  The ultimate miniaturization of electronic devices will likely require local
and coherent control of single electronic wavefunctions. Wavefunctions exist
within both physical real space and an abstract state space with a simple
geometric interpretation: this state space--or Hilbert space--is spanned by
mutually orthogonal state vectors corresponding to the quantized degrees of
freedom of the real-space system. Measurement of superpositions is akin to
accessing the direction of a vector in Hilbert space, determining an angle of
rotation equivalent to quantum phase. Here we show that an individual atom
inside a designed quantum corral can control this angle, producing arbitrary
coherent superpositions of spatial quantum states. Using scanning tunnelling
microscopy and nanostructures assembled atom-by-atom we demonstrate how single
spins and quantum mirages can be harnessed to image the superposition of two
electronic states. We also present a straightforward method to determine the
atom path enacting phase rotations between any desired state vectors. A single
atom thus becomes a real space handle for an abstract Hilbert space, providing
a simple technique for coherent quantum state manipulation at the spatial limit
of condensed matter.
