Transcript Slide 1

Using Atomic Diffraction to Measure the
van der Waals Coefficient for Na and
Silicon Nitride
J. D. Perreault1,2, A. D. Cronin2, H. Uys2
1Optical Sciences Center, University of Arizona, Tucson AZ, 85721 USA
2Physics Department, University of Arizona, Tucson AZ, 85721 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 C3 is determined by fitting a
modified Fraunhoffer optical theory to the experimental data.
Experiment Geometry
•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
x
x
z
z
Na
supersonic
source
.5 μm
skimmer
100 nm period 60 μm diameter
10 μm
diffraction
hot wire
collimating
grating
detector
slits
van der Waals Diffraction Theory
•The far-field diffraction pattern for a perfect grating is given by
2
Ix  

λ dB z 

A

L
x

n



n
d
n  


lineshape


 i ξ 
 ξ 
A n  e rect 
w




 f  n
T ξ 
ξ
d
•The diffraction envelope amplitude An is just the scaled Fourier transform of
the single slit transmission function T(x)
•Notice that T(x) is complex when the van der Waals interaction is
incorporated and the phase following the WKB approximation to leading order
in V(x) is
1
 x    k ( z, x ) dz   2m E  V ( z , x ) dz

3
3
tVξ  tC3 
w
w 

 x  

 ξ     ξ   
v
v 
2
2  

Definitions
I(x): atom intensity
ldB: de Broglie wavelength
An: diffraction envelope amplitude
v: velocity
|An|2: number of atoms in order n
sv: velocity distribution
T(x): single slit transmission function
d: grating period
V(x): vdW potential
w: grating slit width
f(x): phase due to vdW interaction
t: grating thickness
x : grating coordinate
fx : Fourier conjugate variable to x
x: detector coordinate
z: grating-detector separation
L(x): line shape function
n: diffraction order
Intuitive Picture
•As a consequence of the fact that matter propagates like a wave there
exists a suggestive analogy
index : light :: potential: atoms
•The van der Waals interaction makes each slot act as a lens, adding
curvature to the de Broglie wave fronts and thus modifying the far-field
diffraction pattern
optical phase front
negative lens
Measuring the Grating Parameters
•A grating rotation experiment along with an SEM image are used
to independently determine the grating parameters w and t and g.
SEM image:
w = 68.44 ± .01 nm
grating rotation experiment:
2
v = 1015 m/s
100
4
2
10
4
2
-2
-1
0
1
Position [mm]
2
Relative Number of Atoms
Intensity [kCounts/s]
Determining |An|2
1
6
4
v = 2109 m/s
v = 1015 m/s
2
0.1
6
4
2
Intensity [kCounts/s]
0
4
v = 2109 m/s
2
1
2
3
4
Diffraction Order
10
4
2
•Free parameters: |An|2, v, σv
4
•The background noise and lineshape
function L(x) are determined from an
independent experiment
1
2
-1
0
Position [mm]
1
5
C3 = 5.95 ± .45 meVnm3
(statistical error only)
1
v = 2109 m/s
VDW Theory
Optical Theory
0.1
0.01
0.001
0
1
2
3
Diffraction Order
4
5
Relative Number of Atoms
Relative Number of Atoms
Preliminary Results: Best Fit C3
C3 = 3.13 ± .04 meVnm3
(statistical error only)
1
v = 1015 m/s
VDW Theory
Optical Theory
0.1
0.01
0.001
0
1
2
3
4
Diffraction Order
•The relative number of atoms in each diffraction order was fit with only one free
parameter: C3
•Notice how optical theory (i.e. C3=0) fails to describe the diffraction envelope
correctly for atoms.
5
Using Zeroeth Order Diffraction to
Measure C3
0 order phase [rad]
1.0
0.8
0.6
0.6
0.4
0.2
th
0.4
th
0 order intensity [arb. units]
•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
0
2
4
63
C3 [meVnm ]
8
•The ratio of the zeroeth order
to the raw beam intensity
could be used to measure C3
10
0.0
0
2
4
63
C3 [meVnm ]
8
•The phase shift could be
measured in an interferometer
to determine C3
10
Conclusions and Future Work
•A preliminary determination of the van der Waals coefficient C3 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 central diffraction
order on C3 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
References
“Determination of Atom-Surface van der Waals Potentials from
Transmission-Grating Diffraction Intensities” R. E. Grisenti, W. Schollkopf,
and J. P. Toennies. Phys. Rev. Lett. 83 1755 (1999)
“He-atom diffraction from nanostructure transmission gratings: The role of
imperfections” R. E. Grisenti, W. Schollkopf, J. P. Toennies, J. R. Manson,
T. A. Savas and H. I. Smith. Phys. Rev A. 61 033608 (2000)
“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)