Mass-energy equivalence in harmonically trapped particles

Precise understanding of the dynamics of trapped particles is crucial for nascent quantum technologies, including atomic clocks and quantum simulators. Here we present a framework to systematically include quantum effects arising from the mass-energy equivalence in harmonically trapped systems. We find that the mass-energy equivalence leads to squeezing, displacement and frequency changes of harmonic modes associated with different internal energies. The framework predicts new phenomena, notably, the existence of a lower bound to the fractional frequency shift in atomic clocks arising from the interplay between gravitational effects and so-called time dilation shifts. Analogous effects will arise in other trapping potentials, especially in periodic lattices, and may play a role in correlation dynamics and thermalisation process in many-body systems and cold gases.