Edit 3: I jumped to some conclusions about what their actual apparatus would be, so some of this doesn't really address what the authors have written (see the link to the actual paper and preprint below). I'm mostly leaving this because it's already written, right in some parts, and the link at the start of "Edit 2" is worth a glance to anyone interested in an ELI12 on Weber-bar style detectors. I also am much less confident know that I know roughly how a telephone works!

It's more in analogy to MiniBooNE. In that, experimenters shoot a high-energy neutrino beam through a suitable medium, recording the electromagnetic radiation from a Z-boson interaction that gives a kick to a target particle. Sometimes these are electrons that give off flashes of Cherenkov radiation; sometimes the neutrino interaction is somewhat more complicated in the radiation ultimately produced. MiniBooNE's detector is colocated with a major neutrino beam generator.

Gravitationally, ideally one would produce a beam of very small merging black hole binaries to be shot through a detector medium. Binaries throw off gravitational waves with frequency proportional to the size of the orbit; a tiny orbit means high frequency, and a local merger is interesting in other ways. If gravitons are the quanta of such HF gravitational waves, the individual graviton energies should be sufficient to scatter particles in a suitable medium (e.g., a light particle like an electron might throw off Cherenkov radiation, which we could detect). We could also rely upon the https://en.wikipedia.org/wiki/Gravitational_memory_effect to leave fingerprints in the less excited bits of the detecting medium.

Since we don't have that ideal, we are stuck with astrophysical sources of gravitational waves, and so roughly the authors' idea <https://www.nature.com/articles/s41467-024-51420-8> (more conveniently at <https://arxiv.org/abs/2308.15440>) is comparable with what the MiniBooNE experimentalists did in arriving at their particular mineral oil: some media produce a "ladder" of events from a small particle deflection to a detectable signal, and all the ladder rungs rely on discrete energy spectra (in this neutrino detection example, the oils form a natural scintillator that produces light given small movements in the medium of charged particles characteristic of certain weak current interactions; in the gravitational case the idea is to use an ultra-low-temperature condensate). Careful tuning "should" pick out monochromatic gravitons from the astrophysical menu http://www.tapir.caltech.edu/~teviet/Waves/gwave_spectrum.ht... -- then it's a matter of waiting a lonnnnnnng time for a detection.

The Weber Bar was a classical stress-strain measuring device; I don't think Weber was thinking in terms of quantum anything as much as how a shock through a suitable substance could produce electrically- or acoustically-measurable side effects, somewhat like piezoelectric crystals. Ultimately his ideas about (classical) gravitational wave detection were so misguided as to be pseudoscientific. I am not very surprised that in pop sci reporting an analogy was drawn to them rather than to the difficulties in detecting Z-bosons but I'm surprised they get much mention (Weber 1960 is cited!) in the actual paper. I'm an optimist so think it's in part because the good science Weber did was tube engineering for very early masers and lasers, and it's not hard to imagine a (backwards) connection from this paper about stimulating high amplitude coherent states of gravitational waves to grasers <https://en.wikipedia.org/wiki/Gravity_laser>.

(In reality the gravitational waves we get here from verrrrrry extragalactic binary mergers won't be effectively monochromatic, so the detection idea seems to die in Section III: you won't be able to correlate a LIGO/VIRGO-style detection with this experiment with any reliability -- the "beep!" from the device (assuming it's not error) is as likely to be from the background. But maybe there are lots and lots of highly-similar EMRIs out there building up the black hole mass hierarchy, who knows.)

Edit: paid more attention to their Table I and how they got there, and well, their smaller devices (mostly leftmost columns) are basically "build better Weber bars". I wonder if there's any hope of someone getting hands on nine tonnes of niobium and getting it to ~ 1 mK. Anyway, good luck to them, publish the low-hanging-fruit results fast!

Edit 2: https://web.mit.edu/klmitch/classes/8.224/project/resonant.h... (Re the NIOBE resonant bar §1.10 of <https://dcc-llo.ligo.org/public/0125/P1600131/001/optomechan...> (disappointingly not 9 tonnes of niobium!) among others) - but decades of experimentalists trying and no confident result...