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arXiv:2407.11446 · AVS Quantum Science

Feynman diagrams for matter wave interferometry

Jonah Glick, Tim Kovachy

Published May 8, 2026• 0 citations
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Abstract

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.

DOI
10.1116/5.0313350

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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.
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.
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.
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.
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.
A substantial part of this thesis consolidates, summarizes, and deepens the theoretical work that has led to the recent observations of the Kennard phase and the construction of a novel T³ Stern-Gerlach interferometer for matter waves. Our novel device is sensitive to a linear potential as originating from the gravitational field. In comparison to existing atom interferometers, it displays a faster scaling of the interferometer phase with regard to the interferometer time T. The resulting cubic dependence originates from the Kennard phase and benefits from employing state-dependent forces that are induced by magnetic field gradients. In addition, several parts of this thesis exceed this particular topic. By developing a general and easily applicable formalism based on branch-dependent dynamics, we simplify, unify, and generalize the description of matter-wave interferometers. Within our approach, we compare and contrast classical and quantum concepts from a fundamental perspective while concentrating on the contrast and phase shift of matter-wave interferometers. Based on our formalism, we propose schematic rules which enable a straightforward analysis of a given interferometer geometry in the presence of state- and time-dependent forces. With the help of these rules, we pursue the classification of existing and the construction of novel interferometer geometries that display a sensitivity to particular aspects of spatially and temporally varying external fields. Our method is supported by several examples of novel interferometers which still await an experimental implementation.
The wave-particle duality is a key concept of quantum physics and extending quantum superpositions to bodies of increasing mass and complexity has become a key goal to numerous research groups. Such experiments are motivated by the desire to probe and better understand the quantum-classical interface in advanced tests of spontaneous or gravity-induced wave function collapse and in the search for non-linear modifications of quantum theory. High mass interferometry has been proposed to serve in the search for light dark matter via weak recoil decoherence as well as in tests of the weak equivalence principle by comparing a large variety of quantum systems with very distinct internal composition, excitation, mass, shape or angular momentum. Here we discuss the experimental state of the art in high-mass matter-wave interferometry, as well as new perspectives from cluster and nanoparticle physics to push quantum superpositions by 3 to 4 orders of magnitude in mass or up to 6 orders of magnitude in 'macroscopicity'. We discuss different near-field interferometer configurations that shall become useful in near-future demonstrations of quantum phenomena with objects as diverse as proteins, metal clusters and dielectric nanoparticles.