- #1
Malamala
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Homework Statement
(Note: this is the last part of a longer problem, so I hope I won't miss anything important)
We have a laser with 2 energy levels, so the emitted photons will have the energy ##\omega##. For this given energy there are ##m## states in which the photon can be, denoted ##|\psi_i>##. Assume we have ##n## photons in state ##|\psi_1>## and one more in the state ##|\phi>=\frac{1}{\sqrt{m}}(|\psi_1>+|\psi_2>+...+|\psi_m>)## and denote the state as ##|n\phi>##. From a previous part, this can be written as $$|n\phi>=\sqrt{\frac{m}{(m+n)(n+1)}}(|11..1\phi>+|11..1\phi1>+...|\phi1..1>)$$ where, for example ##|11..1\phi>## stands for ##|\psi_1>|\psi_1>...|\psi_1>|\phi>## and the above wave function is normalized and symmetrized (for bosons). Now for the last part we are asked to write the state of the system after ##r## photons leave the ##|n\phi>## state.
Homework Equations
The Attempt at a Solution
I'll try to do it for a simpler case (which should be easily generalized). So for n=3 we have $$|3\phi>=N(|111\phi>+|11\phi1>+|1\phi11>+|\phi111>)$$ where N is a normalization. If we remove a photon, this can be the ##|\psi_1>## or the ##|\phi>## So after one removal the state would be a liner combination of these 2 possibilities, something like: $$N_1(|11\phi>+|1\phi1>+|\phi11>+|111>)$$ After one more, you get $$N_2(|1\phi>+|\phi1>+|11>)$$ after one more: $$N_3(|\phi>+|1>)$$ And of course this can be generalized for arbitrary ##n## and ##r##. But I am not sure if my logic is correct. I am mainly confused about the fact that ##|\phi>## is not an eigenstate of this system, but a linear combination, so if you would think in terms of annihilation operators I am not sure how you would write this removal. Any suggestion would be greatly appreciated. Thank you!