  Quantum mechanics sets fundamental limits on how fast quantum states can be
transformed in time. Two well-known quantum speed limits are the
Mandelstam-Tamm and the Margolus-Levitin bounds, which relate the maximum speed
of evolution to the system's energy uncertainty and mean energy, respectively.
Here, we test concurrently both limits in a multi-level system by following the
motion of a single atom in an optical trap using fast matter wave
interferometry. Our data reveal two different regimes: one where the
Mandelstam-Tamm limit constrains the evolution at all times, and a second where
a crossover to the Margolus-Levitin limit is manifested at longer times. We
take a geometric approach to quantify the deviation from the speed limit,
measuring how much the matter wave's quantum evolution deviates from the
geodesic path in the Hilbert space of the multi-level system. Our results,
establishing quantum speed limits beyond the simple two-level system, are
important to understand the ultimate performance of quantum computing devices
and related advanced quantum technologies.
