The celebrated Hong-Ou-Mandel interferometric effect

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Homework Statement

Determine the expectation values of the product of the number operators

\left\langle{N_{a_{1}}}{N_{b_{1}}}\right\rangle


Explain your result.

Homework Equations

http://en.wikipedia.org/wiki/Hong-Ou-Mandel_effect

The Attempt at a Solution

I've got the expectation value as zero, and I've been told that is correct but i can't explain how. The N_a and N_b refer to the two photons or modes after the beam splitter (i think)(mode zero would be before the beam splitter) so i don't know why they seem to disappear after the beam splitter and why the expectation value (is this the average value of the intensity at one of the detectors?) is zero. I would have expected it to be either:

1 - because of the HOM effect the two input photons will merge into one photon with double intensity and since there are two detectors, and the odds of being detected at either is 50-50, the average value at either is half of 2 = 1.

Or

2 - This is if the expectation value only takes into account one of the detectors when the doubled intensity photon is detected..maybe?

 

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Here is the maple file i have used if anyone wants to look...