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Journal de Physique II

The Feynman path integral approach to atomic interferometry. A tutorial

Pippa Storey, Claude N. Cohen-Tannoudji

Published Nov 1, 1994• 254 citations
Publisher

Abstract

Many problems of current interest in atomic interferometry lend themselves to a path integral treatment. We present a practical guide to solving such problems, taking as examples the gravitational experiments of Kasevich and Chu, and the atomic equivalents of the Sagnac and Aharonov-Bohm effects.
DOI
10.1051/jp2:1994103

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Abstract Among the various areas of neutron optical research, neutron interferometry has provided some of the most challenging and spectacular developments. In most cases, separated coherent beams are used which are produced either by wavefront division (Young’s type) or by amplitude division (Mach− Zehnder type). These beams are subsequently coherently superposed after passing through regions of space where the neutron wavefunction is modified in phase and amplitude by various interactions: nuclear, magnetic, electromagnetic, gravitational, or by geometry and topology (as in Aharonov−Bohm-type experiments). Various types of neutron interferometers that have been tested are shown in Fig. 2.1. The first neutron interferometer was based upon wavefront division and biprism deflection, but it had several constraints for interferometric applications mainly due to the small beam separation (100 μm) (Maier-Leibnitz and Springer 1962). Nevertheless, this method gained further interest for diffraction experiments from macroscopic objects and for the detection of weak interaction effects where the long flight paths (10 m) enhance the sensitivity for the measurements of these effects (Gähler et al. 1980; Klein et al. 1981a). The perfect-crystal interferometer (Rauch et al. 1974) provides widely separated coherent beams and has become a standard method for advanced neutron optical investigations. The development of diffraction and phase grating interferometers extended the Mach-Zehnder interferometer method to very slow neutrons (loffe et al. 1985; Gruber et al. 1989). The superposition of spin-up and spin-down states in the longitudinal direction provides the basis for Larmor and Ramsey interferometry, where no lateral beam splitting is necessary. The well-known spin-echo systems (Mezei 1972) and zero-field spin-echo systems (Gähler and Golub 1987; Dubbers et al. 1989) are examples of such interferometers, which also play an important role in atom interferometry (Bord é 1989; Kasevich and Chu 1991; Riehle et al. 1991).
Motivated by the recent debate on whether the Kasevich-Chu atom interferometer can measure the gravitational redshift, we show that in different representations of quantum mechanics chosen for the calculation, the observed phase shift appears as though it originates from different physical phenomena. In particular, we demonstrate that the decomposition of the total phase shift into three dynamical phases, which emerges in a semiclassical approach and is at the very heart of the redshift controversy, does not appear in an exact treatment based on a representation-free analysis. Here only two phenomena determine the phase shift: the difference of the laser phases and the acceleration of the atom. Hence, the Kasevich-Chu interferometer is an accelerometer or gravimeter.
The Schrodinger equation is not covariant. Nevertheless, quantum field theory is often formulated using the Schrodinger equation to describe the time evolution of the system, which is equivalent to using Feynman path integrals. It is well known that scattering theory gives covariant results provided that the interaction vanishes in the asymptotic limit of t→±∞t \rightarrow \pm \inftyt→±∞, but there are other situations of interest that do not satisfy those conditions. As an example, the Aharonov-Bohm effect is derived here using second-quantized field theory to describe all the electrons as well as the electromagnetic field, which allows the effects of retarded vector potentials to be calculated in a straightforward way using the Feynman propagator. The results are in agreement with the usual expression for the Aharonov-Bohm effect when the retardation of the electromagnetic field is negligible, but they predict a fractional phase shift, a lack of covariance, and violations of causality when retardation effects are significant. These results suggest that quantum field theory based on the Schrodinger equation or Feynman path integrals is incomplete in the sense that it cannot give a correct description of all observable phenomena.