Kim's delayed choice quantum eraser -- Is it really delayed?

[Derek Potter]

So, will you please tell me if I have at last understood the source of the entangled pairs correctly? Pardon the quantum baby-talk.

Photon pairs are created at random times throughout the pumped regions of the BBO but the phase of the pair's probability amplitude is determined by the phase of the pump excitation which is, of course, the same over all the BBO. Thus all the "possible" emissions are in superposition and add up to a proper, spatially-coherent two slit source. The rest follows :)

 

[andrewkirk]

Cthuga I have now carefully worked through your explanation in the other thread. I found it very helpful and feel that I now almost understand what's going on. The idea that there is coherence between the A and B idlers in relation to the corresponding arrivals at D0 (two-photon coherence, which I think of as 'four-photon coherence'), but no coherence between A and B signal photons (one-photon coherence, which I think of as 'two-photon coherence') seems to be the crux of it.

I don't quite understand the relationship between photons and light waves though, in particular when photons are created - or if that question even makes sense. The interpretation that I find easiest to grasp is that photons are created only when observed, so the slits, BBO, mirrors and prisms just shape a wavefunction that generates probabilities of photon detection at D0-D5. So although we say the BBO 'creates' a photon pair we actually mean it transforms the wavefunction into one that gives different probabilities of observing photons, in two different directions - one heading towards D0 and one heading towards D1-D4 in the diagram. Those probabilities only turn into actual photons at the detectors D0-D5.

I can write out a crude, fudgy bra-ket description of this that sort of makes sense to me, and includes the entanglement, and that seems to embody the principles you lay out in the other thread.

But I'm stuck on how one can represent the incoherent light. My representation describes a wavefunction, after exit from the BBO, with form $\frac{1}{\sqrt{2}}\big( |\psi(p_1,\phi_1)\rangle |\psi(p_2,\phi_2)\rangle+|\psi(p_2,\phi_1) |\psi(p_1,\phi_2)\rangle \big)$ where $|\psi(p,\phi\rangle$ denotes a wavefunction for a particle with phase $\phi$ and mean momentum $p$. But that description seems to be only for one 'thing' (photon? moment in time?). I'm not sure how to think about, let alone represent, there being millions of such things happening, all with different values of $\phi_1,\phi_2$. I feel that it's probably a sum or integral of ket expressions of the above type, maybe over time, but I'm not sure how to bring the time dimension into it.

I don't know if this question makes any sense. I'll think about it some more and see if I can express it better.

 

[Cthugha]

So, will you please tell me if I have at last understood the source of the entangled pairs correctly? Pardon the quantum baby-talk.

Photon pairs are created at random times throughout the pumped regions of the BBO but the phase of the pair's probability amplitude is determined by the phase of the pump excitation which is, of course, the same over all the BBO. Thus all the "possible" emissions are in superposition and add up to a proper, spatially-coherent two slit source. The rest follows :)

The phase is not necessarily the same, but it has some fixed and well defined relationship, which is just as good. I am not exactly sure, what you mean by a proper spatially coherent two slit source. I assume that you mean the two-photon state. If so, I guess you are getting the idea. I admit that it is hard to express what is really going on without "talking math".

I don't quite understand the relationship between photons and light waves though, in particular when photons are created - or if that question even makes sense. The interpretation that I find easiest to grasp is that photons are created only when observed, so the slits, BBO, mirrors and prisms just shape a wavefunction that generates probabilities of photon detection at D0-D5. So although we say the BBO 'creates' a photon pair we actually mean it transforms the wavefunction into one that gives different probabilities of observing photons, in two different directions - one heading towards D0 and one heading towards D1-D4 in the diagram. Those probabilities only turn into actual photons at the detectors D0-D5.

As a somewhat tongue-in-cheek response, let me present a quote I like:
"My complete answer to the late 19th century question "what is electrodynamics trying to tell us?" would simply be this: Fields in empty space have physical reality; the medium that supports them does not.
Having thus removed the mystery from electrodynamics, let me immediately do the same for quantum mechanics: Correlations have physical reality; that which they correlate, does not."
N. David Mermin, in "What is Quantum Mechanics Trying to Tell Us?"

As a somewhat more serious remark: One of the problems is that there are no wavefunctions in the common sense for photons. There are for example no position eigenstates for massless particles and when looking at it in more detail one finds that the common wavefunction description does not work. What we are left with are probability amplitudes for processes connecting some initial state and some final state. Indeed we do not know anything about what the photon actually does in between. There is some quantized excitation of the light field, which is the photon. However, it is a particle in the sense of qm, which should never be confused with the classical notion of a localized "ball-like" thing. The exact time of emission is as well defined as the coherence time of the light field. We do not know much more.

I can write out a crude, fudgy bra-ket description of this that sort of makes sense to me, and includes the entanglement, and that seems to embody the principles you lay out in the other thread.

But I'm stuck on how one can represent the incoherent light. My representation describes a wavefunction, after exit from the BBO, with form $\frac{1}{\sqrt{2}}\big( |\psi(p_1,\phi_1)\rangle |\psi(p_2,\phi_2)\rangle+|\psi(p_2,\phi_1) |\psi(p_1,\phi_2)\rangle \big)$ where $|\psi(p,\phi\rangle$ denotes a wavefunction for a particle with phase $\phi$ and mean momentum $p$. But that description seems to be only for one 'thing' (photon? moment in time?). I'm not sure how to think about, let alone represent, there being millions of such things happening, all with different values of $\phi_1,\phi_2$. I feel that it's probably a sum or integral of ket expressions of the above type, maybe over time, but I'm not sure how to bring the time dimension into it.

I don't know if this question makes any sense. I'll think about it some more and see if I can express it better.

Getting the math straight is not trivial. Maybe have a look at the Review paper by Walborn et al. I cited in post #34. Section 6.1. deals with conditional interference fringes and complementarity. This is not exactly the same setup as used by Kim, but there are similarities. Also chapter 4 might be helpful.

 

[Derek Potter]

I assume that you mean the two-photon state. If so, I guess you are getting the idea.

The two-photon state was never an issue for me. The issue was how the A and B regions could cooperate to create an output that would show (two photon) interference. The common-cause explanation (the laser) had eluded me.