  Qudits, with their state space of dimension d > 2, open fascinating
experimental prospects. The quantum properties of their states provide new
potentialities for quantum information, quantum contextuality, expressions of
geometric phases, facets of quantum entanglement and many other foundational
aspects of the quantum world, which are unapproachable with qubits. We here
experimentally investigate the quantum dynamics of a qudit (d = 4) that
consists of a single 3/2 nuclear spin embedded in a molecular magnet transistor
geometry, coherently driven by a microwave electric field. We propose and
implement three protocols based on a generalization of the Ramsey
interferometry to a multilevel system. First, the standard Ramsey interference
is used to measure the accumulation of geometric phases. Then, two distinct
transitions of the nuclear spin are addressed to measure the phase of an iSWAP
quantum gate. Finally, through a succession of two Hadamard gates, the
coherence time of a 3-state superposition is measured.
