Using Atomic Diffraction to Measure the van der Waals Coefficient for Na and Silicon Nitride J. D. Perreault 1,2, A. D. Cronin 2, H. Uys 2 1 Optical Sciences Center, University of Arizona, Tucson AZ, USA 2 Physics Department, University of Arizona, Tucson AZ, USA

Abstract In atom optics a mechanical structure is commonly regarded as an amplitude mask for atom waves. However, atomic diffraction patterns indicate that mechanical structures also operate as phase masks. In this study a well collimated beam of Na atoms is used to illuminate a silicon nitride grating with a period of 100 nm. During passage through the grating slots atoms acquire a phase shift due to the van der Waals (vdW) interaction with the grating walls. This phase shift depends both on atom position and velocity. As a result the relative intensities of the matter-wave diffraction peaks deviate from optical theory and depend on the de Broglie wavelength of the atoms. The vdW coefficient C 3 is determined by fitting a modified Fraunhofer optical theory to the experimental data.

Experiment Geometry Na ξ z x z supersonic source.5 μm skimmer 10 μm collimating slits 100 nm period diffraction grating 60 μm diameter hot wire detector A supersonic Na atom beam is collimated and used to illuminate a diffraction grating A hot wire detector is scanned to measure the atom intensity as a function of x

van der Waals Diffraction Theory The far-field diffraction pattern for a perfect grating is given by The diffraction envelope amplitude A n is just the scaled Fourier transform of the single slit transmission function T( ξ ) Notice that T( ξ ) is complex when the van der Waals interaction is incorporated and the phase following the WKB approximation to leading order in V( ξ ) is

Definitions λ dB : de Broglie wavelength v: velocity σ v : velecity distribution d: grating period w: grating slit width t: grating thickness I(x): atom intensity A n : diffractin envelope amplitude |A n | 2 : number of atoms in order n T(ξ): single slit transmission function V(ξ): vdW potential φ(ξ): phase due to vdW interaction ξ: grating coordinate f ξ : Fourier conjugate variable to ξ x: detector coordinate z: grating-detector separation L(x): lineshape function n: diffraction order

Intuitive Picture As a consequence of the fact that matter propagates like a wave there exists a suggestive analogy The van der Waals interaction acts as a kind of negative lens that fills each slit of the grating, adding curvature to the de Broglie wave fronts and modifying the far-field diffraction pattern optical phase front negative lens

Measuring the Grating Parameters w = ±.0091 nm SEM image A grating rotation experiment along with an SEM image are used to determine w and t grating rotation experiment

Determining |A n | 2 Free parameters: |A n | 2, v, σ v The background noise and linshape function L(x) are determined from an independent experiment

Calculating C 3 The relative number of atoms in each diffraction order was fit with only one free parameter: C 3 Notice how optical theory (i.e. φ →0) fails to describe the diffraction envelope correctly for atoms C 3 = 3.13 ±.04 meVnm 3 C 3 = 5.95 ±.45 meVnm 3

Using Zeroeth Order Diffraction to Measure C 3 Using the previously mentioned theory one can see that the zeroth order intensity and phase depend on the strength of the van der Waals interaction The ratio of the zeroeth order to the raw beam intensity could be used to measure C3 The phase shift could be measured in an interferometer to determine C3

Conclusions and Future Work A preliminary determination of the van der Waals coefficient C 3 for Na on silicon nitride has been presented here for two different atom beam velocities based on the method of Grisenti et. al Using the phase and intensity dependence of the zeroeth diffraction order on C 3 we are pursuing novel methods for the measurement of the van der Waals coefficient The van der Waals phase could be “tuned” by rotating the grating about its k-vector, effectively changing the value of t by some known amount

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