Quadrature squeezing of a vacuum–one-photon superposition ($\mathrm{Var}{X}_1=\frac{1}{4}-\frac{1}{2}{ϵ}^2+{ϵ}^4$)

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.