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Physical Review Letters

Redshift Controversy in Atom Interferometry: Representation Dependence of the Origin of Phase Shift

Wolfgang P. Schleich, Daniel M. Greenberger, Ernst Maria Rasel

Published Jan 2, 2013• 27 citations
Publisher

Abstract

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.

DOI
10.1103/physrevlett.110.010401

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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.
Motivated by a recent claim by Müller et al (2010 Nature 463 926-9) that an atom interferometer can serve as an atom clock to measure the gravitational redshift with an unprecedented accuracy, we provide a representation-free description of the Kasevich-Chu interferometer based on operator algebra.We use this framework to show that the operator product determining the number of atoms at the exit ports of the interferometer is a c-number phase factor whose phase is the sum of only two phases: one is due to the acceleration of the phases of the laser pulses and the other one is due to the acceleration of the atom.This formulation brings out most clearly that this interferometer is an accelerometer or a gravimeter.Moreover, we point out that in different representations of quantum mechanics such as the position or the momentum representation the phase shift appears as though it originates from different physical phenomena.Due to this representation dependence conclusions concerning an enhanced accuracy derived in a specific representation are unfounded.
In recent years there has been enormous progress in matter wave interferometry. The Colella-Overhauser-Werner (COW) type of neutron interferometer and the Kasevich-Chu (K-C) atom interferometer are the prototypes of such devices and the issue of whether they are sensitive to relativistic effects has recently aroused much controversy. We examine the question as to what extent the gravitational redshift and the related twin paradox effect can be seen in both of these atom and neutron interferometers. We point out an asymmetry between the two types of devices. Because of this, the nonvanishing, nonrelativistic residue of both effects can be seen in the neutron interferometer, while in the K-C interferometer the effects cancel out, leaving no residue, although they could be present in other types of atom interferometers. Also, the necessary shifting of the laser frequency (chirping) in the atom interferometer effectively changes the laboratory into a free-fall system, which could be exploited for other experiments.
The computation of the phase shift in a symmetric atom interferometer in the presence of a gravitational field is reviewed. The difference of action-phase integrals between the two paths of the interferometer is zero for any Lagrangian which is at most quadratic in position and velocity. We emphasize that in a large class of theories of gravity the atom interferometer permits a test of the weak version of the equivalence principle (or universality of free fall) by comparing the acceleration of atoms with that of ordinary bodies, but is insensitive to that aspect of the equivalence principle known as the gravitational redshift or universality of clock rates.
Atom interferometers allow the measurement of the acceleration of freely falling atoms with respect to an experimental platform at rest on Earth's surface. Such experiments have been used to test the universality of free fall by comparing the acceleration of the atoms to that of a classical freely falling object. In a recent paper, Müller, Peters and Chu [Nature {\bf 463}, 926-929 (2010)] argued that atom interferometers also provide a very accurate test of the gravitational redshift when considering the atom as a clock operating at the Compton frequency associated with the rest mass. We analyze this claim in the frame of general relativity and of different alternative theories. We show that the difference of "Compton phases" between the two paths of the interferometer is actually zero in a large class of theories, including general relativity, all metric theories of gravity, most non-metric theories and most theoretical frameworks used to interpret the violations of the equivalence principle. Therefore, in most plausible theoretical frameworks, there is no redshift effect and atom interferometers only test the universality of free fall. We also show that frameworks in which atom interferometers would test the redshift pose serious problems, such as (i) violation of the Schiff conjecture, (ii) violation of the Feynman path integral formulation of quantum mechanics and of the principle of least action for matter waves, (iii) violation of energy conservation, and more generally (iv) violation of the particle-wave duality in quantum mechanics. Standard quantum mechanics is no longer valid in such frameworks, so that a consistent interpretation of the experiment would require an alternative formulation of quantum mechanics. As such an alternative has not been proposed to date, we conclude that the interpretation of atom interferometers as testing the gravitational redshift is unsound.
From the principle of equivalence, Einstein predicted that clocks slow down in a gravitational field. Since the general theory of relativity is based on the principle of equivalence, it is essential to test this prediction accurately. Muller, Peters and Chu claim that a reinterpretation of decade old experiments with atom interferometers leads to a sensitive test of this gravitational redshift effect at the Compton frequency. Wolf et al dispute this claim and adduce arguments against it. In this article, we distill these arguments to a single fundamental objection: an atom is NOT a clock ticking at the Compton frequency. We conclude that atom interferometry experiments conducted to date do not yield such sensitive tests of the gravitational redshift. Finally, we suggest a new interferometric experiment to measure the gravitational redshift, which realises a quantum version of the classical clock "paradox".