Topological transitions of the generalized Pancharatnam-Berry phase

Distinctfrom the dynamicalphase,in a cyclicevolution,a system’sstate may acquire an additionalcomponent,a.k.a.geometricphase.Recently, it has beendemonstrated that geometricphasescan be inducedby a sequenceof generalizedmeasurementsimplementedon a singlequbit.Furthermore, it has beenpredictedthat thesegeometricphasesmay exhibita topologicaltransitionas a functionof the measurementstrength.We demon-strate and studythis transitionexperimentallyby usingan opticalplatformwhere the qubitis representedbythe polarization of lightand the weak measurementis performedby meansof couplingwiththe spatial degreeof freedom.Our protocolcan be interpretedin termsof environment-inducedgeometricphases,whosevaluesare topologicallydeterminedby the environment-system couplingstrength.Our resultsshow that the two limitsof geometricphaseinducedby sequencesof eitherweak or projective measurementsare topologicallydistinct.