 Solution

ID: past-exam-of-the-laser-science-and-quantum-information-processing-course-of-the-university-of-oxford/2015/2/d/solution

⟨ \hat{E} (r, t) ⟩_{α} = E α e^{i (k r - ω t)} + E^{*} α^{*} e^{- i (k r - ω t)} = 2∣ E α ∣ cos (k r - ω t + ar g (E α)) .

This is a classical monochromatic electric field: the modulus of

α

fixes the wave amplitude, while its argument fixes the wave phase relative to the chosen mode convention. Direct normal ordering also gives

(Δ E)_{α}^{2} = ∣ E ∣^{2}

, independent of

α

. Hence the fluctuation divided by the peak mean-field amplitude is

1/ (2∣ α ∣)

, small in the bright coherent state limit. A relative fluctuation evaluated exactly at a field zero is not a useful classicality test.

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