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fluctuation of the photon number in a coherent state

Problems 2.11 and 2.12 of the notes.

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Problem 2.11

We can use the expression Eq. 2.2.29 for the density matrix:

rho(n) = 1/(1+n) sum from k = 0 to infinity of r^k |k><k|

where n is the mean photon number and r = n/(1+n). To compute the variance, we need to evaluate:

<(a-dagger a)^2> = Tr[rho (a-dagger a)^2]

We can do this as follows. We first write:

(a-dagger a)^2 = a-dagger a a-dagger a

We can use the commutation relation [a, a-dagger ] = 1 to move all the a-dagger's to the left. We have that a a-dagger = 1 + a-dagger a, therefore:

a-dagger (a a-dagger) a = a-dagger (1 + a-dagger a ) a = a-dagger a + a-dagger^2 a^2.

We thus have:

Tr[rho (a-dagger a)^2] = Tr[rho ...

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