  Steering a quantum harmonic oscillator state along cyclic trajectories leads
to a path-dependent geometric phase. Here we describe an experiment observing
this geometric phase in an electronic harmonic oscillator. We use a
superconducting qubit as a non-linear probe of the phase, otherwise
unobservable due to the linearity of the oscillator. Our results demonstrate
that the geometric phase is, for a variety of cyclic trajectories, proportional
to the area enclosed in the quadrature plane. At the transition to the
non-adiabatic regime, we study corrections to the phase and dephasing of the
qubit caused by qubit-resonator entanglement. The demonstrated controllability
makes our system a versatile tool to study adiabatic and non-adiabatic
geometric phases in open quantum systems and to investigate the potential of
geometric gates for quantum information processing.
