Modeling frequency instability in high-quality resonant experiments

This paper demonstrates that rapid frequency jitter in high-quality resonant systems, such as the Dark SRF experiment, causes minimal power loss, thereby refining dark-photon exclusion bounds to establish world-leading laboratory limits on the photon mass below $6\,\rm \mu eV$.

The Gist

Imagine you are trying to listen to a very faint whisper in a room full of noise. To hear it, you build a special, super-sensitive microphone (a resonator) that is tuned to pick up only that specific whisper.

In the world of physics, scientists build these "microphones" using superconducting metal boxes called cavities. They are so precise that they can detect particles that might make up "dark matter" (the invisible stuff holding galaxies together). One such experiment is called Dark SRF.

The Problem: The Shaky Microphone

The problem is that these perfect microphones aren't perfectly still. Imagine your microphone is sitting on a table, but the table is vibrating slightly because of a nearby refrigerator or people walking by. This causes the microphone's tuning to wobble up and down randomly. This wobbling is called "jitter."

For a long time, scientists were worried about this jitter. They thought:

"If the microphone's tuning wobbles even a tiny bit, it will drift away from the whisper's frequency. It will spend most of its time 'out of tune,' missing the signal entirely. This means we'll lose almost all our ability to hear the dark matter."

They were so worried that they assumed the worst-case scenario: that the jitter would ruin the experiment, reducing their sensitivity by a factor of 100,000.

The Discovery: The Fast Dancer

The authors of this paper, Hao-Ran Cui, Saarik Kalia, and Zhen Liu, decided to look closer. They asked a simple question: How fast is the table shaking?

They realized that the speed of the wobble matters more than the size of the wobble.

Here is the analogy:
Imagine you are trying to balance a broomstick on your hand.

1. Slow Wobble: If your hand moves slowly and stays in one wrong position for a long time, the broomstick falls. This is like the "slow drift" scientists were worried about.
2. Fast Wobble: Now, imagine your hand is shaking incredibly fast, vibrating back and forth so quickly that it's a blur. Even though you are moving, the broomstick doesn't have time to fall because you are constantly correcting it before it gets a chance to tip over.

The paper shows that in the Dark SRF experiment, the "jitter" is shaking very fast (about 45 times a second). Because it shakes so quickly, the resonator doesn't have time to "get confused" or lose its phase. It effectively averages out the wobble and stays perfectly tuned to the signal, just as if the table were perfectly still.

The Result: A Massive Upgrade

Because the jitter is fast, it doesn't kill the experiment's sensitivity.

• Old Belief: The jitter would ruin the signal, making the experiment 100,000 times less sensitive.
• New Reality: The jitter only reduces the signal by about 10%.

This is a game-changer. It means the Dark SRF experiment is actually 10,000 times more sensitive than they previously thought (because sensitivity scales with the fourth root of the power).

Why This Matters

By fixing this math error, the scientists have updated the "rules" for what kind of dark matter particles we can rule out.

• They have now set the world's best limit on the mass of a "dark photon" (a hypothetical particle) for a wide range of masses.
• They have also set the best laboratory limit on the mass of the photon itself (the particle of light), proving it is even lighter than we thought.

The Takeaway

The paper teaches us a valuable lesson about noise: Not all noise is bad. Sometimes, if the noise happens fast enough, it actually helps the system stay stable. It's like how a spinning top stays upright because it's spinning so fast; if it stopped spinning (or wobbled slowly), it would fall over.

Thanks to this new understanding, our "microphones" for the universe are much sharper than we realized, bringing us one step closer to hearing the whispers of the dark universe.

Technical Summary

1. Problem Statement

Modern resonant sensing experiments, such as the Dark SRF (Superconducting Radio Frequency) experiment, utilize high-quality factor ($Q \sim 10^{10}$) cavities to search for new physics (e.g., dark photons). These systems rely on extremely narrow linewidths ($\sim 0.1$ Hz) to accumulate signal power.

However, microscopic deformations in the cavity (e.g., from cooling fluid bubbles) cause stochastic frequency jittering ("microphonics") with amplitudes significantly larger than the linewidth (e.g., $\sim 3$ Hz vs. $0.15$ Hz).

• The Naive Assumption: Previous analyses (e.g., Ref. [7]) assumed that any frequency mismatch, regardless of its timescale, would cause the resonator to spend significant time off-resonance. This led to a conservative model treating the jitter as a constant frequency mismatch, predicting a massive suppression of signal power ($\sim 10^{-5}$) and a corresponding loss in Signal-to-Noise Ratio (SNR).
• The Core Question: Does rapid stochastic jittering actually suppress power accumulation and sensitivity in high-$Q$ resonators, or does the timescale of the jittering mitigate this effect?

2. Methodology

The authors developed a rigorous theoretical and numerical framework to model a driven damped harmonic oscillator with a stochastically varying natural frequency.

• Mathematical Model:
  The system is described by the stochastic differential equation:
  $$\ddot{x}(t) + \gamma \dot{x}(t) + (\omega_0 + \delta\omega(t))^2 x(t) = F(t)$$
  Where $\delta\omega(t)$ is the stochastic frequency shift. The authors modeled $\delta\omega(t)$ as a stationary random process with zero mean, characterized by a Power Spectral Density (PSD) $S_{\delta\omega}(\omega)$ featuring a Lorentzian peak at frequency $\omega_j$ with correlation time $\tau$.

• Statistical Models:
  Two distinct statistical models for the jittering process were analyzed:

  1. Gaussian Process: Physically motivated, where higher-order moments are determined by Wick's theorem.
  2. Dichotomic Markov Process (DMP): A process switching between $\pm \delta\omega_0$, used to derive exact analytic solutions via the Shapiro-Loginov formula.

• Analytical & Numerical Approaches:

  • Perturbative Regime: Derived analytic expressions for expected power and spectral response assuming small power suppression ($\alpha \ll 1$).
  • Numerical Simulation: Implemented a computationally efficient algorithm to solve the SDE by discretizing time on a scale $\Delta t$ such that $1/\omega_0 \ll \Delta t \ll \tau, 1/\omega_j$. This allowed for the simulation of long-time asymptotic behavior and ensemble averaging over thousands of realizations.
  • Spectral Analysis: Computed the mechanical susceptibility PSD and the resulting position PSD to determine the Signal-to-Noise Ratio (SNR) in the frequency domain.

3. Key Contributions

A. The Timescale of Jittering is Critical

The paper establishes that the timescale of frequency jittering is the dominant factor in power suppression, not just the amplitude.

• Mechanism: Power suppression occurs when a relative phase develops between the driving force and the resonator response.
  • Slow Jittering: If the frequency stays off-resonance for a time longer than the cavity's damping time ($1/\gamma$), a large phase shift accumulates, leading to destructive interference and power loss.
  • Fast Jittering: If the frequency oscillates rapidly (timescale $\tau \ll 1/\gamma$ or $\omega_j \gg \delta\omega_0$), the phase shift randomizes and averages out ("washes out"). The system accumulates power as if the frequency were stable.

B. Analytic Conditions for Negligible Suppression

The authors derived a perturbative parameter $\alpha$ governing power suppression:
$$\alpha \equiv \frac{4\delta\omega_0^2}{\gamma^2} \rho$$
where $\rho$ depends on the jittering timescale.

• For broadband jittering ($\omega_j = 0$), suppression is negligible if $\tau \ll \gamma / (2\delta\omega_0^2)$.
• For monochromatic jittering ($\tau \to \infty$), suppression is negligible if $\omega_j \gg \delta\omega_0$.
• Result: In the Dark SRF experiment, the jittering is fast enough that the power suppression is only $\sim 13\%$, not the previously assumed $99.999\%$.

C. Spectral Features and Sidebands

The study reveals that jittering introduces sidebands in the spectral response at frequencies $f_0 \pm f_j$.

• While sidebands carry signal power, the response within these bands is highly non-Gaussian and correlated.
• Consequently, standard SNR integration formulas (which assume Gaussian statistics) cannot be applied to the sidebands to boost sensitivity. The sensitivity remains dominated by the central resonant peak.

4. Results

• Power Accumulation: For the Dark SRF benchmark parameters ($Q \sim 10^{10}$, jitter amplitude $\sim 3$ Hz, correlation time $\sim 0.06$ s), the actual power loss due to jittering is only $\sim 10\%$. This contradicts the conservative estimate of a $10^{-5}$ suppression factor.
• Sensitivity Enhancement: Because the power suppression is minimal, the sensitivity of the Dark SRF experiment is effectively unchanged from the ideal stable case.
• Revised Exclusion Bounds: Applying these findings to the Dark SRF pathfinder run data:
  • The constraint on the dark-photon kinetic mixing parameter $\epsilon$ improves by one order of magnitude.
  • This corresponds to an effective Signal-to-Noise Ratio (SNR) enhancement of four orders of magnitude compared to the previous analysis.
  • The result sets the world-leading constraint on dark photons for masses below $6 \, \mu\text{eV}$.
  • It translates to the best laboratory-based limit on the photon mass: $m_\gamma < 2.9 \times 10^{-48}$ g.

5. Significance

• Paradigm Shift in Resonant Sensing: The work overturns the conservative assumption that any frequency instability larger than the linewidth is fatal to high-$Q$ experiments. It demonstrates that fast stochastic noise can be benign, provided the correlation time is short.
• Immediate Impact on Dark Matter Searches: The re-analysis of existing Dark SRF data significantly tightens constraints on dark photons without requiring new hardware or data collection, effectively unlocking the full potential of current high-$Q$ superconducting cavities.
• General Applicability: The methodology applies to any high-$Q$ resonant system, including gravitational wave detectors, atomic clocks, and other searches for axions or hidden sector particles, where frequency stability is a critical limiting factor.
• Methodological Rigor: The paper provides a complete toolkit (analytic formulas, numerical codes, and statistical frameworks) for modeling stochastic frequency variations, distinguishing between Gaussian and non-Gaussian responses, and correctly calculating SNR in the presence of jitter.