this sounds very much like an entangled state.

The path is a superposition. Whether or not such states should be considered as an entangled one is an old question. However, you do not need HOM to create such a state. You can also take just a single photon, put it on a beam splitter and it will be in a superposition state of being transmitted and reflected. This is the same scenario. Some discussion on whether such a dual-rail qubit should be considered as an entangled state can be found in Reviews of Modern Physics 81, 299 - 332 (2009) in section V.b (Arxiv link: https://arxiv.org/abs/quant-ph/0511044).

Also as I have read, the HOM effect depends on the two initial photons being indistinguishable. If they have definite and opposite polarizations, that would be distinguishable but having different phases would not. How close to simultaneous do the two photons have to hit the half-mirror? One experiment used photons downconverted with a non-linear optic and equal path lengths, but "equal" really means very close - getting them exact with infinite precision isn't possible.

First: single photon states do not have any well-defined phase. They cannot. Besides: Getting the path lengths equal is not a problem in practice. Using mechanical delay lines you can get femtosecond precision or slightly better quite easily. In practice, it is more complicated to get the "duration" of the two single photon wavepackets to be equal. This is given by the Fourier transform of their spectral power density.

Another experiment I read about used quantum dots to emit photons very regularly, and shifted one path to the half-mirror such that a photon and the next would be the ones to meet at the half-mirror (there must have been two quantum dots each emitting a photon). I would think that the phases of those two photons would not match, therefore the HOM effect doesn't require it and different phases don't make photons distinguishable. This would also mean the two photons, while ending up on the same path, don't have the same phase necessarily.

Again: a single photon does not have any preferred phase. If you plot e.g. the Wigner function of a single photon state in phase space, it is perfectly spherically symmetric. Talking about the phase of single photon states is like talking about the momentum of position eigenstates. Besides: taking two quantum dots is possible, but rarely done as the spectral shape of the emitted photons from two QDs is rarely the same. Usually it is easier to use just two consecutively emitted photons from a single QD. The typical waiting time between two consecutive excitation laser pulses is about 13 ns.

If two identical lasers were used as the sources for HOM, what percentage of the outgoing light would be in HOM pairs (in HOMiny)? It seems like it would be very low, since the probability of two photons to hit the mirror simultaneously would be very low. Of course, this depends on just how simultaneous they have to be.

HOM works only, if you have exactly one photon arriving at each entrance port of the beamsplitter, not 2 and 1 not 0 and 1 and not 5 and 6. As the photon number distribution of laser pulses is Poissonian, for pretty much any combination of mean photon numbers, the probability to have a single photon present in both laser pulses simultansously is very close to 0.

What about using a lasing medium such that photons can be sent through it, and the peak probability is that one stimulated emission will occur so that most of the time, two photons come out for each one sent it (the lasing medium would not have mirrors on the side, but just be used for amplification)? Then the resulting beam could be put through a beam splitter that would break roughly half the pairs apart. Those two paths could be the input to the HOM half-mirror and, on average, half would then be path-entangled pairs. That should be a lot better than sending in beams from two different lasers.

That is still horrible, as you will get the 50/50 situation only half of the time and when sending single photons through a lasing medium the probability that only one stimulated emission event will occur is close to 0. Usually many more will take place.

 

[Erik Ayer]

Thank you very much for the reply!

The path is a superposition. Whether or not such states should be considered as an entangled one is an old question. However, you do not need HOM to create such a state. You can also take just a single photon, put it on a beam splitter and it will be in a superposition state of being transmitted and reflected

Yes, this works for single photons. What I was thinking is that, with HOM, there are two photons. Their paths are in a superposition state, but whatever that turns out to be when interacting with stuff, they will be together. The paper you linked is going to beat up my brain but I'll keep plugging away at it.

First: single photon states do not have any well-defined phase.

Now that's interesting, but makes sense. This must be something that gets entangled in that photons won't have a definite phase but are, say, pi shifted from each other. With HOM, one of the reflections will cause a pi phase shift so, if the beams from the two paths are overlapped and interfered, it seems like there will be both interference and anti-interference, adding up to a big mess.

That is still horrible, as you will get the 50/50 situation only half of the time and when sending single photons through a lasing medium the probability that only one stimulated emission event will occur is close to 0. Usually many more will take place.

With a HeNe laser, the average is 1.05 photons per one photon passing through the laser tube once. So the range varies, and of course gas would have a much smaller chance of stimulated emission than a solid. It would seem like this could be engineered tohave a peak probability of 1 stimulated emission, either with a very small solid state lasing medium of, like, 20 HeNe tubes in series.

It sounds like getting this effect (HOM) is very tricky, and possibly the best way is to downconvert a beam into pairs with SPDC.

 

[Cthugha]

Yes, this works for single photons. What I was thinking is that, with HOM, there are two photons. Their paths are in a superposition state, but whatever that turns out to be when interacting with stuff, they will be together. The paper you linked is going to beat up my brain but I'll keep plugging away at it.

Most of the paper is unrelated to the topic at hand. However, the state is not really show entanglement between the photon properties. The photons are classically correlated in that sense. You only get entanglement between a two-photon state in one output port and a vacuum state in the other port. That is all there is.

Now that's interesting, but makes sense. This must be something that gets entangled in that photons won't have a definite phase but are, say, pi shifted from each other. With HOM, one of the reflections will cause a pi phase shift so, if the beams from the two paths are overlapped and interfered, it seems like there will be both interference and anti-interference, adding up to a big mess.

As there are no eigenstates for photons or other massless particles one usually instead calculates probability amplitudes for events taking place. Here, the relative phase between transmission and reflection events enters. However, this is just a question of the phase differences introduced by the setup.

With a HeNe laser, the average is 1.05 photons per one photon passing through the laser tube once. So the range varies, and of course gas would have a much smaller chance of stimulated emission than a solid. It would seem like this could be engineered tohave a peak probability of 1 stimulated emission, either with a very small solid state lasing medium of, like, 20 HeNe tubes in series.

"On average" and "exactly" are two very different things. The photon number distribution after stimulated emission is Poissonian because also the statistics of stimulated emission is Poissonian. In fact, in the case you suggest, the number of stimulated emission events taking place will also follow a Poisson distribution with a mean value of 1. Getting the mean value to one is manageable. Getting it to exactly 1 every time is close to impossible.

It sounds like getting this effect (HOM) is very tricky, and possibly the best way is to downconvert a beam into pairs with SPDC.

At least this is a simple way as you are almost guaranteed to get exactly identical photons out of the process. People have been working on being able to get indistinguishable photons from independent single photon emitters for quite some time, but only with limited success when it comes to things that might work outside of strict lab settings. Using two individual atoms is a nice way because the properties of the photon will usually be well determined, but spontaneous emission is quite slow and it is incredibly hard to have two atoms undergo spontaneous emission at the same moment. For quantum dots, getting them to emit at the same time is somewhat easier, but managing to fabricate two quantum dots that are so identical that they emit indistinguishable photons is incredibly tough. SPDC is indeed the easiest approach. Especially as you can also do postselection to know whether the SPDC process worked.