= Solution
For $X=(a+a^\dagger)/2$ and $Y=(a-a^\dagger)/(2i)$, $[X,Y]=i/2$, so $\Delta X\Delta Y\geq1/4$. Vacuum and <coherent states> have <variance> $1/4$ in either <optical quadrature>. <quadrature squeezing> means that some rotated <optical quadrature> has <variance> below this reference, with compensating noise elsewhere. A reduced second moment about zero is not the correct test when the field's mean is displaced.
For $S(\zeta)=\exp[(\zeta^*a^2-\zeta a^{\dagger2})/2]$ and $\zeta=re^{i\theta}$, the quiet principal axis of the <optical squeezed vacuum> is at angle $\theta/2$ under the squeeze-operator convention used above, and
$$
\boxed{\operatorname{Var}X_{\theta/2}=\tfrac14e^{-2r},\qquad
\operatorname{Var}X_{\theta/2+\pi/2}=\tfrac14e^{2r}.}
$$
The pure <Gaussian quantum state> saturates the uncertainty product. Displacement moves its centre without changing its <covariance ellipse>; <squeezed thermal states> instead have a larger uncertainty product. The <optical squeezed vacuum> has zero mean field, even <photon-number parity> and mean occupation $\sinh^2r$; it need not have sub-Poissonian <photon statistics>. <number squeezing> and <quadrature squeezing> are different conditions.
Degenerate parametric amplification generates single-mode squeezing; a <two-mode squeezed vacuum> instead has correlated occupations and joint <quadrature squeezing>. Either individual mode of an ideal two-mode state is thermal after the other is ignored, not individually squeezed. <balanced homodyne detection> measures the noise versus <local oscillator> phase. Reference phase drift admixes the anti-squeezed <optical quadrature>, while electronic noise and detector imbalance need separate checks.
For an optical loss channel with efficiency $\eta$ and vacuum entering the unused port, <optical squeezing degradation by loss> gives
$$
\boxed{V_{\rm out}=\eta V_{\rm in}+(1-\eta)/4.}
$$
The linear-loss limit follows directly by writing the output field as $\sqrt\eta\,a+\sqrt{1-\eta}\,v$ with independent vacuum mode $v$. Loss therefore places a floor under measured squeezing. Squeezed input can improve phase-sensitive measurements when aligned with the measured <optical quadrature>; it does not reduce every noise source or both conjugate uncertainties at once.
\Image[/past-exam-of-the-nonlinear-optics-and-quantum-states-of-light-course-of-the-university-of-cambridge/2014/paper-1-squeezing-ellipse.png]
{title=Original optical-quadrature covariance ellipses and the effect of vacuum loss on squeezed variance}
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