Investigating the emergence of time in stationary states with trapped ions

Even though quantum systems in energy eigenstates do not evolve in time, they can exhibit correlations between internal degrees of freedom in such a way that one of the internal degrees of freedom behaves like a clock variable, and thereby defines an internal time, that parametrizes the evolution of the other degrees of freedom. This situation is of great interest in quantum cosmology where the invariance under reparametrization of time implies that the temporal coordinate dissapears and is replaced by the Wheeler-DeWitt constraint. Here we show that the emergent character of an internal time variable can be investigated experimentally using the exquisite control now available on moderate-size quantum systems. We describe in detail how to implement such an experimental demonstration using the spin and motional degrees of freedom of a single trapped ion.