Modeling frequency instability in high-quality resonant experiments

TL;DR

This work shows that stochastic frequency fluctuations in ultra-high-$Q$ resonators need not catastrophically suppress power or degrade sensitivity. By modeling jittering with a Lorentzian PSD and decomposing the frequency fluctuations into Gaussian or dichotomic processes, the authors derive both analytic perturbative results and numerical simulations that reveal a counterintuitive regime: fast jittering can erase phase slips and allow power to accumulate almost as if there were no jitter, while still imprinting distinct spectral sidebands. They quantify the power suppression through a perturbative parameter $\alpha$ and a timescale factor $\rho$, demonstrating that Dark SRF operates in a regime where suppression is modest (about 10%), which revises the dark-photon exclusion bounds upward by roughly an order of magnitude. The analysis shows that jittering preserves on-resonance sensitivity while enriching the spectral response, enabling stronger laboratory-based constraints on dark photons and photon mass across a broad mass range. These findings have practical implications for future high-$Q$ resonant experiments, informing data interpretation and guiding design choices to maximize sensitivity in the presence of frequency instability.

Abstract

Modern resonant sensing tools can achieve increasingly high quality factors, which correspond to extremely narrow linewidths. In such systems, time-variation of the resonator's natural frequency can potentially impact its ability to accumulate power and its resulting sensitivity. One such example is the Dark SRF experiment, which utilizes superconducting radio frequency (SRF) cavities with quality factors of $Q\sim10^{10}$. Microscopic deformations of the cavity lead to stochastic jittering of its resonant frequency with amplitude 20 times its linewidth. Naively, one may expect this to lead to a large suppression in accumulated power. In this work, we study in detail the effects of frequency instability on high-quality resonant systems, utilizing the Dark SRF experiment as a case study. We show that the timescale of jittering is crucial to determining its effect on power accumulation. Namely, when the resonant frequency varies sufficiently quickly, the system accumulates power as if there were no jittering at all. This implies that the sensitivity of a jittering resonator is comparable to that of a stable resonator. In the case of Dark SRF, we find that jittering only induces a $\sim 10\%$ loss in power. Our results allow the dark-photon exclusion bound from Dark SRF's pathfinder run to be refined, leading to a constraint that is an order of magnitude stronger than previously reported (corresponding to a signal-to-noise ratio which is four orders of magnitude larger). This result represents the world-leading constraint on dark photons over a wide range of masses below $6\,\rm μeV$ and translates to the best laboratory-based limits on the photon mass $m_γ<2.9\times 10^{-48}\,\rm g$.

Figures (8)

• Figure 1: Realizations of $\delta f(t)=\delta\omega(t)/2\pi$ for various choices of parameters. In each plot, the black curve utilizes the benchmark parameters for Dark SRF shown in Table \ref{['tab:parameters']}, and the colored curves show the behavior when a single parameter is varied. The upper plot shows the dependence on the correlation time $\tau$, the middle plot shows the dependence on the peak jittering frequency $f_j$, and the lower plot compares the Gaussian and DMP models of jittering.
• Figure 2: Real-time evolution of Eq. (\ref{['eq:resonator']}) for the parameters in Table \ref{['tab:parameters']} and both models of $\delta\omega(t)$. We show the cases of Gaussian jittering (blue), DMP jittering (orange), and no jittering (black). We plot the power $|x(t)|^2$ in the resonator, normalized by its asymptotic value in the no-jittering case [see Eq. (\ref{['eq:nojittering_power']})]. The dashed grey line indicates the asymptotic ensemble-averaged power $\langle |x(\infty)|^2\rangle_\infty$ for the cases with nonzero jittering. (The Gaussian and DMP cases give the same value of this quantity to within a percent.)
• Figure 3: Dependence of expected power $\langle|x(t)|^2\rangle_\infty$ on $\tau$ (upper plot) and $f_j$ (lower plot), normalized by the power $|x_0(\infty)|^2$ in the no-jittering case. For all other parameters, we use the Dark SRF values in Table \ref{['tab:parameters']}. In blue and orange, we show numerical estimates of the expected power, for both Gaussian and DMP jittering, respectively, accompanied by shaded bands representing the total error $\sigma_{|x(t)|^2}$ on our estimate. Each data point is estimated using $N=1000$ simulations, each with integration time $T=15\,\mathrm{{}s}$. In black, we show the analytic result in Eq. (\ref{['eq:power']}), which applies when $\alpha\ll1$. The estimate applies well even outside this regime, and the Gaussian and DMP cases agree except in the case of significant power suppression when $f_j$ is very small. (Note that the vertical axis in the lower plot utilizes a logarithmic scale.) The dashed grey lines indicate the Dark SRF values for $\tau$ and $f_j$, respectively, shown in Table \ref{['tab:parameters']}. These lie in the regime where the analytic estimate is reliable.
• Figure 4: Relationship between power accumulation (upper plot) and relative phase $\theta$ (lower plot; see Eq. (\ref{['eq:phase']}) for definition), for two different values of $f_j$. We use the values in Table \ref{['tab:parameters']} for all other parameters, and model the jittering as Gaussian. We show the evolution of the system between $t=15\,\mathrm{{}s}$ and $t=25\,\mathrm{{}s}$. We highlight a few periods of significant power loss using shaded bands. (The band color matches the curve which is exhibiting power loss.) Note that the periods of power loss occur when the relative phase deviates significantly from $\pi/2$. The orange curve uses a smaller value of $f_j$ and so develops a larger relative phase. This, in turn, leads to regions of more severe power loss.
• Figure 5: Probability distribution of $|x(T)|^2$ at $T=15\,\mathrm{{}s}$, normalized by the no-jittering power $|x_0(\infty)|^2$. Here we use the parameter values in Table \ref{['tab:parameters']}. In blue, we show the case of Gaussian jittering, while in orange we show the case of DMP jittering. Each distribution consists of $N=10^4$ samples. The dashed grey line indicates the mean power $\langle|x(t)|^2\rangle_\infty$. Note that both cases exhibit similar distributions, which rise exponentially and then drop rapidly before the no-jittering value $|x_0(\infty)|^2$.