  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.
