2-photon interference question

[Killtech]

It's sufficient to consider this for a single mode. [...]

Thanks, that's easy enough. Okay, now I understand how to read a state like $|n\rangle$ in terms of measurement. What I still don't get though is how do you calculate conditional probabilities within this formalism? Or in other words how do you note any kind of entanglement between the sates? Well, in this case of a coherent beam everything is assumed to be independent so there is no entanglement anyway.

But still still let's take it as a simple example anyway to proof the statement of @Cthugha that

All detection events are statistically completely independent foor coherent beams.

So the simplest case would be to run the state $|C_\alpha\rangle$ through a beam splitter and calculate the conditional probabilities for measuring of beam given the results of measurement on the other. After the splitter can I write the state as
$$e^{-|\alpha_A^2|} \sum_{n=0}^{\infty} \frac {\alpha_A^n} {\sqrt{n!}} |A,n \rangle + e^{-|\alpha_B^2|} \sum_{n=0}^{\infty} \frac {\alpha_B^n} {\sqrt{n!}} |B,n \rangle$$
where $A$,$B$ are supposed to denote the two outgoing arms of the splitter? So again the Born rule would give me the probabilities and given that all vectors $|A,n\rangle$, $|B,n\rangle$ are independent (i.e. orthogonal) the statement is perhaps trivial. Still how do I properly and formally calculate a conditional probability here? And how would I denote a state where the outgoing beams $A$,$B$ were statistically dependent, like for a most simple case of a single photon after passing the splitter?

 

[Killtech]

Ah, I think I was just stupid. writing the state via a product rather then a sum makes a lot more sense.

$$(e^{-\frac 1 2 |\alpha_A^2|} \sum_{n=0}^{\infty} \frac {\alpha_A^n} {\sqrt{n!}} |n \rangle_A)(e^{-\frac 1 2|\alpha_B^2|} \sum_{n=0}^{\infty} \frac {\alpha_B^n} {\sqrt{n!}} |n \rangle_B = e^{-\frac 1 2(|\alpha_A^2|-|\alpha_B^2|)}\sum_{n,m=0}^{\infty} \frac {\alpha_A^n \alpha_B^m} {\sqrt{n! m!}} |n\rangle_A |m\rangle_B$$

It that about right? The product form of the sum coefficients $c_{n,m} = \frac {\alpha_A^n \alpha_B^m} {\sqrt{n! m!}}$ makes it immediate obvious that the beams are independent in this case and more generally i can easily express any possible dependency between $A$ and $B$ via the coefficients $c_{n,m}$. In this form it's kind natural how to conditional expectations/distributions can be expressed/calculated. So is this actually correct?

 

[vanhees71]

Yes, if you have two independent coherent states, that's the correct description.

 

[Killtech]

Yes, if you have two independent coherent states, that's the correct description.

[...]All detection events are statistically completely independent foor coherent beams.

That said, it would seem to me that a laser beams that are statistically dependent could be constructed? I was just looking at the form of state of two independent beams while also having the HOM effect open in another tab...

So let's say both beams have an expected photon number of 0.01 per some time interval, which means for a Poission distribution that $|\alpha_A|^2=|\alpha_B|^2=0.01$. Hence writing only the dominant terms of the beams i'd get
$$ e^{-0.01}( |0,0\rangle_{A,B} + 0.1(|0,1\rangle_{A,B} + |1,0\rangle_{A,B}) + 0.01|1,1\rangle_{A,B} + \frac {0.01} {\sqrt 2} (|2,0\rangle_{A,B} + |0,2\rangle_{A,B}) + ...) $$
and assuming these beams are the input to a beam splitter in a HOM-setup I would be let to believe the term $|1,1\rangle_{A,B}$ in this expansion would participate in the HOM-effect. While this wouldn't change the outgoing beams in a visible way, it should ever so slightly alter their statistics: for one they should deviate from a true Poisson statistic in that in the case of detecting one photon in one of the beams has a slightly increased chance of detecting at least another one there too. On the other hand the chance of detecting a photon at all would somewhat drop. That increased probability of a 0-detection event in one beam should however be compensated by the increase chance for a two photon detection in the other rendering the two outgoing beams a little bit statistically dependent. This effect would be quite small but recording a sufficiently large statistic it should be enough to make stand out over the statistic error.

Hmm, for my purpose strengthening the $|1,1\rangle$ term above all else is what i want while this would actually diminishing it. Still, it might be salvageable.

 

[Cthugha]

It is not that easy as even for weak coherent light fields, you still have off-diagonal elements of the density matrix (which means that the coherent state has a phase), while the phase for single photon Fock states is completely undefined. So in order to provide some meaningful calculations, you need to define what kind of relative phase between the beams you are talking about. Usually people consider phase randomized states, which means that the relative phase varies quickly, but has some well defined fluctuating value for short instants of time. If you then consider all the classical interference patterns you can get, the results will depend on where you place your detectors. You can get some classical equivalent of the HOM dip that goes down to 50% of the visibility of a Fock state HOM dip that way, but it depends on a lot of details.

The math for the standard spatial interference pattern in the classical phase-randomized case was presented by Mandel already in 1983 https://journals.aps.org/pra/abstract/10.1103/PhysRevA.28.929 .
It is not too different to construct the equivalent for a beam splitter geometry along similar lines.

 

[Killtech]

It is not that easy as even for weak coherent light fields, you still have off-diagonal elements of the density matrix (which means that the coherent state has a phase), while the phase for single photon Fock states is completely undefined. So in order to provide some meaningful calculations, you need to define what kind of relative phase between the beams you are talking about. Usually people consider phase randomized states, which means that the relative phase varies quickly, but has some well defined fluctuating value for short instants of time. If you then consider all the classical interference patterns you can get, the results will depend on where you place your detectors. You can get some classical equivalent of the HOM dip that goes down to 50% of the visibility of a Fock state HOM dip that way, but it depends on a lot of details.

The math for the standard spatial interference pattern in the classical phase-randomized case was presented by Mandel already in 1983 https://journals.aps.org/pra/abstract/10.1103/PhysRevA.28.929 .
It is not too different to construct the equivalent for a beam splitter geometry along similar lines.

Unfortunately I have no access to that paper but I figured it isn't that trivial. After I posted that I have tried to google a bit myself and am currently reading this here: https://www.researchgate.net/publication/45921675_Interference_of_dissimilar_photon_sources - though there is a lot to do at work which is slowing me down. Anyhow, it's using somewhat similar approach albeit simplified to just observe the HOM effect whereas I ultimately want to understand if its possible to filter out/detect an exact probability amplitude value (its modulus more then its phase) of a single photon state. And I am more inclined to understanding the capabilities of the underlying mathematical model here rather then the physics, hence I am a bit lenient in overlooking the practical difficulties in preparing the assumptions I sometimes use :D... well, unless they are already theoretically invalid.

So in the case of my latest posts I was just trying to understand how laser beams are properly described mathematically and then using that to check if they could be made usable for my intentions. The first question for that was to answer whether it was possible to construct statistically dependent beams from laser sources at all - and using the HOM effect was just an idea. The exact form of that dependence is perhaps less of an issue for now and therefore the HOM effect needs to be present just so it doesn't effectively cancel out in the end - just so any statistical dependency is achieved. If it roughly worked though I wonder what would happen if one of the the HOM output beams were interfered destructively with one of the original beams - if there were any difference in statistics and dependence between those beams the destructive leg at such a splitter should work as a filter for that dependent component of the beam which might be what I am looking for. Hmm, should I draw a sketch of what I have in mind or does everyone(anyone?) still follow?

 

[Killtech]

Also, as I am not so familiar with optics, can anyone give me a simple link about optical couplers - like the ones used in the paper I linked before? Can't find a wikipedia article on them like there is one about beam splitters. Google gives me a lot of manufacturer sites... and some papers on how to produce even better and more compact ones. I get the concept of what they do, but would feel better when I saw how they are treated in the calculus.