Integration of the Berry curvature on a qubit state manifold by coupling to a quantum meter system

We present a scheme that allows integration of the Berry curvature and thus determination of the Chern number
of a qubit eigenstate manifold. Our proposal continuously couples the qubit with a meter system while it explores
a quasiadiabatic path in the manifold. The accumulated change of one of the meter observables then provides
an estimate of the Chern number. By varying the initial state of the meter, we explore the delicate interplay
between the measurement precision and the disturbance of the qubit. A simple argument yields a correction factor
that allows estimation of the Chern number, even when the qubit is significantly disturbed during the probing.
The Chern number arises from the geometric phase accumulated during the exploration, while we observe the
dynamic phase to produce a broadening of the meter wave function. We show that a protocol, relying on three
subsequent explorations, allows cancellation of the dynamic phase while the geometric phase is retained.