Figure 1

Gravitationally distorted matter-wave cavity and wave packet (a) prior to scattering and (b) after scattering. The matter-wave cavity consists of two Gaussian barriers with height \(V_{b}\) and width \(\sigma _{b}\), at positions \(z = z_{\pm}\), chosen such that the overlap between both barriers is negligible for vanishing cavity length d. (a) The initial wave packet \(\psi _{0}\) is located at \(z_{0}\) and has an initial momentum \(p_{0}\) that corresponds to the kinetic energy \(\mathcal{E}_{0} = p_{0}^{2} / (2 m)\). The gravitational field disturbs the propagation of the wave packet and the matter-wave cavity. To account for the influence of the gravitational field g, we take the kinetic energy \(\mathcal{E} = \mathcal{E}_{0} - m g |z_{0}|\) at the center of the matter-wave cavity as reference. (b) The initial wave packet scatters from the matter-wave cavity, resulting in a superposition of reflected and transmitted wave packet \(|\psi _{L}|^{2}\) and \(|\psi _{R}|^{2}\). To obtain the number of transmitted atoms, we introduce the operator \(\hat{P}_{R}^{2} = \hat{P}_{R}\) that projects on the region to the right of the cavity (shaded in red)