Generation of four-photon hyperentangled state using spontaneous parametric down-conversion source with the second-order term

摘要

Nowadays, the generation of multiphoton entangled states is almost realized by combining the coupled entangled photons emitted from spontaneous parametric down-conversion (SPDC) with the first-order term. In this case, one may focus mainly on the first-order term, and then avoid multipair emission events by restricting experimental parameters. On the other hand, for the higher-order terms in SPDC source, these emitted entangled photons have interesting features. For example, they are entangled maximally not only in photon number for the spatial modes, but also in polarization degree of freedom. In general, two photons, which are entangled in two or more degrees of freedom, are called hyperentangled pair of photons or hyperentangled state. We present a scheme to generate the four-photon hyperentangled state based on four indistinguishable photons emitted from SPDC source with the second-order term. Consider two SPDC sources with equal probability of emission of photons in respective spatial modes. With the passive linear optical devices, i.e., beam splitters, half wave plates, polarizing beam splitters, etc., under the condition of registering a specified four-photon coincidence, we can obtain the four-photon hyperentangled state in which the photons are entangled in both polarization and spatial-mode degrees of freedom. Here, of course, for an arbitrary fourfold coincidence detection, one obtains a canonical four-photon Greenberger-Horne-Zeilinger (GHZ) state. Then we show the results of fourfold coincidence detections and the corresponding probabilities for the four-photon GHZ states, where the generation of the four-photon hyperentangled state is included as long as we are not to distinguish the two detectors located at the same locations. As a result, our scheme has two notable features. When we only consider the second-order emission, since it is not needed for us to distinguish between the two SPDC sources, the present scheme is simple and feasible. Also, based on the postselection with fourfold coincidence detection, our scheme is suitable for the normal first-order emission where we restrict the four photons emitted from the same source. In this sense, our scheme is efficient. In a word, we describe a method to generate the four-photon hyperentangled state with the second-order emission in SPDC source, which may contribute to the exploration of multipair entanglement with higher-order emissions from the SPDC source.