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Physical Review Letters

Tailored Complex Potentials and Friedel's Law in Atom Optics

Claudia Keller, Markus K. Oberthaler, Roland Abfalterer, Stefan Bernet, Jörg Schmiedmayer, Anton Zeilinger

Published Nov 3, 1997• 90 citations
Publisher

Abstract

Tailored complex potentials for atoms can be made of two overlapping standing light waves, one on resonance and one far detuned. The observed diffraction asymmetry of Bragg diffraction of such light structures is due to a corresponding asymmetry of the Fourier components of the potential. In crystal physics this is known as a violation of Friedel's law.

DOI
10.1103/physrevlett.79.3327

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Tailored complex potentials for atoms can be made of two overlapping standing light waves, one on resonance and one far detuned. The observed diffraction asymmetry of Bragg diffraction of such light structures is due to a corresponding asymmetry of the Fourier components of the potential. In crystal physics this is known as a violation of Friedel's law.
Diffraction of atoms by a particular absorbing `crystal of light', that has been studied experimentally, is described by the complex potential . The diffracted beam intensities can be calculated exactly, as a function of (dimensionless) potential strength , angle of incidence , and crystal thickness . Only the beams have nonzero intensity; this `lop-sidedness' is a dramatic violation of Friedel's law. The n th beam is strong at Bragg angles , and the peaks get sharper with increasing ; they correspond to degeneracies of the (non-Hermitian) governing matrix. For normal incidence , the are periodic in , and as n increases they approach a self-similar function of .
Atoms in light crystals formed by a standing light wave are a model system to study the propagation of matter waves in periodic potentials. The encountered phenomena can be described by dynamical diffraction theory which has been extensively studied for x-ray, electron, and neutron scattering from solid state crystals. In this paper we show that an atomic de Broglie wave traversing a standing light wave allows investigation of predictions of dynamical diffraction theory which were previously experimentally not accessible. We present standard diffraction efficiency characterizations for pure absorptive and pure refractive crystals. Additionally we were able to measure directly the total atomic wave field formed inside a refractive and an absorptive crystal and to confirm the predicted absolute position of the atomic wave field with respect to the lattice planes. By superposing two standing light waves of different frequency we tailored a new kind of crystal potential where Friedel's law about usual diffraction symmetries is maximally violated.
Temporal light modulation methods which are of great practical importance in optical technology, are emulated with matter waves. This includes generation and tailoring of matter-wave sidebands, using amplitude and phase modulation of an atomic beam. In the experiments atoms are Bragg diffracted at standing light fields, which are periodically modulated in intensity or frequency. This gives rise to a generalized Bragg situation under which the atomic matter waves are both diffracted and coherently shifted in their de Broglie frequency. In particular, we demonstrate creation of complex and non-Hermitian matter-wave modulations. One interesting case is a potential with a time-dependent complex helicity $[V\ensuremath{\propto}\mathrm{exp}(i\ensuremath{\omega}t)],$ which produces a purely lopsided energy transfer between the atoms and the photons, and thus violates the usual symmetry between absorption and stimulated emission of energy quanta. Possible applications range from atom cooling over advanced atomic interferometers to a new type of mass spectrometer.
Atoms interacting with standing light waves are a model system for the propagation of waves in static and time varying periodic media. We present here experiments studying the coherent motion of atomic deBroglie waves in periodic potentials made from on and off resonant light. We observe anomalous transmission of atoms through resonant standing light waves and experimentally confirm that atoms fulfilling the Bragg condition form a standing matter wave pattern. We furthermore demonstrate how Bragg diffraction of atomic matter waves at a time-modulated thick standing light wave can be used to coherently shift the deBroglie frequency of the diffracted atoms. Our frequency shifter for atomic matter waves is similar to an acousto-optic frequency shifter for photons.
Bragg diffraction of atoms at thick standing light waves requires that the wave-matching condition is fulfilled. This usually means that the atomic beam crosses the light wave exactly at the Bragg angle. Nevertheless, our experiments also demonstrate Bragg diffraction at detuned angles if the amplitude of the standing light wave is temporally modulated with an appropriate frequency. If, on the other hand, the phase of the light wave is modulated no diffraction is observed. Both modulation processes produce frequency sidebands which set up `slowly travelling standing waves' in front of a retro-reflection mirror. Atoms are diffracted at these `almost standing' light waves in a similar way to photons at the travelling sound waves in an acousto-optic modulator. The frequency of the diffracted de Broglie waves is assumed to be shifted by the intensity modulation frequency. The different results using amplitude- and phase-modulated light waves are due to interference between the diffraction contributions of the individual frequency sidebands contained in the standing light wave.