Complementarity relationship between coherence and path distinguishability in an interferometer based on induced coherence

We consider an interferometer based on the concept of induced coherence, where two photons that originate in different second-order nonlinear crystals can interfere. We derive a complementarity relationship that links the first-order coherence between the two interfering photons with a parameter that quantifies the distinguishing information regarding the nonlinear crystal where they originated. We show that the derived relationship goes beyond the single-photon regime and is valid for any photon-flux rate generated. We report experimental results in the low photon-flux regime that confirm the validity of the derived complementarity relationship.