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Deriving the Atom Interferometer Phase Difference for Acceleration, Rotation, and Gravity Gradients

Setthaphat Kongkhun

Published May 5, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.5281/zenodo.21532423

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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.
We present a method for accurate measurements of both the gravity acceleration and its vertical gradient using a dual-atom interferometer, in principle, free from any uncertainty related to the absolute or relative positions of the two atomic samples. The method relies on the use of a dual-lock technique, which simultaneously stirs the chirp rate applied to the frequency difference between the interferometer lasers to compensate the gravity acceleration and the frequency jump applied to the lasers at the mid pulse of the interferometer to compensate for the gravity gradient. This allows one, in the end, to determine the two inertial quantities of interest in terms of frequencies.
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.
Gravity gradients represent a major obstacle in high-precision measurements by atom interferometry. Controlling their effects to the required stability and accuracy imposes very stringent requirements on the relative positioning of freely falling atomic clouds, as in the case of precise tests of Einstein's equivalence principle. We demonstrate a new method to exactly compensate the effects introduced by gravity gradients in a Raman-pulse atom interferometer. By shifting the frequency of the Raman lasers during the central π pulse, it is possible to cancel the initial position- and velocity-dependent phase shift produced by gravity gradients. We apply this technique to simultaneous interferometers positioned along the vertical direction and demonstrate a new method for measuring local gravity gradients that does not require precise knowledge of the relative position between the atomic clouds. Based on this method, we also propose an improved scheme to determine the Newtonian gravitational constant G towards the 10 ppm relative uncertainty.
Matter-wave interferometry has enabled high precision measurements of inertial forces such as gravity and the Coriolis force. This is facilitated by the long-term stability of the physical properties of atoms and lasers. Recent experiments have demonstrated the operation of portable, robust sensors using atom interferometry. This has potential uses in the context of inertial navigation, where conventional devices suffer from long-term drifts due to bias instability. Furthermore, determining position via dead reckoning requires minimisation of dead time between measurements. This thesis presents the development of an atom interferometer for measuring horizontal accelerations. In this configuration, gravity induces motion across the laser wavefront, which constrains the tolerable level of wavefront distortions. Effective control of the experiment allows the interferometer to be operated at a rate of 4 Hz. A cold ensemble of 10^6 atoms in the same internal state is prepared in 150 ms. The interferometer operates using a sequence of three laser pulses separated by T= 25 ms to achieve sensitivity to horizontal accelerations. Combining this with a classical accelerometer provides a method of correcting for vibration-induced noise, as well as determining the interferometer fringe order. After an integration time of 40 s, the sensitivity to horizontal accelerations is better than 3^-6 m s^-2. Effects which limit this sensitivity are discussed.
We investigated the Raman-pulse-duration effect in a gravity gradiometer with two atom interferometers. Since the two atom clouds in the gradiometer experience different gravitational fields, it is hard to compensate for the Doppler shifts of the two clouds simultaneously by chirping the frequency of a common Raman laser. This leads to an appreciable phase shift. The magnitude of the phase shift relative to the differential phase shift of the two interferometers is in an order of $\ensuremath{\tau}/T$, and cannot be neglected in the precision measurements such as measuring the gravity gradient and the Newtonian gravitational constant.