Hello all. I am trying to determine what is the effect of having photons that are distinguishable undergoing a quantum interference process. To do that, I try to generalize the Hong-Ou-Mandel effect, and try to determine what are the terms that appear as a function of the product of the creation operators at the output ports of a Directional Coupler (DC). DCs are integrated waveguide beam-splitters with a Unitary Operator U(adsbygoogle = window.adsbygoogle || []).push({}); _{DC}as in Fig.1 below.

A single photon input at port 1 transforms into a combination of creation operators at output ports 3 and 4, which determine the respective probability of detection according to Eq. 3.

In Eq.3, ε is the coupling factor between the DC waveguides, generally 1/2 for a 50:50 beam-splitter.

Next, I consider a generic Unitary Operator U_{1}(Eq. 4) on the photon before entering the DC (Fig. 2).

Now, a single photon at new input port 1 transforms into a combination of creation operators at output ports of U_{1}, port 3 and 3', according to Eq. 7.

If ports 3 and 3' are combined, what can be done with waveguides or optical fibers, as in Fig. 3, their creation operators become identical (Eq. 8). Eq. 7 then becomes Eq. 9. The single photon at input port 1 is then transformed, at output ports 5 and 6 of DC, according to Eq. 10.

Now, I do the same for the other input port of DC, transforming the second photon of an identical photon pair, according to another generic Unitary Operator U_{2}(Eq. 11).

Eqs. 9 and 10 then become, for this second photon, Eqs. 13 and 14 respectively.

The idea is to try determining what is the effect of U_{1}and U_{2}on the final probability of simmultaneous counting of identical photon pairs at coincidence detectors in output ports 5 and 6. This simmultaneous counting probability is proportional to the multiplier of terms containing a product of creation operators at 5 and 6, according to Eq. 16.

These multipliers should contain a function of U_{1}and U_{2}, to account for the increace of distinguishability of photons in case U_{1}and U_{2}are different. And here is the problem, since the coincidence probability seems to become zero if ε is 1/2, independently of U_{1}and U_{2}(Eq. 17). I am definitely missing something here! Can anybody tell me what is wrong?

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# A Generalizing distinguishability of photons on HOM dip?

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