Past exam of the laser science and quantum information processing course of the University of Oxford / 2022 / 3 / d

For the normalized coherent state,

⟨ a ⟩ = α

and

⟨ a^{†} ⟩ = α^{*}

⟨ E ⟩ = α + α^{*} = 2 Re α, ⟨ B ⟩ = - i (α - α^{*}) = 2 Im α .

The physical electric field and magnetic field strengths have the corresponding proportionality factors and mode functions. In free evolution replace

α

α e^{- iω t}

to recover the classical oscillation.

Expanding the squared operators cancels the

a^{2}

and

(a^{†})^{2}

terms:

E^{2} + B^{2} = 2 (a a^{†} + a^{†} a) = 4 \hat{N} + 2 I .

On the other hand,

(α + α^{*})^{2} + [- i (α - α^{*})]^{2} = 4∣ α ∣^{2}

. Hence

⟨ E^{2} + B^{2} ⟩ = 4∣ α ∣^{2} + 2, ⟨ E ⟩^{2} + ⟨ B ⟩^{2} = 4∣ α ∣^{2} .

The difference is the sum of the variances, not an extra mean field. Indeed

Var (E) = Var (B) = 1

, independent of

α

[E, B] = 2 i I

requires

σ_{E} σ_{B} \geq 1

, and the coherent state attains equality. The residual two units are the vacuum fluctuations of the two optical quadratures.

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