de Broglie wave phase shifts induced by surfaces

20 nm away

Alex CroninJohn PerreaultBen McMorran

Funding from: Research Corporation and NSFNSF

University of Arizona, Tucson AZ, USA

• Nano-Structure Gratings

• Coherent effects of VvdW(r)=C3/r3 on Atom beams

– Diffraction Intensities |An|2 depend on atom velocity

– No missing orders regardless of open fraction

– Interferometer measurement of phase shift in 0th order

– Measure of phase in 1st and 2nd orders

• Electron Optics Experiments on Vimage(r) = C1/r– Asymmetric Diffraction

– Depends on incident velocity and angle

Outline:

Na

z

x

supersonic source

.5 mm skimmer

10 μm collimating

slits

100 nm period diffraction

grating

60 μm diameter hot wiredetector

Atom Diffraction

1

2

4

10

2

4

100

2

4

Ato

m F

lux

( k

/sec

)

-1.0 -0.5 0.0 0.5 1.0detector position (mm)

0

1

3

4

2

6

5

7

Ato

m F

lux

(kC

/s)

Detector Position x (mm)

ninA e

2

nA

Na

z

x’

supersonic source

.5 mm skimmer

10 μm collimating

slits

100 nm period diffraction

grating

60 μm diameter hot wiredetector

Atom Interferometry

Ato

m F

lux

Ato

m F

lux

(kC

/s)

Grating Position x’ (nm)

n

Removable interaction grating

1 um

Nano-Structure Gratings

Nano-Structure Gratingsperiod 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)

•Start with Vvdw in all space •Compute phase shift just after the grating •Propagate to the detector plane

To understand the role of vdWforces on atom diffraction:

Atom-surface (and electron-surface) interactonscause phase shifts for de Broglie waves that aretransmitted through the grating channels.

At 10 nm, 3 eV and 0.3 eV but the same 0.3 rad

Far-Field Diffraction Envelopes for Intensity

Note: 2nd order would have phase shift if C3=0.

vdW Potential Atom phase

))

((

t

v

V

1.0

0.8

0.6

0.4

0.2

0.0

(r

ad)

50403020100position (nm)

gratingbar

-5

-4

-3

-2

-1

0

En

erg

y (

eV

)

50403020100position (nm)

[MDB97] [DJS99] [MDB97e] [SPT93] [ZhS95]

gratingbar

gratingbar

( )

r1 r2atom

( )V

t with

velocity v

Amplitude and phase of nth order change with C3/velocity

dnkiAiA incnn )(exp)exp(

Phase shift in nth order.

velocity = 3171 m/s

velocity = 2219 m/s

velocity = 1091 m/s

velocity = 662 m/s

1

10

100

Inte

nsity

[k/

s]

0.50.0Position [mm]

01

23

54

1

10

100

Inte

nsity

[k/

s]

3210Position [mm]

0

1

23

1

10

100In

tens

ity [

k/s]

210Position [mm]

0

1

2 3

54

1

10

100

Inte

nsity

[k/

s]

1.00.50.0Position [mm]

01

23

54

Best Fit to |An|2

4

6

0.01

2

4

6

0.1

2

4

6

1R

ela

tive

In

ten

sity

543210Diffraction Order

v = 662 m/s v = 1091 m/s v = 2219 m/s v = 3171 m/s vdW theory C3 = 0

with only one free parameter: C3

Rx to determine the strength of thevan der Waals potential V(r)=C3r -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 C3

Imperfect determination of grating geometry w, t and wedge angle. Uncertainty of 1 nm in w dominates the uncertainty in C3. 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

Dirty Gratings

Have beenCleaned

(AFM images)

105

106

107

Inte

nsi

ty

-4 -3 -2 -1 0 1 2 3 4order

C3=0 C3=5 C2=1.2 C4=30 C5=1000

Computed usingPhasor method matching I1 / I0

Different power-law potentialsmake distinct diffraction envelopes.

V(r) = -Cn/rn

r=0n=2n=3n=4n=5

105

106

107

Inte

nsity

-4 -3 -2 -1 0 1 2 3 4order

C3=0 C3=5 C2=1.2 C4=30 C5=1000

Computed usingPhasor method matching I1 / I0

105

106

107

Inte

nsity

-4 -3 -2 -1 0 1 2 3 4order

C3=0 C3=5 C2=1.2 C4=30 C5=1000

Computed usingPhasor method matching I1 / I0

Cn chosen to match I1/I0

• 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 0th order

• Use different surface coating.

Next:

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

0.10

0.08

0.06

0.04

0.02

0.00

Inte

nsity

[I 0

]-1

-10 -5 0 5 10sin() [d]

-1

0.10

0.08

0.06

0.04

0.02

0.00

Inte

nsity

[I 0

]-1

-10 -5 0 5 10sin() [d]

-1

detector

twist axis

atom beam

Experiment 2: Twist the grating to search for missing orders

Model parameters: d=100 nm, w=67 nm, t=116 nm, = 3.5o.

Dashed red lines C3=0 Solid black lines C3=5 meVnm3

Intensity in Each Order vs. Twist

angle (degrees)

1st order

2nd order

3rd order

4th order

0th order

Asymmetric +/- 1 orders

w

d

l

Blazed Gratings for Atom Waves

requirements:•van der Waals •Asymmetric channel walls

0.05

0.04

0.03

0.02

0.01

0.00

Inte

nsi

ty

[I 0]-1

1050-5-10sin() [d]

-1

0

1.00.80.60.40.20.0

Pha

se

-40 -20 0 20 40' (nm)

Region where d)/d = kj most strongly affects jth order

0.10

0.08

0.06

0.04

0.02

0.00

Inte

nsity

[I 0

]-1

-10 -5 0 5 10sin() [d]

-1

0.10

0.08

0.06

0.04

0.02

0.00

Inte

nsity

[I

0]-1

-10 -5 0 5 10sin() [d]

-1

Optical diffractionMuch more symmetric,missing orders possible

Atom diffractionAsymmetric,no missing orders - ever

0.04

0.02

0.00

Inte

nsity

[I 0

]-1

1050-5-10sin() [d]

-1

There are no missing orders in atom diffraction from a material grating.

Theorem:

Corollary:

Atom-surface interactions can be measuredby studying atom diffraction.

Vibration curves can prove “No Missing Orders” Thm

2nd

dw

1

2

2

vdWex (p )

w

w

n gi nk i d

A way to visualize the cumulative integral for n

Arrows from tail to tip represent amplitudes n. In =|n|2

Vibration Curves

(a) diffraction with absorption only

(b) Van der Waals included

Arrows from tail to tip represent amplitudes n. In =|n|2

2

2

Unless C3=0, endpoints of the spiral never overlap.

second diffraction orderw/d = 0.48C3 = 0, 1, 10, 100 meVnm3

Model shown for:

Thereforen is never zero, i.e., there are never missing orders. Q.E.D.

Expt.#3: Use Diffraction Phase to Measure C3

1.0

0.8

0.6

0.4

0th o

rde

r in

ten

sity

[a

rb.

un

its]

1086420C3 [meVnm

3]

• Zeroth order intensity and phase depend on the strength of the van der Waals interaction with the grating bars.

0th order transmission vs. C3 phase shift vs. C3

0.6

0.4

0.2

0.0

0th

ord

er

ph

ase

[ra

d]

1086420C3 [meVnm

3]

Phase 0 due to Interaction Grating

Atom beam

Slits

Interaction Grating

Detector|a>

|b>

bea og xki

og xkCI cos1

140

120

100

80

60

40

20

0

-0.2 0.0 0.2

Preparing an interaction grating to act on one arm of the interferometer

Gap

In-tact Grating Position (mm)

1 um

gapgapgap

In tact grating

Measurement of Phase Shift Induced by the Interaction Grating

0.25

0.20

0.15

0.10

0.05

cont

rast

interaction grating position

4.2

4.0

3.8

3.6

3.4

3.2

3.0

phase [rad]

contrast phase

A B C B A B

A

B

C

Interaction grating position

(raw data 5 sec/ pt)

Phase Shift Induced by the Interaction Grating (averaged data)

• Phase shift induced by grating is φo=.22+/-.03 rad

• 90 seconds of data total• C3 = 4.0 +/- 1.0 meV nm3

60

50

40

30

Inte

nsity

[k/

sec]

-200 -100 0grating position (nm)

A

B

C

-0.2

-0.1

0.0

0.1

0.2

Pha

se

A B C

See talk byJohn Perreault

about theinterferometerexperiment.

L1 L2

-4-2024

200

150

100

500

-4 -2 0 2 4

200150

10050

0

A

B

C

D

Expt. #4: Measure the far-field 1 and 2

Four different interferometers:

Amplitude and phase of nth order change with C3/velocity

Velocity / [km/s] Velocity / [km/s]

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

Predicted Phase shifts in red

Predicted Phase shifts in red

** *

* = preliminary data

Summary of4 Experiments Theory

velocity

twistInteraction grating

4-IFMs

Na + SiNx

Na + Na metal

Na valence + =∞Na with core + e=∞

10

8

6

4

2

0

C3

(meV

nm

3 )

No extra interaction needed to explain the data

Nano-Structure Grating in an Electron Microscope

twist lever

Grating

O. Lens

Grating

4 µm wire 4 µm wire

What About Electron Waves?

Images of a single wire with diffracted e-beam.

4 keV beam

twist = -10±2°

1.5 keV beam

twist = 5±3°

w

d

l

140

120

100

80

60

40

20

0

-50 0 50

120

100

80

60

40

20

0

-50 0 50

=5.4o

=-2o

500 eV electron beam

w

d

l

140

120

100

80

60

40

20

0

-50 0 50

120

100

80

60

40

20

0

-50 0 50

=5.4o

=-2o

500 eV electron beam Envelope from same theory (image chg)

diffraction profiles - comparison

120

100

80

60

40

20

0

-60 -40 -20 0 20 40 60

twist = 5±3°, 500 eV

100

80

60

40

20

0

-60 -40 -20 0 20 40 60

twist = 10±2°, 500 eV

60

40

20

0

-60 -40 -20 0 20 40 60

twist = -10±2°, 500 eV

140

120

100

80

60

40

20

0

-60 -40 -20 0 20 40 60

twist = 0±2°, 500 eV

200

150

100

50

0

-60 -40 -20 0 20 40 60

twist = 5±3°, 1.5 keV

140

120

100

80

60

40

20

0

-60 -40 -20 0 20 40 60

twist = 10±2°, 1.5 keV

100

80

60

40

20

0

-60 -40 -20 0 20 40 60

twist = -10±2°, 1.5 keV

150

100

50

0

-60 -40 -20 0 20 40 60

twist = 0±2°, 1.5 keV

140

120

100

80

60

40

20

0

-40 -20 0 20 40

twist = 10±2°, 4 keV

150

100

50

0

-40 -20 0 20 40

twist = 5±3°, 4 keV

160

140

120

100

80

60

40

20

0

-40 -20 0 20 40

twist = -10±2°, 4 keV

160

140

120

100

80

60

40

20

0

-40 -20 0 20 40

twist = 0±2°, 4 keV

11°:

5°:

-2°:

-12°:

500 eV 1.5 keV 4 keV

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) = e2 (1-)/(1+) r = 1 eV nm / r

Impact of atom-surface & electron-surface V(r)

• measured C3 = 3 (1) meVnm3 four different ways

• Flux is diverted from 0th 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?

105

106

107

Inte

nsi

ty

-4 -3 -2 -1 0 1 2 3 4order

C3=0 C3=5 C2=1.2 C4=30 C5=1000

Computed usingPhasor method matching I1 / I0

Different power-law potentialsmake distinct diffraction envelopes.

V(r) = -Cn/rn

r=0n=2n=3n=4n=5

105

106

107

Inte

nsity

-4 -3 -2 -1 0 1 2 3 4order

C3=0 C3=5 C2=1.2 C4=30 C5=1000

Computed usingPhasor method matching I1 / I0

105

106

107

Inte

nsity

-4 -3 -2 -1 0 1 2 3 4order

C3=0 C3=5 C2=1.2 C4=30 C5=1000

Computed usingPhasor method matching I1 / I0

Cn chosen to match I1/I0

Impact: Performance of interferometer.Figure of Merit = C sqrt(N).

0.015

0.010

0.005

0.000

C (

I/Io

)1/2

20151050C3

w1 = .56w2 = .50w3 = .37

w1 = .70w2 = .80w3 = .37

• Thank you

40 45 50 55 60 65 70 75 800

1

2

3

4

5

6

w

4050

6070

80

10001500

20002500

30000

2

4

6

wv

C3 = 5

Student Version of MATLAB

Model of I2/I3

I2/I3

different velocities

W

Atom Interferometer

)cos(|2|| vdW

)()( vdW

xkLULLUUFlux

eLowerUppere

g

xkitkyi g

Calculation Ref 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 radobserved

Measurement of C3 Using an Atom Interferometer

• Measured C3 consistent with vdW for Na and silicon nitride.

• Stat. Uncertainty in φo can be reduced to 2% in 1 hr.

• Uncertainty in grating geometry will permit 5% level for C3.

φo = .22 +/- .03 rad

C3 = 4.0 +/- 1.0 meV nm3

0.30

0.20

0.10 vdW

[rad

]

86420C3 [meV nm

3]

Questions:

• What about far field phase shifts?

• Can we detect electron-surface interactions?

• is 20% of C3 from core electrons correct?

• What is UvdW for a structure?

• How does radiation modify UvdW ?

• Can there be vdW friction?