  State-of-the-art atom interferometers can keep atoms in a superposition of
heights in Earth's gravitational field for times reaching minute-scale,
allowing for precise measurements of the gravitational potential. Yet, the
phase shifts measured in such experiments can always be explained with a
non-relativistic theory of gravity. There is therefore growing interest in
finding feasible ways to use such new experimental capabilities to go beyond
the non-relativistic regime. Here we propose modifying the existing
experimental setups to probe both the quantum and the general relativistic
effects on the atom's dynamics. Our proposal consists of adding two additional
laser pulses in a trapped atom interferometer that would set up a quantum clock
trapped at a superposition of heights reading a quantum superposition of
relativistic proper times. We derive the phases acquired by different
trajectories in the interferometer and demonstrate that the effect of
superposition of proper times would manifest itself in two ways in the
interference pattern: as visibility modulations, and as a shift of the resonant
frequency of the atom. We argue that the latter might be observable with
current technology.
