Enhanced phase estimation with coherently boosted two-mode squeezed beams and its application to optical gyroscopes

Abstract: Quantum techniques, developed in recent decades, provide new approaches to achieving high-precision measurements beyond the classical bounds. In this paper, we theoretically demonstrate a metrology method for improving the sensitivity of the interferometric optical gyroscope, robust against the loss, by using coherent-light stimulated two-mode squeezed beams as the light source. The detection protocol is based on a simple intensity measurement, and the quantum noise is far below the shot-noise limit. The enhancement factors for different coherent light fields are analyzed in detail. Additionally, the influence of loss during the propagation in the optical path is studied, and the conditions for achieving sub-shot-noise measurement sensitivity are obtained. We also find that the phase sensitivity of the proposed gyroscope scheme becomes closer to the quantum Cramér-Rao bound with increasing of the photon number of the coherent beams.

• The paper shows a quantum protocol that uses coherently boosted two-mode squeezed beams to enhance phase estimation in optical gyroscopes, achieving sub-shot-noise sensitivity.
• It utilizes an optical parametric amplifier to generate strong quantum correlations, achieving an exponential phase variance reduction proportional to e^(-2r).
• The protocol remains robust under moderate optical losses and unbalanced inputs, paving the way for near quantum Cramér-Rao bound measurements in practical settings.

Enhanced Phase Estimation with Coherently Boosted Two-Mode Squeezed Beams for Optical Gyroscopes

Introduction and Problem Overview

The paper "Enhanced phase estimation with coherently boosted two-mode squeezed beams and its application to optical gyroscopes" (2607.05732) addresses quantum metrological enhancements to optical gyroscope sensitivity, leveraging the combination of coherent-light–stimulated two-mode squeezing and intensity-difference measurement. The analysis is situated in the context of Sagnac interferometry, where the measurement of rotation is fundamentally limited by quantum noise, typically quantified by the shot-noise limit (SNL). The central goal is to demonstrate a gyroscopic measurement protocol that exhibits robust, sub-shot-noise phase sensitivity, efficiently approaching the quantum Cramér-Rao bound (QCRB), and show practical feasibility even in the presence of optical losses.

Methodology: Optical Configuration and Theoretical Construction

The proposed scheme utilizes a pair of balanced (or unbalanced) coherent states, which are injected into an optical parametric amplifier (OPA) to generate coherently boosted two-mode squeezed states. These states serve as the inputs to a Sagnac interferometer. The OPA induces strong quantum correlations, characterized by a squeezing gain parameter $r$ and a pump phase $\theta$, between the two photonic modes.

The output beams from the interferometer are detected simply by measuring the intensity difference between the two output ports. Analytical treatments employ the standard two-mode squeezing transformation and account for various input state configurations by varying the relative amplitudes ($\alpha,\ \beta$) and phases of the seeded coherent states.

The analysis is performed both in the ideal case (no loss) and under realistic conditions, modeling transmission losses by fictitious beam splitters with non-unitary transmission coefficients in both optical paths.

Results: Phase Sensitivity, Enhancement Regimes, and Loss Analysis

Ideal Regime and Enhancement Over SNL

For balanced input amplitudes ($\alpha = \beta$), the phase sensitivity achieves a reduction by a factor of $e^{-2r}$ relative to the SNL, under optimal phase matching ($\theta = \pi + 2\gamma$). The SNL itself is reduced by the OPA-induced photon number enhancement, so the overall quantum enhancement is doubly beneficial: increased probe power and reduced quantum noise.

For the optimal case, the phase variance at small measured phase shifts ($\varphi \to 0$) is

$$\Delta \varphi^2 = e^{-2r} \Delta \varphi^2_{\text{SNL}}$$

meaning a strong exponential dependence on the squeezing gain.

Unbalanced input amplitude and phase scenarios (cases II/III in the paper) yield explicit analytic enhancement factors, showing that while phase sensitivity is generally degraded with imbalance, sub-shot-noise performance is still accessible over a broad parameter region, especially for moderate OPA gain and careful phase alignment.

Loss Tolerance and Practicality

Inclusion of loss, modeled as asymmetric or symmetric beam splitter attenuation in the interferometer arms, leads to analytic expressions for the achievable enhancement factor

$$R = e^{-2r} + \frac{t_a^2 r_b^2 + t_b^2 r_a^2}{2t_a^2 t_b^2}$$

where $t_a, t_b$ are field transmission coefficients. The criterion for sub-SNL operation ($\theta$0) is explored as a function of OPA gain, reciprocity, and transmission. The analysis shows that for symmetric loss (i.e., $\theta$1), up to 3 dB loss can be tolerated while still maintaining quantum advantage, provided the gain is sufficiently high. Similarly, moderate imbalance between arms does not preclude significant noise suppression.

The result underscores the practical feasibility for fiber-optic gyroscope architectures, as current technology can realize OPA gains in the required regime. Furthermore, the protocol is resilient to the quantum efficiency of detectors, provided it exceeds 50%.

Quantum Fisher Information and Attainability of the QCRB

A full quantum estimation framework is applied. The intensity-difference measurement protocol asymptotically saturates the QCRB in the high-coherent–photon-number regime ($\theta$2), demonstrating that no information is wasted—unlike protocols relying on parity or higher-order coincidence, which are experimentally more challenging.

For pure two-mode squeezed vacuum input ($\theta$3), the optimal measurement diverges from intensity-difference; phase information vanishes in this measurement and one must resort to non-Gaussian observables.

Comparison with Other Protocols and Numerical Benchmarks

Compared to standard Sagnac-based fiber-optic gyroscopes, the protocol yields a substantial decrease in the minimum resolvable angular velocity ($\theta$4) proportional to $\theta$5 for constant photon number at the output. As an explicit example, for representative parameters ($\theta$6), the quantum-enhanced scheme outperforms the classical configuration by nearly an order of magnitude in angular sensitivity.

The approach avoids the need for bias phase modulation, which typically introduces additional optical loss in classical schemes.

Comparison to previously studied quantum protocols (SU(1,1) interferometry, nonlinear schemes, TMSV-state input) reveals similar or stronger enhancement with lower requirements on squeezing, and a simpler, more robust measurement protocol.

Practical and Theoretical Implications

The demonstrated protocol is implementable with mature OPA and intensity-difference detection hardware. The tolerance to loss and robust performance under unbalanced conditions recommend it for deployment in real-world, high-performance navigation or inertial measurement units. The analysis confirms that the Heisenberg limit, and in some cases the QCRB, are approachable without highly non-Gaussian states or complex detection.

The theoretical implications center on confirming that SU(2) interferometry with coherently boosted Gaussian states and direct detection can close the gap between standard quantum-limited and quantum-enhanced metrological applications, offering a practical path to quantum advantage in macroscale sensors.

Future extensions should consider mode-matching imperfections, time-dependent or stochastic loss, and real-time adaptive feedback to optimize input states under dynamic noise conditions.

Conclusion

This work establishes the viability and quantitative performance of phase estimation protocols based on coherently boosted two-mode squeezed beams for optical gyroscopes, including a rigorous loss analysis and comparison to the quantum Cramér-Rao bound. The findings provide both analytic and practical benchmarks for quantum sensor deployment, substantiate the metrological advantage in experimentally accessible regimes, and serve as a template for future quantum-enhanced interferometric inertial sensors.