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Physical Review A

Creating and probing subwavelength atomic gratings using spatially separated fields

B. Dubetsky, P. R. Berman

Published Nov 1, 1994• 16 citations
Publisher

Abstract

The interaction of an atomic beam with three spatially separated linearly polarized standing-wave fields is considered, taking into account the magnetic degeneracy and hyperfine splitting of the atomic levels. A spatial modulation of the ground-state density matrix occurs, and is not washed out as a result of spontaneous decay from the upper states. The interaction with fields separated by a distance L results in a focusing of harmonics at distances L'=(m/n)L, where m and n are integers; the period of the matter gratings is \ensuremath{\lambda}/2n, where \ensuremath{\lambda} is the wavelength of the radiation field. For a beam having angular divergence \ensuremath{\theta}, one can observe these higher harmonics independently from each other if L\ensuremath{\gg}\ensuremath{\lambda}/\ensuremath{\theta}. The gratings can be probed using a third standing-wave laser field in the focal plane of the harmonic under investigation. General expressions that allow one to calculate the probe absorption for atoms having arbitrary fine and hyperfine structure are derived. Explicit calculations for the alkali metals are presented. It is predicted that spatial modulations having periods 30--500 nm and modulation depths 2.5--50 % can be produced in this manner. The focusing of the spatial harmonics is interpreted in terms of a shadow effect.

DOI
10.1103/physreva.50.4057

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The interaction of an atomic beam with three spatially separated linearly polarized standing-wave fields is considered, taking into account the magnetic degeneracy and hyperfine splitting of the atomic levels. A spatial modulation of the ground-state density matrix occurs, and is not washed out as a result of spontaneous decay from the upper states. The interaction with fields separated by a distance L results in a focusing of harmonics at distances L'=(m/n)L, where m and n are integers; the period of the matter gratings is \ensuremath{\lambda}/2n, where \ensuremath{\lambda} is the wavelength of the radiation field. For a beam having angular divergence \ensuremath{\theta}, one can observe these higher harmonics independently from each other if L\ensuremath{\gg}\ensuremath{\lambda}/\ensuremath{\theta}. The gratings can be probed using a third standing-wave laser field in the focal plane of the harmonic under investigation. General expressions that allow one to calculate the probe absorption for atoms having arbitrary fine and hyperfine structure are derived. Explicit calculations for the alkali metals are presented. It is predicted that spatial modulations having periods 30--500 nm and modulation depths 2.5--50 % can be produced in this manner. The focusing of the spatial harmonics is interpreted in terms of a shadow effect.
We have studied sub-optical wavelength atomic gratings generated by the interference of atomic de Broglie waves in a time-domain interferometer. Three short optical standing- wave pulses, detuned from resonance with a cloud of approximately 100 (mu) K 85Rb atoms, act as phase gratings for atomic matter waves. Shortly after the first pulse, a modulation in the atomic density containing spatial harmonics of period (lambda) /2N appears ((lambda) is the optical wavelength, and N is any integer) and then rapidly vanishes due to the atomic thermal motion. The cloud maintains phase memory of this modulation, however, and the second pulse (applied at time T after the first one) results in a temporal sequence of atomic density gratings of periods (lambda) /2N. The appearance of a grating is referred to as an echo. The third pulse (and a subsequent traveling wave pulse) is used to detect these gratings. By this technique, we have created and detected in real time a grating with period (lambda) /4 at time 1.5T after the first pulse and observed new types of `echoes' caused by interaction of the de Broglie waves with all three phase gratings. This ability to produce sub-wavelength structures makes our techniques applicable to atomic beam lithography.
A theory of matter-wave interference is developed in which resonant, standing-wave optical fields interact with an ensemble of two-level atoms. If effects related to the recoil the atoms undergo on absorbing or emitting radiation are neglected, the total atomic density is spatially uniform. However, when recoil effects are included, spatial modulation of the atomic density can occur for times that are greater than or comparable to the inverse recoil frequency. In this regime, the atoms exhibit matter-wave interference that can be used as the basis of a matter-wave atom interferometer. Two specific atom-field geometries are considered, involving either one or two field-interaction zones. For each geometry, the recoil-induced spatial modulation of the total atomic density is calculated. In contrast to the normal Talbot and Talbot-Lau effects, the spatially modulated density is nota periodic function of time, owing to spontaneous emission; however, after a sufficiently long time, the contribution from spontaneous processes no longer plays a role and the periodicity is restored. With a suitable choice of observation time and field strengths, the spatially modulated atomic density serves as an indirect probe of the distribution of spontaneously emitted radiation.
An atom interferometer based on a Stern-Gerlach beam splitter is proposed. Atom scattering from a combination of magnetic quadrupole and homogeneous magnetic fields is considered. Using Raman transitions, atoms are coherently excited into and de-excited from sublevels having nonzero magnetic quantum numbers. The spatial regions in which the atoms are in such sublevels are small and have magnetic fields designed to have constant gradients. Therefore, the atoms experience position-independent accelerations, and the aberration of the coherently separated and recombined atomic beams remains small. We find that because of these properties it is possible to envision an apparatus producing atomic density gratings with nm-scale periods and large contrasts over 10-100 μμμm. We use a new method of describing the atomic interaction with a pulsed spatially homogeneous field. In our detailed analysis, we calculate corrections caused by the non-linear part of the potential and the finite value of the de-Broglie wave length. The chromatic aberration and the effects of an angular beam divergence are analyzed, and optimal conditions for an experimental demonstration of the technique are identified.%The lin⊥lin\bot %linlin⊥ combination of resonant co-propagating traveling waves splits %atoms between different Zeeman sublevels. %For a given desirable atom %grating period and length, characteristics of the atomic beam and %magnetic field are found from requirements that corrections remain %below 1010%.10
The long range goal of this project was the creation of nanostructures using light fields to manipulate atoms. By passing an atomic beam through one or more standing wave light fields, it is possible to create modulations in the atomic density having periods which are a fraction of the period of the standing wave light fields. Patterns having periods as small as lambda/10 were observed using optical masks and clear evidence for matter-wave effects were established. Several theoretical methods for achieving sub-lambda matter wave patterns were developed, Moreover it was shown that it is possible to create optical lattices having reduced periodicity using Raman transitions in a novel atom-field geometry Experiments were carried out in both magneto-optical traps and atomic beams. In both experimental set-ups, pulsed standing wave optical fields were used to excite the atoms. At some later time the atoms were probed by a traveling wave field, revealing the spatial patterns that had evolved as a result of the atom-field interaction.
We consider the interference of atoms caused by the wave nature of their motion. Coherently interfering beams of particles may be obtained by using the diffraction of atoms from separated standing waves resonant with the transition between the ground and excited states of the atoms. Interference arises with and without spatial separation of the diffracted atomic beams. A problem concerning two-quantum perturbation of the wave function of an atom in the ground state by a standing-wave field has been solved. Both amplitude and phase perturbation are taken into account. We show that in spite of the absence of coherence in the incident beam and a rapid decay of the coherence induced by the standing wave for scattered spatially overlapping beams, the interference can be observed for the conditions of an echo. We first consider the phenomenon of an echo under the quantum-mechanical action of light on the translational degrees of freedom. We show that owing to this phenomenon phase memory can be transferred through the ground state over unlimitedly large distances. Owing to the interference of the density harmonics, a periodic structure in the spatial distribution of the atomic density with a period of less than one wavelength of the light is localized at distances of the order of the distance between the fields. The harmonic amplitude is calculated for instantaneous excitation in the case in which the duration of the excitation is comparable with the backward Doppler linewidth in the beam and for scattering under Bragg conditions. In the latter case we show that the effective temperature of the atoms participating in the interference is less than one recoil energy.
We demonstrate periodic localization of neutral atoms of better than 65 nm behind amplitude, i.e., absorptive masks made of light. With these masks, produced by a standing on resonant light wave, it is possible to create and to probe spatially well-defined atomic distributions. Applications of such absorptive masks range from atom lithography to fundamental atom optical experiments. As two examples we show how to use these gratings as a tool to measure the evolution of an atomic wave field behind a static Bragg crystal and its dependence on the incidence angle of the atomic beam and how to demonstrate the frequency shift of atoms diffracted at a modulated Bragg crystal in a beating experiment.