= Quadrature squeezing of a vacuum–one-photon superposition
{title2=$\operatorname{Var}X_1=\tfrac14-\tfrac12\epsilon^2+\epsilon^4$}
For real $\epsilon$ with $|\epsilon|\leq1$, let $|\psi\rangle=\sqrt{1-\epsilon^2}|0\rangle+\epsilon|1\rangle$. The <Fock state> matrix elements give $\langle a\rangle=\epsilon\sqrt{1-\epsilon^2}$, $\langle a^2\rangle=0$ and $\langle a^\dagger a\rangle=\epsilon^2$. Therefore
$$
\operatorname{Var}X_1=\frac14-\frac12\epsilon^2+\epsilon^4,\qquad \operatorname{Var}X_2=\frac14+\frac12\epsilon^2.
$$
Only the first of these <optical quadratures> is squeezed for this real superposition, with minimum variance $3/16$ at $\epsilon^2=1/4$. The second is never below vacuum noise. This non-Gaussian superposition is distinct from an <optical squeezed vacuum>.
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