Quantum Fisher information and quadrature squeezing in Janus superpositions of squeezed vacua

Abstract

Janus states, defined as coherent superpositions of two single-mode squeezed vacua, provide a simple but genuinely non-Gaussian setting for studying how interference reshapes quantum Fisher information (QFI) beyond the Gaussian squeezed-vacuum picture. Using an exact analytic treatment, we determine the QFI of Janus states and identify the benchmarks under which they can or cannot offer a metrological advantage over the single squeezed vacuum. We find that, under a fair comparison at fixed mean photon number, the single squeezed vacuum remains optimal for principal second-moment squeezing, so no genuine Janus advantage exists at that level. By contrast, within a fixed two-state span, a Janus superposition can simultaneously outperform its constituents in a laboratory quadrature variance and in number-generated phase QFI. We also introduce an operational benchmark based on fixed measured squeezing and show that, at the same observed squeezing level, Janus interference can substantially enhance the QFI for quadratic-generator sensing beyond the pure-Gaussian squeezed-vacuum reference. These results show that the metrological performance of Janus states is controlled not only by quadrature squeezing, but also by higher-order fluctuations and by the benchmark used for comparison.

Figures (7)

• Figure 1: Equal-strength ($r=s$) Janus minimum quadrature noise $(\Delta Q)^2_{\min}$ versus the relative squeezing phase $\Delta=\theta-\phi$, computed from Eq. \ref{['eq:DQmin_equal_strength_final']} for $\Delta\neq 0$ on the slice $\theta=0$ and $\phi=-\Delta$. Solid curves show the optimized two-state-span minimum, while dashed curves show the single squeezed-vacuum benchmark at the same strength, $(\Delta Q)^2=\tfrac{1}{2} e^{-2r}$. The curves are rendered continuous at $\Delta=0$ and $2\pi$ by connecting to the one-dimensional identical-state value.
• Figure 2: Two-parameter landscape of the equal-strength Janus optimum $(\Delta Q)^2_{\min}(r,\Delta)$ for the slice $\theta=0$ and $\phi=-\Delta$. Colors show $(\Delta Q)^2_{\min}$ on a logarithmic scale, with labeled contour lines indicating constant-variance levels. The vertical guide at $\Delta=\pi$ marks the maximally phase-opposed configuration and organizes a broad region of interference-enhanced squeezing.
• Figure 3: Correlation comparison at $r=s=0.34$ and $\Delta=\pi$ (state normalized by Eq. \ref{['eq:janus_norm_constraint']}). The upper panels display the landscapes of $\Delta Q\,\Delta P$ and $\log_{10}g^{(2)}(0)$ over $(|\eta|,\delta)$. The lower-left overlay shows $\log_{10}g^{(2)}(0)$ with $\Delta Q\,\Delta P$ contours, while the lower-right panel shows the induced scatter relation $\log_{10}g^{(2)}(0)$ versus $\Delta Q\,\Delta P$.
• Figure 4: Evolution of the correlation trade-off at $\Delta=\pi$: scatter plots of $\log_{10}g^{(2)}(0)$ versus $\Delta Q\,\Delta P$ for several choices of $r=s$. The vertical dashed line marks the Heisenberg boundary $\Delta Q\,\Delta P=1/2$, and the color encodes $\delta/\pi$.
• Figure 5: Enhancement of the quadratic-generator quantum Fisher information for a Janus probe relative to the pure-Gaussian squeezed-vacuum benchmark evaluated at the same measured squeezing level $u$. The landscape is plotted over the non-degenerate coefficient ratio $t=\eta/\chi$ and the relative squeezing phase $\Delta=\theta-\phi$ for fixed squeezing strengths $x=y=1/2$. The gray region masks the non-squeezed regime ($u > 1$) where the squeezed-vacuum benchmark is not applicable. The solid contour identifies the break-even boundary where the ratio is 1 (logarithm is 0), while the dashed contours track constant values of the measured squeezing $u$. The star indicates the representative state evaluated in Eq. \ref{['eq:quad_advantage_example']}.