Photon Wave Collapse Experiment (Yeah sure; AJP Sep 2004, Thorn )

[nightlight]

That's simply not true: quantum theory is not "different" for QED than for anything else.

Well, than you need some more reading and thinking to do. Let me recall the paper by Haus (who was one of the top Quantum Opticians from lengendary RLE lab at MIT) which I mentioned earlier (http://prola.aps.org/abstract/PRA/v47/i6/p4585_1):

Quantum-nondemolition measurements and the ``collapse of the wave function''

F. X. Kärtner and H. A. Haus
Department of Electrical Engineering and Computer Science and Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

This paper intends to clarify some issues in the theory of quantum measurement by taking advantage of the self-consistent quantum formulation of nonlinear optics. A quantum-nondemolition measurement of the photon number of an optical pulse can be performed with a nonlinear Mach-Zehnder interferometer followed by a balanced detector. The full quantum-mechanical treatment shows that the shortcut in the description of the quantum-mechanical measurement, the so-called ``collapse of the wave function,'' is not needed for a self-consistent interpretation of the measurement process.[/color] Coherence in the density matrix of the signal to be measured is progressively reduced with increasing accuracy of the photon-number determination. The quantum-nondemolition measurement is incorporated in the double-slit experiment and the contrast ratio of the fringes is found to decrease systematically with increasing information on the photon number in one of the two paths. The ``gain'' in the measurement can be made arbitrarily large so that postprocessing of the information can proceed classically.

The point there and in what I was saying is not that QM is wrong, but that the remote "projection postulate" (collapse) is simply not telling you enough in the abstract postulate form which only guarantees the existence[/color] of such projection operation (which implements 'an observable yielding one result'). Specifically, it doesn't tell you what kind of operations are involved in its implementation. If, for example an operational imlementation of an "observable" requires plain classical collection of data from multiple locations and rejections or subtractions based on the values obtained from remote locations, than one cannot base claim of nonlocality on such trivially non-local observable (since one can allways add the same convention and their communications channels to any classical model for the raw counts; or generally -- defining extra data filtering or post-processing conventions cannot make the previosuly classically compatible raw counts into result which excludes the 'classical' model).

The typical leap of this kind is in the elementary QM proofs of Bell's QM prediction and the use of abstract "observable", say, [AB] = [Pz] x [Pm], where factor [Pz] measures polarization along z of photon A, and factor [Pm] measures polarization along m on B. The leap then consists in assuming an existence of "ideal" and local QM detectors implementing the observable [AB] (i.e. it will yield the raw and purely local counts reproducing statistically the probabilities and ocrrelations of the eigenvalues of the observable for a given state).

Therefore any such "theorems" of QM incompatibility with clasical theories based on such "predictions" should be understood as valid 'modulo existence of ideal and local detectors for the observables involved'. If one were to postulate such 'existence' then of course, the theorems for the theory "QM+PostulateX" indeed are incompatible with any classical theory (which could, among other things, also mean that PostulateX is too general for our universe and its physical laws).

The abstract QM projection postulate (for the composite system) doesn't tell you, one way or the other, anything about operational aspect of the projection (to one result), except that the operation exists. But the usual leap (in some circles) is that if it doesn't say anything then it means the "measurement" of [AB] values can be done, at least in principle, with purely local counts on 4 "ideal" detectors (otherwise there won't be Bell inequality violation on raw counts).

The QED/QO derivation in [5] makes it plain (assuming the understanding of Gn of [4]) that not only are all the nonlocal vacuum effects subtractions (the "signal" function filtering conventions of QO, built into the standard Gn() definition), included in the prediction of e.g. cos^2(a) "correlation" but one also has to take upfront only the 2 point G2() (cf. eq (4) in [5]) instead of the 4 point G4(), even though there are 4 detectors. That means the additional nonlocal filtering convention was added, which requires removal of the triple and quadruple detections (in addition to accidentals and unpaired singles built into the G2() they used). Some people, based on attributing some wishful meanings to the abstract QM observables, take this convention (of using G2 instead of G4) to mean that the parallel polarizers will give 100% correlated result. As QED derivation [5] shows, they surely will correlate 100%, provided you exclude by hand all those results where they don't agree.

With QED/QO derivation one sees all the additional filtering conventions, resulting from the QED dynamics of photon-atom interactions[/color] (used in deriving Gn() in [4]), which are needed in order to replicate the abstract prediction of elementary QM[/color], such as cos^2(a) "correlation" (which obviously isn't any correlation of any actual local counts at all, not even in principle).

The abstract QM postulates simply lack information about EM field-atom interaction to tell you any of it. They just tell you observable exists. To find out what it means operationally (which you need in order to make any nonlocality claims via Bell inequalities violations; or in the AJP paper, about violation of classical g2>=1), you need dynamics of the specific system. That's what Glauber's QO detection and correlation modelling via QED provides.

In other words the "ideal" detectors, which will yield the "QM predicted" raw counts (which violate the classical inequalities) are necessarily the nonlocal devices[/color] -- to make decisions trigger/no-trigger these "ideal" detectors need extra information about results from remote locations. Thus you can't have the imagined "ideal" detectors that make decisions locally, not even in principle (e.g. how would an "ideal" local detector know its trigger will be the 3rd or 4th trigger, so it better stay quiet, so that its "ideal" counts don't contain doubles & triples? or that its silence will yield unpaired single so it better go off and trigger this time?). Even worse, they may need info from other experiments (e.g. to measures the 'accidental' rates, where the main source is turned off or shifted in the coincidence channel, data accumulated and subtracted from the total "correlations").

The conclusion is then that Quantum Optics/QED don't make a prediction of violation of Bell inequality (or, as explained earlier, of the g2>=1 inequality). There never was (as is well known to the experts) any violation of Bell inequality (or any other classical inequality) in the QO experiments, either. The analysis of the Glauber's QO detection model shows that no violation can exist, not even in principle, using photons as Bell's particles, since no "ideal" local detector for photons could replicate the abstract (and operationally vague) QM predictions.

The "dynamical model of the detection process" you always cite is just the detection process in the case of one specific mode which corresponds to a 1-photon state in Fock space, and which hits ONE detector.

Well, that is where you're missing it. First, the interaction between EM field and cathode atoms is not an abstract QM measurement (and there is no QM projection of "photon") in this treatment. It is plain local QED interaction, so we can skip all the obfuscatory "measurement theory" language.

Now, the detection process being modeled by the Glauber's "correlation" functions Gn() defines very specific kind of dynamical occurence to define what are the counts that the Gn() functions are correlating. These counts at a given 4-volume V(x) are QED processes of local QED absorption of the whole EM mode[/color] by the (ideal Glauber) "detector" GD, which means the full mode energy must be transferred to the GD, leaving no energy in that EM mode (other than vacuum). The energy transfers in QED are always local[/color], which implies in case of a single mode field (such as our |Psi.1> = |T> + |R>) that the entire flux of EM field has to traverse the GD cathode (note that EM field operators in Heisenberrg picture evolve via Maxwell equations, for free fields and for linear optical elements, such as beam splitters, mirrors, polarizers...).

Therefore, your statement about mode " which hits ONE detector" needs to account that for this "ONE" detector to generate a "Glauber count" (since that is what defines the Gn(), used in this paper, e.g. AJP.8) of 1, it has to absorb the whole mode, the full EM energy of |Psi.1> = |T>+|R>. As you can easily verify from the picture of the AJP setup, DT is not a detector configured for such operation of absorbing the full energy of the mode in question, the |Psi.1>. You can't start now splitting universes and invoking your consciousness etc. This is just plain old EM filed propagation. Explain how can DT absorb the whole mode |Psi.1>, its full energy leaving just vacuum after the absorption? (And without it your Glauber counts for AJP.8 will not be generated.)

It can't just grab it from the region R by willpower, be it Everett's or von Neumann's or yours. The only way to absorb it, and thus generate the Glauber's count +1, is by regular EM field propagation (via Maxwell equations) which brings the entire energy of the mode, its T and R regions, onto the cathode of DT. Which means DT has to spread out to cover both paths.

Therefore, the AJP setup doesn't correspond to any detector absorbing the whole incident mode, thus their setup doesn't implement Glauber's G2() in (AJP.8) for the single mode state |Psi.1> = |T> + |R>. Therefore, the count correlations of DT and DR are not described by eq. (AJP.8) since neither DT nor DR implement Glauber's detector (whose counts are counted and correlated by the AJP.8 numerator). Such detector, whose counts are correlated in (AJP.8), counts 1 if and only if it absorbs the full energy of one EM field mode.

The eq (AJP.8) with their detector geometry, applies to the mixed incident state[/color] which is Rho = |T><T| + |R><R|. In that state, for each PDC pulse, the full single mode switches randomly from try to try between the T and R paths, going only one way in each try, and thus the detectors in their configuration can, indeed, perform the counts described by their eq. AJP.8. In that case, they would get g2=0 (which is also identical to the classical prediction for the mixed state), except that their EM state isn't the given mixed state. They're just parroting the earlier erroneous interpretation of that setup for the input state |Psi.1> and to make it "work" as they imagined they had to cheat (as anyone else will have to get g2<1 for raw counts).

What I mean is: If your incoming EM field is a "pure photon state" |1photonleft>, then you can go and do all the locla dynamics with such a state, and after a lot of complicated computations, you will find that your LEFT detector gets into a certain state while the right detector didn't see anything. I'm not going to consider finite efficiencies (yes, yes...), the thing ends up as |detectorleftclick> |norightclick>...

What you're confusing here is the behavior of their detectors DT and DR[/color] (which will be triggering as configured, even for the single mode field |Psi.1> = |T> + |R> EM field) with the applicability of their eq. (AJP.8) to the counts their DT and DR produce[/color]. The detectors DT and DR will trigger, no-one ever said they won't, but the correlations of these triggers are not described by their (AJP.8).

The G2(x1,x2) in numerator of (AJP.8) descibes correlation of the counts of the "full mode absorptions" at locations x1 and x2 (which is more evident in the form AJP.9, although you need to read Glauber to understand what precise dynamical conditions produce these count entering AJP.8: only the full mode absorption, leaving the EM vacuum for the single mode input, make the count 1). But these cannot be their DT and DR since neither of them is for this input state a full mode absorber. And without it they can't apply (AJP,8) to the counts of their detectors. (You need also to recall that in general, the input state determines how the detector has to be placed to perform as an absorber, of any kind, full absorber of AJP.8, or partial absorber which is what they had, for a given state.)

You can do the same for a |1photonright> and then of course we get |noleftclick> |detectorrightclick>. These two time evolutions:

|1photonleft> -> |detectorleftclick> |norightclick>
|1photonright> -> |noleftclick> |detectorrightclick>

are part of the overall time evolution operator U = exp(- i H t)
Now if we have an incoming beam on a beam splitter, this gives to a very good approximation:

|1incoming photon> -> 1/sqrt(2) {|1photonleft> + |1photonright>

You're replaying von Neumann's QM measurement model, which is not what is contained in the Glauber's detections whose counts are used in (AJP.8-11) -- these contain additional specific conditions and constraints for Glauber's counts (counts of full "signal" field mode absorptions). There is no von Numann's measurement here (e.g. photon number isn't preserved in absorptions or emissions) on the EM field. The (AJP.8) is dynamically deduced relation, with precise interpretation given by Glauber in [4], resulting from EM-atom dynamics.

The Glauber's (AJP.8) doesn't apply to the raw counts of AJP experiment DT and DR for the single mode input |Psi.1> = |T> + |R>, since neither DT nor DR as configured can absorb this full single mode. As I mentioned before, you can apply Glaubers detection theory here, by defining properly the two Glauber detectors, GD1 and GD2 which are configured for the requirements of the G(x1,x2) for the this single mode input state (which happens to span two separate regions of space). You simply define GD1 as a detector which combines (via logical OR) the outputs of DT and DR from the experiment, thus treat them as single cathode of odd shape. Thus DG1 can absorb the full mode (giving you Poissonioan count of photo-electrons, as theory already predicts for single photo-cathode).

Now, the second detector DG2 has to be somewhere else, but the x2 can't cover or block the volume x1 used by DG1. In any case, wherever you put it (without blocking DG1), it will absorb vacuum state (left from DG1's action) and its Glauber count will be 0 (GD's don't count anything for vacuum photons). Thus you will get the trivial case g2=0 (which is the same as semi-classical prediction for this configuration of DG1, DG2).

The upshot of this triviality of DG1+DG2 based g2=0 is that it illustrates limitation of the Glauber's definition of Gn() for these (technically) "nonclassical" state -- as Glauber noted in [4], the most trivial property of these functions is that they are, by definition, 0 when the input state has fewer photons (modes) than there are 'Glauber detectors'. Even though he noted this "nonclassicality" for the Fock states, he never tried to assign its operational meaning to the setup like DT and DR of this "anticorrelation" experiment, he knew it doesn't apply here except in the trivial manner of GD1 and GD2 layout (or mixed state Rho layout), in which it shows absolutely nothing non-classical.

It may be puzzling why is it "nonclassical" but it shows nothing genuinly non-classical. This is purely the consequence of the convention Glauber adopted in his definition of Gn. Its technical "nonclassicality" (behavior unlike classical or mathematical correlations) is simply the result of the fact that these Gn() are not really correlation functions of any sequence of local counts at x1 and x2. Namely, their operational definition includes correlation of the counts, followed by the QO subtractions. Formally, this appears in his dropping of the terms from the QED prediction for the actual counts. To quote him ([4] p 85):

... we obtain for this Un(t,t0) an expression containing n^n terms, which represent all the ways in which n atoms can participate in an n-th order process. Many of these terms, however, have nothing to do with the process we are considering, since we require each atom [his detector] to participate by absorbing a photon once and only once.[/color] Terms involving repetitions of the Hamiltonian for a given atom describe processes other than those we are interested in.

He then goes on and drops all the terms "we are not interested in" and gets his Gn()'s and their properties. While they're useful in practice, since they do filter the most characteristic features of the "signal", with all "noise" removed (formally and via QO subtractions), they are not correlations functions of any counts and they have only trivial meaning (and value 0) for the cases of having more Glauber detectors than the incident modes, or generally when we have detectors which capture only partial modes, such as DG and DT interacting with |Psi.1> state in this experiment.

The Mandel, Wolf & Sudarshan's semiclassical detection theory has identical predictions (Poissonian photo-electron counts, proportionality of photo-electron counts to incident intensity, etc) but it lacks Glauber's "signal" function definition for multipoint detections (MWS do subtract local detector's vacuum contributions to its own count). For multipoint detections, they simply use plain product of detection rates of each detector, since these points are at the spacelike distances, which gives you "classical" g2>=1 for this experiment for the actual counts. And that is what you will always measure.

Glauber could have defined the raw count correlations as well, the same product as the MWS semiclassical theory, and derived it from his QED model, but he didn't (the QED unfiltered form is generally not useful for the practical coincidence applications due to great generality and "noise" terms). The correlation functions here would have the unordered operator products instead of normally ordered (as discussed in our earlier thread).

Note that the fact that semiclassical model (or same for any non-Glauberized QED model) uses products of trigger rates doesn't mean the counts can't be correleted. They can be since the local trigger rates are proportional to the local intensities, and these intensities can correlate e.g. as they do in this experiment between G and T + R rates. The correlation in counts is entirely non-mysterious, due to simple EM amplitude correlations.

{ PS: Grangier, who is your countryman and perhaps nearby, is famous for this experiment. Ask him if he still thinks it is genuinly nonclassical (on raw counts g2<1 and with high enough visibility). Also, whether he has some real QED proof of the existence of any nontrivial anticorrelation in this setup (genuinly nonclassical), since you don't seem to know how to do it (I don't know how, either, but I know that).}

 

[vanesch]

Well, than you need some more reading and thinking to do. Let me recall the paper by Haus (who was one of the top Quantum Opticians from lengendary RLE lab at MIT) which I mentioned earlier (http://prola.aps.org/abstract/PRA/v47/i6/p4585_1):

Yes, the subject of that paper is the application of decoherence in order to analyse in more depth what is usually a "shortcut" when we apply the projection postulate. I'm fully in agreement with that, I even try to "sell" the idea here. But it doesn't invalidate the RESULTS of the projection postulate (if applied late enough!), it justifies them.

If, for example an operational imlementation of an "observable" requires plain classical collection of data from multiple locations and rejections or subtractions based on the values obtained from remote locations, than one cannot base claim of nonlocality on such trivially non-local observable (since one can allways add the same convention and their communications channels to any classical model for the raw counts; or generally -- defining extra data filtering or post-processing conventions cannot make the previosuly classically compatible raw counts into result which excludes the 'classical' model).

It doesn't change the settings. You're right that in order to be completely correct, one should include, in the whole calculation, ALL detectors. But often, this is doing a lot of work just to find out that the ESSENTIAL quantity you're looking after is popping out, in a factorized way, and corresponds with what you are really doing, and that the experimental corrections you are so heavily protesting against, is nothing else but applying these calculations in reverse.
Do it if you like. Calculate the 3-point correlation function, and then find that it factorizes in a 2-point correlation function and a poisson stream. I don't stop you, it is the correct way of proceeding. And you will then probably also realize that what you will have achieved with a lot of sweat is equivalent to the simple experimental calculation that has been done and which you don't like :-)

Here, do it:

The QED/QO derivation in [5] makes it plain (assuming the understanding of Gn of [4]) that not only are all the nonlocal vacuum effects subtractions (the "signal" function filtering conventions of QO, built into the standard Gn() definition), included in the prediction of e.g. cos^2(a) "correlation" but one also has to take upfront only the 2 point G2() (cf. eq (4) in [5]) instead of the 4 point G4(), even though there are 4 detectors. That means the additional nonlocal filtering convention was added, which requires removal of the triple and quadruple detections (in addition to accidentals and unpaired singles built into the G2() they used). Some people, based on attributing some wishful meanings to the abstract QM observables, take this convention (of using G2 instead of G4) to mean that the parallel polarizers will give 100% correlated result. As QED derivation [5] shows, they surely will correlate 100%, provided you exclude by hand all those results where they don't agree.

Do the full calculation, and see that the quantity that is experimentally extracted is given by the 2-point correlation function to a high degree of accuracy.

In other words the "ideal" detectors, which will yield the "QM predicted" raw counts (which violate the classical inequalities) are necessarily the nonlocal devices[/color] -- to make decisions trigger/no-trigger these "ideal" detectors need extra information about results from remote locations. Thus you can't have the imagined "ideal" detectors that make decisions locally, not even in principle (e.g. how would an "ideal" local detector know its trigger will be the 3rd or 4th trigger, so it better stay quiet, so that its "ideal" counts don't contain doubles & triples? or that its silence will yield unpaired single so it better go off and trigger this time?). Even worse, they may need info from other experiments (e.g. to measures the 'accidental' rates, where the main source is turned off or shifted in the coincidence channel, data accumulated and subtracted from the total "correlations").

But if you really want to, you CAN do the entire calculation. Nothing stops you from calculating a 5-point correlation function, and then work back your way to the quantum expectation value of the quantity under study. Only, people have developed a certain intuition of when they don't need to do so ; when the result is given by the quantity they want to calculate (such as the 2-point function) and some experimental corrections, or trigger conditions. You don't seem to have that intuition, so it is probably a good exercise (because it seems to be so important to you) to go through the long calculation.

See, your objections sound a bit like the following analogy.
One could think of somebody objecting against the "shortcuts taken" when naive student calculations calculate Kepler orbits for point particles in a 1/r^2 field, and then associate this to the movement of planets in the solar system.
The objections could be: hey, PLANETS ARE NOT POINT PARTICLES. You could delve into continuum mechanics, plate tectonics, and fluid dynamics to show how complicated the material movement is, no way that this can be considered a point particle ! So it should be clear that those student calculations giving rise to Kepler orbits to "fit the data" are bogus: real Newtonian mechanics doesn't do so !

The "dynamical model of the detection process" you always cite is just the detection process in the case of one specific mode which corresponds to a 1-photon state in Fock space, and which hits ONE detector.

Well, that is where you're missing it. First, the interaction between EM field and cathode atoms is not an abstract QM measurement (and there is no QM projection of "photon") in this treatment. It is plain local QED interaction, so we can skip all the obfuscatory "measurement theory" language.

But there is NEVER an abstract QM measurement. All is just entanglement.

Now, the detection process being modeled by the Glauber's "correlation" functions Gn() defines very specific kind of dynamical occurence to define what are the counts that the Gn() functions are correlating. These counts at a given 4-volume V(x) are QED processes of local QED absorption of the whole EM mode[/color] by the (ideal Glauber) "detector" GD, which means the full mode energy must be transferred to the GD, leaving no energy in that EM mode (other than vacuum). The energy transfers in QED are always local[/color], which implies in case of a single mode field (such as our |Psi.1> = |T> + |R>) that the entire flux of EM field has to traverse the GD cathode (note that EM field operators in Heisenberrg picture evolve via Maxwell equations, for free fields and for linear optical elements, such as beam splitters, mirrors, polarizers...).

This is where you don't understand quantum theory. The "whole mode" can be the "left beam". The "left beam" then interacts LOCALLY with the detector. If you now say that the whole mode is in fact half the left beam plus half the right beam, you can do that, thanks to the superposition principle and the 1-1 relationship between EM modes and 1-photon states.

After all the LR mode can be written as a very good approximation as half an L mode and half an R mode. I know that this is not EXACTLY true, but it is, to a good enough approximation. So the 1-photon state corresponding to the LR mode is the superposition of the 1-photon states corresponding to the almost L mode and almost R mode. They follow exactly the superpositions of the classical EM fields, it is just a change of basis.

One could make all these details explicit, but usually one has enough "physical intuition" to jump these obvious but tedious steps.

Therefore, your statement about mode " which hits ONE detector" needs to account that for this "ONE" detector to generate a "Glauber count" (since that is what defines the Gn(), used in this paper, e.g. AJP.8) of 1, it has to absorb the whole mode, the full EM energy of |Psi.1> = |T>+|R>.

That's where YOU are missing the whole point in quantum theory. You cannot talk about the localisation or not of the energy of a quantum state !
That "full mode" can be written in another basis, namely in the L and R basis, where it is the superposition of two states (or modes). And your detector is locally sensitive to ONE of them, so this basis is the "eigenbasis" corresponding to the measurement ; but if you don't like that language (nor do I!), you say that the detector ENTANGLES with each state L and R: for the L state he is in the "click" state, and for the R state he's in the "no click" state.

As you can easily verify from the picture of the AJP setup, DT is not a detector configured for such operation of absorbing the full energy of the mode in question, the |Psi.1>. You can't start now splitting universes and invoking your consciousness etc. This is just plain old EM filed propagation.

Well, you can object to quantum theory if you want to. But you should first understand how it works! What you have been saying above indicates that you don't. If the detector was absorbing the "entire LR mode" (which could be in principle possible, why not ; it would be a strange setup but ok), then you would NOT measure exactly what we wanted to measure, because you would measure AN EIGENSTATE ! You would always get the same result, namely, well, that there was a photon in the LR mode. Big deal.

Explain how can DT absorb the whole mode |Psi.1>, its full energy leaving just vacuum after the absorption? (And without it your Glauber counts for AJP.8 will not be generated.)

Because we were not working in that basis ! So the system was FORCED to be in one of the eigenstates of the LOCAL detectors, namely L OR R. That was the whole point of the setup.

It can't just grab it from the region R by willpower, be it Everett's or von Neumann's or yours. The only way to absorb it, and thus generate the Glauber's count +1, is by regular EM field propagation (via Maxwell equations) which brings the entire energy of the mode, its T and R regions, onto the cathode of DT. Which means DT has to spread out to cover both paths.

I guess that the Pope wrote that in his testament, so it must be true ?

I think that you've been missing the essential content of quantum theory, honestly. Whether or not you think it is true is another issue, but first you should understand the basic premisses of quantum theory and how you arrive at predictions in it. You are making fundamental mistakes in its basic application, and then confuse the issues by going to complicated formalisms in quantum optics. Newtonian mechanics DOES predict Kepler orbits.

cheers,
Patrick.

 

[nightlight]

This is where you don't understand quantum theory. The "whole mode" can be the "left beam"...

You forgot the context. As general isolated statement, any basis defines modes, so any state is a mode. But it is irrelevant for the usability of (AJP.8) for the given input state. For Glauber's detector to produce a count +1 it needs to absorb dynamically this entire incident mode, its entire energy and leave the vacuum.

The reason the numerator G2 of (AJP.8,9) goes to zero is precisely because of the 2nd abosrber has no mode to absorb, because it was absorbed by the 1st absorber (in the G2). Either of the two absorbers of (AJP.9) is therefore absorbing this particular mode, |Psi.1> and each leaves vacuum when it completes its interaction with |Psi.1> = |T> + |R>. The actual detector DT (or DR) in their setup does not correspond to this type of absorption. DT of their setup with this input state will never generate Glauber count +1 counted in the (AJP.8,9) (DT in experiment generates counts, of course, but this is not the "count" being counted by Glauber detector of AJP.9), since it will never leave the vacuum as result of the interaction with the mode it absorbs, the |Psi.1> (since it is not placed to interact with its field in the region R).

The "left beam" then interacts LOCALLY with the detector. If you now say that the whole mode is in fact half the left beam plus half the right beam, you can do that, thanks to the superposition principle and the 1-1 relationship between EM modes and 1-photon states.

You're explaining why the actual DT detector in the setup will trigger. That has nothing to do with the Glauber count in the numerator of AJP.8. Each "count" being correlated in AJP.8) counts of full absorbtions of mode |Psi.1>, not just any general mode you can imagine. That's why the numerator of (AJP.8) turns zero -- the first anihilation operator a_t (cf. (AJP.9)) leaves vacuum for the mode GT of (AJP.8) absorbs, then the 2nd operator a_r yields 0 since it acts on the same mode |Psi.1>. But these are not the absorptions performed by the actual DT and DR of the setup since these absorb each its own mode.

The (AJP.8) has the desired behavior only for the mixed state Rho =1/2(|T><T| + |R><R|), where in each try DT or DR will aborb the single mode (DT will absorb whole mode |T> and |DR> mode |R>) and leave the EM state vacuum, so the other absorber acting on this vacuum will yield 0. But here the anticorrelation is also present in the field amplitudes, so that classical g2 is 0 as well (the single "photon" goes one definite path in each try).

Your whole argument here is based on mixing up the general applicability of (AJP.8) to any conceivable input states (such as separate modes T and R) with the specific conditions of this setup, where (AJP.8) is required to yield 0 in the numerator when used with the state |Psi.1> = |T> + |R> for the expectation value of (AJP.8). In order to make the numerator of (AJP.8) vanish when applied to this input state, the |Psi.1>, which is what their result g2=0 requires, you need both of its anihilation operators to act on the same mode.

The only way the actual DT and DR counts (interpreted as Glauber full mode absorption counts used in AJP.8) of the setup will yield 0 in the numerator of (AJP.8,9), as the paper's g2=0 conclusion demands, is if the input state is Rho, the mixed state.

For the superposed state |Psi.1> = (|T> + |R>)/sqrt(2), with DT absorbing mode |T> and DR absorbing mode |R>, as you are intepreting it, the numerator of (AJP.8-9) yields 1/4, resulting in g2=1. But for the mixed input state Rho, using the same DT as aborber of |T> and DR as absorber of |R>, the (AJP.8-9) yields 0 (since each term in Rho via trace yields 0 against the product a_r*a_t in AJP.9).

Your defense appears as if tryng to weasel out using the trivial non sequitur generalities and the slippery Copenhagen photon-speak. So let's get fully specific and concrete in the two questions below:

Question-a)[/color] Given actual DT and DR of their setup, thus DT absorbing mode |T> (hence a_t |T> = |0>, a_t |R> = 0) and DR absorbing mode |R>, what input state (used for <...> in AJP.9) do you need in the numerator of (AJP.9) to obtain their g2=0? Is that input state same as their "one photon" state |Psi.1> = (|T> + |R>)/sqrt(2) ?

My answer: only the mixed state Rho =1/2 (|T><T| + |R><R|) gives g2=0 with their actual DT and DR absorbers in (AJP.9). Their actual input state |Psi.1> used with their DT and DR absorbers in (AJP.9) gives g2=1.

Question-b)[/color] To avoid confusion with (Question-a), take their (AJP.9) and replace the subscripts T and R with generic 1 and 2. Then, given their actual "one photon" state |Psi.1> and given their g2=0 conclusion, tell me what kind of absorbers (what is the basis defining the anihilation operators) a1 and a2 do you need to obtain 0 in the numerator of (AJP.9) on the |Psi.1> (used for the averaging)? What is the spatial extent of the Glauber detector realizing such absorber? { As a check, consider placing just that first detector by itself at your place chosen and examine its effect on |Psi.1>, i.e. after the QED absorption of |Psi.1>, does |Psi.1> become |0> or some other state? E.g. DT of their setup doesn't produce the required effect, the |0> as the final field state after acting on |Psi.1>. Also check your design for GD1 detector (implementing this absorption effect) that does not violate QED evolution equations for the field operators in the region R, which will be the Maxwell equations.}

My answer: To get 0, a1 (or a2) must be an absorber of mode |Psi.1>, hence a2 a1|Psi.1> = a2 |0> = 0. The corresponding Glauber detector performing a1 absorption must extend spatially to receive the entire EM flux of |Psi.1>, which happens to cover T and R paths.

Note also their cheap trick used to mask the triviality of the "anticorrelation" obtained in (b) through their (AJP.11): they transformed the mode absorbers from the post-splitter fields (n_t and n_r) to the pre-splitter fields (n_i). This corresponds to placing a detector in front of the beam splitter in order for a single detector to absorb that single mode via n_i (and yield their g2=0). They don't want a student to wonder about the actual spatial layout of the absorbers of (AJP.9) which have to absorb this single mode after the beam splitter and give g2=0 in (AJP.9). That is also why I asked in (Q-b) that you put non-suggestive labels on the absorbers in order to decide what they ought to be to yield 0 in the numerator of (AJP.9).

Their related interference experiment used the same trick (cf. their web page) -- they stick the interferometer in front of the beam splitter not after, so they don't test whether they have genuine 50:50 superposition (that is in their case a minor loophole, given the main cheat with the gratuitous third coincidence unit).


It doesn't change the settings. You're right that in order to be completely correct, one should include, in the whole calculation, ALL detectors. But often, this is doing a lot of work just to find out that the ESSENTIAL quantity you're looking after...

The whole front section of your responses has lost entirely the context of the discussion. You are arguing that the filtering rules of the Glauber's conventions for Gn() are useful. Indeed they are.

But that usefulness has nothing to do with the topic of non-classicality we were discussing.

The first point of my comments was that the non-local (formally and operationally) filtering defining the G2(), which in turn yields the QED version of Bell's QM prediction, turns the claim that "the cos^2() value of G2 filtered out from the counts data implies nonlocality" into a trivial tautology -- of course that cos^2() interpreted as a genuine "correlation" (despite the fact that it isn't, as the QED derivation makes it transparent, while it is obscured in the QM abstract "measurement") -- it is defined as a non-local "signal" function, filtered out from the actual counts via the non-local QO procedures (or extracted formally in [4] by dropping non-local coincidence terms). So, yes, it is trivial that the "correlation" of counts which is computed using nonlocal operations (QO subtractions) from the measured counts may not be in general case also obtainable as a direct correlation of the purely local counts (thus it will violate Bell's inequality). So what?

This is the same kind of apples vs oranges "discoveries", when claiming you can get g2<1 on adjusted data, so the effect must be nonlocal. The error in this is that you're comparing the classical g2c of (AJP.2) which does not define g2c as a filtered value, against the g2q of (AJP.8) which Glauber's conventions require to filter out the unpaired singles (DG trigger with no DT and DR trigger, which decreases the numerator of AJP.14, thus decreases the g2q) and the accidental coincidences. Then you "discover" the quantum magic by labeling both g2c and g2q as g2 (as this paper does), and act surprised that the second one is smaller. Duh. There is as much of "discovery" and "mystery" here as in getting excited after "discovering" that three centimeters are shorter than three inches.

The second point being made with Haus' quote and my comments was that the dynamical model of detecton explains the "collapse" of "photon" state, giving the exact operational interpretation of the abstract QM projection postulate (which merely assures existence of some such operation without telling you what kind of operation it is, expecially regarding the locality of the procedures that implement the abstract QM observable).

Therefore the dynamical approach is quite relevant in the discussion of whether some formal result implies non-locality -- to find that out you need very precise operational mapping between the formal results and the experimental procedures and their data. The existence fact that the abstract QM projection postulate gives, provides you no information to decide whether cos^(a) correlation it yields implies any non-locality or not. You need to find out precisely how does such cos^2(a) map to experiment and its data for given type of a system, such as photons. The non-relativisitic elementary QM doesn't have an adequate model of the EM fields to flesh out the precise operational requirements for the realizations of its abstract observables of the EM fields. That's what is provided by the QED analysis and the dynamical model of the photo detection & coincidence measurements, as demonstrated in [4].

And the conclusion from that, as shown in the previous note, is that QED doesn't predict either the Bell inequality violation or the g2>=1 violation for photons. In both cases, the operational mapping provided by the QED models (of [4]) shows that the locality violating results obtained in QM have a trivial source of non-locality already built into the procedures needed to actually implement the abstract QM observables in a manner compatible with QED.

 

[vanesch]

Each "count" being correlated in AJP.8) counts of full absorbtions of mode |Psi.1>, not just any general mode you can imagine.

Yes, and that's the essential, basic part of quantum theory that you refuse to see, because you make an identification with the classical case, I presume.

I really don't need all that Glauber stuff, which only renders this basic reasoning opaque, so I'm not going to plunge into it for the moment. It is just a way of putting on a more formal basis the very simple reasoning I'm trying, without result, to explain you here.

To every (complex) classical field configuration ("mode"), up to an arbitrary (complex) amplitude coefficient, corresponds a 1-photon state ; but they are not the same physical situation of course. Only, there is a bijective relationship between them, so we can LABEL each 1-photon state with the classical field configuration. "modes" are a BASIS of the vectorspace of classical field configurations, and the equivalent 1-photon states are a basis of the Hilbert space of 1-photon states.

So GIVEN A 1-photon state, we can always expand it in any basis we like.

A physical detector, of small size, will split the hilbert space of 1-photon states in two orthogonal eigenspaces: one will contain all the modes that ARE COMPLETELY ABSORBED, the other will contain all modes that COMPLETELY MISS THE DETECTOR.
Let us call those two orthogonal eigenspaces E1 and E0 of the detector.
So each mode that belongs to E1 will give, with certainty if the detector is 100% opaque, a "detection" ; E0 will give, with certainty, NO "detection".

WHAT IS SEMICLASSICALLY DONE, and which corresponds to the full quantum calculation, is that we take a mode in E1 (one that will hit the detector fully).
We don't bother about E0, we know that it will miss the detector.

Let us now consider a second detector, somewhere else. It too has associated with it, two eigenspaces, let us call them F1 and F0, which are orthogonal and "span the entire hilbert space" of 1-photon states.

We now have 4 subspaces of the hilbert space: E0 and E1, F0 and F1.
Clearly, if the detectors are in different locations, E1 and F1 have no element in common: no beam which fully hits D1 also fully hits D2. However, E0 and F0 (which are much bigger) do overlap.
This means that from these 4 sets, we can construct 3 ORTHOGONAL sets:
E1, F1 and EF0.

Now let us consider our actual incoming beam and beamsplitter. This corresponds to a certain EM field configuration, psi1.
Because E1, F1 and EF0 are orthogonal and complete, it is possible to write psi1 as a sum of 3 terms, each in one of the above.

So in all generality:
|psi1> = a |e1> + b|f1> + c |ef0>

However, if the beams are well-aligned, they *always end into one of the two detectors*, so the |ef0> part can be discarted. If the thing is balanced, moreover, we have:

|psi1> = 1/sqrt(2) (|R> + |T>) with R in E1, T in F1.

We have split up the hilbert space in 3 orthogonal components: E1, F1 and EF0, and our measurement apparatus (the 2 detectors) will react in the following way:

in E1: we have D1 clicks, D2 doesn't.
in F1: we have D1 doesn't click, D2 does.
in EF0: we have nor D1, nor D2 click.

Note that there is no state in which D1 and D2 can click, and that's because E1 and F1 are orthogonal (the detectors are in different locations).

If we now submit a state like |psi1> to this measurement, we apply the general procedure in quantum theory, which is the same, for quantum mechanics, for QED and every other application of quantum theory:

We write the state in the eigenbasis of the measurement (is done here), and we assign probabilities to each of the eigenstates, corresponding to each of the terms' coefficients, squared:

50% probability that D1 clicks, D2 doesn't ;
50% probability that D2 clicks, D1 doesn't.

Here, we applied the projection postulate, which is justified, but if you want to do decoherence, you also can do so. The detector state eigenspace can be split in two orthogonal subspaces: D0 and D1, where D0 corresponds to about all physical states of the detector with "no click" and D1 with "a click".
We have the two detectors De and Df.
We take it that initially the detectors reside somehow in their D0 space.

You consider a time evolution operator U in which:
each mode in E1, |e1>, makes a De0 state evolve in a De1 state ;
each mode in E0 makes a De0 state evolve into another De0 state;
each mode in F1, |f1>, makes a Df0 state evolve in a Df1 state;
each mode in F0, makes a Df0 state evolve into another Df0 state.

The photons end up absorbed so we return to vacuum mode |0>

So we start out with the state (EM state) x (D1 state) x (D2 state):

|psi1> |De0> |Df0> = 1/sqrt(2) {|R> + |T>}|De0> |Df0>

Applying the (linear) time evolution operator of the measurement interaction, we end up in:

1/sqrt(2) {|0>|De1>|Df0> + |0>|De0>|Df1>}

= |0> 1/sqrt(2) { |De1>|Df0> + |De0> |Df1> }

So, there is one branch in which D1 clicked and D2 didn't, and there is another branch in which D1 didn't click, but D2 did.

I didn't need any Glauber or other formalism, which does just the same, but in a more formal setting, in order to deal with more complicated cases. THIS is elementary quantum theory, but there's nothing "naive" or "toy" about it. It is fundamental.

I should add a point, because it is the very essence of your misunderstanding of quantum theory (at least that's how I see it from your arguments). It is the following: I said above:

A physical detector, of small size, will split the hilbert space of 1-photon states in two orthogonal eigenspaces: one will contain all the modes that ARE COMPLETELY ABSORBED, the other will contain all modes that COMPLETELY MISS THE DETECTOR.
Let us call those two orthogonal eigenspaces E1 and E0 of the detector.
So each mode that belongs to E1 will give, with certainty if the detector is 100% opaque, a "detection" ; E0 will give, with certainty, NO "detection".

Now, it should be clear that, although E0 and E1 are orthogonal eigenspaces, they don't of course COVER all of hilbert space (in the same way that the X axis and the Y axis don't cover the plane). So there are many modes of which we don't say how they "interact" with the detector, and this is the typical case you're complaining about. But don't forget that 1-photon states are NOT "em fields". They are all the quantum states of the EM quantum field which are eigenstates of the (free) hamiltonian with energy hbar omega above the ground state (if we limit ourselves to one frequency), and it just happens that there is a bijective relationship with the classical EM field configurations.
And here now comes in the very fundamental postulate of quantum theory: the superposition principle. Namely that if we KNOW WHAT HAPPENS FOR A CERTAIN SET OF BASIS STATES, then we know what happens for any superposition. So we happen to know what happens for those specific field configurations (basis states) which FULLY hit the detector, or which completely MISS the detector, and THAT FIXES ENTIRELY the behaviour for ALL 1-photon states because these "fully hit" and "completely miss" configurations SPAN the whole state space.
This is fundamental, and related to the superposition principle in quantum theory, and this is entirely different in classical field theory where "what happens to the lump of field at A" has normally nothing to do with "what happens to the lump of field at B", and what you are always referring to. The classical fields are just handy bookkeeping devices for the state space of 1-photon states, as can be seen in the following way:
for a 1-photon state, which is an eigenstate of the EM free field hamiltonian, we know that the AVAILABLE ENERGY in the entire quantum field equals hbar omega. This is true for ALL 1-photon states, in its entire hilbert space (limiting, again, ourselves to the monochromatic case). So no matter what happens, if we go from any 1-photon state to the vacuum state, the EM quantum field looses exactly hbar omega in energy. And that is the reason why it can only interact with ONE SINGLE detector ; why geometrically different detectors correspond to orthogonal "fully absorb" eigenspaces, and, in fine, why there is this correspondence between classical EM field configurations and 1-photon states.

Now, something else: if there is a 1-1 relationship between the 1-photon state space and classical EM field configurations, does that mean that the best quantum description of a classical field is its corresponding 1-photon state ? The answer is no. The best quantum description is a coherent state which is a sum over n-photon states.
This indicates already that in the case of 1-photon states, you do not have a classical description, and hence your "grabbing the (classical) field energy at the other side" doesn't make much sense in 1-photon settings.

It can be shown that starting from a classical beam (a coherent state), and using a process which generates a transition from 1-photon to 2-photon states (such as a PDC), detecting a hit of a 2-photon state on 1 detector gives you an experimental situation, in this short time interval, which is very close to a 1-photon state. There is still some contamination of coherent states and so on, but not much. And THIS is what is reproduced with the G detector. Now you can rederive all this stuff again, or you can assume this known, use your G-detector to select the time intervals, and then consider that on the other side you have an almost pure 1-photon state. THIS is what is done in the AJP article.
And once the almost pure 1-photon state is in the RT arm of the experiment, you don't need to go through all the glauber stuff which is fairly complicated: my analysis above is completely correct, and can be traced back to all that creator / annihilator stuff.

cheers,
Patrick.

 

[nightlight]

I didn't need any Glauber or other formalism, which does just the same, but in a more formal setting, in order to deal with more complicated cases. THIS is elementary quantum theory, but there's nothing "naive" or "toy" about it. It is fundamental.

I see, you can't answer the two direct questions on how to get zero in the numerator of the (AJP.9) along with its coherent operational mapping to the conditions of the experiment (including the given input field and the detectors placements).

The nonrelativistic QM handwaving just won't do, no matter how much Copenhagen fog you blow around it, since the nonrelativistic QM doesn't model the constraints of the EM field dynamics (such as the field operator evolution equations, commutations at the spacelike regions, EM field absorption etc.).

If you have a bit of trouble interpreting the (AJP.9), ask your friendly Quantum Optician at your work, see if with his help the two of you can answer questions Q-a and Q-b, and thus show me how the QED (and not the elementary nonrelativistic QM) does predict such genuine anticorrelation effect, by producing zero in the numerator of (AJP.9). Or ask Grangier if he can do it, he is famous for this experiment, so he must have believed back in 1986 the (genuine) anticorrelation effect existed and thus he has a valid QED proof of its existence in this experiment.

... my analysis above is completely correct, and can be traced back to all that creator / annihilator stuff.

Your analysis again deals with question how does DT or DR trigger, which is not at issue. They do trigger, we agree. The question is what does QED predict[/color] for their correlations (its formal answer is contained in AJP.9). I said the QED predicts g2>=1 for their detectors DT and DR and the input state |Psi.1> (see [post=536215]Question-a[/post]).

That's why, to avoid such non sequiturs, I had put the the main issue into a very specific form, [post=536215]the two direct and concrete questions[/post] on how to get zero in the numerator of the (AJP.9) and provide a coherent operational mapping (consistent with Glauber's definitions and conventions [4], used in AJP.9) of the formal result to the conditions of the experiment.

I gave you the two answers there, both showing that G2=0 is trivially classical in both cases, thus no genuine anticorrelation exists as a QED effect. You give it some thought and find your third way, if you can, to get zero there, thus G2=0, and let's see if it holds.

 

[vanesch]

The nonrelativistic QM handwaving just won't do, no matter how much Copenhagen fog you blow around it, since the nonrelativistic QM doesn't model the constraints of the EM field dynamics (such as the field operator evolution equations, commutations at the spacelike regions, EM field absorption etc.).

What I've written down is NOT non-relativistic QM. It is quantum theory, for short. QED is a specific application of quantum theory, and the Fock space (and its subspace of 1-photon states) is entirely part of it.
Non-relativistic quantum theory is the application of the same formalism to the configuration space of an n-particle system, and with a time evolution operator U which has as its derivative the non-relativistic hamiltonian.
QED is the application of quantum theory to the configuration space of the EM field (and the dirac field of the electron, but we only use the EM part here), and with a time evolution operator which describes all EM interactions.
But, understand this very well, QED is based on exactly the same quantum theory which contains:
- a hilbert state space
- a linear time evolution operator based upon the hamiltonian of the system
- the superposition principle
- the Born postulate linking hilbert states to probabilities of observation, as related to the eigenspaces of the operator corresponding to the measurement.

The above stuff is part of EVERY quantum theory, be it non-relativistic quantum mechanics, quantum field theories such as QED, string theory, or anything else: whatever is a quantum theory takes on the above form.

The EM field dynamics (all the things you mention above) you are always talking about comes in in the specific structure of the hamiltonian, but that doesn't change a iota to the superposition principle, or to the linearity of the time evolution operator, and these were the only aspects I needed.

I specifically don't want to plunge into the Glauber formalism because it is not needed. Forget about that g2. I have simply shown you where the anticorrelation of detection in 1-photon states comes from, and it is NOT a shortcut, it is not handwaving, and it is not non-relativistic quantum mechanics. It is full QED, with some abstract notations of the state vectors. The very fact that you don't understand this means that you have a fundamental misunderstanding of quantum theory in general, and de facto about QED in particular.
As I said, there's a difference between contesting the validity of a certain theories' predictions, and misunderstanding the predictions of the theory. You're in the second case.

cheers,
Patrick.

 

[nightlight]

I specifically don't want to plunge into the Glauber formalism because it is not needed. Forget about that g2.

Oh, well, Ok. Thanks, though, for the spirited battle which made me dig some more into the Glauber's lectures [4] (which is the holy of holies of the Quantum Optics) to clarify few of my own misconceptions.

 

[vanesch]

Oh, well, Ok. Thanks, though, for the spirited battle which made me dig some more into the Glauber's lectures [4] (which is the holy of holies of the Quantum Optics) to clarify few of my own misconceptions.

Well, I can tell you the following: one of the reasons why I don't want to delve into it is that I don't have a very profound understanding of it myself. I more or less see how it is handled, but I think it would be too dangerous for me to go and talk much about it.
However, what I tried to point out (and it is no "pedagogical weaseling out") is that for THIS PARTICULAR CASE (the anti-correlation of detections predicted by QED), you really don't need that machinery. I'm really convinced that what I've written down is correct, as a QED prediction, because I really need only very few elements of quantum theory to make the point, and I'm really amazed to see you contest it each time.

That said, I'll try to plunge deeper into the Glauber formalism to try to address the points you raise.

cheers,
Patrick.

 

[vanesch]

Question-a)[/color] Given actual DT and DR of their setup, thus DT absorbing mode |T> (hence a_t |T> = |0>, a_t |R> = 0) and DR absorbing mode |R>, what input state (used for <...> in AJP.9) do you need in the numerator of (AJP.9) to obtain their g2=0? Is that input state same as their "one photon" state |Psi.1> = (|T> + |R>)/sqrt(2) ?

This is a particularly easy question to answer.

The numerator is < a_t^dagger a_r^dagger a_r a_t >

and the state is 1/sqrt(2) (|t> + |r>)

This is nothing else but the norm of the following vector:

a_r a_t 1/sqrt(2) (|t> + |r>)

apply distributivity (linearity of the a operators):

1/sqrt(2) ( a_r a_t |t> + a_r a_t |r> )

Now take the first term: a_t |t> = |0> and a_r |0> = 0

the second term: note that a_t |r> = 0, so a_r a_t |r> = 0

So it seems that we have the null vector. Its norm is 0.

QED

cheers,
Patrick.

 

[vanesch]

Question-b)[/color] To avoid confusion with (Question-a), take their (AJP.9) and replace the subscripts T and R with generic 1 and 2. Then, given their actual "one photon" state |Psi.1> and given their g2=0 conclusion, tell me what kind of absorbers (what is the basis defining the anihilation operators) a1 and a2 do you need to obtain 0 in the numerator of (AJP.9) on the |Psi.1> (used for the averaging)? What is the spatial extent of the Glauber detector realizing such absorber? { As a check, consider placing just that first detector by itself at your place chosen and examine its effect on |Psi.1>, i.e. after the QED absorption of |Psi.1>, does |Psi.1> become |0> or some other state? E.g. DT of their setup doesn't produce the required effect, the |0> as the final field state after acting on |Psi.1>. Also check your design for GD1 detector (implementing this absorption effect) that does not violate QED evolution equations for the field operators in the region R, which will be the Maxwell equations.}

My answer: To get 0, a1 (or a2) must be an absorber of mode |Psi.1>, hence a2 a1|Psi.1> = a2 |0> = 0. The corresponding Glauber detector performing a1 absorption must extend spatially to receive the entire EM flux of |Psi.1>, which happens to cover T and R paths.

This is also simple to answer. ANY single-photon mode, with any detector setup, will give 0.

The reason is quite simple: you have TWO ANNIHILATION OPERATORS in (9). A single-photon state (no matter which one) is always annihilated by two annihilation operators.

The reason is this: imagine a basis of EM modes, labeled q1...
Now, a_q5 which annihilates the 5th mode has the following behaviour:

a_q5 |mode 5> = |0>
a_q5 |mode not 5> = 0

Now, write a general 1-photon state in the basis of the |q> modes, and let's say that our two annihilators are a_7 a_5. Now, or psi contains a term with mode 5, say c |mode5>. Then a_7 a_5 will lead to a_7 c |0> (all the other terms are already put to 0 by a_5).
Next a_7, acting on |0> gives you 0.
Or psi does not contain a term with mode 5. Then a_5 |psi> will already be 0.

So this comes down to saying that in a 1-photon mode, you can't have 2 detectors detect them together. Or, as Granier put it: you can detect one photon only once.

So, for a 1-photon state, no matter which one, g2(0) is always 0.
Simply because you have 2 annihilators.

BTW, for fun, compare this to my orthogonal E1 and F1 spaces :-)

cheers,
Patrick.

EDIT: it seems that your mistake is somehow that you think that an annihilator operator, working on a state, only produces the vacuum state if the state is its own mode ; and that if it works on another mode, it is "transparent". Although the first part is correct, the second isn't: an annihilator of a certain mode, working on a state of another mode, just gives you the null vector.

a_q |q> = |0> but a_q |r> = 0 and not, as you seem to think, a_q|r> is |r>.

 

[vzn]

hi all I skimmed thru most of this thread. quite
amazing burst of dialogue. I hope to chat with "nite" 1-on-1 in
the future.. nite, why the mysterious anonymity? are you at
a university currently? want to protect your reputation? who is the
mysterious priesthood anyway, wink..

I have been looking at loopholes in QM for close to a decade.
lately I've realized that it will take an extremely brilliant person
to "get past" QM if that will ever happen. this person will have to
have a brilliant grasp of both theory AND experiment. nite is
the closest I've ever seen to this in many,many years.

I admire
marshall/santos work like nite, however they are hardcore
theorists. note that many of einsteins 1905 papers had NO references,
although he seemed to allude to michelson/morley in one. stunning!
einstein was an unadulterated theorist. the EPR paper is the closest
you can see to einstein actually "getting his hands dirty"...

imho nite really strong in-the-know challenge to the thorn et al
experiment (& even classic predecessors like grangier)
is electrifying, I just went over his criticism very carefully
with the original in front of me, and I would like to delve into
that further at some pt

(I see thorn et al circuitry is indeed quite different
from kwiat paper nite cites, and makes me wonder-- there might be
a lot of variation in detector electronics across experiments and yet
its always reported in papers as a "black box".. I share nite's
frustration over this! all the way back to aspects original papers!
I understand space limitations and all that, blah blah blah, but its
the 21st century, and let's throw away those
archaic & useless conventions pre-mass-digital space. how about
writers put up full schematics online, & summaries in journals? could
it be we are only having this conversation because of add'l detail
of more modern papers makes more careful analysis possible?
but.. clearly.. only approaching the minimum required level of detail to
discriminate two virtually identical theories?).

I can/may write much,much more on this topic just off the top of my
head & years of notes & musings & hope to contribute much more
over time here.

meantime I would like to invite everyone to my (almost 4-yr old)
mailing list to
discuss this thread in particular (outside all the other physics forum threads)

http://groups.yahoo.com/group/theory-edge/


next, some pts for nite. (I will nickname "nite" for now)

nite: I am really rooting for you & have been involved in the same
"research project" you are advancing, namely looking for LHV theories
maximally compatible with QM, or a minor revision of QM.
I agree with your critics however
that you should try to expand your criticism. why is it there can be so
much controversy over what should be _conceptually_
very simple experiments? the problem with LHV advocates is that
they cannot point to any EXPERIMENTS that back their position. not at
all! they point to experiments that are designed to reveal NONLOCALITY.

so if nonlocality is bogus, let's TURN THE TABLES. make experiments and
parameters in which LOCALITY is demonstrated even where NONLOCALITY
is expected. I have yet to see marshall/santos ET AL _EVER_ propose
NOVEL EXPERIMENTS designed to reveal the real problem with QM--
this is a feat bell managed that almost nobody has ever topped.

even better, RUN THEM YOURSELF! I hope that a genuine LHV
advocate gets ahold of an experimental apparatus soon! I have
met at least one (more on that later).. why is the LHV crowd so
devoid of any _experimentalist_ supporters?

what is an experiment that would leave QM supporters scratching their
heads? nite, in particular, here are some things that have never been
done:

a) if bell experiments don't demonstrate nonlocality wrt QFT
predictions, then where is the paper that shows that bell made
a mistake in the theoretical prediction? I in fact have found such
a paper that suggests he is getting photon number operators
mixed up or not clear on them-- I think there is good evidence
that bell didnt understand the photon number operator concept
much at all in his writing. (get his book of collected writings
and look for it! where is it?)

it will take me awhile to dig up
this paper (peer reviewed & published in highly reputable journal)
if there is interest, but it is indeed out there. unfortunately it
is only a beginning, it does not rederive the entire bell thm based
on this new perspective. how about YOU write this paper??
of anyone I have ever seen write, you are about the closest
to filling in all the blanks.

b) you say that QM is just subtracting "accidentals". then how
about this: design an experiment that will maximize the
effect of accidentals. QM has very little to say, or maybe nothing,
about quantitative measurements of accidentals, right?
can we create an experiment
that is entirely focused on "accidentals" which QM is somewhat
blind to, or considers them NOISE,
such that we can force a prediction using semiclassical
(glauber et al) theories, for which QM is MUTE in its prediction?

c)

nite, you are giving up too easily here! (but shame on your
critics for not discovering how to do this themself,
and contributing themself to this foremost goal,
not admitting that it is THEIR responsibility also, if they
want to be serious scientists and not just reactive anklebiters, just
as they accuse nite of being!)

I propose something along these lines, you virtually wrote
it up yourself, and I have long, long been
trying to get info on this experiment. Imagine a single
(semi) classical wavefront going thru N detectors. as I
read vanesch post #64, and we are dealing mostly
in this thread with N=2, look at what he says.
he says QM is only talking about these cases:

(a) only detector R clicks
(b) only detector T clicks.
(c) neither click
(d) both click.

now isn't the entire point (as vanesch seems to be writing) QM
applies only to predicting (a),(b) based on _collapse of the wavefn_
and has NOTHING TO SAY about cases
(c),(d)? and isn't it true however that semiclassical theory can give you
predictions about about all FOUR cases? imho, you have fallen
for the sleight of hand yourself without realizing it!
forget about (a),(b) and focus on (c),(d) which QM is indeed
apparently blind to! ie, exactly as einstein asserted.. INCOMPLETE

for those who advocate QM theory here (eg vanesch),
can you predict the following?
given N detectors, what is the possibility of detecting M<N of them
at a time? nite has given an EXACT PREDICTION FOR THIS
on binomial/poisson statistics, which is ENTIRELY CONSISTENT
with semiclassical predictions (see his earlier post..
I will look up the exact # later).

as I understand it, QM can only predict that only ONE of the N
detectors clicks at a time, if the collapse of the wavefunction
is a real physical phenomenon! so the experiments so far are
focused on N=2, but a really nice experiment would look at a
detector bank and show how a single E/M wavefront, as it
moves thru the detector array, creates a "probability wave" of
clicks! ie (contradicting the existing copenhagen dogma/theory)
the sch. wavefn is a real physical entity!

more later

 

[vanesch]

for those who advocate QM theory here (eg vanesch),
can you predict the following?
given N detectors, what is the possibility of detecting M<N of them
at a time? nite has given an EXACT PREDICTION FOR THIS
on binomial/poisson statistics, which is ENTIRELY CONSISTENT
with semiclassical predictions (see his earlier post..
I will look up the exact # later).

as I understand it, QM can only predict that only ONE of the N
detectors clicks at a time, if the collapse of the wavefunction
is a real physical phenomenon! so the experiments so far are
focused on N=2, but a really nice experiment would look at a
detector bank and show how a single E/M wavefront, as it
moves thru the detector array, creates a "probability wave" of
clicks! ie (contradicting the existing copenhagen dogma/theory)
the sch. wavefn is a real physical entity!

I have one very clear advice for all those trying to explain that QM is bogus, and that is: learn it first!

First of all, these experiments don't "show that collapse is a real phenomenon". I for one, don't believe in such a collapse, as many other people (not all) interested in foundational issues. That discussion (about collapse) has nothing at all to do with what we all agree on: the PREDICTIONS OF OUTCOME OF EXPERIMENT of quantum theory.

As to your specific question: is quantum theory not able to predict several coincident detections ? Answer: you must be kidding. Of course it is. For instance, the simultaneous detection of idler and signal photons from a PDC xtal.
There are two ways to see it: one is the "student" way, the other is the "more professional" way.
The student way: PDC xtals generate 2-photon states. Applying two annihilation operators to a 2-photon state gives you the vacuum state which has a finite norm.
The more professional way: PDC xtals have an effective time evolution operator which couples the 1 UV photon -> idler photon + signal photon and the incoming beam (a UV laser) is well described by a coherent state.
Apply the time evolution operator to that incoming beam, and you generate a lot of photon states, but with a surplus (as compared to a coherent state) of two-photon states (= the student case). The rest generates you the non-matched clicks (the 1-photon states that remain).
See, the "professional way" takes into account more "side effects" but the student way has the essential effect already.

But if your aim is to critique QM, all the above should be EVIDENT to you.

As should be evident the answer to your specific question:
If we have an incoming 1-photon state (such as is essentially produced by taking a 2-photon state from a PDC and triggering on the idler), and you give it the possibility to hit (through beamsplitters) several detectors, then there will only be ONE that hits indeed. Of course there could be a slight contamination in the beam of other, multi-photon states (remnants of the original coherent beam, or light leaking in or...), which can give rise to occasional double hits.

How about the following: if you send the signal beam on a hologram, and that image then onto a position-sensitive photodetector, then your N detectors are the N pixels of the position sensitive detector. Each time the idler clicks, at most ONE pixel should be hit. And it should built up the image of the hologram over time. I have vague souvenirs of such an experiment, but I don't know if they looked at the right things. I'll try to look it up.

As to the binomial distribution in the classical case: quantum theory also predicts that, if the incoming beam is a coherent state. You can do a lot of calculations, but there is a neat trick to see it directly:
Coherent states are eigenstates of the annihilation operator:
a |coh> = some number x |coh>

This means that applying an annihilation or not, to a coherent state, doesn't change its photon content, which comes down to saying that the statistics of detecting a photon somewhere is _independent_ of having detected a photon somewhere else or not. Detections are statistically independent, which immediately, as in the classical case, gives rise to the same binomial distributions.

cheers,
Patrick.

 

[nightlight]

Question-a) Given actual DT and DR of their setup, thus DT absorbing mode |T> (hence a_t |T> = |0>, a_t |R> = 0[/color]) and DR absorbing mode |R>,...[/color]

This is a particularly easy question to answer.

It surely was, especially with the little hint (a_t |R> = 0) added to "help" you (sorry:) decide whether to "plunge into" the head on answer.

Unfortunately, the low-res analysis you provided (shared also by the "quantum prediction" of AJP.8-11 and the typical textbooks they copied it from) does not resolve well enough to distinguish between an absorber (=detector) covering both paths T,R and an absorber covering just path T, or for that matter, an absorber set somewhere else altogether, away from T and R paths. The reason it can't resolve these different absorbers is not because all these cases yield the same outcomes (they obviously don't; e.g. think of different apperture sizes of DT and consider its singles rates) but because the low-res treatment lacks any formal counterpart for the detector size. And the reason for this omission is that the low-res approach assumes one fixed size, an infinite absorber absorbing an infinite plane-wave modes, thus in the low-res it is indeed true that a_t |R> = 0. In our case of (Q-a), though, neither the absorbers nor the EM fields are of this kind.

In order to model the difference, say, of the singles on DT as we change its size or position relative to T and R beams (or similarly for the coincidences of DT and DR), we would need a formalism which does not assume one fixed size, much less infinite, of the absorbers or the infinite modes -- thus the formalism which has the formal counterparts for these parameters[/color] (which are the key in discussing matters of non-locality).

As luck would have it, Mandel had already done it in 1966 (cf. [8]) and a simplified form is in his QO textbook [9]. The main simplification in [9] (sect. 12.11) is that [8] considers general multi-time operators while [9] deals only with the single-time operators. For our present questions Q-a,b, though, that doesn't matter (it would matter if we were discussing effects such as the possible [post=533012]polarizer flip-flop effect[/post] I mentioned earlier). In [9], Mandel breaks the problem of the finite absorbers to cases of infinite (12.11.1-4) and finite (12.11.5) EM field modes. The upshot of his analysis is that in the latter case (relevant to our problem Q-a,b), no general relations of the kind a_t |R> = 0 holds for the finite absorbers (his V(r,t), eq 12.11-1) acting on finite modes (his Phi(r,t), eq. 12.11-28), although the Glauber's mode absorption rates are still formally given by the same type of normally-ordered expectation values of localized mode creators & annihilators averaged over localized states. Thus, one can formally look at the (AJP.9) as still being valid, except that now the annihilators a_t and a_r must be considered as localized absorbers acting on localized modes.

Mandel emphasized the difference between his first case of infinite modes (where the absorption rates and the photon intensities are roughly interchangeable, almost as freely as in the more elementary analysis) and the second case of finite absorbers and finite modes (where the two are not interchangeable cf. eqs 12.11-30 vs 12.11-40). Pointing to this difference, he warns of the pitfalls[/color] (for the low-res analysis relying on naive photon image and photon numbers to deduce detection probabilities) in the precise case we are discussing, where we have localized modes/photons and the localized absorbers, due to the particular distinctions between the photon numbers & intensities (as photon fluxes) and the detector counts & the absorption rates ([9] p 639):

From these considerations it is apparent that the concept of the photon as a localized [approximately, as defined in eq 12.11-30] particle traveling with velocity c can be quite inappropriate and misleading under some circumstances, even though it works in other cases.

To obtain the classicality of the coincidence rates via (AJP.9) in the case (Q-a), where we have the localized detectors DT and DR, as given in the actual experiment, and on the actual input state |Psi.1> = (|T> + |R>)/sqrt(2), as explained by Mandel in [8],[9], we need to use generalized absorbers for the a_t and a_r. To get these absorbers, we will go back to their origin as "absorbers" (whose counts G2(DT,DR) in numerator of AJP.9 "correlates") in the Glauber's perturbative treatment ([4] Lect's iv,v) and augment them there in accordance with the finite interaction volumes given by DT and DR.

The interaction Hamiltonian for the absorber+EM field in dipole approximation is H_i = Sum(j){q_j(t) E(r,t)} (cf [4] eq 4.1, ignoring constants), where q_j are dipole moments of j-th electron in the absorber and E is the incident Electric field operator. To obtain the time evolution of the field-detector system Glauber uses interaction picture, where the combined state |Psi> = |Psi_em> |Psi_a> (where subscripts em refers to EM field and 'a' to absorber) evolves as: i d|Psi>/dt = H_i |Psi> (note that here the state evolves via the interaction H_i only, while the EM field operators, including those in H_i, evolve via the free field Hamiltonian H_f, thus via Maxwell equations). In the 1st order perturbation he obtains for the ionization rates (or the electron transition "probabilities" which already include his subtraction conventions): <i|E(-)E(+)|i>, where |i> is the incident field state and the E=E(+)+E(-) (positive & negative frequency decomposition of Electric field operator in the H_i). He identifies (E+) as 'photon' annihilation operator (and (E-) as creation), thus in our earlier notation we can write this as <i|(a+)(a)|i>. The multiple absorbers ([4] lect 5) yield in n-th order perturbation the regular Glauber "correlation" functions Gn() (shown as G2() in numerator of AJP.9), with his subtractions already built into their definition.

Those derivation did not assume any localization of the absorbers, thus any restrictions on the H_i interaction. In our case of localized detectors, we need to limit the effect of the H_i to the space region of the detector. We can do this by attaching a factor with H_i which is 0 outside of the detector and 1 inside. In his analysis [8] & [9] of space limited detectors, Mandel had also introduced this factor as U(r,V) where U=0 if r outside of volume V and 1 if r is inside V (cf. [9], p 633). With that factor included (which will follow E of H_i throughout derivations in [4]), the interaction part evolution operator U_i (cf. [4] eq 4.3 for single detector and 5.2 for n detectors) remains unchanged from the earlier analysis inside V and becomes identity outside V (in interaction picture). In Schrodinger picture the latter becomes free evolution of field states |Psi> outside the volume V(DT) of the detector DT, and the absorption described via 'limited' annihilator A_t(V) inside the detector volume V(DT). The Glauber's Gn() functions remains formally same as before, except for the replacements of the 'unlimited' operators a_t and a_t(+) with the 'limited' operators A_t(V) and A_t(+,V).

Thus we can now interpret the (AJP.9), when applied to the actual limited space detectors DT and DR used in the experiment, as containing the 'limited' versions of the operators, simply labeled as a_t and a_r (with the understanding that, e.g. a_t includes the volume parameter, which defines its action as identity outside the volume and absorption inside V).

Now we arrive to the critical question: how does this 'limited' a_t act on the space limited state |R> (this is precisely the type of case that Mandel warned about when using the naive photon number reasoning in predicting detection rates)? Since the region R is outside the volume V(DT), in Schrodinger picture the state |R> will evolve via the free field equations, as it did before its "interaction" with the 'limited' absorber a_t. Therefore a_t |R> = |R>. As you already noticed, this implies (we can call it, I guess, our joint conclusion) that QED prediction is g2>=1 for the space limited detectors DT and DR and the input state Psi.1> = (|T>+|R>)/sqrt(2) of this experiment.



--- Ref

[8] L. Mandel "Configuration-Space Photon Number Operators in Quantum Optics"
Phys. Rev 144, 1071-1077 (1966)

[9] L. Mandel, E. Wolf "Optical Coherence and Quantum Optics"
Cambridge Univ. Press., Cambridge (1995)

 

[vanesch]

Therefore a_t |R> = |R>.

That's silly. You know the solution for that, don't you ? An eigenstate of an annihilation operator: it is a coherent state. So you just showed that |R> is a coherent state :-)

cheers,
Patrick.

 

[seratend]

as I read vanesch post #64, and we are dealing mostly
in this thread with N=2, look at what he says.
he says QM is only talking about these cases:

(a) only detector R clicks
(b) only detector T clicks.
(c) neither click
(d) both click.

now isn't the entire point (as vanesch seems to be writing) QM
applies only to predicting (a),(b) based on _collapse of the wavefn_
and has NOTHING TO SAY about cases
(c),(d)? and isn't it true however that semiclassical theory can give you
predictions about about all FOUR cases? imho, you have fallen
for the sleight of hand yourself without realizing it!
forget about (a),(b) and focus on (c),(d) which QM is indeed
apparently blind to! ie, exactly as einstein asserted.. INCOMPLETE

It seems that you have some problems with probability in general and the meaning of random variables and observables. Let take a basic example:
If in a classical probability problem you choose, for example 2 random variables X and (-X), where X has only two values {-1,+1}, do you think that case (c) and (d) are possible for these 2 random variables?
If you think it is impossible, does that mean that Kolgomorov probability is incomplete or blind?
Do you really understand what is, mathematically, the sample space of an observable?

Seratend.

 

[vanesch]

Now we arrive to the critical question: how does this 'limited' a_t act on the space limited state |R> (this is precisely the type of case that Mandel warned about when using the naive photon number reasoning in predicting detection rates)? Since the region R is outside the volume V(DT), in Schrodinger picture the state |R> will evolve via the free field equations, as it did before its "interaction" with the 'limited' absorber a_t. Therefore a_t |R> = |R>. As you already noticed, this implies (we can call it, I guess, our joint conclusion) that QED prediction is g2>=1 for the space limited detectors DT and DR and the input state Psi.1> = (|T>+|R>)/sqrt(2) of this experiment.

In order to illustrate a bit further your confusion here, let us consider one of those infinite absorbers you like, which are absorbing perfectly IR photons, and are completely transparant to, say, UV photons. This is in fact the best possible definition associated to the annihilators of the plane wave basis. (you still didn't understand that you can change basis, and that that is all that matters here).
So, with this absorber, which absorbs ONE fock space mode, namely certain IR photons exactly along the z-axis, corresponds perfectly a_IRmode, and our detector DOES NOT absorb UV photons.
So, applying the same reasoning, a_IRmode |UV state> = |UV state>, right.
Well, that's wrong. a_IRmode |UVstate> = 0.
Because now we are with the orthodox creation and annihilation operators.

I can tell you that you are digging something up you don't understand yourself: what Mandel is addressing is those cases where you have such tiny detectors, so close to one another, that you cannot have modes which are solutions to the EM field, hitting fully, or missing fully, the tiny detectors. But from the moment they have a size of several wavelengths, and are zillions of wavelengths apart, this doesn't matter ; for all purposes you can approximate the modes by the plane wave modes, in spatial wave packets. This is tedious, complicates matters, and doesn't change the result. It only leads to confusion. You are again talking about plate tectonics to show that planets are not point particles and that Newtonian theory doesn't predict Kepler orbits.
One thing is sure: your conclusion that for a finite detector, a_detector|other stuff>= |other stuff> is obviously plain wrong, because it means that ALL other stuff is a coherent beam of a certain intensity!
You've been digging in plate tectonics because you've read somewhere that someone said that the mass distribution within a planet has gravitational effects (which is true, tiny effects are due to that), and you came to the conclusion that this means that planets go in square orbits.

cheers,
Patrick.

 

[nightlight]

That's silly. You know the solution for that, don't you ? An eigenstate of an annihilation operator: it is a coherent state. So you just showed that |R> is a coherent state :-)

The limited volume operator a_t isn't the same thing as the annihilation operator. Additionally, for Mandel's case two (of limited volume modes, such as |R>) most analogies with regular algebra of a,a+ doesn't work, and many usual implications dont' follow. Check the cited sections in ref [9], where I quoted Mandel's warning.

(I've seen not long ago another paper which aimed to establish different QO measurement theory, where the "single photon" Fock states appear statistically as coherent states, which agrees with my conclusion as well.)

-- EDIT

Ok, I found that paper (quant-ph/0307089), and indeed Fock state in their model produces Poissonian photo-count distribution (which is an entirely different claim than claiming Fock state has Poissonian photon distribution, as you tried to interpret it, illustrating thus perfectly the very warning by Mandel, I cited):

M. C. de Oliveira, S. S. Mizrahi, V. V. Dodonov
"A consistent quantum model for continuous photodetection processes"

Abstract

We are modifying some aspects of the continuous photodetection theory, proposed by Srinivas and Davies [Optica Acta 28, 981 (1981)], which describes the non-unitary evolution of a quantum field state subjected to a continuous photocount measurement. In order to remedy inconsistencies that appear in their approach, we redefine the `annihilation' and `creation' operators that enter in the photocount superoperators. We show that this new approach not only still satisfies all the requirements for a consistent photocount theory according to Srinivas and Davies precepts, but also avoids some weird result appearing when previous definitions are used.

CONCLUSION

Summarizing, in this paper we proposed modifications
in the SD photocount theory in order satisfy all the precepts,
as proposed by Srinivas and Davies for a consistent
theory. Our central assumption was the choice of
the exponential phase operators E− and E+ as real ‘annihilation’
and ‘creation’ operators in the photocounting
process, instead of a and a†. The introduction of those
operators in the continuous photocount theory, besides
eliminating inconsistencies, leads to new interesting results
related to the counting statistics. A remarkable
result, which is responsible for all the physical consistency
of the model, is that in this new form an infinitesimal
photocount operation JE really takes out one photon
from the field,[/color] if the vacuum state is not present. Consequently,
the photocounting probability distribution for
a Fock field state is Poissonian[/color], evidencing again the direct
correspondence of number of counted photons and
number of photons taken from the field.
We also have investigated the evolution of the field
state when photons are counted, but with no readout,
leading to the pre-selected state. The mean photon number
change shows now (in contrast to the exponential
law obtained for the amplitude damping model) a nonexponential
law, which only depends on the condition
that photons are present in the field, independently of
their mean number.
An advantage of the proposed model is its mathematical
consistence. Since many of its predictions, especially
those related to multiphoton events, are significantly different
from the predictions of the SD theory, it can be
verified experimentally. One of the first questions which
could be answered is: whether the decrease of number of
photons in the cavity due to their continuous counting
always obeys the exponential law (61) (i.e., the rate of
change is proportional to the instantaneous mean number
of photons), or nonexponential dependences can be also observed (for example, in the case of detectors with
large dead times)? ...

.

 

[vanesch]

The limited volume operator a_t isn't the same thing as the annihilation operator. Additionally, for Mandel's case two (of limited volume modes, such as |R>) most analogies with regular algebra of a,a+ doesn't work, and many usual implications dont' follow.

I think you'll agree that you can write a_detector (your "limited volume" modes) as an algebraic combination of the a_{planewave}, right ? Or not ?
Note that all operators are writable in a_{planewave} and a^dagger_planewave.
So some expression is possible. Please do.

Say I have a detector that has 1cm^2 absorption surface for 500nm photons, is placed in the XY plane {points {0,0,0}, {0,1,0},{1,1,0} and {1,0,0}}.
It doesn't absorb anything else.

Can you write me, according to Mandel, what a_{squaredetector 500nm} is, as a function of the plane wave annihilation operators ?

cheers,
Patrick.

 

[vanesch]

I think you'll agree that you can write a_detector (your "limited volume" modes) as an algebraic combination of the a_{planewave}, right ? Or not ?
Note that all operators are writable in a_{planewave} and a^dagger_planewave.
So some expression is possible. Please do.

I looked at the Mandel paper Phys. Rev. 144, of 1966 and in fact that's exactly what he does (and what I'd had more or less in mind).

Ok, look at equation, say (32) in that paper. For our correlation function of detection of two photons, we'd have a linear combination (integrals) of terms of the kind: <A_i1(x1,t1) dagger A_i2 (x2,t2) dagger, A_i2 (x2,t2) A_i1(x1,t1) >

So this is a sum of products of two "A-dagger" operators, and two "A-operators".

Now, if you look at equation (1), you see that an A-operator is written as a linear combination of TRUE PLANE WAVE ANNIHILATION operators, and by conjugation, A - dagger is a linear combination of true plane wave creation operators.

Substituting, you will thus find that we have a linear combination of terms which take on the form:

< adagger_s a_dagger_r a_r a_s >

And my story goes again: in each term, we have two annihilation operators, acting on a 1-photon state, which will always give you 0.

But as I said, this is hopelessly complicating the issue, because given the size of the detectors (huge compared to the wavelength), in equation (2) you ESSENTIALLY PICK OUT plane wave annihilators, combined in a wave packet, as I told you. And as our detectors are hugely distant the relations (23) and (24) come close to my "low resolution" treatment.

Kepler orbits.

cheers,
Patrick.

 

[nightlight]

I looked at the Mandel paper Phys. Rev. 144, of 1966 and in fact that's exactly what he does (and what I'd had more or less in mind).

Ok, look at equation, say (32) in that paper. For our correlation function of detection of two photons, we'd have a linear combination (integrals) of terms of the kind: <A_i1(x1,t1) dagger A_i2 (x2,t2) dagger, A_i2 (x2,t2) A_i1(x1,t1) >

So this is a sum of products of two "A-dagger" operators, and two "A-operators".
...

That paper covers only the Mandel's first case, which is the finite detector with infinite space modes. The second case, when modes and detectors are finite is only in the sect 12.11.5 of [9], and that is the case of interest. Physically, in the infinite mode case, the detector does interact with all modes and the detection behavior is approximately similar to ordinary infinite detector case.

The [8] is not directly relevant for the problem discussed, except that it was the paper which recognized the problem (which you denied to exist at all). He introduced there only the limited size detector model and the notation which [9], sect 12.11.5 develops further for the case we are discussing (which is: what happens when the actual DT detector, which is limited, is detecting field in state |R>). Unfortunately, in 12.11.5, he pursues entirely different objectives (analogies with particle wave functions his Phi and Psi, and their difference, eqs. (30) vs (40)).

Thus, to obtain the properties of interest of the correlation functions, it turns out it was more convenient to go back to Glauber's model [4] and obtain formally the limited-space Gn()'s by directly limiting the interaction volume of the H_i and replaying his derivation of Gn()'s with this limitation carried along, as I sketched earlier. It can probably be replicated with Mandel's quantum detection model in [9], chapter 14 (that's left as an excercise for the reader).

The conclusion is also physically perfectly satisfactory since, forgetting the detector and all the QM baggage that goes along, just put an atom (Glauber's model for his "ideal" detector) at DT place and prepare narrow beam, mode |R>, as the EM field state. Its evolution will be entirely unaffected by the presence of the atom (or of the entire detector DT cathode, if you wish), as you can easily verify by experiment.

--- Edit:

I don't doubt that you also realize, without having to do any experiment, it is exactly what will hapen. Then, the subsequent superposition of this |R> with |T>, which is also localized in its own region, doesn't change anything with the EM fields in region R (other than normalization constant if your convention is to keep the total energy fixed). The superposition in this case (with the spacelike interval between T and R at the time of detection) merely means that EM fields at T and R have a common phase, their oscillations are synchronized (with at most some delay), thus should you deflect them and bring them back to a common detector you will observe interference.

The presence of such synchronization surely doesn't somehow allow some kind of magic effect by the T branch of EM interacting with DT, to make any difference at all in the spacelike region R (in any experiment performed in R region on the R EM branch). The only pseudo-magic that can happen is if you perform the experiments on R in sync with the experiments on T, then you will obtain various forms of synchronized efects, such as beats in the results, of these two sets of experiments.

 

[vanesch]

The [8] is not directly relevant for the problem discussed, except that it was the paper which recognized the problem (which you denied to exist at all).

There was nothing to deny, you are, as usual, overcomplicating the issue. Now that [8], which "showed" according to you, that a_det |other mode> = |other mode> doesn't hold, and that my very good approximation of a_det1 and a_det2 as annihilation operators which are to be brought in relationship with essentially orthogonal modes comes out, you say that it isn't relevant, and that I have to switch to its "simplified student treatment" [9].
And there again, you are going to play the "finite space mode" game, which is just a narrow wavepacket of plane wave modes, because, again, the sizes involved are so huge as compared to the wavelengths that it doesn't matter.

But go on, complicate the issue, in the end I'll have to give up because you'll point out that I didn't take into account the gravitational attraction of the moon in my toy model.

I will tell you that it won't work, with finite modes either, and that is because the finite modes involved here (beam goes to the left, and beam goes to the right), will be written as superpositions of plane waves which are 1) all very close to one another in k-space, for one beam, and 2) very remote from each other for the left beam and the right beam.
So these "finite beams" use orthogonal modes. And as there is no overlap in the used modes, everything still holds. No big deal.

cheers,
Patrick.

 

[nightlight]

There was nothing to deny, you are, as usual, overcomplicating the issue.

Of course, you denied there was any difference that mode sizes and detectors size make, when you said:

This is also simple to answer. ANY single-photon mode, with any detector setup, will give 0.

As you now realize (or will realize, when you had time to think it through), using the finite mode sizes and finite detector sizes makes big difference in the results. Among other things, instead of g2=0 you get g2>=1.

Now that [8], which "showed" according to you, that a_det |other mode> = |other mode> doesn't hold,

Which no one claimed to hold under the assumptions in [8], either.

you say that it isn't relevant, and that I have to switch to its "simplified student treatment" [9].

That was already done in my initial post on [8], [9]. Except for the multiple-time results, the paper [8] is three decades older than [9], too. That's about as many decades as there were in the failed attempts, euphemisms aside, to demonstrate Bell inequalities violation. Not a single one worked. And there is no theoretical reason from QED to believe any will ever work.

And there again, you are going to play the "finite space mode" game, which is just a narrow wavepacket of plane wave modes, because, again, the sizes involved are so huge as compared to the wavelengths that it doesn't matter.

It matters because it affects what detection rates you will get. With the DT and DR setup they had, with their state |Psi.1>, the infinite mode treatment is absurd, it gives absurd result (which they could "confirm" experimentally only by dropping almost all the triple coincidences through their timing trick).

But go on, complicate the issue, in the end I'll have to give up because you'll point out that I didn't take into account the gravitational attraction of the moon in my toy model.

It does make big difference whether the mode |R> overlaps detector DT or not. The infinite mode treatment simply doesn't apply. The finite modes complicate matter only mathematically. But the infinite modes complicate matters conceptually and logically, since you get tangled in the web of absurd results, for which your "simple" solution is to imagine splitting universes (how often?), and finally declaring that the whole universe, along with all its innumerable MWI replicas, is a figment of your mind (solipsism to which you fell on as your last defense in the previous discussion).

And now you complain that I am complicating matters by insisting, what ought to be obvious, that the detectors DR and DT and the modes |T> and |R> are restricted in size and that this restriction makes difference in correlations compared to infinite detectors and modes. You make an example of pot calling snow black.


I will tell you that it won't work, with finite modes either, and that is because the finite modes involved here (beam goes to the left, and beam goes to the right), will be written as superpositions of plane waves which are 1) all very close to one another in k-space, for one beam, and 2) very remote from each other for the left beam and the right beam.

You can't have infinite extent of |R> transverally either. Which means you will nead many transversal components to build a narrow beam |R>. Plus, you need to have a very large box, so that the periodicity of the expansion doesn't leave nonzero parts in the region DT.

So these "finite beams" use orthogonal modes. And as there is no overlap in the used modes, everything still holds. No big deal.

No it doesn't hold, not with finite detectors (which you apparently forgot again), with spatial limitation of DR and DT.

--- Edit

I think the discussion of the last several messages has been running in circles. Again, thanks for the good challenge, and we'll be at again in some other thread.

 

[vzn]

hi guys. yes I think I now see an experiment that could
prove that QM is incomplete based on nite's arguments & the replies
of his critics. so far
even nite does not seem to be aware of this possibility.

lets look at a PDC "ring" of correlated photons emitted
from a crystal illuminated by a laser, a photo of this can
be seen in the kwiat et al paper that nite cited here.

existing QM experiments tend to be focused on looking
at only two branches or detection locations of this emission.

lets just look at multiple "branches" (to borrow charged manyworld
terminology, but only at nite's lead) with N>2. unfortunately
this would translate into very expensive experiments
because good photodetectors tend to cost $5K or more each. but
maybe it could be shown with cheaper detectors. (and in fact
imho, the experimenters are going somewhat
in the wrong direction, at least wrt these types of
experiments, by trying to get the most expensive
detectors possible, translating into low N..)

QM experiments are oriented around "signal" and "idler" photons.
but this is a misnomer in the sense that both branches are fully
physically equivalent/symmetric. there is no physical distinction.
it is just an N=2 detector experiment. both branches are equivalent.

according to the predictions of QM, if I have say N=3 detectors
and NOT assigning any particular detector as "signal vs idler"
(as is natural in the symmetry of the physics),
the following events are mutually exclusive:

detector 1 clicks only.
detector 2 clicks only.
detector 3 clicks only.

IN CONTRAST the prediction of semiclassical theory is that you will always
get a meaningful, nonrandom distribution of
coincident clicks in any of the detectors.
nite quotes this result in post #16 on this thread where he talks about
the binomial vs poisson statistics. (possibly due to a paper of glauber,
based on semiclassical theory, I would like nite to clarify where that
came from.. also I wish nite would describe in short what a "glauber detector" is...)

and yes, I have pointed out in the past & vanesch notes--
an experiment using
a CCD camera (array) would be natural to use (N very large) &
might be able to show this effect across detectors.
however one would have
to get "flat" planar wavefronts hitting the front of the array, which is
difficult because optics always makes them spherical. however some kind
of collimator might be possible.

vanesch replies that QM makes the prediction about
coincident poisson/binomial statistics in multiple detectors. how about
giving me a reference on that? or let's see the derivation! this after
he denies that it talks about coincident clicks! I don't see it.

let us call this "noncoincident vs coincident" clicks to try to contrast
QM vs semiclassical.

semiclassical states (as I crudely understand it..if not just call
it "vzn-classical"),
there will be NO WAY to narrow the time window such that all events
are mutually exclusive as QM predicts. however if we LIMIT OUR SAMPLE
of incoming data
(either in the electronics or post experimental data selection)
only to events in which we don't have coincident events, then we get
EXACTLY the QM predictions.
in other words, QM makes no INCORRECT predictions, but it is
INCOMPLETE; semiclassical theory is more COMPLETE because it makes
the same predictions as QM for noncoincident events, but also can
talk meaningfully about COINCIDENT events, which QM is mute on.

QM is mute on COINCIDENT events because they are simply nonexistent
by the "collapse of the wavefn", which is the semantical shorthand
which refers to the projection operation in the mathematical formalism
(ie collapse of wavefn is in the theory, and its effect is that QM cannot
speak about coincident events) & also its model of the probability spaces
(yes that's what I am saying seratend)..

actually I am being generous to QM, as vanesch replied,
if QM DENIES there can be anything other than random coincident
events, as he has stated & interpreted ("statistically independent"),
then it must be INCORRECT and I have
given an experiment above to prove it. (yes vanesch I would be
interested in your "vague souvenirs" of this experiment)

acc to nite, semiclassical can speak naturally on the case N>2 detectors,
whereas QM has no such prediction

actually, let me revise this experiment to be as open as possible.
I propose an experiment that
just tries to work with an N very large, and then GRAPH the distribution
over time, not ASSUMING any particular distribution (poisson or binomial
or whatever) and then showing how well it fits to a binomial or poisson
distribution.. can anyone show me that in the literature anywhere??
wouldnt it be a nice experiment scheme that would
tend to discriminate semiclassical from QM type theories, without
any experimenter/experimental bias? its one I've proposed a long time
ago & wanted to carry out myself..

ps guys it is true that I am not a specialist in the QM formalism. please
do not crucify me on this

ps nite, you complain about physics corrupting the mind of physics
students. I am one of those students. I am all ears..
when are you going to teach me? or would you rather spear at the infidels
or priesthood some more? after about 4 yrs I have lined up group of
N>200 "detectors" waiting for your "signal".. please reply to my email
wink

 

[vanesch]

So these "finite beams" use orthogonal modes. And as there is no overlap in the used modes, everything still holds. No big deal.

No it doesn't hold, not with finite detectors (which you apparently forgot again), with spatial limitation of DR and DT.

I still remain with my initial claim, that no matter how you combine, in wave packets, plane wave modes into finite-size beams, and their corresponding plane wave annihilation operators and creation operators in "finite volume" number operators, that the expectation value:

< 1-photon state | A+(1) A+(2) A(2) A(1) | 1-photon state>

is ALWAYS 0.

For always the same reasons: these "finiteness" just combines the "infinite plane wave" quantities (number operators, 1-photon states) LINEARLY together, in what I've been calling spatial wave packets, and in the end you become a huge linear combination of terms which CAN be expressed in the "plane wave" quantities (by bringing all these weighting functions and their integrals outside the in-product). ALL these terms take on the form:

<plane wave 1 photon state | a+(r) a+(s) a(t) a(u) | plane wave 1 photon state>
with a(r), a(s), a(t) and a(u) the plane wave mode annihilation operators.

And now we're home, because we have TWO ANNIHILATION operators acting on a single photon state, GIVING 0.

All these terms are 0, no matter how you combine them in linear superpositions.

What is however uselessly complicating the issue is that such considerations could be important if the distances and sizes involved were of the order of the wavelength. However, when they are of the order of mm or even cm, there is no point at all not to work directly with the idealized plane wave situation directly.

Nevertheless, you didn't show an explicit calculation yourself, reducing your quantities to plane waves and plane wave creators and annihilators, that you obtained 1 for this quantity, and not 0. You always said that the calculation that was presented didn't take this, or that, into account, but you never presented a clear calculation yourself, for 1cm^2 detectors, at 1 m distance apart, with light of 650 nm, that you had another result. You only claimed that we COULDN'T do certain things.

I still want to see you derive a kind of A_detector and a finite 1-photon mode, as expressed in plane wave quantities so that there's no discussion, so that you have A_detector |finite 1-photon mode> = |finite 1-photon mode>.

cheers,
Patrick.

 

[vanesch]

lets look at a PDC "ring" of correlated photons emitted
from a crystal illuminated by a laser, a photo of this can
be seen in the kwiat et al paper that nite cited here.

End of the game already: there is no ring of correlated photons.
Depending on the angular conditions, let us assume that we placed ourselves (in order to come as close as possible to what you think is happening) in the lambda -> 2 lambda + 2 lambda condition, then there are many 2-photon states that are emitted, and because of the cylindrical symmetry of the setup, these two "arms" of the 2-photon states can take any orientation ; however, within one "pair", they are oppositely aligned. So you have many independent "pairs".
There's no special correlation between different pairs.

existing QM experiments tend to be focused on looking
at only two branches or detection locations of this emission.

That's because it is in that way that there is some hope of "catching the two arms of the same pair".

lets just look at multiple "branches" (to borrow charged manyworld
terminology, but only at nite's lead) with N>2. unfortunately
this would translate into very expensive experiments
because good photodetectors tend to cost $5K or more each. but
maybe it could be shown with cheaper detectors. (and in fact
imho, the experimenters are going somewhat
in the wrong direction, at least wrt these types of
experiments, by trying to get the most expensive
detectors possible, translating into low N..)

I just build a 128-branch neutron detector, for a total worth of about $800000,- so this is probably not the argument :-)

QM experiments are oriented around "signal" and "idler" photons.
but this is a misnomer in the sense that both branches are fully
physically equivalent/symmetric. there is no physical distinction.
it is just an N=2 detector experiment. both branches are equivalent.

Let's change the names then in "signal1" and "signal2" :-)

Problem solved ?

according to the predictions of QM, if I have say N=3 detectors
and NOT assigning any particular detector as "signal vs idler"
(as is natural in the symmetry of the physics),
the following events are mutually exclusive:

detector 1 clicks only.
detector 2 clicks only.
detector 3 clicks only.

Absolutely not. If these detectors are not specifically aligned to see the "two arms of the 2-photon pairs", they will detect independently different pairs. The "arrival sequence" of these independent pairs is determined by the pump beam ; but if it is a coherent laser beam, then these pairs can be thought of to be generated Poisson like. As each detector will see one arm of an arbitrary pair, it will click in independent Poisson series. As in classical optics.

IN CONTRAST the prediction of semiclassical theory is that you will always
get a meaningful, nonrandom distribution of
coincident clicks in any of the detectors.

No, classical theory will also predict independent Poisson clicks.

and yes, I have pointed out in the past & vanesch notes--
an experiment using
a CCD camera (array) would be natural to use (N very large) &
might be able to show this effect across detectors.

Good luck with the time resolution of a CCD camera :-))) (about a few ms ?)

vanesch replies that QM makes the prediction about
coincident poisson/binomial statistics in multiple detectors. how about
giving me a reference on that? or let's see the derivation! this after
he denies that it talks about coincident clicks! I don't see it.

He must be real stupid. In fact, depending on the incoming state of the field, the predictions change ! If it is a 1-photon state you can only detect 1 photon, and suddenly if it is a coherent beam containing a superposition of all number of photon states, it can be binomially distributed. How strange...
As if the time of orbit around the sun of a planet depended on its distance to the sun and the sun's mass! Crazy. Never ever the same results.

semiclassical states (as I crudely understand it..if not just call
it "vzn-classical"),
there will be NO WAY to narrow the time window such that all events
are mutually exclusive as QM predicts. however if we LIMIT OUR SAMPLE
of incoming data
(either in the electronics or post experimental data selection)
only to events in which we don't have coincident events, then we get
EXACTLY the QM predictions.
in other words, QM makes no INCORRECT predictions, but it is
INCOMPLETE; semiclassical theory is more COMPLETE because it makes
the same predictions as QM for noncoincident events, but also can
talk meaningfully about COINCIDENT events, which QM is mute on.

It is well known that any theory that doesn't predict pink flying elephants is incomplete. Proof: consider a pink flying elephant. Try to describe it with the theory. QED.

QM is mute on COINCIDENT events because they are simply nonexistent
by the "collapse of the wavefn", which is the semantical shorthand
which refers to the projection operation in the mathematical formalism
(ie collapse of wavefn is in the theory, and its effect is that QM cannot
speak about coincident events) & also its model of the probability spaces
(yes that's what I am saying seratend)..

QM predicts coincident events in certain cases, and it predicts absense of coincidence in others. It predicts absense of detection of 2 photons in 1-photon states, and it predicts coincidence of detection of 2 photons in 2-photon states. And... it can even predict 3 coincidences if the incoming state contains a 3-photon component. More: if we have 4-photon states as incoming state, QM predicts the simultaneity of 4 photon detections.

Exercise: what incoming state is needed for QM to predict the coincidence of 5 photons ... ?



Answer: ... a 5-photon state :-)

After this deep philosophical debate, it is useful to point out that coherent light is a superposition of ALL n-photon states.

actually I am being generous to QM, as vanesch replied,
if QM DENIES there can be anything other than random coincident
events, as he has stated & interpreted ("statistically independent"),
then it must be INCORRECT and I have
given an experiment above to prove it.

And if QM doesn't deny it, then it could be correct.
It is statistically independent if the incoming state is a coherent state.

ps guys it is true that I am not a specialist in the QM formalism. please
do not crucify me on this

ps nite, you complain about physics corrupting the mind of physics
students. I am one of those students. I am all ears..
when are you going to teach me? or would you rather spear at the infidels
or priesthood some more? after about 4 yrs I have lined up group of
N>200 "detectors" waiting for your "signal".. please reply to my email
wink

What is amazing is that for over 4 years, you have been working to disprove a theory of which you don't understand a iota ?
Others complete a PhD on the subject in such a lapse of time...

Hey, I think I'll start a group to disprove the existence of irregular Spanish verbs. Although I don't speak much spanish, that shouldn't be a problem :-))

You're right, you'll need some fresh input to find a guy or a gall that will get you guys beyond QM

At least, nightlight knows some QM. His problem is more that he has read much more than he has understood, and knows miriads of references and derivations which are sophisticated and of which he only understood half, but which can serve at first to "show you that you didn't consider the question deep enough". That means that he can send you from paper to paper, from consideration to consideration, without an end, and it gives the impression of a senior scientist that knows what he's talking about. But it has an advantage: up to a point, it obliges the other also to read through that stuff, which can be hard, and in doing so, I learn also a lot.

cheers,
Patrick.

 

[vzn]

ok look vanesch I am going to wave a white flag
momentarily, but challenge you to be a little
more proactive than a reactive kneejerking anklebiter.
how about you come up with an experiment, please
describe it for me. you say in your last post it is possible.

(Im sure I am aware of some useful/relevant
refs you haven't heard
of either, like nite, but you're such an intense/bitter/thankless
"anticrank", at the moment I doubt its worth my time.)

anyway, anyone else here other than vanesch with an open mind,
consider this writeup on the significance of nite's
"glauber detectors"
http://groups.yahoo.com/group/qm2/message/9870


for vanesch: ok, let me learn a little qm from you. assuming you
believe in the concept of educating here & are not just a posturing
poseur on the idea. (if you feel you
have nothing to teach or for me to learn, then how about saving
me the trouble, & not replying? & I will listen to someone else)

in your earlier posts you describe how to measure
anticorrelated pairs, basically via the classic grangier
et al experiment which nite started this thread critiqing in the
undergraduate experiment.

earlier you denied that measurement of the collapse of the wavefn
is possible, but seem to misunderstand.. THAT IS EXACTLY THE
POINT OF THESE EXPERIMENTS.

so, (you say this is possible in your last post)
I propose to you, describe to me an experiment that will
will show anticorrelation (in the sense of mutual exclusion demanded
by the projection postulate, informally referred to as "collapse of wavefn") between multiple detectors for N>2.

I mean, suppose I have 3 detectors. describe to me an
experiment in which I can send lightwaves thru the apparatus
& detect only 1 of the 3 clicking at all-- never 2, never 3.
next please generalize to N>3.

 

[vanesch]

for vanesch: ok, let me learn a little qm from you. assuming you
believe in the concept of educating here & are not just a posturing
poseur on the idea. (if you feel you
have nothing to teach or for me to learn, then how about saving
me the trouble, & not replying? & I will listen to someone else)

Education is a certain form of communication. You need a sender, and a receiver. If either the sender or the receiver is broken, it cannot take place.

So I will take your second option, and watch you listen to someone else.

cheers,
patrick.

 

[nightlight]

I still remain with my initial claim, that no matter how you combine, in wave packets, plane wave modes into finite-size beams, and their corresponding plane wave annihilation operators and creation operators in "finite volume" number operators, that the expectation value:

< 1-photon state | A+(1) A+(2) A(2) A(1) | 1-photon state>

is ALWAYS 0.

The problem is that the Gn() doesn't correspond to coincidence rates for most of the 'plane waves' you stick into the formula. It has relatively narrow range of validity due to numerous approximations used in deducing it as a coincidence counting expression (dipole approximation, interaction cross sections, limited EM intensites, retaining only E+ based precisely on frequency range assumptions, misc. wavelength assumptions, ... etc, cf [4] 78-88, just note all the invocations of the assumption that (a >> b) holds for various quantities a and b).

Therefore, using the plane wave expansions, especially for these cases of finite detectors and finite modes, you start producing terms which, while mathematically legitimate, have no relation with the coincidence rate for such plane wave. Thus, your calculation has ceased to be a modelling of the coincidence rate for the experiment.

The questions Q-a and b are asking you to use QED/QO via AJP.9 to model the actual experiment and show that it does model this experiment and yields g2=0. I said it doesn't. Your expansion applied in AJP.9 for our finite modes & finite detectors, as actually layed out, ceases to model the experiment right upfront, by using G2() beyond the range of its applicability as a model of any photocounts, much less their coincidences.

There are many ways in physics you can reach absurd or contradictory conclusion by that kind of procedures -- of formally rewriting the parameters into mathematically equivalent sums, then applying the basic formula term by term, even when the parameters in some terms are outside of the valid range for the formula used, then summing the terms and wondering how you got different prediction (Jaynes' neoclassical ED fell into that trap in his prediction of chirp for spontaneous emissions in the early 1970s, which caused him to drop his theory for over 15 years, until Barut & Dowling http://prola.aps.org/abstract/PRA/v36/i2/p649_1 in 1987 and produced the correct prediction).

Therefore, your plain wave expansion does not show that the QED expression AJP.9 for this experiment (with finite detectors and finite modes as layed out), yield the DT and DR coincidence rate of 0. The reason for the failure to show DT and DR coincidence rate is 0, was the misuse of the formal expression G2() outside of its range of applicability to this problem (you need to predict detector counting rates and their correlation, but your terms stop being any such, cf. [4], 78-88).

That doesn't mean you can't use expression of the G2() form to model the experiment. You can, as shown earlier, assuming the finite volumes for the interaction Hamiltonian H_i in [4], in which case the mode annihilators become different kind of operators from the usual lowering operators of harmonic oscillator, but the normal products expectation value form remains. That is how I deduced that g2>=1 for QED treatment of the problem. You try it or replicate it with Mandel's [9], chap 14 model, and see what you say that QED predicts for coincidence rates.

Note also that the Poissonian distribution of photo-ionizations for the Fock state that my approach yields (as well as the other approaches, such as the recent preprint I cited) is perfectly consistent with treating DT and DR as a single cathode and finding Poissonian distribution of photo-ionizations on this large cathode (a known QO result for the distribution of photo-electron emissions). Alternatively, you could, in principle build a large cathode which extends across the T and R regions and then use regular, non-controversial prediction of QO that the number of photo-electron emissions will be Poissonian. Now you split (conceptually) this large cathode into two, by counting separately the photoelectrons in R half and T half and you will arrive at the g2=1 as a QED prediction. As hinted earlier, the same reasoning invalidates all experimental claims of Bell inequality violations with photons (not that any had ever obtained any violation, anyway) since QED doesn't predict such violation.

The basic problem you're having with this conclusion is that it violates your, apparently hardwired and subconsciousat this point, association between "photon number" and photo-detection count. You may benefit from Mandel's [9] 12.11.5 which shows some pitfalls (different than our case) of such mixup.

 

[vanesch]

The problem is that the Gn() doesn't correspond to coincidence rates for most of the 'plane waves' you stick into the formula. It has relatively narrow range of validity due to numerous approximations used in deducing it as a coincidence counting expression (dipole approximation, interaction cross sections, limited EM intensites, etc, cf [4] 78-88, just note all the invocations of the assumption that (a >> b) holds for various quantities a and b).

I'm sorry but as a *photon number* operator, there are no approximations involved. You seem to be talking about a specific model of how a detector is responding to the EM field, and how one can deduce that it is dependent on the photon number. In other words, how detector response is a function (or not) of g2. Ok, as I said, you are going to find something to complicate the issue such that in the end, it will be hard to argue :-)
But that doesn't change the fact that using photon number operators, no matter how you combine them in linear superpositions, the expression:

|1-photon state> = integral g(k) |1-photon state k> dk
nv1 photon number operator a la Mandel in volume v1
nv2 photon number operator a la Mandel in volume v2

then <1-photon state | :nv1 nv2: | 1-photon state>

is always 0.

Therefore, using the plane wave expansions, especially for these cases of finite detectors and finite modes, you start producing terms which, while mathematically legitimate, have no relation with the coincidence rate for such plane wave. Thus, your calculation has ceased to be a modelling of the coincidence rate for the experiment.

The questions Q-a and b are asking you to use QED/QO via AJP.9 to model the actual experiment and show that it does model this experiment and yields g2=0. I said it doesn't.

So you have to show now that the specific model of the detector, for the plane wave modes that occur in the |1-photon state> above, are not based upon g2.
Because that's in fact your claim. Your claim is not so much that g2 = 1, your claim is that the photon detectors correspond to measurement operators that are not simply a function of the local number operator a la Mandel.
I would be surprised to see your explicit calculation, because that would mean that it depends on something else but E x E(cc) (the intensity of the local electric field).
Remember (from the very beginning of this discussion) that QED being a quantum theory, the superposition principle holds.
Show me your explicit model, of a 1cm^2 detector, and its associated operator.

Therefore, your plain wave expansion does not show that the QED expression AJP.9 for this experiment (with finite detectors and finite modes as layed out), yield the DT and DR coincidence rate of 0. The reason for the failure to show DT and DR coincidence rate is 0, was the misuse of the formal expression G2() outside of its range of applicability to this problem (you need to predict detector counting rates and their correlation, but your terms stop being any such, cf. [4], 78-88).

So, g2 wasn't 1 after all, it was the expectation value of the product of the two operators corresponding to finite detectors, and that happened not to be the number operator. Right. We changed again the goal :-)
g2, as an expectation value of the product of photon number operators is, I think you understood that there is no weaseling out, equal to 0 for a 1-photon state. Now it is up to you to show me that your detector model gives something that doesn't depend on the photon number.

That doesn't mean you can't use expression of the G2() form to model the experiment. You can, as shown earlier, assuming the finite volumes for the interaction Hamiltonian H_i in [4], in which case the mode annihilators become different kind of operators from the usual lowering operators of harmonic oscillator, but the normal products expectation value form remains.

Ok, that's chinese to me, and I don't speak chinese.
I guess you're talking about the interaction hamiltonian of your detector model with the EM field. Now show me explicitly a calculation where you relate your new annihilation operators to the standard plane wave annihilation and creation operators ; after all the standard annihilation and creation operators span algebraically the entire operator space of QED, and show me how you come to that result. I would be *highly* surprised :-)

That is how I deduced that g2>=1 for QED treatment of the problem. You try it or replicate it with Mandel's [9], chap 14 model, and see what you say that QED predicts for coincidence rates.

Hehe, YOU claim, so YOU do.
If it is well done, you can even publish it. I'll try to read it, honestly.

The basic problem you're having with this conclusion is that it violates your, apparently hardwired and subconsciousat this point, association between "photon number" and photo-detection count. You may benefit from Mandel's [9] 12.11.5 which shows some pitfalls (different than our case) of such mixup.

I will read that. Indeed, I associate photodetection and photon number, especially in the simple case of our set up.

My honest impression is that you haven't understood the superposition principle in quantum theory (of which QED is a specific application) ; meaning that you can't somehow accept that if you know the response of a measurement setup to a set of basis states, that this fixes entirely the response of any superposition. That's what I've been trying to point at from the beginning, and it is the reason why I'm convinced that detailled modelling WITHIN QED have nothing to do with the issue.
Given that the two beams out of the beam splitter are a QM superposition of the one beam left, and the one beam right, I ONLY need to know how my detector setup (the entire setup, with the correlations) for the one beam left, and one beam right, and I AUTOMATICALLY know how it reacts for the superposition. This is fundamental quantum theory, which also applies to QED.
This means, in your case, that we will have correlation counts EVEN IF THERE IS ONLY THE LEFT BEAM, or IF THERE IS ONLY THE RIGHT BEAM.
Indeed, if, when there is only the left beam, there is no correlation for sure, then this is an eigenstate of the "correlation measurement operator" with eigenvalue 0;
when there is only a right beam, then this is ALSO an eigenstate of the correlation measurement operator with eigenvalue 0,
This means that any linear combination of both states will ALSO be an eigenstate with eigenvalue 0. So any superposition will also have correlation 0.

See, that's BASIC QUANTUM THEORY. It has nothing to do with the specific model. The only way out is that you now have a model of your detectors that gives us correlated counts when there is only the left beam.

I'm waiting.

cheers,
Patrick.

 

[nightlight]

My honest impression is that you haven't understood the superposition principle in quantum theory (of which QED is a specific application) ;

Well, my impression that you haven't understood the reply (or the earlier related posts) at all. And, sorry, but I won't write your papers for you in here (or waste time on homeworks, which only show how much you missed the argument altogether, but which you keep trying to assign me) having given you more than enough info and references on how to replicate the conclusions on your own if you wish.

The photo-ionization is inherently nonlinear process. That is achieved in [4] by applying series of approximations, which allow the linear formalism to simulate such nonlinearity and obtain the square-law for the detection rates (and the related Gn() as adjusted coincidence rates). But, as explained, the approximations in [4] require numerous restrictions on the fields, which some generic superposition (trying to simulate the finite volumes of DR, DG, R and T beams) will violate for some plane waves, making them invalid as the contributions to the photo-ionization counts. You can check the cited places in [4] and verify whether all your plane waves satisfy all the a>>b type of assumptions made there to justify the approximations, and thus the usage of a particular plane-wave component in the Gn() expressions (while retaining their operational mapping to the photo-detection counting rates; e.g. the high-frequency components arising in Fourier expansions of finite volume fileds and interactions, will violate frequency restrictions in [4], which were used to justify the dipole approximation and the dropping of the E- terms, thus the resulting contribution is not valid counting rate at all, and the sum using such terms can't be assumed as valid either).

Given that the two beams out of the beam splitter are a QM superposition of the one beam left, and the one beam right, I ONLY need to know how my detector setup (the entire setup, with the correlations) for the one beam left, and one beam right, and I AUTOMATICALLY know how it reacts for the superposition. ...

This is where your disregard of the finite detectors leads you into an elementary error. If you have a finite detector, than it is obvious you can easily superpose two components and get 0 counts by the finite detector (e.g. you change the phase of one component, so the detector is in the dark fringe), even though changing phase of a component with an infinite detector would show unchanged counts (since the total EM energy is preserved). Correlations of such counts of multiple finite detectors consequently also depend on the detector sizes. Your assertion that detector sizes make no difference in their counts or coincidence rates is plainly absurd. (It makes the essential difference in this experiment, as already explained via Q-a,b and the finite detector followups.)

Thus, you can't know the counts (much less their correlations) result of the superposition in general, unless you take into account the sizes of the detectors[/color], for which the plane-wave version of G2() in (AJP.9) lacks any formal counterparts [/color], thus it can't possibly account for such differences. It doesn't even show the existence of any difference (thus you don't see any), much less tell you what effect it would have. I already pointed you to Mandel's [8], [9] to help you at least realize that there is a difference and that it can be accounted for by the formalism, and that in our example of finite fields and finite detectors, the detector sizes make the most drastic difference (that should be obvious anyway) which, as Mandel explicitly warns, can lead to errors with the naive photon number reasoning (as it did in your example of supplementing QM with such imagery). You have just cornered yourself into a hopelessly wrong position.