Measuring the Chern Number of a Superconducting Qubit from Nonadiabatic Response

The accumulation of Berry's phase in superconducting qubits under cyclical evolution has been well studied,\footnote{P. J. Leek \emph{et al.}, Science \textbf{318}, 5858 (2007)}$^,$\footnote{S. Berger \emph{et al.} PRA \textbf{87}, 060303(R) (2013)} typically requiring fully adiabatic evolution. We demonstrate an alternative means of accessing the topology of a qubit, based on nonadiabatic manipulation.\footnote{V. Gritsev and A. Polkovnikov, PNAS \textbf{109}, 6457 (2012)} Integrating the measured Berry curvature over the Bloch sphere yields the Chern number ($\mathrm{ch}_{\mathrm{I}}$), which we find to be quantized. By adding an effective static field to the qubit, we demonstrate a topological transition from $\mathrm{ch}_{\mathrm{I}} = 1$ to $\mathrm{ch}_{\mathrm{I}} = 0$. This simple example of extracting the Chern number may be easily scaled to larger systems.