Anewdeterminationof withcoldrubidium atoms P. Clad S. - - PowerPoint PPT Presentation

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

• S. Guellati-Khélifa
• C. Schwob
• F. Nez
• L. Julien
• F. Biraben
• P. Cladé
• M. Cadoret
• E. De Mirandes

Anewdeterminationofα withcoldrubidium atoms

Determination of the finestructureconstant

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

X r r

m h e A X A c R c e × × =         =

∞

) ( ) ( 2 4

2 2 2

• πε

α quantum Hall effect

137.035 990 137.036 000 137.036 010

hfs muonium h / m(neutron) h / m(Cs)

QED h / m Solid state physics

g – 2 of the electron (UW)

α-1

Γ Γ Γ Γ’p,h-90 h / m(Rb) RK=h/e2=

= = =µ0c/2α

ae = f (α/π) mv=h/λ g – 2 of the electron (Harvard) vr=ħk/m

• G. Gabrielse et al, PRL, 97, 030802, 2006

Principleofourexperiment:measurement

• ftherecoilvelocity

σvr =σv /(2N) N × 2ħk

coherent acceleration measurement (Raman transition) selection (Raman transition) MOT + molasses selection of an initial sub-recoil velocity class

J.L. Hall, Ch.J. Bordé, K. Uehara, PRL 37 (1976) 1339

87Rb

5S1/2 5P3/2 F=2 F=1 ∆ coherent acceleration : N Bloch oscillations,

momentum transfer 2Nħk

measurement of the final velocity class

Blochoscillations

Acceleration ⇔ Bloch oscillations in the fundamental energy band

k

• 2

Only one hyperfin level involved : coherent acceleration, per cycle

M.BenDahan etal,PRL,76 76 76 76 (1996)4508.

Accelerated frame Laboratory frame

E=P2/2m p

δ’=10kvr δ’=6kvr δ’=2kvr

ν1 ν2

k

• 2

k

• 4

k

• 6

Two possibilities with verticalbeams

Acceleration

c h N mgt m ν 2 v × − = ∆

up and down accelerations + differential measurement Measurement of h/m independent of g

Vertical standing wave

The atoms oscillate at the same place with the frequency

k mg

B

• 2

= ν

measurement de h/m gravimeter

m

g

• G.Ferrarietal,PRL,97

97 97 97 (2006)060402.

4000 oscillations in 7 s!

Experimentalsequence

Acceleration in both opposite directions : ) N N ( 2 V V v

down up down up r

+ ∆ − ∆ = m k v

B r

• =

( ) ( ) ( ) B

2 1 down up down meas sel up meas sel

k k k ) N N ( 2 m + + δ − δ − δ − δ =

• MOT

+ molasses detection blow away beam selection π-pulse (δsel fixed) measurement π-pulse (δmeas tunable) acceleration deceleration

( ) ( )

2 1

k k V

meas sel

+ − = ∆ δ δ

• We measure (Doppler effect) :

about 450 Bloch oscillations up and down → 1800 recoils

Results

Transfer efficiency > 99.95% per oscillation (2 recoils)

Cladé et al, PRL, 96 (2006) 033001

statistical uncertainty on α = 4.4×10-9 total uncertainty on α = 6.7 ×10-9 α-1 = 137.035 998 84 (91) measurements performed in April 2005

10-7

61

1 point = 4 spectra 1 point = 1 sequence

Errorbudget

Source Correction Uncertainty (α-1)(ppb) (α-1)(ppb) Laser frequencies 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) 0.2 Light shift (two photon transition)

• 0.5

0.2 Light shift (Bloch oscillations) 0.46 0.4 Index of refraction (cold atomic cloud) <0.1 0.3 Index of refraction (background vapor) - 0.37 0.3 Global systematic effects

• 5.49

5.0 Statistical uncertainty 4.4 TOTAL 6.7

Cladé et al (submitted to PRA) α-1 = 137.035 998 84 (91)

) v v ( 2 ) (

r R a C b C T C

kT E E

R

+ = − = ∆

• φ

Interferometric measurement of the recoil velocity

Ramsey interferometer TR π/2 π/2 F=2 F=1

v0 v0+2vr

R laser

T δ φ = ∆

r R C

kT v 4 − = ∆φ

independent of v0 measure 2vr

Ramsey-Bordé interferometer

v0 v0+2vr v0+4vr

π/2 π/2 π/2 π/2

N π-pulses

r R C

T N k v ) 1 ( 4 + − = ∆φ

measure 2N vr

• A. Wicht, J.M. Hensley, E. Sarajlic and S.Chu, Phys. Scr. T102, 82 (2002)

120 recoils transferred uncertainty on α = 7.4× 10-9

Blochoscillationsand atomic interferometry

• 10
• 5

5 10

• 15
• 10
• 5

5 10 15

π/2 π/2 π/2 π/2

selection F=2 → F=1 measurement F=1 → F=2 acceleration deceleration detection blow away beam TR TR

π/2 π/2 π/2 π/2 TR TR

N Bloch oscillations

v0 v + 2 N v

r

v -2Nv

r

Up to 480 oscillations !

Preliminary tests

TR=3.4 ms π/2-pulse duration = 0.3 ms Raman = 250 GHz and Bloch= 40 GHz = π-pulse duration typically : 350 oscillations statistical uncertainty for 5 determinations of α = 7.5×10-9 4 spectra in « Rabi » configuration h/mRb at 6.6×10-8 4 spectra in « Ramsey » configuration h/mRb at 2.9×10-8 promising!

Further improvements

Oscillations de Bloch (at the present time N ~ 480) The number of Bloch oscillations is limited by the atomic longitudinal motion (500

• scillations & 12 ms , 6 cm).

Velocity measurement (at the present time σv ~ 10-4 vr in 10 minutes)

• a new vacuum cell and a 2D-MOT to increase the initial number of atoms.
• an actively stabilized anti-vibration plateforme to reduce vibrations.

Statistical uncertainty Systematic effects

• a µ-metal shielding to reduce residual magnetic fields
• a Shack-Hartmann wave front analyser to control the beams curvature

~10-9

2N

v vr

σ σ =

Towards aredefinition ofthekilogram

The kilogram is the only SI base unit defined in terms of a material artefact It is not invariable at a level of 10- 8

!"#

Realization of the kg using the watt balance which allows to compare :

• a mechanical power (displacement of a mass in the gravity field)
• to an electrical power

One possible way : This realization is based on the validity of the relations :

α µ 2

2

c e h RK = =

h e K J 2 =

Josephson constant and Von Klitzing constant Need to be tested !

Fix the Planck constant h and relate mass and time units

2

mc h E = = ν

• h

K R UI

J K

4

2 =

∝

v Mg

Conclusion

Highly precise frequency measurements allow very accurate determinations

• f fundamental constants leading to a lot of rich developments…

Another possibility Fix the Avogadro constant (or the atomic mass unit)

• At the present time, NA is measured through the molar volume of a Si sphere

!"# Morever The watt balance gives h/Mmacro Recoil measurements give h/Matom both together can give a competitive value of NA

Recent proposal Fix both h and NA !

• $%&

!!!'( !(# (on going debate in the community of metrologists)

ρ: density Γ: natural width ∆: detuning

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

( )

3

2 1 n       π λ ∆ Γ ρ π = −

• 2

2 n k Γ σ = ∆ For the background vapor density: 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 atoms Initial atomic density : 1011 atoms/cm3 (n-1) ~ 4.10-10

Refractive index

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 Dispersive medium Atoms

n : index of refraction

Bloch oscillations :

if η= 100% Index of refraction

Accelerated atoms dispersive medium

ω’= ω - kvatom

• therwise ~(1-η)(n-1)

+ (n-1)kvmedium n

Raman transition :

vmedium vatom

Atomic cloud

L ω’= ω - kvatom+ (n-1)k(vmedium-vatom) dL/dt = 0 ⇔ vmedium=vatom no effect

k N N k n k n p final

• 2

) 1 ( 2 2

1 =

− + =

2(1-n)N1/N0 k

• 2n k

Refractive index

Phase of the light (1) at the position of the atom i (xi) : Φ1(xi) Two photon transition : Φ =Φ1-Φ2 Assum: without dispersive media : Φ(x) = 2 k x inside the medium : dΦ(x)/dx = 2 nk uniform medium (N atoms), xm of the center of the medium : xm = Σi xi/N at the position xm of the center of the medium effect of refractive index cancel from 1st and 2nd beam One Bloch oscillation :

• atom
• medium

Raman transition : Doppler effect

kx 2 ) x x ( k ) 1 n ( 2 ) x (

m +

− − = Φ k n 2 N k ) n 1 ( 2 k n 2 dx ) x ( d

i i

• ≈

− + = Φ N k ) n 1 ( 2 dx ) x ( d

j i

• −

= Φ ) v v ( k ) 1 n ( 2 kv 2 dt ) t ), t ( x ( d

'

− − + − ω = ω → Φ

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 rad θB=1.6×10-4 rad

Correction on α-1: -( 2 ± 2 )×10-9

θr θB kB kB kR kR

ur(α) = 8 × 10-10

Systematic effects Gravity gradient :

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

Level shifts :

• Light shift

F=2 F=1 5P3/2

k2 k1

ω ω ω ωSHF

∆=1050GHz Expansion 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 ↔ k1

R

Correction on α-1 ~ (6.6 ± 2) × 10-9

Quadratic magnetic force :

Correction on α-1~ ~ 10-10

R t g 2 −

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

r

v N 2 t ) M / F (

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 2 2 2 2 2 //

c k c k ω ω < − =

⊥

( ) ( )

( )

( ) ( )

z , r z , r k avec k p p e r E z , r E

z eff eff z , r i

φ ∂ = + → =

φ

• (

) ( )

R 2 r k z z k z , r

2 G

+ − = φ φ

( )

k dz dR R r z w k k dz d keff × × − − = =

2 2 2

2 2 φ

Gouy phase and wave front curvature

Possible realization of Avogadro constant (ccsd-00084607)

u r u r u r u u A

M ) X ( A ) X ( m h h 1 M ) X ( A m ) X ( A h h 1 m M N = = =

u 87 r ) w ( ) a ( ) a ( 87 ) w ( K 2 J ) 1 ( A

M ) Rb ( A g g g ) Rb ( m h 4 g R K N                   =

Watt balance Bloch oscillations (stationary) Relative gravimeters

u 87 r 87 K 2 J ) 2 ( A

M ) Rb ( A ) Rb ( m h 4 R K N             =

Watt balance h/m experiments

Cold atom experiment and Watt balance → realization of NA (g or h/m) (h if RK=h/e2 and KJ=2e/h are exact)

Velocity Energy m k1

1 1 1

k2

2 2 2

Light shift

Quadratic Zeeman shift

∆ ∆ ∆ ∆Zeeman

∆ ∆ ∆ ∆ v

Reduction of constant systematic shifts

Velocity Energy m k2

2 2 2

k1

1 1 1

Compensation of energy shifts by inverting the direction of Raman beams

∆ ∆ ∆ ∆Zeeman

∆ ∆ ∆ ∆ v

Two spectra →

• ne velocity measurement

2£10-3 vr

Light shift

Quadratic Zeeman shift

Reduction of systematic shifts