Accélération d’atomes ultra-froids :vers une mesure de h/M et de α

Laboratoire Kastler Brossel, Paris (ENS, CNRS, UPMC)

BNM - Institut National de Métrologie, Paris (CNAM)

P. Cladé, R. Battesti, S. Guellati, C. Schwob,B. Grémaud, F. Nez, L. Julien, F. Biraben

Determination of the fine structure constant

Rydberg constant(7 × 10-12)

(2 × 10-10)

atomic massproton mass

(4.6 × 10-10)

proton masselectron mass

h/MA α

Ae

p

p

A

Mh

mm

mM

cR

×××= ∞22α22

21 αcmchR e=∞

Hydrogen atom

• measurement of h/MA at 6 × 10-8

determination of α at 3 × 10-8

• total dispersion of α measurements: 2.4 ×10-7

(CODATA 1998)

Measurements of αCodata98

quantum Hall effectCodata02

muonium

h / M(neutron)h / M(Cs)

QED

h / M

Solid-statephysics

g – 2 of the electron

α-1

2e/h RK=h/e2=µ0c/2α

s.h.f

ae = f (α/π)

Mv=h/λatomic interferometry

10-7

Γ’ (lo)p,h-90

137,035 990 137,036 000 137,036 010

The principle of the experiment :measurement of the atomic recoil

velocity

a

b

ωħk MA MA vrecoil

AMkh

=recoilv 16v87 −⋅≈ smmRbrecoil

measurement of vrecoil and k ⇒ h/MA

Raman transitions (1) : velocity selection

MA

•coherent momentum transfert

k1 k2

v

e

ω1ω2

ab

δ

∆

r

( )21v kkM A −=∆ h

( ) ( )A

ab Mkkkkr 2

v22

212121

rr

hrrr −

−−⋅−−−= ωωωδπ

( )rab kkkkr vv2 and 0 2121 ++≈−⇒−≈== ωωωδ

•control of δr control of v•two hyperfine levels involved velocity selection and measurement

Raman transitions (2) : acceleration of atoms

p

E=p /2M2

2hk 4hk 6hk

ω 1 ω 2

δ’=2kvδ’=6kvδ’=10kvr

r

r

ω , k11 ω , k22

ω1 ω2

a

∆

MA

e

1)(2N2kvδ r −=′

bδ’

Acceleration ⇔ Bloch oscillations

87Rb

δR

D2780 nm

5S1/2

5P3/2

F=2F=1

∆Temporal sequence

initial velocitydistribution

subrecoilvelocity

class selection

coherentacceleration

measurementof the final

velocitydistribution

F=2

F=1

F=2→F=1 F=1→F=2

MOT andoptical molasses

«Raman» «Bloch» «Raman»

detectiontime of flight

t

blow away beam

12 ννν −=sel fixed '12 ννν −=meas tuned

Horizontal acceleration of atoms (1)

221 2).(221

cMhN

MkkkN

A

RB

ABlochmeassel

ννπ

νν =−=−hrrr

RamanBloch g

r

For , atoms miss the detection area.4≥N

MA

Problem !

Detection chamber Probe

Horizontal acceleration of atoms (2)

F=2→F=1 F=1→F=2

MOT andoptical

molasses

«Bloch»initial acceleration

«Raman»

«Bloch»acceleration

«Raman»

detectiontime of fight

t

gr

BlochRaman MA

Detection chamber Probe

Experimental curves (50 oscillations)to reduce systematic errors alternate and symmetric accelerations

in horizontal opposite directions

δ0= + (1 506 799,5 ± 3,1) Hz

νmeas – νsel (MHz)1.506 1.507

δ0= - (1 506 531,2 ± 3,1) Hz

-1.507 -1.506νmeas – νsel (MHz)

deduced recoil : 15 066, 690 (23) Hz

«+100 vr» «-100 vr»

uncertainty: 6105,1 −×

~ 1 kHz

Horizontal geometry : results

137,035930 137,035970 137,036010

QuantumHall effect

Muoniumge-2

h/mnh/MCs

h/MRb

α−1

CODATA 02

Γ’ (lo)p,h-90

h/MRb relative uncertainty : 4.2 × 10-7

Gap between our valueand the expected one : 6.1 × 10-7

χ2 = 99 for 43 measurements.

η (per Bloch oscillation) = 99.5%

R.Battesti et al, Phys. Rev. Lett. 92, 253001 (2004)

Vertical configurations (1)

Vertical acceleration with Bloch oscillations

Bloch Raman

MAgr

chNgtMM AAν2v ×−=∆

( )ν2

v cN

gtMh

A

∆−=

differential measurement («up» and «down» accelerations)independent of g

Vertical configurations (2)

Vertical beams so that the force due to the Raman transitions exactly compensates gravity

Bloch Raman

MAgr

atoms submitted to a verticalstanding wave

E

p

ν ν

-ħk + ħk

In the standing wave, theatom oscillates at the sameplace with the frequency:

kgM A

Bh2

=ν

Present work: vertical geometry

N/(N+N)

212

vertical acceleration of the atoms

• 80 recoil transfers

• careful investigations of experimental limiting factorsand systematic effects

observation of Bloch oscillations in the vertical standing wave

Experimental curves in vertical geometry

0

0,05

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

0,5

-2110500 -2110000 -2109500 -2109000 -2108500 -2108000

δ0= - (2 109 260,6 ± 1,7) Hz

center of the curve: 1,7 Hz 9000vr in 10 minutes80 oscillations

( )Hzmeassel νν −

~ 1 kHz

Inverting the sens of photons: a way to reduce systematic effects

6102 −×(h/M)relative

e

ν1ν2

ab

with ∆(x,t) a systematic effect (light shift, quadratic Zeeman effect (mF=0)..)

( ) [ ])),((2vmeas12 txkk HFS ∆+−=⋅− νδπ

( ) [ ])),((2v'meas21 txkk HFS ∆+−′=⋅− νδπ

( )txh

EEHFS

ab ,∆+=− ν

Raman «plus»

MA

ν 1 , k 1

ν2 , k2

v

ν 1 , k 1

Raman «minus»

MA

ν2 , k2

'v

( )δδπ−′=

k2vmeas independent of ∆(x,t)

kkk =≈ 21

Zeeman effect

-B0 -B0B0

-B0

B0

6102 −×(h/M

)

ħk2 ħk1

ba

MAħk2 ħk1

v

MA ħk2ħk1

v

∆Zee=K(2)B2 ; K(2) ~ 514 Hz/G2

6102 −×

500mG

(h/M

)

∆Zee(zselec)-∆Zee(zmes)~ B0 ∇B ∆z

Applied magnetic field(quantification axis) Residual gradiant

(50mG/cm)

Vertical motion(~1cm)

Zeevδ ∆=⋅rrr

)k-k( 12

Zeevδ ∆=⋅rrr

)k-k( 21

∆HFS+∆Zee

B0

ProspectsFuture : new cell + slow atoms source● better vacuum● better efficiency to load the MOT● magnetic shielding

µ-metal boxes

2D MOT

3D MOT +molasses

Uncertainties (1)

Lasers wavelenghtsuncertainty of 10 MHz on the Raman and Bloch laser frequencies

5.2 × 10-8

Angle between the different laser beamsestimation based on the fiber diameter (5 µm) and the focal lenghts (40 mm) → 63 µrd 2 × 10-9

RCWave fronts curvature

radius of curvature RC > 20 m

vertical motion in 20 ms ∼ 2 mm → θ ∼ 10-4 rd 5 × 10-9

Uncertainties (2) 5P3/2

F=2

∆

F=1

ω1ω2

ωHFSdifferential effect (I1=I2 and π pulse):

8 × 10-6τ ∼ 1.6 ms ∆ ∼ 340 GHz → 26 Hz

differential effect between the measurements +/- 50 oscillations 8 × 10-8

Light shifts

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+∆

+∆

Γ==

HFSs

III

Fωπ

δ 212

8211

( ) ( )∆

−==−= HFSFF ωτ

δδ 212(N=50)

Magnetic fields fluctuations

transition between mF = 0 sub-levelsmagnetic field of 150 mG → 9,6 HzFluctuations [(Bsel − Bmes)+50 − (Bsel − Bmes) − 50] ∼ 1mG → 0,13 Hz

4 × 10-8

Comparison with S. Chu’s experiment

F=2

F=1

π

π

F=2

OBOB

Rubidium

π/2π

π

π/2

F=4

F=3Césium

vertical geometry

atomic interferometry : fringes width of 8 Hz (500 Hz in ourexperiment)

transfert by resonant adiabatic passages betweenthe F=3 and F=4 levels: efficiency 94% (99,5 % in our experiment)

interferometry: sensitivity to gradients (magnetic fields, intensity…)Bloch oscillation in the gravity field: sensitivity to gravity fluctuations

N2π pulse

caracteristic width ~ 1 kHz (vrecoil / 15)

sub-recoil velocity class → vrecoil/300 2 4-2-4

0.1

0.2

0.3F=2 F=1

π

π

F=2

[δνmes-δνselec](kHz)Ramsey fringes

-10 -5 0 5 10(kHz)

π/2 π/2

π/2

π/2F=2

F=1