  The direct measurement of Berry phases is still a great challenge in
condensed matter systems. The bottleneck has been the ability to adiabatically
drive an electron coherently across a large portion of the Brillouin zone in a
solid where the scattering is strong and complicated. We break through this
bottleneck and show that high-order sideband generation (HSG) in semiconductors
is intimately affected by Berry phases. Electron-hole recollisions and HSG
occur when a near-band gap laser beam excites a semiconductor that is driven by
sufficiently strong terahertz (THz)-frequency electric fields. We carried out
experimental and theoretical studies of HSG from three GaAs/AlGaAs quantum
wells. The observed HSG spectra contain sidebands up to the 90th order, to our
knowledge the highest-order optical nonlinearity observed in solids. The
highest-order sidebands are associated with electron-hole pairs driven
coherently across roughly 10% of the Brillouin zone around the \Gamma point.
The principal experimental claim is a dynamical birefringence: the sidebands,
when the order is high enough (> 20), are usually stronger when the exciting
near-infrared (NIR) and the THz electric fields are polarized perpendicular
than parallel; the sideband intensities depend on the angles between the THz
field and the crystal axes in samples with sufficiently weak quenched disorder;
and the sidebands exhibit significant ellipticity that increases with
increasing sideband order, despite nearly linear excitation and driving fields.
We explain dynamical birefringence by generalizing the three-step model for
high order harmonic generation. The hole accumulates Berry phases due to
variation of its internal state as the quasi-momentum changes under the THz
field. Dynamical birefringence arises from quantum interference between
time-reversed pairs of electron-hole recollision pathways.
