Topological transitions of the generalized Pancharatnam-Berry phase

Distinct from the dynamical phase, in a cyclic evolution, a system’s state may acquire an additional component, a.k.a. geometric phase. Recently, it has been demonstrated that geometric phases can be induced by a sequence of generalized measurements implemented on a single qubit. Furthermore, it has been predicted that such geometric phases may exhibit a topological transition as a function of the measurement strength. We demonstrate and study this transition experimentally by employing an optical platform where the qubit is represented by the polarisation of light and the weak measurement is performed by means of coupling with the spatial degree of freedom. Our protocol can be interpreted in terms of environment-induced geometric phases, whose values are topologically determined by the environment-system coupling strength. Our results show that the two limits of geometric phase induced by either sequences of either weak or projective measurements are topologically distinct.Â