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arXiv:2405.05150

Principal Component Analysis for Spatial Phase Reconstruction in Atom Interferometry

Stefan J. Seckmeyer, Holger Ahlers, Jan-Niclas Kirsten-Siemß, Matthias Gersemann, Ernst Maria Rasel, Sven Abend, Naceur Gaaloul

Published May 8, 2024• 1 citation
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Abstract

Atom interferometers are sensitive to a wide range of forces by encoding their signals in interference patterns of matter waves. To estimate the magnitude of these forces, the underlying phase shifts they imprint on the atoms must be extracted. Up until now, extraction algorithms typically rely on a fixed model of the patterns' spatial structure, which if inaccurate can lead to systematic errors caused by, for example, wavefront aberrations of the used lasers. In this paper we employ an algorithm based on Principal Component Analysis, which is capable of characterizing the spatial phase structure and per image phase offsets of an atom interferometer from a set of images. The algorithm does so without any prior knowledge about the specific spatial pattern as long as this pattern is the same for all images in the set. On simulated images with atom projection noise we show the algorithm's reconstruction performance follows distinct scaling laws, i.e., it is inversely-proportional to the square-root of the number atoms or the number of images respectively, which allows a projection of its performance for experiments. We also successfully extract the spatial phase patterns of two experimental data sets from an atom gravimeter. This algorithm is a first step towards a better understanding and complex spatial phase patterns, e.g., caused by inhomogeneous laser fields in atom interferometry.

DOI
10.48550/arxiv.2405.05150

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Atom interferometers are sensitive to a wide range of forces by encoding their signals in interference patterns of matter waves. To estimate the magnitude of these forces, the underlying phase shifts they imprint on the atoms must be extracted. Up until now, extraction algorithms typically rely on a fixed model of the patterns' spatial structure, which if inaccurate can lead to systematic errors caused by, for example, wavefront aberrations of the used lasers. In this paper we employ an algorithm based on Principal Component Analysis, which is capable of characterizing the spatial phase structure and per image phase offsets of an atom interferometer from a set of images. The algorithm does so without any prior knowledge about the specific spatial pattern as long as this pattern is the same for all images in the set. On simulated images with atom projection noise we show the algorithm's reconstruction performance follows distinct scaling laws, i.e., it is inversely-proportional to the square-root of the number atoms or the number of images respectively, which allows a projection of its performance for experiments. We also successfully extract the spatial phase patterns of two experimental data sets from an atom gravimeter. This algorithm is a first step towards a better understanding and complex spatial phase patterns, e.g., caused by inhomogeneous laser fields in atom interferometry.
Abstract Atom interferometers are employed for numerous purposes such as inertial sensing. They measure forces by encoding their signal in phase shifts between matter waves. Signal extraction algorithms typically require the resulting interference patterns to feature a priori known spatial distributions of intensity and phase. Deviations from these assumed spatial distributions, such as those caused by inhomogeneous laser wave fronts, can lead to systematic errors. For long interrogation times, such as for space operation, these distributions can display highly complex structures. We present an extraction algorithm designed for interference patterns featuring arbitrary and unknown temporally stable spatial phase profiles utilizing Principal Component Analysis. It characterizes complex phase profiles and thereby turns effects into a measured quantity which caused systematic errors in previous algorithms. We verify the algorithm’s accuracy and assess the statistical reconstruction error in the presence of atom projection noise as a function of the number of atoms and images. Finally, we extract the spatial phase profiles from experimental data obtained by an atom gravimeter.
Point source atom interferometry (PSI) uses the velocity distribution in a cold atom cloud to simultaneously measure one axis of acceleration and two axes of rotation from the spatial distribution of interferometer phase in an expanded cloud of atoms. Previously, the interferometer phase has been found from the phase, orientation, and period of the resulting spatial atomic interference fringe images. For practical applications in inertial sensing and precision measurement, it is important to be able to measure a wide range of system rotation rates, corresponding to interferograms with far less than one full interference fringe to very many fringes. Interferogram analysis techniques based on image processing used previously for PSI are challenging to implement for low rotation rates that generate less than one full interference fringe across the cloud. We introduce a new experimental method that is closely related to optical phase-shifting interferometry that is effective in extracting rotation values from signals consisting of fractional fringes as well as many fringes without prior knowledge of the rotation rate. The method finds the interferometer phase for each pixel in the image from four interferograms, each with a controlled Raman laser phase shift, to reconstruct the underlying atomic interferometer phase map without image processing.
Point source atom interferometry (PSI) uses the velocity distribution in a cold atom cloud to simultaneously measure one axis of acceleration and two axes of rotation from the spatial distribution of interferometer phase in an expanded cloud of atoms. Previously, the interferometer phase has been found from the phase, orientation, and period of the resulting spatial atomic interference fringe images. For practical applications in inertial sensing and precision measurement, it is important to be able to measure a wide range of system rotation rates, corresponding to interferograms with far less than one full interference fringe to very many fringes. Interferogram analysis techniques based on image processing used previously for PSI are challenging to implement for low rotation rates that generate less than one full interference fringe across the cloud. We introduce a new experimental method that is closely related to optical phase-shifting interferometry that is effective in extracting rotation values from signals consisting of fractional fringes as well as many fringes without prior knowledge of the rotation rate. The method finds the interferometer phase for each pixel in the image from four interferograms, each with a controlled Raman laser phase shift, to reconstruct the underlying atomic interferometer phase map without image processing.
When the initial size of an atom cloud in a cold atom interferometer is negligible compared to its size after free expansion, the interferometer is approximated to a point-source interferometer and is sensitive to rotational movements by introducing an additional phase shear in the interference sequence. This sensitivity on rotation enables a vertical atom-fountain interferometer to measure angular velocity in addition to gravitational acceleration, which it is conventionally used to measure. The accuracy and precision of the angular velocity measurement depends on proper extraction of frequency and phase from spatial interference patterns detected via the imaging of the atom cloud, which is usually affected by various systematic biases and noise. To improve the measurement, a pre-fitting process based on principal component analysis is applied to the recorded raw images. The contrast of interference patterns are enhanced by 7-12 dB when the processing is present, which leads to an enhancement in the precision of angular velocity measurements from 6.3 μrad/s to 3.3 μrad/s. This technique is applicable in various instruments that involve precise extraction of frequency and phase from a spatial interference pattern.
Interference upon free expansion gives access to the relative phase between two interfering matter waves. Such measurements can be used to reconstruct the spatially-resolved relative phase, which is a key observable in many quantum simulations of quantum field theory and non-equilibrium experiments. However, in 1D systems, longitudinal dynamics is typically ignored in the analysis of experimental data. In our work, we give a detailed account of various effects and corrections that occur in finite temperatures due to longitudinal expansion. We provide an analytical formula showing a correction to the readout of the relative phase due to longitudinal expansion and mixing with the common phase. Furthermore, we numerically assess the error propagation to the estimation of the gases’ physical quantities such as correlation functions and temperature. We also incorporate systematic errors arising from experimental imaging devices. Our work characterizes the reliability and robustness of interferometric measurements, directing us to the improvement of existing phase extraction methods necessary to observe new physical phenomena in cold-atomic quantum simulators.
Atom interferometry is an advanced optical manipulation tool of atoms in precision measurement field. Wavefront aberrations of the Raman beam have become one of the major obstacles impeding the improvement of measurement accuracy. Beforehand measurement of laser wavefront is impractical due to the further wavefront deterioration during optical mounting. In this work, we present a general method for evaluating the effective Raman wavefront that atoms experience and the corresponding phase shift of interferometric fringes. The method extracts the effective Zernike polynomial terms and reconstructs the wavefront using optimal estimation theory. The evaluation accuracy and convergence speed are discussed by simulation. The results predict the method adaptability and provide strong support on analytical and numerical reference for wavefront error compensation.