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Question-a) 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>,...

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 (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 polarizer flip-flop effect 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 (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):
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)