This contribution is an update of a previous presentation of 5D matter- wave optics and interferometry with a correction of some algebraic errors. 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 Bordé- Ramsey interferometers is given as an illustration.
This contribution is an update of a previous presentation of 5D matter- wave optics and interferometry with a correction of some algebraic errors. 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 Bordé- Ramsey interferometers is given as an illustration.
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 ve dimensions reduces both cases to 5D-optics of massless particles. The ordinary methods of optics (eikonal equation, Kirchho¤ 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, and the quantum Langevin twin paradox for atoms, are then easily explained. A general phase-shift formula for interferometers is derived in ve dimensions, which applies to clocks as well as to gravito-inertial sensors.
Relativistic models for two concepts for detecting gravitational waves with atom interferometers are presented and their features discussed. We calculate the phase difference due to a gravitational wave of an atom interferometer in a Ramsey–Bordé configuration. A general expression for the phase is presented including the case that the center of mass of the device and the center of symmetry of the matter wave trajectories do not coincide. The sensitivity of atom interferometers and laser interferometers are then compared for the detection in the low and high frequency regimes. Finally, we calculate the phase difference due to a gravitational wave of an atom gravimeter, and discuss possible strategies to solve the problem of the phase noise and of the measurement frequency.
Atom interferometry is currently developing rapidly, which is now reaching sufficient precision to motivate laboratory tests of general relativity. Thus, it is extremely significant to develop a general relativistic model for atom interferometers. In this paper, we mainly present an analytical derivation process and first give a complete vectorial expression for the relativistic interferometric phase shift in an atom interferometer. The dynamics of the interferometer are studied, where both the atoms and the light are treated relativistically. Then, an appropriate coordinate transformation for the light is performed crucially to simplify the calculation. In addition, the Bord\'e ABCD matrix combined with quantum mechanics and the ``perturbation'' approach are applied to make a methodical calculation for the total phase shift. Finally, we derive the relativistic phase shift kept up to a sensitivity of the acceleration $\ensuremath{\sim}1{0}^{\ensuremath{-}14}\text{ }\text{ }{\mathrm{m}/\mathrm{s}}^{2}$ for a $10\text{\ensuremath{-}}\mathrm{m}$-long atom interferometer.
In this paper, the quantum dynamics was obtained in the framework of the general theory of relativity, where a quantum particle is described by a distribution of matter, with amplitude functions of the matter density, in the two conjugate spaces of the spatial coordinates and of the momentum, called wave functions. For a free particle, these wave functions are conjugate wave packets in the coordinate and momentum spaces, with time-dependent phases proportional to the relativistic Lagrangian, as the wave velocities in the coordinate space are equal to the distribution velocity described by the wave packet in this space. From the wave velocities of the particle wave functions, Lorentz’s force and the Maxwell equations were obtained. From the wave/group equation in the momentum space describing the Lorentz force, the expressions of the electric and magnetic fields as functions of the electric potential conjugated to time and of the vector potential conjugated to the coordinates in the particle-field Lagrangian were obtained. With these expressions, the electric and magnetic fields that satisfy the Faraday-Maxwell law of electromagnetic induction and the two Gauss-Maxwell laws of these fields were obtained. The Ampère-Maxwell law is obtained only by taking into account the physical consistency of the matter-field interaction of the equality of the propagation field velocity with the maximum relativistic velocity c. For a quantum particle in the electromagnetic field, dynamic equations in the coordinate and momentum spaces and the particle and antiparticle wave functions were obtained. It was shown that the electromagnetic potentials as functions of the coordinates describing the matter distribution of the quantum particle do not alter this distribution – under the action of an electromagnetic a quantum particle moves as a whole. The scattering or tunneling rate in an electromagnetic field, for the two possible cases, with the spin conservation, or inversion, were obtained. This description of a quantum particle as a distribution of matter with a density amplitude/wavefunction of the form of a wave packet, with the time-dependent phase proportional to the relativistic Lagrangian as a function of the metric tensor including also the gravitational field, enables the application of this theory in quantum gravity and quantum field theory in agreement with general relativity.
In a previous paper, we showed that the Hamilton equations of motion of a quantum particle are obtained as group velocities of the wave packet describing this particle only if the time dependent phase of a particle wave is proportional to the Lagrangian, not to the Hamiltonian as in a solution of the conventional Schrodinger equation. When a Lagrangian of relativistic form is considered, the wave packet of a quantum particle takes a physical form, with a finite spectrum of a cut off velocity (c). Based on a relativistic quantum principle, asserting the invariance of the time dependent phase for an arbitrary change of coordinates, we obtained the relativistic kinematics and dynamics, the electromagnetic field equations, the spin and the electromagnetic and gravitational interactions. When the Lagrangian is considered as a function of the Hamiltonian, we obtain a Schrodinger type equation with an additional term depending on the velocity and the momentum operator. Based on this equation, we investigate the dynamics of a relativistic quantum particle. In this framework, such a particle is described as a continuous distribution of conservative matter, according to the general theory of relativity. In an electromagnetic field, any time dependent phase variation is modified with a term proportional to a vector potential conjugated to the spatial coordinates and a scalar potential conjugated to time. However, this classical description tells us nothing about the structure of the physical world. Only Quantum Mechanics tells us something about the structure of this world. Namely, that this world is composed of species of identical quantum particles. On one hand, experimentally, it has been found that these particles are of a wavy nature. On the other hand, one could find that the simplest way to define a quantum particle is by a wave packet, with momenta conjugated to coordinates and energy conjugated to time Consider a quantum particle as a wave packet, and find that the group velocities in the coordinate and momentum spaces are in agreement with the Hamilton equations only when the Lagrangian is considered in the time dependent phases, instead of the Hamiltonian in the conventional forms of these waves as solutions of the Schrodinger equation. We define a relativistic quantum principle, and derive a wave equation for a relativistic quantum particle, the relativistic kinematics and dynamics of the particle waves, the Maxwell equations and the Lorentz force of a field interacting with the particle waves, the relativistic transform of such a field, and the spin as a characteristic of the particle waves. We consider a quantum particle as a distribution of conservative matter propagating according to the General Theory of Relativity. We obtain the dynamics of this matter in a gravitational field, the propagation in plane waves perpendicular to geodesic tracks, and equations of conservation. We describe the matter dynamics as a positively defined density and show that, according to the general theory of relativity, such a distribution can be conceive only as of a fragment of matter with a finite mass equal to a mass , as a characteristic of the matter dynamics, – the matter quantization. The group velocities of the Fourier conjugate representations in the coordinate and momentum spaces describe the dynamics of a quantum particle in agreement with the Hamiltonian equations. Under the action of an external (non-gravitational) field, the acceleration of the quantum matter has two components: 1) A component perpendicular to the velocity, given by the relativistic mechanical component of the time dependent phase, and 2) A component given by the additional field terms of this phase. A free quantum particle is described by a non-dispersing wave function , contrary to the solution of the Schrodinger equation. A coherent electromagnetic field, in resonance with a system of active quantum particles in a Fabry-Perot cavity, has a wave vector approximately proportional to the metric elements, as the resonance frequency is approximately constant – a gravitational wave can be detected by the transmission characteristics of an active Fabry-Perot cavity. In a constant gravitational field, a quantum particle undertakes a velocity and an acceleration, which, at the boundary of a black hole are null – absorption and evaporation processes at the boundary of a black hole arise only by gravitational perturbations. Generally, a quantum particle is described by a time-space volume, called graviton, with a spin 2, and a distribution of a specific matter in this volume, with a half-integer spin for Fermions and an integer spin for Bosons. A graviton Lagrangian is obtained as a curvature integral on a graviton volume, and a Hamiltonian tensor is obtained for the gravitational coordinates and velocities. In a gravitational field, the time space coordinates are deformed. In such a field, any plane wave remains perpendicular geodesic, while an additional acceleration is possible in the wave plane.
Electromagnetic optical phenomena are generally understood via Maxwell’s equations for the electric and magnetic vector fields. Neutron optical phenomena are based upon de Broglie matter waves and the Schrödinger wave equation which involves a scalar wave function. Nevertheless, a description of interference effects requires the solution of a Helmholtz scalar wave equation in both cases. Gravitational, inertial, and motional effects lead to rather large phase shifts in neutron interferometry. This chapter describes how the quantum phase shift is calculated, usually by evaluating the action as an integral over the Lagrangian. A connection to relativity theory is given when the Compton frequency depending on the mass of the neutron is taken as a physical quantity. It concludes with a list of the approximately 40 neutron interferometry experiments carried out in various laboratories worldwide, which form the subject of this book.