Gaussian quantum state

A bosonic state with Gaussian phase-space characteristic function, specified by its first and second optical quadrature moments. Vacuum, coherent states, optical squeezed vacuum and thermal states are examples. Gaussian here describes phase-space statistics, not a Gaussian optical pulse envelope.

Here number-conserving means invariant under a global particle-number quantum phase rotation: the field mean and anomalous averages $⟨\psi \psi ⟩$ vanish. It does not mean a sharp total particle number. All normal correlations are obtained by Wick pairings of the one-body density matrix. Independent thermal bosonic modes satisfy this rule; generic nonvacuum bosonic number states do not. A Slater determinant is the analogous fermionic pairing state.

Applying a single-mode squeeze operator to a thermal state gives the displayed principal optical-quadrature variances. Their uncertainty product exceeds the pure-state minimum when $n_{\mathrm{th}}>0$. Sufficient squeezing is needed to put the quiet optical quadrature below the vacuum reference.