  The Bloch sphere is a generic picture describing a coupled two-level system
and the coherent dynamics of its superposition states under control of
electromagnetic fields. It is commonly employed to visualise a broad variety of
phenomena ranging from spin ensembles and atoms to quantum dots and
superconducting circuits. The underlying Bloch equations describe the state
evolution of the two-level system and allow characterising both energy and
phase relaxation processes in a simple yet powerful manner.
  Here we demonstrate the realisation of a nanomechanical two-level system
which is driven by radio frequency signals. It allows to extend the above Bloch
sphere formalism to nanoelectromechanical systems. Our realisation is based on
the two orthogonal fundamental flexural modes of a high quality factor
nanostring resonator which are strongly coupled by a dielectric gradient field.
Full Bloch sphere control is demonstrated via Rabi, Ramsey and Hahn echo
experiments. This allows manipulating the classical superposition state of the
coupled modes in amplitude and phase and enables deep insight into the
decoherence mechanisms of nanomechanical systems. We have determined the energy
relaxation time T1 and phase relaxation times T2 and T2*, and find them all to
be equal. This not only indicates that energy relaxation is the dominating
source of decoherence, but also demonstrates that reversible dephasing
processes are negligible in such collective mechanical modes. We thus conclude
that not only T1 but also T2 can be increased by engineering larger mechanical
quality factors. After a series of ground-breaking experiments on ground state
cooling and non-classical signatures of nanomechanical resonators in recent
years, this is of particular interest in the context of quantum information
processing.
