  A central goal of condensed-matter physics is to understand how the diverse
electronic and optical properties of crystalline materials emerge from the
wavelike motion of electrons through periodically arranged atoms. However, more
than 90 years after Bloch derived the functional forms of electronic waves in
crystals [1] (now known as Bloch wavefunctions), rapid scattering processes
have so far prevented their direct experimental reconstruction. In high-order
sideband generation [2-9], electrons and holes generated in semiconductors by a
near-infrared laser are accelerated to a high kinetic energy by a strong
terahertz field, and recollide to emit near-infrared sidebands before they are
scattered. Here we reconstruct the Bloch wavefunctions of two types of hole in
gallium arsenide at wavelengths much longer than the spacing between atoms by
experimentally measuring sideband polarizations and introducing an elegant
theory that ties those polarizations to quantum interference between different
recollision pathways. These Bloch wavefunctions are compactly visualized on the
surface of a sphere. High-order sideband generation can, in principle, be
observed from any direct-gap semiconductor or insulator. We thus expect that
the method introduced here can be used to reconstruct low-energy Bloch
wavefunctions in many of these materials, enabling important insights into the
origin and engineering of the electronic and optical properties of condensed
matter.
