qBounce: Systematic shifts of transition frequencies of gravitational states of ultra-cold neutrons using Ramsey gravity resonance spectroscopy

qBounce is using quantum states of ultra-cold neutrons in the gravitational field of the Earth to investigate gravitation in the micrometre range. We present current measurements taken in 2021 at the Institut Laue-Langevin (ILL) to determine energy differences of these states by mechanically induced transitions. This allows a determination of the local acceleration $g$ using a quantum measurement. The data presented here results in $g=9.8120(18)m\mathrm{/}{s}^2$. The classical local value at the experiment is $g_c=9.8049m\mathrm{/}{s}^2$. We present an analysis of systematic effects that induces shifts of the transition frequency of order 100 mHz. The inferred value for $g$ at the experiment shows a systematic shift of $\delta g\approx 3.9\sigma$.