  Anticipated breakthroughs in solid-state quantum computing will rely on
achieving unprecedented control over the wave-like states of electrons in
crystalline materials. For example, an international effort to build a quantum
computer that is topologically protected from decoherence is focusing on
carefully engineering the wave-like states of electrons in hybrid devices that
proximatize an elemental superconductor and a semiconductor nanostructure[1-6].
However, more than 90 years after Bloch derived the functional forms of
electronic waves in crystals[7](now known as Bloch wavefunction) rapid
scattering processes have so far prevented their direct experimental
reconstruction, even in bulk materials. In high-order sideband generation
(HSG)[8-15], electrons and holes generated in semiconductors by a near-infrared
(NIR) laser are accelerated to high kinetic energy by a strong terahertz field,
and recollide to emit NIR sidebands before they are scattered. Here we
reconstruct the Bloch wavefunctions of two types of holes in gallium arsenide
by experimentally measuring sideband polarizations and introducing an elegant
theory that ties those polarizations to quantum interference between different
recollision pathways. Because HSG can, in principle, be observed from any
direct-gap semiconductor or insulator, we expect the method introduced in this
Article can be used to reconstruct Bloch wavefunctions in a large class of bulk
and nanostructured materials, accelerating the development of
topologically-protected quantum computers as well as other important electronic
and optical technologies.
