  Quasi-periodically driven quantum systems are predicted to exhibit quantized
topological properties, in analogy with the quantized transport properties of
topological insulators. We use a single nitrogen-vacancy center in diamond to
experimentally study a synthetic quantum Hall effect with a two-tone drive. We
measure the evolution of trajectories of two quantum states, initially prepared
at nearby points in synthetic phase space. We detect the synthetic Hall effect
through the predicted overlap oscillations at a quantized fundamental frequency
proportional to the Chern number, which characterizes the topological phases of
the system. We further observe half-quantization of the Chern number at the
transition between the synthetic Hall regime and the trivial regime, and the
associated concentration of local Berry curvature in synthetic phase space. Our
work opens up the possibility of using driven qubits to design and study
higher-dimensional topological insulators and semi-metals in synthetic
dimensions.
