  Quantum frequency conversion (QFC) of photonic signals preserves quantum
information while simultaneously changing the signal wavelength. A common
application of QFC is to translate the wavelength of a signal compatible with
the current fiber-optic infrastructure to a shorter wavelength more compatible
with high quality single-photon detectors and optical memories. Recent work has
investigated the use of QFC to manipulate and measure specific temporal modes
(TMs) through tailoring of the pump pulses. Such a scheme holds promise for
multidimensional quantum state manipulation that is both low loss and
re-programmable on a fast time scale. We demonstrate the first QFC temporal
mode sorting system in a four-dimensional Hilbert space, achieving a conversion
efficiency and mode separability as high as 92% and 0.84, respectively. A
20-GHz pulse train is projected onto 6 different TMs, including superposition
states, and mode separability with weak coherent signals is verified via photon
counting. Such ultrafast high-dimensional photonic signals could enable
long-distance quantum communication with high rates.
