A note on spontaneous emission of gravitons by a quantum bouncer

Spontaneous emission of graviton rates for the quantum bouncer states are evaluated. The quantum problem of a ball bouncing above a ideal mirror was considered a long time ago, however just as a mere exercise of elementary quantum mechanics. Things did change since the quantization of energy of Ultra Cold Neutrons (UCN) bouncing above a mirror in the Earth’s gravitational field had been demonstrated in an experiment performed at the Institute Laue Langevin (ILL) [1, 2, 3]. However, this effect does not demonstrate any quantum behavior of the gravitational field itself. In analogy with electrodynamics, the observation of spectral lines in atoms shows the quantum behavior of electrons, but does not provide any clue concerning the possible quantization of the electromagnetic field. What does provide a clue is the observation of spontaneous decay of an excited state, for instance, which can only be explained in terms of photon emission. So the observation of spontaneous decay of an excited state in the ILL experiment would be of interest, since it would be a Planck-scale physics effect. Nevertheless, the decay rate is expected to be low, and the purpose of this letter is to estimate it. First we will set notations of the quantum bouncer problem, focusing on its physical implementation, that is, neutrons falling on the Earth’s gravitational field above a perfect mirror. We then derive the spontaneous emission rate in a semi-classical approach. The stationary Schrödinger equation for the vertical motion (z axis) of the mass m quantum bouncer is: