A�new�determination�of� α with�cold�rubidium� atoms P. Cladé S. Guellati-Khélifa M. Cadoret C. Schwob E. De Mirandes F. Nez L. Julien F. Biraben Laboratoire Kastler Brossel (ENS, CNRS, UPMC) Institut National de Métrologie (CNAM)
Determination of the fine�structure�constant� = µ 0 c /2 α R K =h/e 2 = = = quantum Hall effect Solid state Γ Γ ’ p,h-90 Γ Γ physics hfs muonium QED a e = f ( α / π ) g – 2 of the electron (UW) g – 2 of the electron (Harvard) mv=h/ λ h / m h / m(neutron) 2 2 e 2 R A ( X ) h h / m(Cs) α = = × × ∞ 2 r v r = ħ k/m πε � 4 c c A ( e ) m h / m(Rb) 0 r X α -1 137.035 990 137.036 000 137.036 010 CODATA 2002 P. Mohr and B. Taylor, RMP, 77 , n°1, p. 1, january 2005 G. Gabrielse et al, PRL, 97 , 030802, 2006
Principle�of�our�experiment�:�measurement� of�the�recoil�velocity N × 2 ħ k measurement selection (Raman transition) (Raman transition) coherent acceleration MOT + molasses 87 Rb 5P 3/2 ∆ � selection of an initial sub-recoil velocity class � coherent acceleration : N Bloch oscillations, momentum transfer 2N ħ k F=2 5S 1/2 F=1 � measurement of the final velocity class σ vr =� σ v /�(2 N ) J.L. Hall, Ch.J. Bordé, K. Uehara, PRL 37 (1976) 1339
Bloch�oscillations Accelerated frame Laboratory frame E�=�P 2 /2m δ ’ =�10�k�v r ν 2 ν 1 δ ’ =�6�k�v r δ ’ =�2�k�v r p � � � 2 k 4 k 6 k � Only one hyperfin level involved : coherent acceleration, per cycle � 2 k � Acceleration ⇔ Bloch oscillations in the fundamental energy band M.�Ben�Dahan et�al�,�PRL,�76 76 (1996)�4508. 76 76
Two possibilities with vertical�beams m Acceleration Vertical standing wave � g ν 2 h ∆ = − × v m mgt N c The atoms oscillate at the same place with the frequency mg up and down accelerations ν = + B � 2 k differential measurement Measurement of h/m gravimeter measurement de h/m independent of g G.�Ferrari�et�al�,�PRL,�97 97 97 (2006)�060402. 97 4000 oscillations in 7 s!
Experimental�sequence� deceleration acceleration MOT detection + molasses selection blow away measurement π -pulse π -pulse beam ( δ sel fixed) ( δ meas tunable) ( ) δ − δ � ∆ = sel meas We measure (Doppler effect) : V ( ) + k k 1 2 ∆ up − ∆ down V V = Acceleration in both opposite directions : v r up + down 2 ( N N ) � k = B v r m ( ) ( ) δ − δ up − δ − δ down � = sel meas sel meas ( ) B up + down + m 2 ( N N ) k k k 1 2
Results Transfer efficiency > 99.95% per oscillation (2 recoils) about 450 Bloch oscillations up and down → 1800 recoils measurements performed in April 2005 1 point = 4 spectra 10 -7 61 statistical uncertainty on α = 4.4×10 -9 1 point = 1 sequence total uncertainty on α = 6.7 ×10 -9 α -1 = 137.035 998 84 (91) Cladé et al, PRL, 96 (2006) 033001
Error�budget Source 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) 0.46 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)
Interferometric measurement of the recoil velocity Ramsey interferometer ∆ φ = − = + T b a ( E E ) 2 kT ( v v ) R C � C C R 0 r F=1 ∆ φ = δ v 0 +2v r T laser R F=2 v 0 π /2 π /2 T R ∆ φ = − 4 kT v C R r Ramsey-Bordé interferometer independent of v 0 N π - pulses measure 2v r v 0 +4v r ∆ φ = − + 4 k ( N 1 ) T v v 0 +2v r C R r measure 2 N v r v 0 π /2 π /2 π /2 π /2 120 recoils transferred uncertainty on α = 7.4× 10 -9 A. Wicht, J.M. Hensley, E. Sarajlic and S.Chu, Phys. Scr. T102 , 82 (2002)
v N r 2 Bloch�oscillations�and atomic + v 0 interferometry π /2 π /2 v 0 T R π /2 π /2 v -2Nv 0 T R r N Bloch oscillations π/ 2 π/ 2 π/ 2 π/ 2 acceleration deceleration T R T R detection selection measurement blow away F=2 → F=1 F=1 → F=2 beam -10 -5 0 5 10 -15 -10 -5 0 5 10 15
Preliminary tests = π -pulse duration T R =3.4 ms π /2-pulse duration = 0.3 ms � Raman = 250 GHz and � Bloch = 40 GHz Up to 480 oscillations ! typically : 350 oscillations statistical uncertainty for 5 determinations of α = 7.5×10 -9 promising! h/m Rb at 6.6×10 -8 4 spectra in « Rabi » configuration 4 spectra in « Ramsey » configuration h/m Rb at 2.9×10 -8
Further improvements σ σ = v Statistical uncertainty v r 2N Oscillations de Bloch (at the present time N ~ 480) The number of Bloch oscillations is limited by the atomic longitudinal motion (500 oscillations & 12 ms , 6 cm). Velocity measurement (at the present time σ v ~ 10 -4 v r 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. Systematic effects - a µ-metal shielding to reduce residual magnetic fields - a Shack-Hartmann wave front analyser to control the beams curvature ~10 -9
Towards a�redefinition of�the�kilogram 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 � ������������ ����������� ������������ ����� ��������������� ������������������������������ �� �������� !��"# One possible way : ● = ν = Fix the Planck constant h and relate mass and time units 2 E h mc Realization of the kg using the watt balance which allows to compare : - a mechanical power (displacement of a mass in the gravity field) Mg v 2 = - to an electrical power 4 ∝ UI R K K J h This realization is based on the validity of the relations : µ h c 2 e = = = Need to be tested ! 0 R K and K J α 2 2 e h Von Klitzing constant Josephson constant
Another possibility ● Fix the Avogadro constant (or the atomic mass unit) ������������� !��"# At the present time, N A is measured through the molar volume of a Si sphere Morever The watt balance gives h/M macro both together can give a competitive value of N A Recoil measurements give h/M atom Recent proposal ● Fix both h and N A ! � ������������ ���������������$��������%�����������& � ����������������������� �� ��!!��!'( !��(#� (on going debate in the community of metrologists) Conclusion Highly precise frequency measurements allow very accurate determinations of fundamental constants leading to a lot of rich developments…
Refractive index Recoil transmitted by one Bloch oscillation : 2 ~ k or 2n ~ k ? Doppler effect for the Raman transitions : 2kv or 2nkv ? ρ : density 3 σ Γ Γ λ n 2 ( ) ∆ = − = π ρ Γ : natural width k n 1 ∆ π � 2 2 ∆ : detuning For the cold atoms Initial atomic density : 10 11 atoms/cm 3 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 background vapor density: 8.10 8 atoms/cm 3 (n-1) ~ 4.10 -10
Index of refraction PRL 94 170403 (2005) (MIT): Photon Recoil Dispersive medium Atoms Momentum in Dispersive Media N tot N 0 N 1 Observation : modification of recoil energy in a dispersive medium (BEC). n : index of N 1 << N tot � � 2(1-n)N 1 /N 0 k 2n k refraction Bloch oscillations : 1 = N if η = 100% = + − � � � 2 2 ( 1 ) 2 p final n k n k k N 0 otherwise ~(1- η )(n-1) Accelerated atoms � dispersive medium Raman transition : ω ’ = ω - kv atom Atomic cloud n + (n-1)kv medium v medium L ω ’ = ω - kv atom + (n-1)k(v medium -v atom ) v atom dL/dt = 0 ⇔ v medium =v atom no effect
Refractive index � Phase of the light (1) at the position of the atom i (x i ) : Φ 1 (x i ) � Two photon transition : Φ = Φ 1 - Φ 2 � Assum: � without dispersive media : Φ (x) = 2 k x � inside the medium : d Φ (x)/dx = 2 nk � uniform medium (N atoms), x m of the center of the medium : x m = Σ i x i /N � at the position x m of the center of the medium effect of refractive index cancel from 1st and 2nd beam Φ = − − m + ( x ) 2 ( n 1 ) k ( x x ) 2 kx One Bloch oscillation : Φ � d ( x ) k = + − ≈ - atom i � � � 2 n k 2 ( 1 n ) 2 n k dx N i Φ � d ( x ) k = − i � 2 ( 1 n ) - medium dx N j Raman transition : Doppler effect Φ d ( x ( t ), t ) → ω = ω − + − − ' 2 kv 2 ( n 1 ) k ( v v ) 0 dt
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