⚛️ general relativity

Quantum noise and torque correlations in a torsion-bar Sagnac speed meter

This paper demonstrates that in the CHRONOS torsion-bar Sagnac speed meter, vacuum fluctuations entering through weakly transmitting mirrors behind the test masses dominate low-frequency quantum noise and significantly degrade sensitivity, revealing that strong torque cancellation alone is insufficient to ensure optimal detector performance.

Original authors: Yuki Inoue, Mario Onglao, Daiki Tanabe, Masashi Hazumi, Vivek Kumar, Apimook Watcharangkool

Published 2026-10-15

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CC BY 4.0

Original authors: Yuki Inoue, Mario Onglao, Daiki Tanabe, Masashi Hazumi, Vivek Kumar, Apimook Watcharangkool

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Deep in the quiet of the universe, invisible ripples in the fabric of space-time travel across the cosmos, carrying the stories of colliding black holes and the birth of stars. To hear these whispers, scientists build instruments so sensitive they can detect a change in distance smaller than a single atom over a span of kilometers. These devices, known as gravitational-wave detectors, work by splitting a laser beam and sending it down two long paths. If a gravitational wave passes through, it stretches one path and squeezes the other, changing the way the light waves recombine. While current detectors have already opened a new window on the universe, they struggle to hear the lower, deeper notes of the cosmic symphony. To listen to these lower frequencies, researchers are designing a new kind of detector that does not measure the stretching of space directly, but instead measures the speed at which heavy, suspended bars twist in response to the passing waves.

The challenge with these new instruments is that the very act of measuring them with light creates a disturbance. The laser light itself is made of tiny packets of energy that arrive in a jittery, unpredictable stream. When these packets hit the heavy bars, they push on them, creating a random shaking that can drown out the faint signal of a gravitational wave. This is known as quantum noise. To overcome this, the designers of a proposed detector called CHRONOS use a clever trick involving a specific arrangement of mirrors and light paths, known as a Sagnac interferometer. This setup is designed to cancel out the random pushing of the light, allowing the detector to listen more clearly. However, a new study by a team of physicists reveals that this clever cancellation is not the whole story. Even when the main source of this random pushing is suppressed, a hidden source of noise remains, lurking in the tiny gaps of the mirrors themselves.

The researchers focused on the mirrors that sit at the ends and the middle of the light paths within the detector's rings. In the ideal design, these mirrors are almost perfectly reflective, letting only a tiny fraction of light pass through. The team realized that the light that does pass through, or the empty space that exists behind these mirrors, allows invisible quantum fluctuations to enter the system from the back. These fluctuations are not just a minor background hum; they are a dominant force. In their detailed analysis, the team found that at a frequency of 0.1 hertz, these rear-entry fields contribute nearly 90 percent of the total quantum noise power. If a designer were to ignore these fields and only look at the main laser input, they would severely underestimate the noise, thinking the detector was far more sensitive than it actually is. Specifically, omitting these rear fields would lead to an error where the noise level is underestimated by a factor of more than three.

To understand why this happens, one must look at how the forces act on the suspended bars. The detector works by measuring the twist, or torque, of a bar caused by the difference in force between its two ends. The Sagnac design is excellent at canceling out the forces that push equally on both ends, but the quantum fluctuations entering from behind the mirrors do not always push in a way that cancels out perfectly. The researchers calculated how these random forces from the rear vacuum fields travel through the system, affecting the bar's rotation and the final measurement. They found that even though the forces on the left and right sides of the bar might seem to balance, the complex timing and correlation of the light fluctuations mean that a significant amount of noise still gets through. It is a bit like trying to silence a room by closing the front door, only to realize that the back door is wide open and letting in a loud wind that the front door was never meant to block.

The team also tested whether changing the mirrors could solve the problem. They asked if replacing the tiny amount of light that passes through the mirrors with a tiny amount of light that is absorbed by the mirrors would help. In this scenario, the mirrors would reflect the same amount of light, but instead of letting a little bit through to the other side, they would soak it up. The researchers found that this change made almost no difference to the noise level. The quantum noise from the absorption was nearly identical to the noise from the transmission. This means that simply making the mirrors more absorptive does not fix the underlying issue; the noise is inherent to the way the light interacts with the mechanical system, regardless of whether it passes through or is absorbed.

The study provides a clear roadmap for building these future detectors. It shows that to achieve the necessary sensitivity for listening to the low-frequency universe, engineers cannot rely solely on the main laser input or the cancellation of forces at the front of the mirrors. They must account for the vacuum fields entering from behind the mirrors at the ends and the middle of the light paths. The researchers developed a precise mathematical model that tracks these fields and their effects, confirming their findings with independent computer simulations. Their work establishes that the rear vacuum fields are a limiting factor for the sensitivity of this type of detector. For the design to succeed, the mirrors and the readout systems must be engineered with these specific, often overlooked, quantum contributions in mind. Without addressing this hidden source of noise, the detector will never reach the quiet needed to hear the deep, slow ripples of the cosmos.

Technical Summary: Quantum Noise and Torque Correlations in a Torsion-Bar Sagnac Speed Meter

Problem Statement
The paper addresses the quantum noise limitations of the Cryogenic sub-Hz cROss torsion bar detector with quantum NOn-demolition Speed meter (CHRONOS), a proposed torsion-bar Sagnac speed meter designed for low-frequency gravitational-wave detection (0.1–10 Hz). While Sagnac speed meters are theoretically capable of suppressing radiation-pressure back action through correlations between successive optical interactions, the authors identify a critical, often overlooked noise source: vacuum fields entering from behind the End Test Masses (ETMs) and Middle Test Masses (MTMs).

Previous studies suggested that strong torque cancellation between the left and right ends of the torsion bars would mitigate back action. However, this work demonstrates that even with mirror power transmissions as low as $10^{-7}$ , the vacuum fields entering from the rear of the test masses can dominate the low-frequency quantum noise. Neglecting these fields leads to a significant underestimation of the detector's noise floor, potentially by a factor greater than three at 0.1 Hz.

Methodology
The authors develop a comprehensive analytical input–output model to track quantum fluctuations through the complex optical and mechanical system. Key methodological elements include:

Operator Formalism: The model utilizes creation and annihilation operators for seventeen distinct vacuum input channels: one main input at the output beam splitter (OBS), and sixteen rear inputs (eight from ETMs and eight from MTMs across two axes and two circulation directions).
Common Denominator Representation: The optical equations, coupled with the rotational dynamics of the torsion bars, are solved to derive a common denominator ( $M$ ). This allows the signal field and noise contributions to be expressed as a sum of input-specific coefficients ( $C_q$ ) and a signal vector ( $D$ ).
Torque Correlation Analysis: The model explicitly calculates the correlation between radiation-pressure forces acting on the left and right ends of the bars. It distinguishes between the temporal correlations inherent to the Sagnac topology and the spatial correlations of forces at the bar ends, showing that strong torque cancellation does not automatically guarantee improved sensitivity if rear vacuum fields are unaccounted for.
Comparative Validation: The analytical results are rigorously compared against independent numerical simulations using FINESSE 3 (version 3.0.2). The comparison covers two distinct operating points selected by Power-Law Integrated (PI) curve optimization and various detuning scenarios.
Transmission vs. Absorption: The study investigates whether replacing the small mirror transmission ( $T \approx 10^{-7}$ ) with absorption (at fixed reflectivity) alters the noise performance. This tests the hypothesis that the noise arises from the vacuum coupling mechanism rather than the specific transmission loss.

Key Results

Dominance of Rear Vacuum Fields: At the reference operating point (minimizing the PI curve) and a frequency of 0.1 Hz, rear vacuum fields from ETMs and MTMs contribute 89.7% of the total quantum noise power spectral density (PSD).
Magnitude of Underestimation: Omitting these rear fields results in an underestimation of the amplitude spectral density (ASD) by a factor of 3.11 at 0.1 Hz. The error decreases at higher frequencies (factor of 2.76 at 1 Hz, 1.16 at 10 Hz) but remains significant.
Torque Cancellation Limits: While the model confirms that forces at opposite ends of the bar are highly correlated (cancellation ratio $\approx 0.085$ for rear fields), this cancellation is insufficient to suppress the noise contribution from the rear vacuum inputs. The residual torque noise from these fields remains a limiting factor.
Transmission vs. Absorption: Replacing the mirror transmission with absorption (keeping reflectivity fixed) leaves the excess quantum noise essentially unchanged. The relative change in total quantum ASD is approximately $2.5 \times 10^{-8}$ (0.025 ppm). This indicates that the noise is driven by the coupling of vacuum fields to the optical system, regardless of whether the coupling occurs via transmission or absorption.
Model Validation: The analytical model shows excellent agreement with FINESSE 3 simulations, with maximum relative differences of $1.04 \times 10^{-8}$ for the complex transfer function and $4.23 \times 10^{-10}$ for the total quantum ASD across tested operating points and detunings.

Significance and Claims
The paper establishes that rear vacuum fields are a fundamental limiting noise source in torsion-bar Sagnac speed meters, challenging the assumption that low mirror transmission alone ensures negligible readout noise.

Design Implications: The authors claim that mirror specifications and readout designs must explicitly account for vacuum fields coupled through transmission or absorption. Simply optimizing for torque cancellation or minimizing transmission is insufficient; the propagation of these rear fields to the bar rotation and the final readout must be modeled.
Theoretical Clarification: The work clarifies that strong torque cancellation does not inherently ensure a corresponding improvement in detector sensitivity. The sensitivity is determined by the interplay of optical correlations, mechanical feedback, and the specific injection points of vacuum fluctuations.
Operational Constraints: The results provide quantitative requirements for the CHRONOS detector, emphasizing that the "rear vacuum" contribution must be included in the noise budget to accurately predict sensitivity for low-frequency gravitational-wave detection, dark matter searches, and geophysical observations.

The authors remain modest regarding the scope, noting that their conclusions apply to the specific symmetric configuration and small-loss model studied, and that thermal effects of absorption were excluded from the comparison. The primary contribution is the rigorous quantification of these previously neglected noise terms and the validation of an input-resolved analytical framework.

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