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Physical Review A

Quantum theory of correlated-atomic-position measurements by resonance imaging

John E. Thomas, L. J. Wang

Published Jan 1, 1994• 7 citations
Publisher

Abstract

Recently demonstrated methods for resonance imaging of moving atoms in the potential due to a spatially varying light shift have yielded spatial resolution of 200 nm, with linearity of a few percent over several micrometers. With single-atom detection and a potential uncertainty-principle limited resolution of a few nanometers, this method can be used to perform spatial correlation experiments in beams and traps, with ultrahigh spatial resolution. In this paper we present a theoretical treatment for methods to extend the resonance imaging technique to include position measurements of moving atoms at multiple points, and thereby to measure appropriately defined atomic correlation functions which depend generally on both position and momentum. Results are given for both fermions and bosons and simple examples show how correlated position measurement by resonance imaging reveals the atomic statistical properties at the long length scales which arise at low temperature.

DOI
10.1103/physreva.49.558

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Recently demonstrated methods for resonance imaging of moving atoms in the potential due to a spatially varying light shift have yielded spatial resolution of 200 nm, with linearity of a few percent over several micrometers. With single-atom detection and a potential uncertainty-principle limited resolution of a few nanometers, this method can be used to perform spatial correlation experiments in beams and traps, with ultrahigh spatial resolution. In this paper we present a theoretical treatment for methods to extend the resonance imaging technique to include position measurements of moving atoms at multiple points, and thereby to measure appropriately defined atomic correlation functions which depend generally on both position and momentum. Results are given for both fermions and bosons and simple examples show how correlated position measurement by resonance imaging reveals the atomic statistical properties at the long length scales which arise at low temperature.
A simple quantum theory of recently suggested optical techniques for ultrahigh-resolution position measurement and localization of moving atoms in beams is presented. Both the internal and center-of-mass motion are treated quantum mechanically so that the limitations on the ultimate position resolution due to atomic motion and wave-mechanical diffraction are included in the analysis. The techniques utilize a miniaturized form of Raman-induced resonance imaging in which optical fields are used to make transitions from a long-lived initial state to a long-lived final state. The final state is shifted by the spatially varying potential of an applied force field in order to correlate the atomic position with its resonance frequency. Spatially varying level shifts are obtainable by using very large Zeeman field gradients or spatially varying light shifts in a small interaction volume. This results in very high spatial resolution. The atomic transit time across the optical-field region is limited by focusing to an ideal diameter that minimizes the spatial resolution length. The results of the analysis show that nanometer spatial resolution of the initial-state position distribution is attainable. Under appropriate conditions, the final-state spatial wave function can take the form of a minimum-uncertainty Gaussian wave packet obeying \ensuremath{\Delta}x\ensuremath{\Delta}p=\ensuremath{\Elzxh}/2. Such states may prove useful in studying one-dimensional wave-packet motion in applied potentials.
All optical techniques for position measurement and localization of moving atoms have been developed recently, which achieve suboptical wavelength precision. These techniques, which we refer to as‘atom imaging', are based on new methods of resonance imaging. The attainable accuracy ultimately will scale to the uncertainty principle limit, yielding nanometre spatial resolution for single moving atoms. Diverse applications include fundamental studies of Bose and Fermi statistics and direct-writing neutral-atom lithography.
We are currently developing Raman-induced resonance-imaging techniques for ultrahigh-resolution position measurement and localization of moving atoms.1 We obtain transit-time-limited frequency resolution in intense spatially varying potentials, which correlate the atomic position with the Raman resonance frequency. The use of optical fields for the Raman region permits easy application to very small interaction volumes where extremely large spatially varying Zeeman shifts or light shifts (109-1012 Hz/cm) can be employed to achieve position resolution as high as 7 nm. This leads to quantum-mechanical limitations on the measurement accuracy.
Recently, we have begun to develop optical techniques for ultra‐high resolution position measurement and localization of moving atoms. The methods employ Raman‐induced resonance imaging in which a spatially varying potential correlates an atomic resonance frequency with the atomic position. Optical methods are ideally suited for high resolution resonance imaging because they permit the study of very small volumes, which facilitates the use of spatially varying potentials with very large gradients and eliminates the need to place mechanical surfaces in the region to be studied. Hence, the methods have many applications to atom optics experiments in both beams and traps, where they can be used to create and measure small scale atomic spatial distributions and to perform spatial correlation experiments. The current experiments demonstrate a new method of imaging employing a spatially varying light‐shift to achieve spatial resolution of 200 nanometers, with a few percent linearity over several microns. Centroids of localized atomic distributions are determined from the χ2 fits with an accuracy of ±20 nanometers. New experiments employing still larger gradients and single atom detection are expected to achieve nanometer spatial resolution, limited by the uncertainty principle.