  The synthetic dimension opens new horizons in quantum physics and topological
photonics by enabling new dimensions for field and particle manipulations. The
most appealing property of the photonic synthetic dimension is its ability to
emulate high-dimensional optical behavior in a unitary physical system. Here we
show that the photonic synthetic dimension can transform technical problems in
photonic systems between dimensionalities, providing unexpected solutions to
technical problems that are otherwise challenging. Specifically, we propose and
experimentally demonstrate a photonic Galton board (PGB) in the temporal
synthetic dimension, in which the temporal high-speed challenge is converted
into a spatial fiber-optic length matching problem, leading to the experimental
generation of tera-sample-per-second arbitrary waveforms. Limited by the speed
of the measurement equipment, waveforms with sampling rates of up to 341.53
GSa/s are recorded. Our proposed PGB operating in the temporal synthetic
dimension breaks the speed limit in a physical system, bringing arbitrary
waveform generation into the terahertz regime. The concept of dimension
conversion offers possible solutions to various physical dimension-related
problems, such as super-resolution imaging, high-resolution spectroscopy, time
measurement, etc.
