Quantum theory of atomic position measurement using optical fields

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 ΔxΔp=ħ/2. Such states may prove useful in studying one-dimensional wave-packet motion in applied potentials.