Observing photonic de Broglie waves without the maximally-path-entangled |N,0〉+|0,N〉 state

Abstract

The photonic de Broglie wave, in which an ensemble of $N$ identical photons with wavelength $\ensuremath{\lambda}$ reveals $\ensuremath{\lambda}/N$ interference fringes, has been known to be a unique feature exhibited by the photon-number-path--entangled $|N,0\ensuremath{\rangle}+|0,N\ensuremath{\rangle}$ state or the $N00N$ state. Here, we report the observation of the photonic de Broglie wave for a pair of photons, generated by spontaneous parametric down-conversion, that are not photon-number-path entangled. We also show that the photonic de Broglie wave can even be observed for a pair of photons that are completely separable (i.e., no entanglement in all degrees of freedom) and distinguishable. The experimental and theoretical results suggest that the photonic de Broglie wave is, in fact, not related to the entanglement of the photons, rather it is related to the indistinguishable pathways established by the measurement scheme. The phase sensitivity surpassing the standard quantum limit, however, is shown to be closely related to the $N00N$ state.