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Laboratoire Kastler Brossel (ENS, CNRS, UPMC) Institut National de Métrologie (CNAM)

PermanentsS. Guellati-KhélifaC. SchwobF. NezL. JulienF. Biraben

Ph.D StudentsP. CladéM. CadoretE. De Mirandes

A new determination of the fine structure constant based on Bloch oscillations of 87 Rb atoms in a vertical optical lattice

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quantum Hall effect

137.035 990 137.036 000 137.036 010

hfs muonium

h / m(neutron)h / m(Cs)

QED

h / m

Solid statephysics

g 2 of the electron (UW)

α-1

Γp,h-90

Codata98

Codata02

h / m(Rb)

Determinations of the fine structure constant α

c4e

0

2

hπε=α

CODATA 2002 P. Mohr and B. Taylor, RMP, 77, n°1, p. 1, january 2005

RK=h/e2=µ0c/2α

ae = f (α/π)

mv=h/λg 2 of the electron (Harvard)

( )( )Xmh

)e(AXA

cR2

r

r2 ∞=α

3

Two photon transition : defined momentum transfer

Low spontaneous emission

Doppler sensitive

Controllable width of the transition (pulse duration) :

F=2

F=1

5P3/2

mk2 k1

ωSHF

∆

∆ΩΩ

∝Ω 21eff

(M. Kasevich et al, Phys. Rev. Lett 66, (1991) 2297)

Recoil velocity Raman transition

a

b

νħk m

vr m2kmv

21E

mkv

222rr

r

h

h

==

=

4

Principle of our experiment

Final uncertainty : σvr = σv / (2N)

N × 2ħk

Coherent acceleration

measurement(Raman transition)

selection(Raman transition)

MOT +molasses

! selection of an initial sub-recoil velocity class! coherent acceleration : N Bloch oscillations, momentum transfer 2Nħk! measurement of the final velocity class

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Momentum

Ener

gyV

eloc

ity d

istri

butio

n

|bi

~δcenter

Ener

gy

Momentum

Determination of the center of velocity distribution

F=1

Raman detuningδcenter

⇒

1) Subrecoil velocity selection 3) Velocity measurement

F=2

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F=1

MOT +Optical molasses

Selection: F=2 → F=1

δsel

π-pulse

Tuned FrequencyDetection

Measurement:F=1 → F=2

δmeas

π-pulse

-5vr 0 +5 vr-5vr 0 +5 vr-5vr 0 +5 vr

F=2

F=1

Blow awaybeam

5S1/2

F=2

5P3/2

87RbVelocity sensor

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"intensity: 100mW.cm-2 ; detuning ∆ ~ 1050 GHz

"1,8 s per point (loading of the MOT)

"300 points (~10 minutes)

~vr/50

N 2/(

N 1+

N 2)

Velocity measurement : results

15000vHz1 r

v →≈σ

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Velocity

Ener

gymk1 k2

# Light shift

# Quadratic Zeeman shift

∆Zeeman

∆ v

Reduction of constant systematic shifts

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Velocity

Ener

gymk2 k1

Compensation of energy shifts by inverting the direction of Raman beams

∆Zeeman

∆ v

Two spectra →one velocity measurement

2×10

-3v r

# Light shift

# Quadratic Zeeman shift

Reduction of systematic shifts

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Coherent acceleration of atoms

Raman transitions in a same internal state

E = P2/2M

p2~k 4~k 6~k

δ= 10 k vr

δ= 6 k vr δ= 2 k vr

ν1

ν2

Laboratory frame

( ) tak222 21' =−= υυπδπ

coherent acceleration, 2 ~ k per cycle

E

p

ν1 ν1

+hk−hk

ν1

ν1

g

νBmg2 k

=h

Acceleration Stationary

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E

p

n, n

n-1, n+1 n-2, n+2 n-3, n+3

0 2hk 4hk

Coherent acceleration of atoms

Bloch oscillations in the fundamental energy band

M. Ben Dahan et al , Phys. Rev. Lett. 76 (1996) 4508.

E. Peik et al, Phys. Rev. A 55 (1997) 2989.

Dressed atom

Accelerated frame

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t1, N1t2, N2

! Differential measurement to cancel systematic effects

! Atomic cloud dont move between selection and measurement → g measurement at the same place

Stationary experiment

Measurement of g/vr à 10-6

Cladé et al, Europhys Lett. (2005) p.730

( )

( ) ( )21

21r21

r

Biii

ttNN2vvv

vg

Ntgv

−−+−=

τ−=

τB=1,2 ms and U0 = 2,7 Ermg

k2B

h=τ E

p

ν1ν1

+hk−hk

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MOTOptical molasses

87RbDetection

blow awaybeam

Selection: F=2 → F=1

δsel

π-pulse

Measurement: F=1 → F=2

δmeas

π-pulse

Acceleration accelerationDeceleration

Velocity variation

t : time between the two Raman pulsesN: Number of Bloch oscillations

Acceleration in both opposite directions

Experimental sequence (acceleration)

Acceleration

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Blow-awaybeam

Bloch beam

Raman beam 1and

Raman beam 2

Bloch beam

Detection beams

Signal from F=2Signal from F=1

Raman beam 2

Co ld atomic cloud

Repumping beam

Partially reflect ingplate

λ/2 waveplate

gUpper optical fiber

Lower op tical fiber

Experimental set up

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about 450 Bloch oscillations in each direction : 1800 recoils

ResultsTransfer efficiency : > 99.95% per oscillation (2 recoils)

measurements performed in April 2005

α-1 = 137.035 998 78 (91) 10-7

-1

uncertainty 6.7 × 10- 9

Cladé et al, PRL 96, 033001 (2006)

1 point = 4 spectra

Statistical uncertainty on α = 4.4×10-9

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Systematic effects

!Lasers frequencies : FP cavity → uncertainty 300kHz →

!Beams misalignment : Optical fibers to couple Raman/Bloch beams into the cell

maximum misalignment :

θr = 3×10-5 radθB=1.6×10-4 rad

Correction on α-1: -( 2 ± 2 )×10-9

θr θB

kB

kB

kR

kR

ur(α) = 8 × 10-10

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Systematic effects!Gravity gradient :

R : Earth radiust : spacing time / sel-meas = 12 ms

!Level shifts :- Light shift

F=2

F=1

5P3/2

k2

k1

ωSHF

∆=1050GHzExpansion of the cloud ∆I=10% when k²R ↔ k1

R

ur(α) = 3 × 10-10

-Magnetic field gradient = trajectory effect

up acc.

down acc.

meas.

meas.

sel.

sel.

∆z=0.3mm when k²R ↔ k1R

Correction on α-1 ~ (6.6 ± 2) × 10-9

!Quadratic magnetic force :

Correction on α-1~ ~ 10-10

Rtg 2

−

Correction on α-1~ ~(-1.3 ± 0.4) × 10-9

rvN2t)M/F(

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! What is the momentum transferred to the atoms by laser beams ?

Momentum transferred = gradient of the phase

Gaussian beam : Plane waves superposition :

! Gaussian beam:

Gouy phase

Curvature radius

keff can be measured with a wave front analyzer (R, w)

2

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2

22// c

kc

k ωω<−= ⊥

( ) ( ) ( ) ( ) ( )z,rz,rkaveckpperEz,rE zeffeffz,ri φ∂=+→= φ h

( ) ( )R2

rkzzkz,r2

G +−= φφ

( )k

dzdR

R2r

zwk2kk 2

2

2eff ××−−=

Gouy phase and wave front curvature

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ρ: densityΓ: natural width∆: detuning

Recoil transmitted by one Bloch oscillation : 2~k or 2n~k ?Doppler effect for the Raman transitions : 2kv or 2nkv ?

( )3

21n

πλ

∆Γ

ρπ=−∆

22

nk Γσ=∆

For the background vapordensity: 8.108 atoms/cm3

Raman beams : ∆= 1050 GHz : (n-1)= 4.10-10 (selection)(n-1)<10-12 (measure)

Bloch beams : ∆ = 40 GHz: (n-1)=2.10-10 (selected atoms)

For the cold atomsInitial atomic density : 1011 atoms/cm3

(n-1) ~ 4.10-10

Refractive index

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PRL 94 170403 (2005) (MIT): Photon Recoil Momentum in Dispersive Media

Observation : modification of recoil energy in a dispersive medium (BEC).

N1 << Ntot

Ntot N0 N1

2n~k2(1-n)N1/N0~k

Dispersive medium Atoms

n : index ofrefraction

Bloch oscillations :

pfinal = 2n~k + 2(1-n) ~k Ntot/Ntot = 2~k if η= 100%

Index of refraction

Accelerated atoms $ dispersive medium

ω= ω - kvatom

otherwise ~(1-η)(n-1)

+ (n-1)kvmediumn

Raman transition :

vmedium

vatom

Atomic cloud

Lω= ω - kvatom+ (n-1)k(vmedium-vatom)

dL/dt = 0 ⇔ vmedium=vatom no effect

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Error budgetSource Correction Uncertainty

(α-1)(ppb) (α-1)(ppb)

"Laser frequencies 0 0.8"Beams alignment - 2 2"Wave front curvature and Gouy phase - 8.2 4"2nd order Zeeman effect 6.6 2"Quadratic magnetic force - 1.3 0.4"Gravity gradient - 0.18 0.02"Light shift (one photon transition) 0 0.2"Light shift (two photon transition) - 0.5 0.2"Light shift (Bloch oscillations) -0.41 0.4"Index of refraction (cold atomic cloud) <0.1 0.3"Index of refraction (background vapor) - 0.37 0.3

Global systematic effects - 6.36 5.0Statistical uncertainty 4.4

TOTAL 6.7

Cladé et al (paper in preparation) α-1 = 137.035 998 84 (91)

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2D MOT

µ-metal boxes

MOT +molasse

Improvements in progress in Paris

" anti vibration platform" magnetic shielding" new cell and slow atom source"

Goal : 10- 9 uncertainty