1. de Broglie wave phase shifts induced by surfaces 20 nm away Alex Cronin John Perreault Ben McMorran Funding from: Research Corporation and NSF NSF University of Arizona, Tucson AZ, USA

2. Nano-Structure Gratings Coherent effects of V vdW (r)=C 3 /r 3 on Atom beams –Diffraction Intensities |A n | 2 depend on atom velocity –No missing orders regardless of open fraction –Interferometer measurement of phase shift in 0 th order –Measure of phase in 1 st and 2 nd orders Electron Optics Experiments on V image (r) = C 1 /r –Asymmetric Diffraction –Depends on incident velocity and angle Outline:

3. Na  z x supersonic source.5 mm skimmer 10 μm collimating slits 100 nm period diffraction grating 60 μm diameter hot wire detector Atom Diffraction 0 1 3 4 2 6 5 7 Atom Flux (kC/s) Detector Position x (mm)

4. Na  z x’x’ supersonic source.5 mm skimmer 10 μm collimating slits 100 nm period diffraction grating 60 μm diameter hot wire detector Atom Interferometry Atom Flux Atom Flux (kC/s) Grating Position x ’ (nm) Removable interaction grating

5. 1 um Nano-Structure Gratings

6. period d = 100 nm, window size w ~ 50 nm “Large-area achromatic interferometric lithography for 100nm period gratings and grids” T. A. Savas, M. L. Schattenburg, J. M. Carter and H. I. Smith. Journal of Vacuum Science and Technology B 14 4167-4170 (1996)

7. Start with V vdw in all space Compute phase shift just after the grating Propagate to the detector plane To understand the role of vdW forces on atom diffraction:

8. Atom-surface (and electron-surface) interactons cause phase shifts for de Broglie waves that are transmitted through the grating channels. At 10 nm, 3  eV and 0.3 eV but the same  0.3 rad

9. Far-Field Diffraction Envelopes for Intensity Note: 2 nd order would have  phase shift if C 3 =0.

10. vdW Potential Atom phase grating bar grating bar grating bar r1r1 r2r2 atom with velocity v

11. Amplitude and phase of n th order change with C 3 /velocity Phase shift in n th order.

12. velocity = 3171 m/s velocity = 2219 m/s velocity = 1091 m/s velocity = 662 m/s

13. Best Fit to |A n | 2 with only one free parameter: C 3

14. R x to determine the strength of the van der Waals potential V(r)=C 3 r -3 : 1. Measure physical grating parameters: w, t, d 2. Fit diffraction pattern to determine flux in each order |An| 2 3. Fit |An| 2 to determine C 3

16. Imperfect determination of grating geometry w, t and wedge angle. Uncertainty of 1 nm in w dominates the uncertainty in C 3. Lineshape used to fit the raw diffraction pattern Gaussian works poorly, empirical lineshape works better. What is the potential V everywhere in and near the grating? structure is partially coated with sodium Slot walls are not semi-infinite planes  NOT YET ADDRESSED Systematic Uncertainties

17. Dirty Gratings  Have been Cleaned  (AFM images)

18. Different power-law potentials make distinct diffraction envelopes. V(r) = -C n /r n r=0 n=2 n=3 n=4 n=5 C n chosen to match I 1 /I 0

19. Rotate the grating through “50% open fraction” and note second order is never suppressed. measure absolute transmission into zeroth order (it should change with C3/velocity) Use an interferometer to measure phase shift in 0 th order Use different surface coating. Next:

20. Missing orders would occur when zeros from single-slit diffraction coincide with constructive interference from many-slits. missing orders ± 3, ± 6,... E.g. w/d = 1/3

21. detector twist axis atom beam Experiment 2: Twist the grating to search for missing orders

22. Model parameters: d=100 nm, w=67 nm, t=116 nm,  = 3.5 o. Dashed red lines C 3 =0 Solid black lines C 3 =5 meVnm 3 Intensity in Each Order vs. Twist angle (degrees) 1 st order 2 nd order 3 rd order 4 th order 0 th order

23. Asymmetric +/- 1 orders

24.  w d l Blazed Gratings for Atom Waves requirements: van der Waals  Asymmetric channel walls 0 Region where d  )/d  = kj most strongly affects jth order

25. Optical diffraction  Much more symmetric,  missing orders possible Atom diffraction  Asymmetric,  no missing orders - ever

26. There are no missing orders in atom diffraction from a material grating. Theorem: Corollary: Atom-surface interactions can be measured by studying atom diffraction.

27. Vibration curves can prove “No Missing Orders” Thm 2 nd d w  11 A way to visualize the cumulative integral for  n Arrows from tail to tip represent amplitudes  n. I n =|  n | 2

28. Vibration Curves (a) diffraction with absorption only (b) Van der Waals  included Arrows from tail to tip represent amplitudes  n. I n =|  n | 2 22 22

29. Unless C 3 =0, endpoints of the spiral never overlap. second diffraction order w/d = 0.48 C 3 = 0, 1, 10, 100 meVnm 3 Model shown for: Therefore  n is never zero, i.e., there are never missing orders. Q.E.D.

30. Expt.#3: Use Diffraction Phase to Measure C 3 Zeroth order intensity and phase depend on the strength of the van der Waals interaction with the grating bars. 0 th order transmission vs. C 3 phase shift vs. C 3

31. Phase  0 due to Interaction Grating Atom beam Slits Interaction Grating Detector |a> |b>

32. Preparing an interaction grating to act on one arm of the interferometer Gap In-tact Grating Position (mm)

33. 1 um gap In tact grating

34. Measurement of Phase Shift Induced by the Interaction Grating A B C B A B A B C Interaction grating position (raw data 5 sec/ pt)

35. Phase Shift Induced by the Interaction Grating (averaged data) Phase shift induced by grating is φ o =.22+/-.03 rad 90 seconds of data total C3 = 4.0 +/- 1.0 meV nm3

36. See talk by John Perreault about the interferometer experiment.

37. L1L1 L2L2 AA BB CC DD Expt. #4: Measure the far-field  1 and  2 Four different interferometers:

38. Amplitude and phase of n th order change with C 3 /velocity Velocity / [km/s]

39. Contrast from Four interferometers is resolved (thanks to 100 nm period gratings and 2 m IFM)

40. Predicted Phase shifts in red

41. * * * * = preliminary data

42. Summary of 4 Experiments Theory velocity twist Interaction grating 4-IFMs Na + SiN x Na + Na metal Na valence +  =∞ Na with core + e=∞ No extra interaction needed to explain the data

43. Nano-Structure Grating in an Electron Microscope twist lever Grating O. Lens Grating 4 µm wire What About Electron Waves?

44. Images of a single wire with diffracted e-beam. 4 keV beam twist = -10±2° 1.5 keV beam twist = 5±3°

45.  w d l  =5.4 o  =-2 o 500 eV electron beam

46.  w d l  =5.4 o  =-2 o 500 eV electron beam Envelope from same theory (image chg)

47. diffraction profiles - comparison 11°: 5°: -2°: -12°: 500 eV 1.5 keV 4 keV

48. Electron Diffraction Results Asymmetric due to grating tilt More symmetric at higher energy Both explained by diffraction theory with image charge potential and  = 4 V(r) = e 2 (1-  )/(1+  ) r = 1 eV nm / r

49. Impact of atom-surface & electron-surface V(r) measured C3 = 3 (1) meVnm 3 four different ways Flux is diverted from 0 th order no missing orders blazed gratings similar effects for 500 eV electron beams power law of potential can be tested limitations for smaller gratings / slower atoms. Decoherence? Retardation?

50. Different power-law potentials make distinct diffraction envelopes. V(r) = -C n /r n r=0 n=2 n=3 n=4 n=5 C n chosen to match I 1 /I 0

51. Impact: Performance of interferometer. Figure of Merit = C sqrt(N). w1 =.56 w2 =.50 w3 =.37 w1 =.70 w2 =.80 w3 =.37

52. Thank you

54. Model of I 2 /I 3 I 2 /I 3 different velocities W

55. Atom Interferometer

57. CalculationRef Predicted Phase Shift (radians) Short-range van der Waals and ideal surface DJS99 0.45 Long-range C4 and ideal surface MDB97 0.80 Casimir Polder (C.P.) and ideal surface, 1 electron only MDB97 0.36 C.P. and Dielectric half- sapce SpT93 0.29 C.P. and thin dielectric walls ZhS95 0.25 C.P. for a grating bar*theory not yet available 0.22 rad observed

58. Measurement of C 3 Using an Atom Interferometer Measured C 3 consistent with vdW for Na and silicon nitride. Stat. Uncertainty in φ o can be reduced to 2% in 1 hr. Uncertainty in grating geometry will permit 5% level for C 3. φ o =.22 +/-.03 rad C 3 = 4.0 +/- 1.0 meV nm 3

59. Questions: What about far field phase shifts? Can we detect electron-surface interactions? is 20% of C 3 from core electrons correct? What is U vdW for a structure? How does radiation modify U vdW ? Can there be vdW friction?