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arXiv:2508.08475

Wigner representation enables the exact derivation of the atom interferometer phase, unlike the path integral approach

B. Dubetsky

Published Aug 11, 2025• 0 citations
arXivHTMLPDF

Abstract

An exact expression for the phase of an atomic interferometer located in a non-inertial reference frame (platform) moving along an arbitrary trajectory and with an orientation that changes arbitrarily over time is obtained. This expression takes into account precisely gravitational, Coriolis, centrifugal, and gravity-gradient forces, which arise during the rotation of the gravity source at a permanent rate. To achieve this result, we utilized the equations for the atomic density matrix in the Wigner representation. Starting from the exact formula, we derived three new terms in the well-known limit of small rotation angles and short interrogation time, which are attributed to the rotation and translational motion of the platform.

DOI
10.48550/arxiv.2508.08475

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An exact expression for the phase of an atomic interferometer located in a non-inertial reference frame (platform) moving along an arbitrary trajectory and with an orientation that changes arbitrarily over time is obtained. This expression takes into account precisely gravitational, Coriolis, centrifugal, and gravity-gradient forces, which arise during the rotation of the gravity source at a permanent rate. To achieve this result, we utilized the equations for the atomic density matrix in the Wigner representation. Starting from the exact formula, we derived three new terms in the well-known limit of small rotation angles and short interrogation time, which are attributed to the rotation and translational motion of the platform.
In the presence of Earth gravity and gravity-gradient forces, centrifugal and Coriolis forces caused by the Earth rotation, the phase of the time-domain atom interferometers is calculated with accuracy up to the terms proportional to the fourth degree of the time separation between pulses. We considered double-loop atom interferometers and found appropriate condition to eliminate their sensitivity to acceleration to get atomic gyroscope, or to eliminate the sensitivity to rotation to increase accuracy of the atomic gravimeter. Consequent use of these interferometers allows one to measure all components of the acceleration and rotation frequency projection on the plane perpendicular to gravity acceleration. Atom interference on the Raman transition driving by noncounterpropagating optical fields is proposed to exclude stimulated echo processes which can affect the accuracy of the atomic gyroscopes. Using noncounterpropagating optical fields allows one to get a new type of the Ramsey fringes arising in the unidirectional Raman pulses and therefore centered at the two-quantum line center. Density matrix in the Wigner representation is used to perform calculations. It is shown that in the time between pulses, in the noninertial frame, for atoms with fully quantized spatial degrees of freedom, this density matrix obeys classical Liouville equations.
Long-time atom interferometry is instrumental to various high-precision measurements of fundamental physical properties, including tests of the equivalence principle.Due to rotations and gravity gradients, the classical trajectories characterizing the motion of the wave packets for the two branches of the interferometer do not close in phase space, an effect which increases significantly with the interferometer time.The relative displacement between the interfering wave packets in such open interferometers leads to a fringe pattern in the density profile at each exit port and a loss of contrast in the oscillations of the integrated particle number as a function of the phase shift.Paying particular attention to gravity gradients, we present a simple mitigation strategy involving small changes in the timing of the laser pulses which is very easy to implement.A useful representation-free description of the state evolution in an atom interferometer is introduced and employed to analyze the loss of contrast and mitigation strategy in the general case.(As a by-product, a remarkably compact derivation of the phaseshift in a general light-pulse atom interferometer is provided.)Furthermore, exact results are obtained for (pure and mixed) Gaussian states which allow a simple interpretation in terms of the alignment of Wigner functions in phase-space.Analytical results are also obtained for expanding Bose-Einstein condensates within the time-dependent Thomas-Fermi approximation.Finally, a combined strategy for rotations and nonaligned gravity gradients is considered as well.
Abstract With the rapid development of atom interferometry, atom interferometers have been widely used in inertial navigation, resource exploration, basic physics and other fields. For precise measurement experiments with multi-loop atom interferometer configuration, it is urgent to establish a high-precision theoretical model charactering the relationship between target measurement quantities and measurement results. Based on this, we adopt the idea of co-moving reference frame, so as to transfer the gravity, gravity gradient and Earth’s rotation effects to the laser light field interacting with atoms by translating, rotating and stretching the background spacetime. With this method, we not only can simplify the analysis process of the experimental principle, but also give the analytical results of the high-order interference phase of the multi-loop atom interferometer containing the gravity gradient and Earth’s rotation. This work will lay a certain theoretical foundation for higher-precision experiment, and will also help to clearly guide the implementation of the experiment.
We compare expressions for the atom interferometer phase obtained using the path integral approach and the approach based on the density matrix equation in the Wigner representation. The power series of these expressions over the Planck constant, truncated to the ℏ2\hbar ^{2}ℏ2 term, coincide. This coincidence is achieved only after the Q−Q-Q−term is included.
Gravity gradient is known as a serious systematic effect in atomic tests of the universality of free fall, where the initial central position and velocity of atoms need to be exactly controlled. In this paper, we study quantum free fall with high-order gravity gradients. It is shown that the cubic terms in the Newtonian potential shall generate a new phase shift in atom interferometers, which depends on the position and velocity uncertainties of the incident atoms. We further investigate the nonclassicality of free fall and show that, due to the cubic potential, the gravitational Wigner equation in phase space of position and velocity is different from the classical Liouville equation. There exists a mass-dependent correction in the dynamical equation regardless of initial state. Nevertheless, this is just a quantum mechanical effect of microparticles, which does not violate equivalence principle that inertia mass is equal to the gravitational mass.
This paper derives the phase difference produced by an atom interferometer, building from the phase of a matter wave and the momentum and phase that each laser pulse transfers to an atom in a sequence. It derives this phase difference first for a constant acceleration, where it takes the form k_eff·a·τ². It then uses this form to derive the phase difference from rotation and a gravity gradient. It also derives the proportion of atoms in one output state, the quantity the atom interferometer measures, and its relation to the phase difference.