# Methods

In various experiments within the qBounce project, two essential methods to investigate gravitationally bound quantum states of ultra-cold neutrons were developed. The first one enables us to map the probability distribution of the „Quantum Bouncing Ball“ and with the second, the so-called gravity-resonance-spectroscopy-method, we are able to probe the eigenenergies of bound states in the gravitational potential.

## Gravity-resonance-spectroscopy

- State preparation

The first step to perform experiments in the quantum regime is to prepare the system of interest in a well-defined state. At qBounce, we prepare the system in the lower bound states by sending a beam of ultra-cold neutrons through a system of two neutron mirrors.

They are arranged on top of each other with a distance of ca. 30 µm and an upper, rough surface. Neutrons in high energetic states are scattered out of the system, thus only the lowest states are populated. - State transition

To excite neutrons in higher states, the neutron mirrors are vibrated with a frequency ω_{pq}. Because of the vibrating surfaces, the boundary conditions of the system change periodically so that the Hamiltonian is time dependent and allows state transitions. By applying ω_{13}one can drive transitions from state |1> to state |3> and thus effectively measure the energy difference between the two states by measuring the applied frequency. - State analysis

By repetition of region I, the occupation of state |1> may be analyzed. - Time-resolved neutron detection

To check if a transition frequency is met one measures the population of state |1>. If the applied frequency was resonant, there should be a drop in transmission. So one only has to measure the neutron flow for different applied frequencies. This is realized by a proportional counter, that works by converting neutrons to ionizing radiation by a boron coated aluminum foil and intensify the signal by collisions with gas behind the foil.

## Quantum Bouncing Ball with neutrons

- Velocity selection (not in picture 3)

Neutrons with a too high velocity are blocked from the experiment by an aperture system so that the horizontal velocity distribution is restricted to a convenient region. - State preparation

Two neutron mirrors prepare the neutrons in a superposition of the lowest states. This happens similar to the method that was used for the gravity-resonance-spectroscopy. - The prepared superposition passes a step at the transition to region [III] of several ten microns and evolves further on in time.
- Spatially resolved neutron detection

To map the probability distribution of the „Quantum Bouncing Ball“, track detectors of Boron-coated CR-39 plastics were used. Ultra-cold neutrons hitting the boron layer initiate a nuclear reaction that leads to two decay products. One of those enters the plastics and leaves a track of defects, which can be enlarged by etching to the point of easy visibility.

[1] J. Felber, R. Gähler, and C. Rausch, *Matter waves at a vibrating surface: Transition from quantum-mechanical to classical behavior*, Physical Review A Vol. 53 Nr. 1

[2] Tobias Jenke, *qBounce - vom Quantum Bouncer zur Gravitationsresonanzspektroskopie*, Dissertation (2011)