High-Frequency Thermal Noise in Michelson Interferometers

This paper develops and validates general models for various thermal noise sources in Michelson interferometers that are necessary for accurately characterizing weak, high-frequency signals in the MHz band where traditional quasistatic approximations fail, specifically applying these models to the Holometer and the GQuEST experiment.

The Gist

The Big Picture: Listening for a Whisper in a Storm

Imagine you are trying to hear a very faint, mysterious whisper (a signal from quantum gravity) coming from a distant room. To do this, you have built a super-sensitive listening device called a Michelson Interferometer. It works like a giant ruler made of light, measuring tiny changes in distance.

For a long time, scientists thought the main thing blocking them from hearing this whisper was "static" from the light itself (called shot noise). They built new experiments to get rid of that static. But once they turned off the static, they realized there was another, louder noise source they hadn't fully understood: Thermal Noise.

Think of thermal noise like the "hiss" of a crowded room. Even if the room is quiet, the people inside are constantly shuffling, breathing, and moving. In a mirror, the atoms are constantly jiggling because of heat. This jiggling makes the mirror vibrate, which messes up the light measurement.

The problem? The old rules for calculating this "room noise" were written for low frequencies (slow movements). But these new experiments are listening to high frequencies (very fast vibrations, in the MHz range). The old rules don't work anymore because they assume the mirror moves like a slow, heavy rock. In reality, at high speeds, the mirror acts more like a drum skin that ripples and resonates.

This paper writes a new rulebook to accurately predict how much this "heat jiggling" will mess up the experiment.

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The Three Main Types of "Heat Noise"

The authors break down the noise into three main categories, like three different ways a drum can make noise:

1. Mechanical Noise (The "Drum Skin" Vibration)

• The Old View: Scientists used to think the mirror was a solid, infinite block. They assumed the light just pressed on the surface and the whole block moved slowly.
• The New Reality: At high frequencies, the mirror isn't a solid block; it's a thin slab. When the light hits it, it creates ripples (like throwing a pebble in a pond). These ripples travel through the mirror and bounce off the edges.
• The Analogy: Imagine hitting a drum. If you hit it slowly, the whole drum moves. If you hit it very fast, you create a standing wave pattern that vibrates in specific spots. The paper calculates exactly how these "ripples" in the mirror's material (both the glass/silicon body and the special coating on top) create noise.
• Key Finding: For the Holometer (a past experiment), the main noise wasn't the coating (the paint on the drum), but the substrate (the drum skin itself). This was a surprise because previous models predicted the coating would be the loudest.

2. Thermoelastic Noise (The "Hot and Cold" Expansion)

• The Concept: When a material gets slightly hotter, it expands; when it cools, it shrinks. Even tiny, random temperature fluctuations cause the mirror to stretch and squeeze.
• The New View: The old models assumed heat moved slowly through the mirror. But at high frequencies, heat doesn't have time to spread out evenly. It creates a "thermal diffusion length" (how far heat can travel in a split second).
• The Analogy: Imagine trying to warm up a thick winter coat by holding a hairdryer to one spot. If you hold it for a long time, the whole coat warms up. If you blast it for a split second, only the tiny spot under the nozzle gets hot. The paper calculates how these tiny, rapid "hot spots" cause the mirror to expand and contract, creating noise.

3. Thermorefractive Noise (The "Heat Haze" Effect)

• The Concept: Heat doesn't just change the size of the mirror; it also changes how light travels through it (the refractive index). Think of the "shimmer" you see above a hot road.
• The New View: The light beam doesn't just hit the surface; it penetrates slightly into the coating layers. The paper models how heat fluctuations deep inside these layers change the "speed" of the light, messing up the measurement.
• The Analogy: Imagine looking through a window that has a wavy, uneven temperature inside it. The view gets distorted. The paper calculates how much this "wavy heat" distorts the light beam inside the mirror's coating.

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How They Tested It: The "Holometer" Check

To make sure their new math was right, the authors looked at data from a real experiment called the Holometer.

• The Test: They compared their new, complex "ripple" models against the actual data recorded by the Holometer.
• The Result: The new models matched the data perfectly. They could explain the "sawtooth" patterns in the noise graph (the peaks and valleys) that the old models couldn't.
• The Discovery: They found that the "valleys" (the quiet spots between the noise peaks) were actually lower than the old models predicted. This means the experiments are cleaner than we thought, but the "peaks" (resonances) are higher.

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The Future: GQuEST

The paper then applies these new rules to a new experiment called GQuEST, which is currently being built.

• The Goal: GQuEST is designed to look for quantum gravity signals.
• The Optimization: Because the authors now know exactly how the "drum skin" (substrate) and the "paint" (coating) vibrate at high speeds, they can design the mirrors to avoid the loudest frequencies.
• The Outcome: They found that for GQuEST, the noise from the mirror body and the mirror coating are now roughly equal. This is a crucial detail for engineers trying to build the most sensitive detector possible.

Summary

In short, this paper says: "We used to think mirrors were slow, solid rocks. But at high speeds, they act like rippling drums. We wrote new math to describe these ripples, proved it works with real data, and used it to help build a better machine to listen for the secrets of the universe."

Technical Summary

Technical Summary: High-Frequency Thermal Noise in Michelson Interferometers

Problem Statement
New experiments utilizing Michelson interferometers are being developed to detect weak, high-frequency signals, such as those potentially arising from quantum gravity, by employing readout schemes that eliminate the background of quantum shot noise. In the absence of shot noise, classical thermal noise from the interferometer's optics becomes the dominant limitation. However, existing models for thermal noise rely on approximations that are invalid in the megahertz (MHz) frequency band targeted by these new experiments. Specifically, the quasistatic approximation (assuming measurement frequencies are below the first mechanical eigenmode of the mirror) and the assumption that the beam penetration depth is smaller than the thermal diffusion length break down at MHz frequencies. Consequently, accurate characterization of signals and the design of sensitive experiments require more general models that account for the dynamic behavior of mechanical modes and thermal diffusion in this regime.

Methodology
The authors develop generalized analytical models for various thermal noise sources, moving beyond the quasistatic limit. The modeling framework relies on the Fluctuation-Dissipation Theorem (FDT) and Levin's "direct" method, which calculates noise by determining the power dissipated from a fictitious drive rather than summing over mechanical eigenmodes (which becomes computationally prohibitive at high frequencies due to complex modal structures).

The paper derives models for:

1. Mechanical (Brownian) Noise:
   • Substrate: The authors model the substrate as an infinite slab and a finite cylinder to capture bulk-acoustic wave dynamics. They derive a closed-form expression for the noise spectrum at high frequencies, identifying periodic minima ("valleys") between longitudinal eigenmodes and Lorentzian peaks at resonances.
   • Coating: Using Levin's method, the coating is treated as a thin layer on the substrate. The model accounts for the high loss angle of coatings relative to substrates and derives high-frequency approximations where substrate dynamics become negligible away from resonances.
2. Thermoelastic Noise:
   • Substrate: The model extends previous quasistatic work to frequencies near and above the first mechanical eigenfrequency, calculating power dissipation from thermoelastic damping using the expansion scalar derived from the elastic wave equation.
   • Coating: The authors relax the assumption that the thermal diffusion length is larger than the beam penetration depth. They solve the heat equation with driving terms related to both thermoelastic expansion and thermorefractive index changes, incorporating the standing wave profile of the laser within the coating (Bragg profile).
3. Beamsplitter Noise: The models are adapted for beamsplitters, accounting for transmission and reflection paths, photoelastic effects, and the specific geometry of the optical path.

Key Contributions and Results

• Validation with Holometer Data: The new models were validated against data from the Holometer experiment (40-m arm length, fused silica optics). The analytical models successfully reproduce the "sawtooth" features in the power spectral density (PSD) caused by substrate mechanical noise and beamsplitter mechanical noise. The analysis reveals that for the Holometer, the dominant noise source is substrate mechanical noise, not coating mechanical noise as predicted by quasistatic models. The agreement confirms that the aggregation of bulk acoustic modes produces a predictable average spectral density suitable for calibration.
• Application to GQuEST: The models were applied to the Gravity from the Quantum Entanglement of Space-Time (GQuEST) experiment, currently under construction. GQuEST is designed with specific frequency bands (around 15 MHz) to minimize substrate mechanical noise.
  • In these optimized bands, the combined beamsplitter and substrate mechanical noise is approximately equal to the coating mechanical noise.
  • The updated total classical noise budget at 15 MHz is calculated to be $8 \times 10^{-22} \, \text{m}/\sqrt{\text{Hz}}$.
  • This represents a significant reduction compared to the $1.4 \times 10^{-21} \, \text{m}/\sqrt{\text{Hz}}$ estimated using previous literature models that relied on the quasistatic approximation.
• High-Frequency Behavior: The paper demonstrates that near longitudinal eigenfrequencies, the updated models predict significantly more noise than quasistatic models. Conversely, in the "valleys" between resonances, the noise floor is lower and depends on specific geometric and material parameters (e.g., beam size, mirror thickness, loss angles).

Significance and Claims
The paper asserts that accurate classical noise modeling is a prerequisite for detecting weak signals in the MHz band. By developing and validating these general models, the authors provide a necessary tool for characterizing the sensitivity of next-generation quantum gravity experiments like GQuEST.

The authors claim that their work corrects previous overestimates of noise in certain frequency bands and identifies substrate mechanical noise as a critical factor in the Holometer, contrasting with coating-dominated noise in lower-frequency detectors like LIGO. They suggest that for future broadband detectors (including potential upgrades to kilometer-scale interferometers like Cosmic Explorer), reducing the mechanical loss angle of coatings in the MHz band is as critical as it is in the audio band. The paper concludes that these models are essential for designing experiments where classical noise must be engineered to be subdominant to the signal, particularly for techniques aiming to avoid the quantum shot noise background.