Between coherence intervals, the UDM field amplitude is expected to fluctuate [20]; here $a_0$ is defined such that $⟨a_{\mathrm{DM}}^2⟩=g^2{\mathrm{\Delta }}_{12}^2{a}_0^2/2$ [21].

This formula is generally valid in the frequency domain, where $x_{\mathrm{DM}}$ and $a_{\mathrm{DM}}$ are interpreted as Fourier amplitudes at ${\omega }_m$. In the time domain, it is valid only if $Q_{\mathrm{DM}}\gg Q_{\mathrm{m}}$.

Here and throughout, we use “$\sim$” to mean “roughly” or “on the order of.”

We note that the Eöt-Wash (EW) torsion-balance experiment [48] and other equivalence-principle tests such as MICROSCOPE [50] probe composition-dependent, DM-mediated forces between SM test bodies, but do not access the temporal coherence of the UDM field. A direct B-L UDM search has been made with the EW balance at sub-Hz frequencies [51].

Commercial equipment, instruments, and materials are identified in this paper in order to specify the experimental procedure adequately. Such identification is not intended to imply recommendation or endorsement by the National Institute of Standards and Technology or the United States government, nor is it intended to imply that the materials or equipment identified are necessarily the best available for the purpose.

The double-membrane cavity finesse $\mathcal{F}\approx 2.2$ is determined by the membrane reflectivity [45] and can be enhanced using photonic crystal patterning [55] or the membrane-in-the-middle approach [56]. Here we operate with low finesse to simplify alignment and analysis of the differential acceleration sensitivity.

Cold damping does not improve sensitivity [58, 61], but reduces nonlinearities and mitigates the effect of spectral diffusion over the course of the measurement [62], especially relevant for high-$Q$ oscillators.

Measurement time $\tau =800\mathrm{s}$ was limited by practical considerations such as the open stability of our laser and the manual procedure for frequency scanning. Future, longer searches will require more stringent stabilization and automation.

In [34] we observed local photoabsorptive heating as little as ${10}^{-5}\mathrm{K}/\mathrm{\mu }\mathrm{W}$ in trampolines of similar dimensions at $\lambda =850\mathrm{nm}$ and room temperature, while in [45, 65] we observed modal heating at the level of ${10}^{-2}\mathrm{K}/\mathrm{\mu }\mathrm{W}$. The observed increase in modal heating ($0.4\mathrm{K}/\mathrm{\mu }\mathrm{W}$) at 4 K is consistent with a reduction of both radiative and conductive heat transfer in the trampoline [34]; however, the strong dependence of modal heating on temperature profile [66] calls for further investigation.

In this work, we adopt a Bayesian framework for limit setting because it allows us to encode physical constraints such as $D>0$ through the prior [68]. We note, however, that the positivity constraint $D>0$ can also be enforced straightforwardly within a frequentist framework. In [21] we show that for our Gaussian matched-filter estimator at 95% confidence level, the two approaches yield limits that agree within 10%.

The second generation (“Gen 2”) device discussed in the Supplemental Materials [21] uses a modeshape-optimized ${\text{Si}}_3{\mathrm{N}}_4$ trampoline with $m=1.5\mathrm{\mu }\mathrm{g}$, $f=7.3\mathrm{kHz}$, and $Q=7\times {10}^7$.

We note that over this duration, a DM signal is expected to experience sidereal modulation due to Earth’s rotation, providing a cross-check of its astrophysical origin [20, 76].

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