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Journal de Physique II

Atom interferometry with arbitrary laser configurations : exact phase shift for potentials including inertia and gravitation

Jürgen Audretsch, Karl-Peter Marzlin

Published Nov 1, 1994• 32 citations
Publisher

Abstract

We study in a general treatment the influence of a class of perturbations acting on the atoms of an atomic interferometer. An exact expression for the resulting shift of the interference pattern is given for an arbitrary number of laser zones with travelling or standing waves. The assumption made is that the two-level model is appropriate for the description of atoms and that the perturbations can be neglected in the laser zones. As an important application we calculate the influence acceleration, rotation, and space-time curvature on the Ramsey interferometer and the atomic fountain. The measurability of the respective phase shifts is discussed.

DOI
10.1051/jp2:1994248

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We study in a general treatment the influence of a class of perturbations acting on the atoms of an atomic interferometer. An exact expression for the resulting shift of the interference pattern is given for an arbitrary number of laser zones with travelling or standing waves. The assumption made is that the two-level model is appropriate for the description of atoms and that the perturbations can be neglected in the laser zones. As an important application we calculate the influence acceleration, rotation, and space-time curvature on the Ramsey interferometer and the atomic fountain. The measurability of the respective phase shifts is discussed.
The interaction of an atomic beam with four traveling-wave laser beams leads to optically induced Ramsey fringes in the atomic beam. This experimental setup has been used as an optical frequency standard as well as a matter-wave interferometer. We calculate for two-level atoms the shift of the fringes due to the influence of fairly general external forces in lowest order of a perturbation approach. A comparison with the phase shift found in a corresponding WKB treatment of the interaction geometry is made. The consequences of beam reversal and cooling of the atomic beam are studied. As an application, we treat the linear acceleration and rotation of the interferometer and discuss the influence of space-time curvature.
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.
We demonstrate that for atom interferometers with lasers serving as beam splitters state independence (including freedom from Doppler effects) of the phase shift is obtained if the paths of the two interfering beams are closed and have obtained the same momentum transfer. This is also the case when the center-of-mass motion of the atoms is subject to a constant force field (acceleration). Among these interferometers the one with the simplest geometry is deduced for which the phase shift does not depend on acceleration. Its remaining sensitivity to rotation is discussed. Finally the geometry of a state-independent interferometer that is practically insensitive to both acceleration and rotation is worked out. As an application its relevance for the detection of the influence of space-time curvature on the atomic motion is studied. \textcopyright{} 1996 The American Physical Society.
The influence of space-time curvature on quantum matter which can be theoretically described by covariant wave equations has not been experimentally established yet. In this paper we analyze in detail the suitability of the Ramsey atom beam interferometer for the measurement of the phase shift caused by the Riemannian curvature of the Earth or alternatively of two parallel oriented lead blocks. It appears that for the lead blocks the detection should be possible with realistic modifications of existing devices within the near future. For the Earth's gravitational field the experimental difficulties are too big. The paper is divided into two parts. The first one is concerned with the derivation of general relativistic correction terms to the Pauli equation starting from the fully covariant Dirac equation and their physical interpretation. The inertial effects of acceleration and rotation are included. The calculation makes use of Fermi coordinates. In the second part we calculate the shift of the Ramsey fringes for the two different sources of curvature and examine various possibilities to enlarge the sensitivity of the apparatus to space-time curvature. Since the two parts may be more or less interesting for physicists with different research fields they are written in such a way that each one may be read without much reference to the other one.
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 ABCDABCDABCD 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.