Quantum Optical Metrology of Correlated Phase and Loss

Optical absorption measurements characterize a wide variety of systems from atomic gases to in vivo diagnostics of living organisms. Here we study the potential of nonclassical techniques to reduce statistical noise below the shot-noise limit in absorption measurements with concomitant phase shifts imparted by a sample. We consider both cases where there is a known relationship between absorption and a phase shift, and where this relationship is unknown. For each case we derive the fundamental limit and provide a practical strategy to reduce statistical noise. Furthermore, we find an intuitive correspondence between measurements of absorption and of lossy phase shifts, which both show the same analytical form for precision enhancement for bright states. Our results demonstrate that nonclassical techniques can aid real-world tasks with present-day laboratory techniques.

Figure 1

Phase-space representation of the transformation of initial state $\varrho$ to $\tilde{\varrho }$: after passing through the channel $\mathrm{\Lambda }$ with transmission $\eta$ and a phase shift of $\theta .\varrho$ is squeezed in the optimal direction aligned with ${\partial }_{\chi }\tilde{\mathit{d}}$. The red curve is the homodyne signal when the phase of the local oscillator is optimized for the measurement.

Figure 2

Comparing strategies for CPLE and DAE with their respective quantum limits: Within each plot the red dashed line shows the SQL and all plots are normalized to the quantum limit ${\mathcal{Q}}_{\chi }$ or ${\mathcal{Q}}_{\eta }$. Plots (a) and (b) display the amount of statistical information $\mathcal{D}$ and $\mathcal{N}$ encoded onto the displacement vector (mean number of photons) of a squeezed coherent state for CPLE and DAE, respectively. The inset shows that even for very low absorption these states approach the quantum limit for modest energies. Statistical information plotted for varying input mean photon number operating with the optimal squeezing value presented in (a) Eq. (9) and (b) Supplemental Material E [33]. (c) Amount of statistical information $\mathcal{D}$ ($\mathcal{N}$) encoded onto the displacement vector of a squeezed coherent state for CPLE (DAE) when $\alpha$ is large. We note that for (a)–(c), inset figures display the same $y$-axis scale as the main plots, while having an $x$-axis scale as labeled at the top of the plot.