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Physical review. D/Physical review. D.

Relativistic effects in atom gravimeters

Yu-Jie Tan, Cheng-Gang Shao, Zhong-Kun Hu

Published Jan 4, 2017• 18 citations
Publisher

Abstract

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.

DOI
10.1103/physrevd.95.024002

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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∼10\ensuremath−14  m/s2\ensuremath{\sim}1{0}^{\ensuremath{-}14}\text{ }\text{ }{\mathrm{m}/\mathrm{s}}^{2}\ensuremath∼10\ensuremath−14  m/s2 for a 10\ensuremath-m10\text{\ensuremath{-}}\mathrm{m}10\ensuremath-m-long atom interferometer.
Atom interferometry is a sensitive tool for measuring relativistic effects, but there are no known non-trivial exact solutions for relativistic atom interferometer phase shifts. Here we derive relativistically exact expressions within the usual semiclassical approximation for a wide range of experimentally interesting atom interferometer pulse sequences in flat spacetime, including Mach-Zehnder, resonant, and large momentum transfer interferometer geometries. As an example, the leading order phase shift ωagT2/cω_a g T^2/cωa​gT2/c for a Mach-Zehnder clock atom interferometer is found to become ωa(1+ωa2m)(e−gT/c−1)2c/gω_a \left(1 + \frac{ω_a}{2m}\right)(e^{-gT/c}-1)^2 c/gωa​(1+2mωa​​)(e−gT/c−1)2c/g when all relativistic kinematics are included. We calculate exact phase shifts for both clock (single-photon) interferometers and Raman or Bragg (two-photon) interferometers.
Light-pulse atom interferometers serve as tools for high-precision metrology and are targeting measurements of relativistic effects. This development is facilitated by extended interrogation times and large-momentum-transfer techniques generating quantum superpositions of both interferometer arms on large distances. Due to the finite speed of light, diffracting light pulses cannot interact simultaneously with both arms, inducing phase perturbations that compromise the accuracy of the sensor -- an effect that becomes progressively important as spatial separations increase. For a consistent framework, we develop a theory for finite-speed-of-light effects in atom interferometers alongside with other relativistic effects such as the mass defect. Our analysis shows that their magnitude depends crucially on the diffraction mechanism and the specific interferometer geometry. We demonstrate that the velocity of the atomic cloud at the mirror pulse of a Mach-Zehnder interferometer is less critical than the precise tuning of the lasers for resonant diffraction. Finally, we propose an experiment to test our predictions based on recoilless transitions and discuss mitigation strategies to reduce the bias in gravimetric applications.
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.
The use of a Raman laser generated by modulation for a light-pulse atom interferometer allows for a laser system that is compact and robust. However, the additional laser frequencies generated can perturb the atom interferometer. In this article, we present a precise calculation of the phase shift induced by the additional laser frequencies. The model is validated by comparison with experimental measurements on an atom gravimeter. The uncertainty of the phase-shift determination limits the accuracy of our compact gravimeter to $8\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}8}\phantom{\rule{0.28em}{0ex}}\text{m}/{\text{s}}^{2}$. We show that it is possible to reduce this inaccuracy considerably with a better control of experimental parameters or with particular interferometer configurations.
The wave properties of matter in quantum mechanics first postulated by de Broglie in 1923 as well as Einstein’s theory of general relativity have radically changed our perception of the world at the beginning of the twentieth century. While each theory is extremely successful and well tested within its range of validity, a unification of both theories has so far resisted any attempt. However, the advances in precision of modern matter-wave interferometers have paved the way to designing experiments at the interface of gravity and quantum mechanics. Indeed, quantum mechanical devices are on the brink of becoming sensitive enough to challenge predictions of general relativity such as the weak equivalence principle or set bounds on alternative gravitational theories. Reaching sensitivities required for these experiments necessitates a careful assessment of deleterious effects some of which might be atom-atom interactions or the influence of the gravitational potential of the laboratory setup itself. Estimation of the size of such effects calls for refined theoretical tools for the description of light-pulse atom interferometry which is the subject of the present thesis.