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arXiv:gr-qc/0406096

Differing Calculations of the Response of Matter-wave Interferometers to Gravitational Waves

Achilles D. Speliotopoulos, R. Y. Chiao

Published Jun 23, 2004• 2 citations
arXivHTMLPDF

Abstract

There now exists in the literature two different expressions for the phase shift of a matter-wave interferometer caused by the passage of a gravitation wave. The first, a commonly accepted expression that was first derived in the 1970s, is based on the traditional geodesic equation of motion (EOM) for a test particle. The second, a more recently derived expression, is based on the geodesic deviation EOM. The power-law dependence on the frequency of the gravitational wave for both expressions for the phase shift is different, which indicates fundamental differences in the physics on which these calculations are based. Here we compare the two approaches by presenting a series of side-by-side calculations of the phase shift for one specific matter-wave-interferometer configuration that uses atoms as the interfering particle. By looking at the low-frequency limit of the different expressions for the phase shift obtained, we find that the phase shift calculated via the geodesic deviation EOM is correct, and the ones calculated via the geodesic EOM are not.

DOI
10.48550/arxiv.gr-qc/0406096

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There now exists in the literature two different expressions for the phase shift of a matter-wave interferometer caused by the passage of a gravitation wave. The first, a commonly accepted expression that was first derived in the 1970s, is based on the traditional geodesic equation of motion (EOM) for a test particle. The second, a more recently derived expression, is based on the geodesic deviation EOM. The power-law dependence on the frequency of the gravitational wave for both expressions for the phase shift is different, which indicates fundamental differences in the physics on which these calculations are based. Here we compare the two approaches by presenting a series of side-by-side calculations of the phase shift for one specific matter-wave-interferometer configuration that uses atoms as the interfering particle. By looking at the low-frequency limit of the different expressions for the phase shift obtained, we find that the phase shift calculated via the geodesic deviation EOM is correct, and the ones calculated via the geodesic EOM are not.
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 introduce a new theoretical framework based on Feynman diagrams to compute phase shifts in matter wave interferometry. The method allows for analytic computation of higher order quantum corrections, beyond the traditional semi-classical approximation. These additional terms depend on the finite size of the initial matter wavefunction and/or have higher order dependence on ℏ. We apply the method to compute the response of matter wave interferometers to power law potentials and potentials with an arbitrary spatial dependence. The analytic expressions are validated by comparing to numerical simulations, and estimates are provided for the scale of the quantum corrections to the phase shift response to the gravitational field of the earth, anharmonic trapping potentials, and gravitational fields from local proof masses. We also find that for certain experimentally feasible parameters, these corrections are large enough to be measured and could lead to systematic errors if they are not mitigated. We find that to first order in a spatially dependent potential, quantum corrections vanish when the initial matter wavepacket has spherical symmetry and the potential satisfies Laplace's equation. We anticipate these quantum corrections will be especially important for trapped matter wave interferometers and for free-space matter wave interferometers in the presence of proof masses. These interferometers are becoming increasingly sensitive tools for mobile inertial sensing, gravity surveying, tests of gravity and its interplay with quantum mechanics, and searches for dark energy.
Many different formalisms exist for computing the phase of a matter-wave interferometer. However, it can be challenging to develop physical intuition about what a particular interferometer is actually measuring or about whether a given classical measurement provides equivalent information. Here, we investigate the physical content of the interferometer phase through a series of thought experiments. In low-order potentials, a matter-wave interferometer with a single internal state provides the same information as a sum of position measurements of a classical test object. In high-order potentials, the interferometer phase becomes decoupled from the motion of the interferometer arms, and the phase contains information that cannot be obtained by any set of position measurements on the interferometer trajectory. This phase shift in a high-order potential fundamentally distinguishes matter-wave interferometers from classical measuring devices.
The calculation of the phase shift of a matter-wave interferometer looks different for different coordinate choices or the addition of a uniform gravitational field, but the final result of the observable phase shift does not depend on the coordinate choices or the uniform gravitational field. We find that the phase shift observed in ``Observation of the quantum equivalence principle for matter-waves'' is a magnetic recoil phase shift, similar to the recoil phase shift in light-pulse interferometers. Recoil phase shifts are inversely proportional to the inertial mass and do not depend on the gravitational mass. Our analysis highlights the difference between magnetic and gravitational acceleration in quantum mechanics.
It is often argued that the sensitivity of atom interferometer depends on the geometry of the interfering atom paths, conventionally, atom interferometer are usually configured to be sensitive to all deformations including tensor and scalar field deformations, it is a promising and robust tool for obtaining a highly sensitive and accurate measurements of gravitational signals, as such they are potentially capable of testing a wide range of fundamental physics questions including gravitational decoherence. Therefore, motivated by the recent search to improve the sensitivity of the next generation atom interferometer, we derived a broad class of equations that obeys a specific geometric configuration for the interfering paths for a 2-level atom interferometer model, and in doing so, we further analysed various configurations for the geometric interpretation, in addition to interferometric influence phase shift and a possible decoherence factor which could lead to a systematic theoretical framework for future sensing of weak forces, due to, for example, gravitational waves and light dark matter.
We present an atom interferometer that differs from common atom interferometers as it is not based on the spatial splitting of electronic wave functions, but on orienting atoms in space. As an example we present how an orientational atom interferometer based on highly charged hydrogen--like atoms is affected by gravitational waves. We show that a monochromatic gravitational wave will cause a frequency shift that scales with the binding energy of the system rather than with its physical dimension. For a gravitational wave amplitude of $h={10}^{\ensuremath{-}23}$ the frequency shift is of the order of $110 \ensuremath{\mu}\mathrm{Hz}$ for an atom interferometer based on a 919191-fold charged uranium ion. A frequency difference of this size can be resolved by current atom interferometers in 1s1 \mathrm{s}1s.