  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 (MT) and the Margolus-Levitin (ML) 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 MT limit
constrains the evolution at all times, and a second where a crossover to the ML
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.
