Image Theory for the Single Bounce Quantum Gravimeter

TL;DR

This work addresses how to interpret and optimize interference in a single-bounce quantum gravimeter by developing an image-theory framework that uses a continuous-energy basis to describe freely falling waves and a energy-dependent scattering phase to model the bounce. The approach yields a post-bounce image wave whose free-fall propagation determines the detector signal, enabling clear semi-classical and far-field descriptions of the interference pattern. The authors provide an Airy-uniform analytic model that reproduces the observed fringes, derive a simple far-field Fisher-information-based estimate for the acceleration accuracy, and analyze how the accuracy scales with experimental parameters such as flight time $T$, initial height $z_0$, and velocity spread $\sigma_v$. The results offer practical guidance for optimizing single-bounce gravimeters, including potential applications to antihydrogen and exotic species, and suggest extensions to account for non-ideal reflections and Casimir-Polder losses.

Abstract

We develop an image theory for the recently proposed single-bounce quantum gravimeter. Free fall and quantum bounce of a matter wave-packet are described through decompositions over a basis of continuous energies. This leads to a much clearer interpretation of the origin of quantum interferences, associated to semi-classical estimations. We then give new tools to explore the space of parameters, and discuss the expected accuracy of the free-fall acceleration measurement.

Figures (6)

• Figure 1: Squared modulus of the initial wave-function $\Psi_0$ (in green) and of the reflected wave-function $\Psi_1$ (in red) shown as density plots versus altitude $z$ and time $t$. The orange line represents the mirror plate at altitude $z=0$.
• Figure 2: Top plot : Square modulus $\vert{\Psi}_1(Z,T)\vert^2$ of the exact wave-function (blue solid line) and its model to be described in \ref{['sec:semiclassical']} (orange crosses), drawn versus position $Z$ at detector, with the parameters used for Fig.3 in Guyomard2025: $z_0=1mm$, $v_0 = -91.5mm\per s$, $\sigma_v = 79mm\per s$, $T=300ms$. Bottom plot: Square modulus $\vert\widetilde{\Psi}_1(v_1,0)\vert^2$ of the image wave-function (red solid line) and its model to be described in \ref{['sec:semiclassical']} (green crosses), drawn versus velocity $v_1$ at $t=0$.
• Figure 3: Classical trajectories associated with the interferometer described in \ref{['fig:sourceandimage']}. The two solid lines represent classical trajectories going from the source $z_0,t=0$ to a detection point $Z,T$. The dashed line corresponds to the limit of coincidence of these two trajectories and arrival at the branchpoint $Z^\ast\xspace$. No classical trajectory reaches the detection plate at $Z$ below the branchpoint $Z^\ast\xspace$.
• Figure 4: Variation of square root of Fisher information versus propagation time $T$, with all other parameters the same as for Fig.\ref{['fig:comparisonsquaremoduli']}. The simple prediction $\sqrt{\mathcal{I}\xspace_S}$ (solid orange line) and the exact quantity $\sqrt{\mathcal{I}\xspace_Z}$ (blue dots) agree reasonnably well, and the accuracy is improved when increasing $T$ (the red diamond corresponds to the value $T=300\,\mathrm{ms}$ chosen for Fig.\ref{['fig:comparisonsquaremoduli']}).
• Figure 5: Variation of square root of Fisher information versus initial velocity dispersion $\sigma_v$, with all other parameters the same as for Fig.\ref{['fig:comparisonsquaremoduli']}. The simple prediction $\sqrt{\mathcal{I}\xspace_S}$ (solid orange line) and the exact quantity $\sqrt{\mathcal{I}\xspace_Z}$ (blue dots) agree reasonnably well, and the accuracy is improved when increasing $\sigma_v$ (the red diamond corresponds to the value $\sigma_v = 79mm\per s$ chosen for Fig. \ref{['fig:comparisonsquaremoduli']}). The upper value for $\sigma_v$ is chosen so that the whole initial distribution corresponds essentially to one and only one bounce on the mirror, as discussed in Guyomard2025.