Squeezing Enhanced Sagnac Sensing based on SU(1,1) Quantum Interference

TL;DR

The paper introduces a squeezing-enhanced Sagnac interferometer using SU(1,1) interference with a single loop OPA to automatically squeeze counter-propagating beams, enabling sub-shot-noise rotational sensing. Two readout schemes are analyzed: direct detection and parametric homodyne detection with a measurement OPA, the latter offering robustness to detector inefficiency and nearing Heisenberg-like scaling under realistic losses. The results show strong quantum enhancement relative to the SNL across loss regimes, with performance governed by parametric gain g, seed power α, and detection strategy, and extend the framework to non-degenerate two-mode squeezing for fiber-based implementations. The work provides a simple, robust architecture compatible with standard Sagnac setups and highlights practical paths to quantum-enhanced rotation sensing in photonic systems, including lock-free non-degenerate configurations and potential multi-pass cavities for further improvement.

Abstract

We present a simple and robust design for a squeezing-enhanced Sagnac interferometer that employs the concept of SU(1,1) interference to significantly surpass the classical sensitivity limit (shot-noise limit - SNL) in rotational sensing. By strategically placing an optical parametric amplifier (OPA) inside the Sagnac loop, light is automatically squeezed in both forward and backward directions of the loop, which enhances the detectability of a small phase. For measuring the squeezed quadrature, we explore two approaches: Direct detection of the output intensity, which is simple, but requires a high-efficiency photo-detector; and parametric homodyne with an additional OPA, which accepts practical detectors with no efficiency limitation, but is technically more complex. Our analysis demonstrates super-classical sensitivity under most realistic conditions of loss and detector inefficiency, thereby leveraging the resources of squeezing and the principles of SU(1,1) interference, while maintaining compatibility with standard Sagnac configurations.

Figures (8)

• Figure 1: The Sagnac Interferometer: Traditional Configuration. A light beam is split and sent in opposite directions around the Sagnac loop. The two beams recombine to form a destructive interference pattern at the detector. When stationary, there's no phase shift between the beams, resulting in a consistent pattern. However, rotation causes a phase shift due to the differing path lengths traveled by the beams, altering the interference pattern. This shift is proportional to the rotation speed, making the Sagnac interferometer a precise tool for detecting rotational movements.
• Figure 2: Proposed squeezing-enhanced Sagnac sensor with two alternative detection strategies: (a) Direct detection: We incorporate into the standard classical Sagnac loop an OPA that squeezes the circulating fields in both directions. These squeezed beams experience a phase-difference in the loop (due to rotation) and recombine at the BS. Due to the interference between the two squeezed beams, the output at the dark port is intensity-squeezed (as we analyze and exemplify in Fig. \ref{['fig:quadrature_illustration']}), allowing a direct readout of the rotation-induced phase information with sub-shot-noise sensitivity. This approach is technically simple, but requires an efficient photo-detectors, since detection losses directly degrade the observed squeezing. An advantage of this configuration is that the relative phase between the pump phase should be stabilized only at the input. If the pump for the loop OPA in both directions is combined through the Sagnac BS (either with the input seed or through the dark port), the squeezing phase is automatically maintained for both directions (as discussed in the text). (b) Parametric homodyne detection: To alleviate the limitation of detector-efficiency, we add a second OPA at the dark output port for parametric homodyne detection. This measurement OPA is phased to amplify the squeezed quadrature of the output field, just before detection, which is highly robust against detector inefficiency, achieving super-classical sensitivity even with practical, lossy detectors.
• Figure 3: Quadrature evolution in the squeezing-enhanced degenerate Sagnac interferometer. The figure illustrates the transformation of the optical quadratures $\hat{X}$ and $\hat{Y}$ at different stages of the interferometer. (i)Input Ports: A coherent seed with amplitude $\alpha$ enters the seeded port $\hat{a}$, while vacuum $\lvert 0 \rangle$ enters the dark port $\hat{b}$. (ii)Sagnac Beams: The input fields are split by the Sagnac BS into clockwise ($\hat{c}_{1}$) and counterclockwise ($\hat{d}_{1}$) propagating beams. The reflected beam acquires a $\pi$ phase shift relative to the transmitted beam (due to the input-output relations of a passive BS), leading to the destructive interference of the $\hat{X}$ quadrature at recombination. (iii)Squeezed Beams: Both beams pass through the loop OPA, which squeezes the $\hat{Y}$-quadrature and amplifies the $\hat{X}$-quadrature. The quantum uncertainty distributions transform from circles into ellipses, reflecting the squeezing effect. (iv)Phase Shift: The counter-propagating beams accumulate opposite phase shifts $\pm \phi$ due to the Sagnac effect, yielding $\hat{c}_{3}$ and $\hat{d}_{3}$. (Propagation losses, not shown here for clarity, are modeled as effective BSs that couple the fields to vacuum modes, resulting in attenuated fields $\hat{c}_{4}$ and $\hat{d}_{4}$). (v)Output Ports: Upon recombination at the BS, the output field at the dark port $\hat{f}$ is intensity-squeezed along the $\hat{Y}$ quadrature, whereas the bright output $\hat{h}$ (emitted back to the seed laser) is phase-squeezed along the $\hat{X}$ quadrature. Specifically, the $\hat{X}$ quadratures of fields $\hat{c}_{4}$ and $\hat{d}_{4}$ destructively interfere at the dark port, leaving the output $\hat{f}$ dominated by the squeezed $\hat{Y}$ quadrature, whose intensity carries the phase information (induced by the rotation).
• Figure 4: Calculated phase sensitivity ($\Delta^2\phi_{PH}$) of our squeezing-enhanced Sagnac interferometer for an ideal, lossless configuration with parametric homodyne detection. The Y- axis is log-scaled (dB) relative to the power-equivalent SNL, and the X-axis is the phase working-point of the interferometer. (a) Seeded input: We assume coherent seeding of $N_{seed}=10$ for different values of the parametric gain $g = g_m$ (balanced). (b) Unseeded input: ($N_{seed}=0$) with generation gain fixed at $g = 2$ and varying measurement gain $g_m$. The sensitivity is enhanced according to the generation gain, as expected. In the unseeded case (only), balanced gain is important, as imbalance degrades the enhancement and shifts the optimal working-phase away from zero
• Figure 5: Effect of internal loss (symmetric & asymmetric): Calculated sensitivity ($\Delta^2\phi_{PH}$ in dB relative to the SNL) with parametric homodyne detection, as a function of the phase working-point $\phi$. All graphs assume balanced gain $g=g_m=2$ and a coherent seed of $N_{seed}=100$ for two loss distributions within the Sagnac loop: (a) symmetric loss, where the loop loss before the loop OPA and after it are equal for both beams in the loop, and (b) maximum asymmetric loss (solid line), where the loop OPA is placed near the BS, such that the forward (backwards) beam experiences all the loss before (after) the loop OPA. The dashed lines show the result with equivalent symmetric loss for comparison. For very large loop losses (we show 30% and 99% total loop loss) the maximally asymmetric loss is slightly better, but for reasonable (lower) loss situations the sensitivity becomes insensitive to the loss distribution.