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    fluctuation of the photon number in a coherent state

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    Problems 2.11 and 2.12 of the notes.

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    https://brainmass.com/physics/photons/fluctuation-of-the-photon-number-in-a-coherent-state-551048

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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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