  The concept of Fourier synthesis is heavily employed in both consumer
electronic products and fundamental research. In the latter, pulse shaping is
key to dynamically initialize, probe and manipulate the state of classical or
quantum systems. In nuclear magnetic resonance, for instance, shaped pulses
have a long-standing tradition and the underlying fundamental concepts have
subsequently been successfully extended to optical frequencies and even to
implement quantum gate operations. Transferring these paradigms to
nanomechanical systems requires tailored nanomechanical waveforms. Here, we
report on an additive Fourier synthesizer for nanomechanical waveforms based on
monochromatic surface acoustic waves. As a proof of concept, we electrically
synthesize four different elementary nanomechanical waveforms from a
fundamental surface acoustic wave at $ f_1 \sim 150$ MHz using a superposition
of up to three discrete harmonics $f_n$. We employ these shaped pulses to
interact with an individual sensor quantum dot and detect their deliberately
and temporally modulated strain component via the opto-mechanical quantum dot
response. Importantly, and in contrast to the direct mechanical actuation by
bulk piezoactuators, surface acoustic waves provide much higher frequencies (>
20 GHz) to resonantly drive mechanical motion. Thus, our technique uniquely
allows coherent mechanical control of localized vibronic modes of
optomechanical crystals, even in the quantum limit when cooled to the
vibrational ground state.
