Optical quadrature ($X_{\vartheta }=(a{e}^{-i\vartheta }+a^{†}{e}^{i\vartheta })/2$)

A phase-selected Hermitian field observable. With annihilation operator $a$ and $[a,a^{†}]=1$, define $X_1=(a+a^{†})/2$ and $X_2=(a-a^{†})/(2i)$. Then $[X_1,X_2]=i/2$ and the Fock vacuum has variance $1/4$ in either quadrature. This dimensionless convention gives $X_{\vartheta }=X_1\cos \vartheta +X_2\sin \vartheta$. Balanced homodyne detection selects $\vartheta$ through the reference wave phase.

For one optical mode, $E=E_0(a+a^{†})$ and $[a,a^{†}]=1$. A coherent state obeys $a|\alpha ⟩=\alpha |\alpha ⟩$, so normal ordering gives $⟨E^2⟩=E_0^2[(\alpha +{\alpha }^{\ast }{)}^2+1]$. Subtracting the squared mean leaves $E_0^2$, independent of the coherent displacement. The extra one is the canonical commutation relation, not noise from a random coherent amplitude. A rotated optical quadrature has the same coherent-state variance with the same amplitude normalization.

For a single-mode Gaussian state with no quadrature covariance, the covariance contour has semiaxes $\sqrt{\mathrm{Var}X}$ and $\sqrt{\mathrm{Var}P}$. A coherent state has both equal to $1/2$ in the convention $[X,P]=i/2$. A pure squeezed vacuum state has semiaxes $e^{-r}/2$ and $e^r/2$; the uncertainty area remains fixed while one noise width decreases. An ellipse of means alone does not specify a general non-Gaussian state.^[ref]

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.

For real $ϵ$ with $|ϵ|\le 1$, let $|\psi ⟩=\sqrt{1-{ϵ}^2}|0⟩+ϵ|1⟩$. The Fock state matrix elements give $⟨a⟩=ϵ\sqrt{1-{ϵ}^2}$, $⟨a^2⟩=0$ and $⟨a^{†}a⟩={ϵ}^2$. ThereforeOnly the first of these optical quadratures is squeezed for this real superposition, with minimum variance $3/16$ at ${ϵ}^2=1/4$. The second is never below vacuum noise. This non-Gaussian superposition is distinct from an optical squeezed vacuum.