Figure 4

Transmission (top) of two wave packets of different initial widths Δz (left and right) under the influence of gravity and the relative uncertainty (bottom) of such a gravimeter. Initially, the wave packet receives a momentum kick \(p_{0}\) and subsequently scatters from the matter-wave cavity (\(\sigma _{b} = {1}\) μm, \(V_{b}=1.42\times10^{-25}{\text{ J}}\), and \(d={15}\) μm). We chose the final time of numerical evolution \(t_{f}={1}{\text{ s}}\) to ensure a negligible fraction of atoms remain inside the cavity. To take into account the influence of the gravitational field g prior to scattering, we used the kinetic energy \(\mathcal{E} = \mathcal{E}_{0} - m g |z_{0}|\) at the center of the cavity as a reference, where \(\mathcal{E}_{0} = p_{0}^{2} / (2 m)\) describes the initial kinetic energy and \(z_{0}\) the initial position of the wave packet. The resonances in the transmission (top) occur for the same momenta as for momentum eigenstates, but are less prominent due to the finite width Δz of the wave packet. A similar effect is induced by gravity, so that the resonances wash out for \(g > 0\), while they are more prominent for \(g < 0\). The relative uncertainty (bottom) estimates the sensitivity of the matter-wave cavity with respect to gravity by a measurement of the fraction of transmitted atoms. Here, we omitted the term \(| m g z_{0} \partial _{\mathcal{E}} T_{R} |\) in Eq. (5) to isolate the effect of the matter-wave cavity. For small gravitational accelerations the relative uncertainty diverges. This effect is represented by white, visualizing relative uncertainties that exceed the maximum value of the colorbar. The regions of minimal uncertainty (dark blue) define the desired working points of the sensor. While \(\delta g_{R}\) denotes the relative uncertainty for an experiment with N particles and ν repetitions, we plot the quantity \(\sqrt{N}\sqrt{\nu}\delta g_{R}\) which is the single-particle uncertainty without repetitions, assuming shot-noise limited measurements with non-interacting particles