Quadrature squeezing ($\mathrm{Var}{X}_{\vartheta }<1/4$)

With $X_{\vartheta }=(a{e}^{-i\vartheta }+a^{†}{e}^{i\vartheta })/2$, the coherent-state and vacuum variances are $1/4$. A state is squeezed in the chosen quadrature when its variance about its own mean is smaller. A second moment about zero is not the variance when that mean is nonzero. balanced homodyne detection measures a phase-selected quadrature distribution.

Mixing a mode with independent vacuum on a beam splitter of intensity transmission $\eta$ gives the displayed optical quadrature variance when vacuum variance is $1/4$. It approaches vacuum noise as transmission falls. Phase noise instead mixes squeezed and anti-squeezed axes, and cannot in general be represented by the same efficiency alone.