In-vacuum and out-vacuum

If a background is stationary in the remote past and future, positive-frequency modes in those two regions define an in-vacuum and an out-vacuum. The intervening time dependence generally makes the two mode bases differ by a Bogoliubov transformation.

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Out-vacuum ( $b_{i} ∣ 0_{out} ⟩ = 0$ )

The state annihilated by every annihilation operator of the positive-frequency mode basis chosen in the asymptotic future. It need not coincide with the in-vacuum on a time-dependent spacetime.

In-vacuum ( $a_{i} ∣ 0_{in} ⟩ = 0$ )

The state annihilated by every annihilation operator of the positive-frequency mode basis chosen in the asymptotic past. A Bogoliubov transformation can make it contain particles when tested by future out-mode number operators.

Particle number from Bogoliubov coefficients

If

u_{i}^{out} = \sum_{j} (α_{ij} u_{j}^{in} + β_{ij} u_{j}^{in *}),

then the expected out-particle number in mode

i

in the in-vacuum is

⟨ 0_{in} ∣ N_{i}^{out} ∣ 0_{in} ⟩ = \sum_{j} ∣ β_{ij} ∣^{2} .

Out-mode number fluctuations in the in-vacuum ( $⟨ N_{s}^{2} ⟩ = 2 n_{s}^{2} + n_{s} + ∣ c_{s} ∣^{2}$ )

If

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}

. The canonical commutation relations give

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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