Proposition: Finite Gaussian packet state

The fixed-background state restricted to these modes is a normal, zero-mean, gauge-invariant Gaussian state determined by the full positive matrix $K$. Its displacement characteristic function is

After a passive mode transformation diagonalizing $K$, the state is a product of thermal oscillator states with mean occupations equal to the eigenvalues of $K$.

Proof. The boundary state is quasifree, and linear scattering and restriction to a finite CCR subspace preserve quasifreeness. Stationarity and positive-frequency gauge invariance give vanishing anomalous covariance. Wick's rule, or expansion of the Weyl exponential, gives (eq:weylcharacteristic). Diagonalize $K$ by a unitary transformation of the annihilation operators. Each diagonal mode has characteristic function $\exp [-(1/2+\mu ){\left|z\right|}^2]$, which is that of

These factors are trace class, and their finite tensor product is normal. Their characteristic function equals (eq:weylcharacteristic), proving the assertion.