Matter wave interferomery with poorly collimated beams

x [µm]

Ben McMorran, Alex CroninDepartment of Physics

x [µm]

main idea

can get matter wave interference fringes with uncollimated beams but:

• grating position matters

• spatial coherence matters

• beam divergence matters

• grating alignment matters

we’ve got a way to model this

outline

1. partial coherence in grating interferometers

2. examples of grating matter wave interferometersMach-Zehnder atom interferometer

Talbot-Lau C60 interferometer

Lau electron interferometer

3. grating alignment sensitivity

4. ideas for g measurement using uncollimated beam

partially coherent optical field

partially coherent optical field

partially coherent optical field

intensity I(x)

complex degree of coherence µ(x)

… We simulate (1) the Talbot effect, (2) far-field diffraction, (3) Mach Zehnder interferometers(4) Talbot-Lau Interferometers(5) Lau interferometers …

A Model for Partial Coherence and Wavefront Curvature in Grating Interferometers

PRA (June 2008)

Mutual Intensity Function:

Intensity:

z

(ρa ,z)

(ρb ,z)

Mutual Intensity Function:

Intensity:

GSM:

w0

σ0

ρaρb

Mutual Intensity Function:

Intensity:

GSM:

Mutual Intensity Function:

Intensity:

GSM:

partially coherent Fresnel optics

a second grating in the far field

a second grating in the far field

Atom Interferometer

Objective: Pioneer new techniques using matter-wave interference to make precision measurements.

• Study quantum decoherence,• Matter-wave index of refraction,• Atomic polarizability.

Approach: 3 nano-fabricated diffraction gratings.

• Mach-Zhender interferometer for atom-waves..

Interferometer Performance:

• Up to 50% contrast.• Small phase drift (< 2 rad / hr).• Layout is easily changed for new experiments.• Macroscopic (100 m) path separation.

gratings for matter waves

100nm

1.5µm

Second Grating

Optical Grating

atom beam

Na

skimmer

S1 = 10µm`

L = 1m

S2 = 10µm

v = 1km/s λ = 17pm

α

α = (S1+S2)/L ~ 10-5

θdiff = λ/d ~ 10-4

ℓ = λL/S1 ~ 1µm

atom beam

Na

skimmer

S1 = 10µm`

L = 1m

S2 = 10µm

α

“Gaussian Schell Source as Model for Slit-Collimated Atomic and Molecular Beams”

McMorran, Cronin arXiv:0804.1162 (2008)

ℓ > d coherent diffractionθdiff / α = 10 resolved diffraction

but

β = ℓ/S1 ~ 0.1 partially coherent

1

2

4

10

2

4

100

2

4

Ato

m F

lux

( k

/se

c )

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

Atom Diffraction:

Atom Interference Fringes:

01

3

42

6

5

7

100

80

60

40

20

0

inte

nsity

[kco

unts

/sec

]

4003002001000-100grating position [nm]

54

22 cos( )gFlux U U L L U L ik x

( ) gik xi ky wte Upper Lower e

Atom Interferometer

100

80

60

40

20

0inte

nsity

[kco

unts

/sec

]

4002000position [nm]

C = 24.7%

Atom fringes

inte

nsity

add a second grating

add a second grating

“Talbot-Lau fringes”

“Matter-Wave Interferometer for Large Molecules”Brezger, Hackermüller, Uttenthaler, Petschinka, Arndt, Zeilinger

Physical Review Letters 88 100404-1 (2002)

S1 = 1.2mm

S2 = 0.5mmL = 1.38m

α ~ 10-3

θdiff ~ 10-6

ℓ ~ 10nm

add a second grating

add a second grating

coarse fringes in the far field:

“Lau fringes”

1µm

electron interferometery with two gratings

aperture

magnetic lens

stationary beam

grating 1

imaging detector

grating 2

Cronin and McMorran, PRA 74 (2006) 061602(R)

α ~ 10-3 θdiff ~ 10-4 ℓ ~ 5nm

Lau interferometer

G1 G2

incoherent source

• each opening of G1 acts as a point source for a diffraction pattern from G2

• at certain grating separations, diffraction patterns overlap

z12

A

z

ML S

G1G2 x

y

z

CCD

Lau interferometer – fringe contrast vs. grating separation

Lau interferometer – fringe contrast vs. grating separation

2.5

2.0

1.5

1.0

0.5

G1

-G2

se

pa

ratio

n (

mm

)

0.80.60.40.20.0CCD pixel location (mm) Cronin and McMorran, PRA 74 061602(R) (2006)

Lau interferometer –twist gratings to measure coherence

z0

G1

G2

xy

z

z1

z2

z3

θ

GSM source

Lau fringe visibility

23

3max

sin

2

3

23

)(z

deV yd

z

y

Lau interferometer – fringe contrast vs. grating rotation

alignment sensitivity depends on coherence parallel to grating slits

ℓ0 ≈ 10 nm

some figures:

antihydrogen incident on 1µm period gratings 1m from source:

v = 10 km/s: λ = 0.4Å S1 < 40µm for coherent diffraction (ℓ > d) v = 5 km/s: λ = 0.8Å S1 < 80µmv = 1 km/s: λ = 4.0Å S1 < 400µm

T = 4K Δvx = 260m/s

position echo interferometer

position echo interferometer?

Mach-Zehnder position echo

position echo interferometer?

• fine-spaced interference fringes

precision for measuring deflection

• uncollimated

more counts from wider slits

• integrate across w

more counts looking at all paths

position echo interferometer?

position echo interferometer?

NEEDS FURTHER STUDY WITH REALISTIC PARAMETERS

conclusion

• simulations + experiments:

matter wave fringes can be formed with uncollimated beams

• necessary to think about partial coherence

• less coherence parallel to slits contrast sensitive to grating misalignment

• position echo behind 2 gratings useful for measure g?

• we have a tool to model this