Beam Splitter Physics: |45>,-|45> State Output

[StevieTNZ]

If we have a beam splitter, and we fire two photons from either side towards it (both meeting and interfering at the same time):

If the state of the photon coming in from the left is |45>, and the one coming in from the right is -|45>, would the output be:

-i|45>(l)|45>(l) - i|45>(r)|45>(r) - i|45>(l)|i45>(r) - |45>(l)|45>(r)

Where the 3rd term is both reflected, the last term both transmitted, and the first two where they exit on the same side of the beam splitter output.

The attached image might give some clarity.

 

[Cthugha]

Why did you put the i in the middle of the ket for the third term?

If you include it and normalize, you will find that the third and fourth term have equal magnitude, but different sign. So they cancel out. As a consequence exactly identical and indistinguishable single photons entering different ports of a beam splitter will always leave as a pair of two photons via the same exit port. This is the Hong-Ou-Mandel effect.

 

[StevieTNZ]

Why did you put the i in the middle of the ket for the third term?

Because both reflected.

 

[Cthugha]

Oh, sorry. Maybe my post was not clear. I was rather puzzled why you wrote |i45> instead of i|45>.

 

[StevieTNZ]

Oh, sorry. Maybe my post was not clear. I was rather puzzled why you wrote |i45> instead of i|45>.

My bad! Thanks for spotting that out for me. Should be i|45>.