Quantized light lenses for atoms: The perfect thick lens B. Rohwedder and M. Orszag Facultad de Fı ´sica, Pontificia Universidad Cato´lica de Chile, Casilla 306, Santiago 22, Chile ~Received 16 July 1996! A cylindrical light lens for atoms is studied in the limit of high detuning. The dynamics of atomic motion in a quantized electromagnetic field is shown to be separable into strictly classical and purely quantum- mechanical aspects when an ideal lens of arbitrary thickness is assumed. New insight is gained in the thick-lens regime, both in the classical and quantal domain. Different sources of aberration in nonideal lenses are studied. Their inclusion in a subsequent feasibility discussion for the observation of focal structures caused by field quantization sets experimental tolerances for an eventual measuring apparatus. @S1050-2947~96!08112-7# PACS number~s!: 03.75.Be, 32.80.Lg, 42.50.Vk I. INTRODUCTION Recent advances in the field of atom optics have demon- strated several techniques for manipulating neutral atom beams. In particular, since the first realization of a conver- gent lens for atoms @1#, a wide range of focusing schemes has been conceived and concretized experimentally, among them radiation pressure force- @2#, Fresnel- @3#, and magnetic hexapole @4# lensing. The first demonstration of atomic beam focusing with laser light was however based on the dipole force and made use of a red detuned TEM 00 laser beam that caused the copropagating near-resonant atoms to be attracted to its center @5#. A similar technique using a blue detuned TEM 01 * mode has been proposed in order to minimize spon- taneous forces by attracting the atoms to the zero field at the donut-laser core @6#. Alignment problems are less severe when the atomic beam intersects a standing laser field or- thogonally, but the ideal, i.e., parabolical shape of the optical potential will only be realized close to the maxima ~minima! of the red ~blue! detuned standing light wave. Such a cylin- drical lens potential can have quite a large period, if formed above a mirror by reflection at a near-grazing incidence angle. With a single period of such a lens, imaging of a microstructure was demonstrated for the first time @7#. Mul- tiple periods can be used as an array of cylindrical lenses, which focus the incoming atomic beam into a parallel set of lines. By depositing these onto a substrate, a lithographic technique was created @8–10#. For an appropriate choice of relative phases and polarizations, a pair of orthogonal lin- early polarized standing waves forms a two-dimensional ar- ray of atom lenses and could be used for depositing a high number of identical, arbitrary patterns onto a substrate @11#. The recent observation of multilevel effects in direct- write lithography using Cr atoms @12# shows that the present state of the art is in need of a deeper understanding of the basic interaction of atoms with near-resonant light. As an example for a useful extension of two-level atomic models, we mention here the recently proposed possibility of improv- ing the parabolic shape of standing-wave optical potentials by making use of a three-level configuration @13#. The de- scription of the external degree of freedom can be improved as well, by going beyond ordinary particle optics. For some lens configurations, both corpuscular and wave-mechanical descriptions have already been developed @6,10#, thus allow- ing the estimation of diffractive limits for the focal spot size. Also the modeling of the light field has been enhanced. In their seminal paper @14#, Averbukh, Akulin, and Schleich predict a focal substructure caused by the quantal nature of a thin light lens for two-level atoms in the limit of high detun- ing. In the present paper we show that even under extreme experimental conditions, the influence of light quantization on the detailed focal shape of perfect atomic light lenses can be completely ignored in the classical limit. Here is a brief outline. After specifying the system we are interested in and proposing a dynamic model for its descrip- tion, the essentials of the atomic focusing process are studied in detail. Under these idealized conditions a clean separation of ‘‘classical’’ ray-optical and ‘‘quantum’’ wave optical properties is possible, thus establishing a very intuitive rela- tion among the usual atom-optical viewpoint @16# and the results presented in @14,15#. Expressions describing the size of the ‘‘classical’’ focus ~disregarding details imposed by the given photon statistics! are derived as well as the geometry of the ‘‘quantal’’ focal distribution. Since we deal with a parabolic lens of arbitrary thickness ~with obvious restric- tions on the interaction time imposed by diffusive aberration! it becomes possible to study the thick-lens regime. On the classical level and contrary to what one would naively expect by extrapolating thin-lens results, we find that our atom lens should become divergent for high enough laser intensities. On the quantum level, an interesting ‘‘eight’’-shaped distri- bution of foci over the focal plane is found. In Sec. V the strict physical conditions imposed so far are relaxed and various sources of aberration are studied. The stage is then set for discussing the observability of focal details imposed by the quantum nature of light. A simple criterion is given and tested using very extreme, recently achieved experimen- tal parameters @25#. After a quantitative estimation of the maximal tolerances that an eventual measuring apparatus would have to fulfill, the paper closes with the proposition of several observational schemes. In order to ease the comparison with prior work, we de- cided to keep as close as possible to the notational conven- tions introduced in @15#. II. THE MODEL We consider a tightly collimated beam of neutral atoms of mass M that moves in the x - z -plane along x 5k.0. We will PHYSICAL REVIEW A DECEMBER 1996 VOLUME 54, NUMBER 6 54 1050-2947/96/54~6!/5076~9!/$10.00 5076 © 1996 The American Physical Society