 Out-mode number fluctuations in the in-vacuum

ID: out-mode-number-fluctuations-in-the-in-vacuum

b_{s} = \sum_{j} (A_{s j} a_{j} + B_{s j} a_{j}^{†})

and the state is the in-vacuum, put

n_{s} = \sum_{j} ∣ B_{s j} ∣^{2}

and

c_{s} = \sum_{j} A_{s j} B_{s j}

N_{s}^{2} = b_{s}^{† 2} b_{s}^{2} + N_{s}

. Acting with

b_{s}^{2}

on the vacuum produces

c_{s} ∣0 ⟩

and a two-particle vector whose squared norm is

2 n_{s}^{2}

, because its two possible vacuum contractions have equal contributions. This proves the displayed second moment and

Var (N_{s}) = n_{s} (n_{s} + 1) + ∣ c_{s} ∣^{2}

. A thermal-mode formula requires

c_{s} = 0

; a general squeezed vacuum state need not satisfy that restriction.

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