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.