We first give a short historical introduction to the fundamentals of atom interferometry with internal excitation, based on density matrix diagrams and a Liouville space approach. Then, a new framework is proposed to compare and unify photon and atom optics, which rests on the quantization of proper time. A common wave equation written in five dimensions reduces both cases to 5D optics of massless particles. The ordinary methods of optics (eikonal equation, Kirchhoff integral, Lagrange invariant, Fermat principle, symplectic algebra and ABCD matrices…) are used to solve this equation in practical cases. The various phase shift cancellations, which occur in atom interferometers, are then easily explained. A general phase-shift formula for interferometers is derived in five dimensions, which applies to clocks as well as to gravito-inertial sensors. This contribution is an update of a previous presentation of 5D matter-wave optics and interferometry. Electromagnetic interactions are explicitly added in the 5D metric tensor in complete analogy with Kaluza's work. The 5D Lagrangian is rederived and an expression for the Hamiltonian suitable for the parabolic approximation is presented. The corresponding equations of motion are also given. The 5D action is shown to cancel for the actual trajectory which is a null geodesics of the 5D metric. This presentation is mainly devoted to the classical aspects of the theory and only general consequences for the quantum phase of matter-waves are outlined. The application to Borde-Ramsey interferometers is given as an illustration.
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