Quantum time and the time-dilation induced interaction transfer mechanism

Given a bipartite quantum system in an energy eigenstate, the dynamical description for one component can be derived via entanglement using the other component as a clock. This is the essence of the Page and Wootters mechanism. Moreover, if the clock is subject to a gravitational-like interaction, relative time evolution is then described by a time-dilated Schrödinger equation, in which the so-called redshift operator describes a purely quantum effect, analog to gravitational time-dilation. Here we adopt a nonperturbative approach and present a finite-dimensional generalization of this mechanism, expressing the quantum time-dilation effect as an effective interaction involving previously noninteracting system components. We name this a time-dilation induced interaction transfer (TiDIT) mechanism and discuss an example using two coupled spins as a quantum clock model. Our approach is suitable for implementations in current quantum technology and provides a new tool for exploring gravity at the intersection with quantum physics.