Optimal baseline exploitation in vertical dark-matter detectors based on atom interferometry

Several terrestrial detectors for gravitational waves and dark matter based on long-baseline atom interferometry are currently in the final planning stages or already under construction. These upcoming vertical sensors are inherently subject to gravity and thus feature gradiometer or multi-gradiometer configurations using single-photon transitions for large momentum transfer. While there has been significant progress on optimizing these experiments against detrimental noise sources and for deployment at their projected sites, finding optimal configurations that make the best use of the available resources is still an open issue. Even more, the fundamental limit of the device's sensitivity is still missing. Here, we fill this gap and show that (a) resonant-mode detectors based on multi-diamond fountain gradiometers achieve the optimal, shot-noise limited, sensitivity if their height constitutes 20% of the available baseline; (b) this limit is independent of the dark matter oscillation frequency; and (c) doubling the baseline decreases the ultimate measurement uncertainty by approximately 65%. Moreover, we propose a multi-diamond scheme with less mirror pulses where the leading-order gravitational phase contribution is suppressed and compare it to established geometries and demonstrate that both configurations saturate the same fundamental limit.