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Gravitationally induced quantum transitions

In this letter, we calculate the probability for resonantly induced transitions in quantum states due to time dependent gravitational perturbations. Contrary to common wisdom, the probability of inducing transitions is not infinitesimally small. We consider a system of ultra cold neutrons (UCN), which are organized according to the energy levels of the Schrödinger equation in the presence of the earth's gravitational field. Transitions between energy levels are induced by an oscillating driving force of frequency $ω$. The driving force is created by oscillating a macroscopic mass in the neighbourhood of the system of neutrons. The neutrons decay in 880 seconds while the probability of transitions increase as $t^2$. Hence the optimal strategy is to drive the system for 2 lifetimes. The transition amplitude then is of the order of $1.06\times 10^{-5}$ hence with a million ultra cold neutrons, one should be able to observe transitions.