TAILORED COMPLEX POTENTIALS AND FRIEDEL'S LAW IN ATOM OPTICS

It is often found that concepts of photon optics can be adapted to matter-wave optics. In our article we choose the conjugate approach. We use the simplicity of the interaction between light and matter waves to design complex periodic potentials for the matter waves and reveal optical concepts. As an example, we investigate a violation of Friedel’s law due to fundamental optical principles in a very controlled system. Typically, diffraction phenomena are invariant under an inversion of the crystal, even when the elementary cell of the crystal possesses no symmetry. This empirical rule is generally referred to as Friedel’s law [1]. However, violations of this rule are known from diffraction experiments of x rays or electrons at solid state crystals [2], for example, due to the presence of “anomalous” (absorptive) scatterers. In this Letter we present a violation of Friedel’s law in a very different system, where atomic matter waves are diffracted at specially designed “crystals” of light [3,4]. The diffraction asymmetry is due to the interaction of both “normal” and anomalous scattering at superposed refractive and absorptive subcrystals, respectively [5]. This mechanism even works although our light crystal obviously cannot be really absorptive for the atoms, but only changes their internal state. In our experiment (Fig. 1), we detect atom intensities depending on the atom’s incidence angle at the light crystals, and their diffraction angle. In the case of spatial coincidence between the refractive and the absorptive parts of the crystals we obtain symmetric diffraction, as shown in Fig. 1(b). This corresponds to the normal situation in solid state crystals, where Friedel’s law is obeyed. However, a violation of Friedel’s law is demonstrated in Figs. 1(a) and 1(c), where Bragg diffraction is dominant in one direction. There, the absorptive and refractive index parts of the crystals are arranged such that they are out of phase by 6py2. In the remainder of this Letter, we will show that this diffraction asymmetry can be understood by evaluating the Fourier composition of the resulting complex potential, and employing dynamical diffraction theory. This means specifically that the effect of the total crystal potential cannot be separated into the individual actions of its components. However, in the weak diffraction limit a