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.
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