Wheeler's delayed-choice experiment highlights strange features of quantum theory such as pre-sensing of the experimental setup by the quantum object and the role of time. A recent proposal for such an experiment with an interferometer having a quantum beam splitter (QBS) [R. Ionicioiu and D. R. Terno, Phys. Rev. Lett. 107, 230406 (2011)] and its subsequent experimental implementations through photonics and NMR have produced results including the modification in the concept of complementarity. Here we propose a matter-wave Mach-Zehnder-Bragg cavity-QED interferometric setup with final QBS engineered through a cavity field that is taken initially in the superposition of zero and one photon. The setup operates through first-order off-resonant Bragg diffraction of the neutral atoms from the cavity fields with the matter wave's particle (wave) nature marked through the absence (presence) of a photon in the final cavity. The proposal, addressing the issue through atomic de Broglie waves, can be executed within the present cavity-QED experimental scenario with appreciable success probability and fidelity.

The conventional teleportation protocol requires a state entangled in only one degree of freedom (DOF), while hyperteleportation requires more than single DOF to complete the task. The hyperteleportation schematics are demonstrated only for the photonic systems, where in the present paper we extend the idea to a hyperteleportation protocol involving the atomic internal and external states. The protocol is deterministically engineered through resonant and off-resonant atomic Bragg diffraction involving two-level neutral atoms under standard cavity-QED working environment. Moreover, the longer interaction time Bragg’s regime with well separated transverse momenta states as an output of the neutral atoms guarantees the high enough engineering fidelities with reduced decoherence rates. The experimental parameters for the demonstration of the proposed scheme are also elucidated briefly describing the optimistic feasibility for the experimental execution of the proposed schematics.