Squeezing-Enhanced Rotational Doppler Metrology

TL;DR

This work addresses estimating the angular velocity $\Omega$ via the rotational Doppler effect using quantum-enhanced light. It derives the classical RDE from paraxial boundary conditions, then quantizes the process with a Bogoliubov transformation and a continuous-variable metrology protocol employing squeezed and displaced Laguerre-Gaussian modes read out by homodyne detection. The key finding is that, in the noiseless limit, the quantum protocol achieves Heisenberg scaling for the quantum Fisher information ($F_Q \propto N^2$), with a robust quantum advantage persisting under realistic noise when energy is optimally allocated between squeezing and displacement; two surface models (metasurface and complex rough surface) illustrate the practical regimes and trade-offs. The results indicate a feasible path toward quantum-enhanced gyroscopes and rotation sensing in structured-surface environments, highlighting the balance between squeezing resources and loss in real devices.

Abstract

A rotating surface can induce a frequency shift in incident light by changing its angular momentum, a phenomenon known as the rotational Doppler effect. This effect provides a means to estimate the angular velocity of the rotating surface. In this work, we develop a continuous-variable quantum protocol for estimating the angular velocity of a rotating surface via the rotational Doppler effect. Our approach exploits squeezed and displaced Laguerre-Gaussian modes as quantum resources, which interact with a rotating metallic disc with surface roughness. The frequency shift induced by the rotational Doppler effect is then measured using a homodyne detection scheme. By analyzing the Fisher information, we demonstrate that the proposed squeezing-enhanced protocol achieves Heisenberg scaling in the ideal noiseless regime. Furthermore, we investigate the influence of noise and consider different surface models to assess their impact on the protocol's performance. While Heisenberg scaling is degraded in the presence of noise, we show that optimizing the energy allocation ratio between displacement and squeezing of the probe ensures that the quantum strategy consistently outperforms its classical counterpart.

Figures (5)

• Figure 1: Schematic recreation of our proposed squeezing-enhanced protocol for the estimation of the rotational Doppler effect. An emitter E can implement single-mode squeezing $\hat{S}$ and/or displacement $\hat{D}$ on some modes, producing a monochromatic propagating beam with field operators $\hat{a}^\text{in}_{n^\prime, l^\prime, p^\prime }(\omega)$, with the discrete subindices associated to a Laguerre-Gauss (LG) basis and where specifically $l$ corresponds to the quantized orbital angular momentum. The beam elastically interacts with a rough, flat surface characterised by some points such that $f(\mathbf{r}_\text{S})=0$ with the function $f$ depending on the type of surface, which rotates with angular velocity $\Omega$ along the $z$ axis. For graphical convenience the propagation and the rotation axes are not shown parallel, which is the simplified version studied in the main text. The outgoing, or scattered modes will be shifted in frequency through the change in OAM: $\Delta l \Omega$. Although the state after the interaction consists of several populated modes, only one of them is measured via a mode selector (MS). For simplicity, the measurement is homodyne detection (HD), whose outcomes are classically processed and an estimator $\widecheck{\Omega}$ for the angular velocity $\Omega$ is obtained.
• Figure 2: Contour plot of the ratio $R$ as a function of the squeezing and displacement photon numbers, $N^{\text{Sq}}$ and $N^{\text{Coh}}$, respectively. The ratio $R$ summarizes the quantum advantage of our squeezing-enhanced protocol for the rotational Doppler effect with a metasurface that induces a definite change in the incident OAM (see Eq. \ref{['eq:RatioMeta']}). We choose a realistic noise parameter $\eta=0.1$. Note that for a given $N^{\text{Coh}}$ increasing the squeezing photon number $N^{\text{Sq}}$ could be detrimental for the advantage. The energy employed in the squeezing helps reducing the noise when measuring the displacement that is proportional to $N^{\text{Coh}}$. But if $N^{\text{Coh}}$ is small, it is not optimal to use a lot of energy to reduce its noise.
• Figure 3: Ratio $R$ between the classical Fisher information of our quantum strategy versus the optimal classical one, as a function of the total photon number $N$ --with optimal squeezing and displacement allocation--, and plotted for different noise levels $\eta$, including the noiseless case. $R$ increases fast for small photon number, showing a constant asymptotic behavior when $N$ is large. This maximum achievable value of $R$ for large $N$ decreases with $\eta$. However, as one should expect, this is not the case for the noiseless case, where $R$ shows the Heisenberg scaling described in the main text.
• Figure 4: Optimal energy allocation, represented by the ratio $N^\text{Coh}/N$, as a function of the total number of photons $N$ for five different path transmittivities, $\eta$, including the ideal noiseless case. The optimal $N^\text{Coh}$ is determined from the optimization presented in Eq. \ref{['eq:RatioOptimization']}. We observe that for all plotted values of $N$ and $\eta$, the optimal probe state allocates more energy to displacement than to squeezing. This trend is particularly pronounced for higher values of $\eta$, indicating that the classical strategy --that is, without squeezing-- becomes nearly optimal in noisy environments. Interestingly, there seems to be a region around $N \approx 3$ for which squeezing allocation becomes crucial. The ideal noiseless scenario differs in the bright beam situation: for beams with $N \gtrsim 10$ the trend is that $N^\text{Coh}/N \rightarrow 0.5$, meaning that half of the resources should be put in squeezing and half in displacement.
• Figure 5: Gain maximization $R$, that is, the optimized ratio of Fisher information for the quantum vs the classical strategies (see Eq. \ref{['ratiocomplex']}) as a function of the transmittitivity $\eta$ (0 for noiseless, 1 for completely noisy) and the surface parameter $\varepsilon$ plotted for an expected total photon number $N=N^\text{Sq} + N^\text{Coh}=20$. The parameter $\varepsilon$ is related to the energy of the scattered light that is diffused, and we assume $\varepsilon \in [0,0.2]$, more details about the model can be found in Appendix \ref{['Roughmodel']}. The shaded region represents $R=1$, that is, no quantum advantage. As expected, $R$ grows inversely to both $\eta$ and $\varepsilon$.