Recent results on high precision experiments : determination of the 1S-3S transition in hydrogen determination of the fine structure constant

F. Biraben, R. Bouchendira, P.Cladé, S.Guellati, L.Julien

and F. Nez

http://www.lkb.ens.fr/-Metrologie-Quantique- 1/29

a bit of history : QED -Theory-Experiments Electron in a magnetic field :• Cyclotron frequency : ωcyc

= eB/m• Larmor

frequency: ωlar

= γB

Landé

g-factor : ge

= ωlar

/ ωcyc

Dirac

equation predicts ge

= 2. In 1947, Kusch

& Foley performed a measurement of ge

. The result is not compatible with Dirac

equation.

Prediction of QED (Schwinger, 1948):ge

/2 = 1+ α/2π+…

α

is the coupling constant of QED, α

~

1/137

DiracBohr

n= 1

n= 2

Lamb1S1/2

2S1/2

, 2P1/2

2P3/2

2P1/2

2S1/2

2n nhcRE

QED

(3/8)mc²α²

~mc²α4

~mc²α5

1947 Lamb shift

Since the 50's, the best precision tests of QED are still realized with :

Atomic spectroscopy : Precision spectroscopy of hydrogen (hydrogen atom 1S-3S @LKB, muonic

atoms@PSI

CREMA Collaboration)

Anomalous magnetic moment of the electron measurement by Gabrielse

et al. Calculation by Kinoshita (Recoil determination Rb

@LKB)

2/29

hydrogen theory

E(n,l,j) = Dirac

+ recoil + L(n,j)

hcR∞

f(α, me

/mp

, n,l,j)

exactnot well known

• QED corrections (1/n3)• relativistic

recoil• charge radius of

the

proton (1/n3)

Ltheo(1S1/2

) = 8172.903 (4) (50) MHzQED

scattering proton size

r

0.15 MHz

1.2 MHz

e-

spinrelativity

177 MHz

Dirac

43.5 GHz

hfs

1.4 GHz

Bohr

n= 1

n= 2

n= 3

Lamb

8.2 GHz

energy

1S1/2

2S1/2

, 2P1/2

2P3/2

2P1/2

2S1/2

F=0

F=1

F=1

rp

2n nhcRE QED proton

spinprotonsize

re-

p

rp

Potentialenergy

3/29

hydrogen spectroscopy experiments : R∞

and L1S

determination

Linear combinations

R∞

, Lexp(1S)

L

exp(1S) = 8172.840 (19) kHz + QED

rp

(1%)

cR∞

= 3 289 841 960 360.9 (21.9) kHz (6.6μ10-12)

E(n,l,j) = hcR∞

f(,, me

/mp

, n,l,j) + recoil + L(n,j,rp

) ≈

+ L(n,rp

)R∞n2

Pachucki K.calculatedpreciselyL(2S)L(1S)Karshenboi S.

L(8S)L(2S))R641

41(8S)(2SParisLKB

L(2S)L(1S))R41(12S)(1SGarchingMPQ

8

ν

ν

m

1S-3S experiment4/29

Two photon spectroscopy : no first order Doppler effect

Two photon spectroscopy

1S-3S transition

at LKB/MPQ (

= 205 nm)G. Hagel

et al, PRL 89, 203001 (2002)

1S-2S transition

at MPQ (Garching) (

= 243 nm) (2 μ

10-14)M. Niering

et al, PRL 84, 5496 (2000) (Impressive new result see next talk)

2S-nS/nD transitions

at LKB/ NPL

(

= 778 nm) (8 μ

10-12)B. de Beauvoir

et al, Eur. Phys. J. D 12,61 (2000)

E0

1S

2S

3S

h(1+ ---)h(1-

---)vc

v v

c

5/29

1S-3S hydrogen spectroscopy

TiSa

frequency

3S

1S

2P

205 nm

205 nm

656 nm

• 1S atomic beam (1014

at/cm3) (~1011at/s)2S atomic beam 17 at/cm3

! (2μ106

at/s)

• transition probability fi

f 2

1S-3S:

=2.14 a.u.f

= 1 MHz

2S-8S:

=14.921 a.u.f

= 144 kHz

1S-2S:

=7.85 a.u.f

= 1.2 Hz• 205 nm laser (<1mW) ( 820 nm

410 nm

205 nm

)

2S-8S

778 nm 1.6W !

• Velocity distribution measurement :

No “easy”

optical transition for Doppler spectroscopy

121 nm

6/29

Second order Doppler effect compensation

Principle: quadratic Stark effect

for opposite parity levels (ex. S and P)

v

B

h(1-

-

)vc

v2

2c2+h(1 -

)v

cv2

2c2 E = v Bv

◉

dop =-atv2

2c2E2

SP

v2B2

SP

Stark = =

F. Biraben, L. Julien, J. Plon

and F. Nez, Europhys. Lett., 15 (1991) p.831 :"Compensation of the second Doppler effect in two photon spectroscopy of atomic hydrogen". 7/29

Partial compensation →

distribution velocity measurement

Second order Doppler effect compensation

ΔmF

=0

Level crossing

G. Hagel, R. Battesti, F. Nez, L. Julien

and F. Biraben, Phys. Rev. Lett. 89 (2002) p.203001 : "Observation of a motional Stark effect to determine the second order Doppler effect".

B (G)

mF

=-1

mF

=+1

mF

=±1v= 3km/s

0

100

200

-

100

-

200

-

300

-

400

-

500130 150 170 190 210 230

Line position (kHz)

mF

=0 B calibration

8/29

Experimental

set-up

3S

1S

2P

TiSa

frequency

B

LBO cavity

BBO cavity

820 nm (2W)

410 nm (1W)

205 nm (<1mW)

Detection

@656 nm (100 ph/s)

FP

DL/Rb

Frequency comb

TiSa

laser

H(1S)

Cs Maser HLNE Syrte 3km

opt. fib.

9/29

BBO cavity mirror motion

CCD

PM

PM

1S -

3S line shape

Relative intensity of mF

=±1

mF

=+1

mF

=-1

v=0

B = 171 G

B = 171 G

= 1.6 km/s

= 1.6 km/s

10/29

Hydrogen velocity determination

f(v,) = v3

e- ---v2

22

MkT

with

(km/s)

B = 160.1 G

B = 171.2 G

B = 191.5 G

B = 0.29 G

= 1.646(89) km/s

= 2 922 742 936. 7275 (120) MHz (4.1μ10-12)

Least square

11/29

Error budget • Frequency measurements 8 μ

0.3 kHz• Light shift 0.3 kHz• Pressure shift 1.2 kHz• Velocity distribution 3.0 kHz• Scan of BBO cavity 2.6 kHz• Statistic 12.0 kHz

Results

[1S1/2

-3S1/2

(F=1)] = 2 922 742 936. 7292 (130) MHz (4.5μ10-12)

LKB

[1S1/2

-3S1/2

] = 2 922 743 278.6783 (130) MHz (4.6μ10-12)

NIST data base [1S1/2

-3S1/2

] = 2 922 743 278.6716 (14) MHz (4.8μ10-13)

1S-2S and 1S-3S

c R∞

= 3 289 841 960. 467 (204) kHz (6.4 μ10-11)rp

= 0.911 (65) fm

O. Arnoult

et al, Eur. Phys. J. D 60 p.243 (2010) 12/29

LKB prospects

Sum frequency generation : 205 nm 1S-3S

532 nm 266 nm

896 nm

205 nm

Liquid N2

cooled atomic beam

Sum frequency generation : 194.5 nm 1S-4S

532 nm 266 nm

724 nm

194.5 nm

13/29

Determinations of the fine structure constant

c4e

0

2

α dimension less

scale electromagnetic interaction

137.035 990 137.036 000 137.036 010

h/m

(neutron)

-1

quantum Hall effectSolid state

physics’p,h-90

hfs muonium

QEDg –

2 of the electron

(UW)g –

2 of the electron

(Harvard)

h/m

(Cs)

h / mh/m (Rb)

20062008

He fine structure

2010

mv=h/λDB

vr

=ћk/m

14/29

Determination of the fine structure constant α

from h/m

Rydberg

constant in terms of energy : 22e αcm

21Rch

Rbm

heARbA

cR2α 87

r

87r2 -

Rydberg

constant : 7 x 10- 12

-

atom-to-proton mass ratio :1.4 x 10- 10

-

electron-to-proton mass ratio : 4.2 x 10- 10

Recoil effect h/m

The recoil velocity is directly related to the h/M ratio J.L. Hall et al, : PRL 37,1339 (1976)

Spontaneous emission

Raman two photon transition

mkvr

and can be measured very precisely in terms of frequency (Doppler effect)

a

E=hp=ћk

b

m

c

b

4Er

Same internal state

a

c

b

Two different internal states

am

v=2μ----ħkm

15/29

Principle of our experiment

vr

= v

/ (2N)

N

2ħk

coherent

acceleration

measurement(Raman transitionor Ramsey fringes)

selection(Raman transition

or Ramsey fringes)

MOT +molasses

selection of an initial sub-recoil velocity class

87Rb

5S1/2

5P3/2

F=2F=1

coherent acceleration : N

Bloch oscillations,momentum transfer 2Nħk

measurement of the final velocity class

16/29

Succession of stimulated Raman transitions(same hyperfine level)

F=1

2m

2vr

Momentum

Ene

rgy

h

h2

0

rvk2

rvk10

k2

rvk6

k6k4k2

k2

Coherent acceleration of atoms : simple approach

t 21

Addiabatic

passage : acceleration of the atoms

The atom is placed in an accelerated standing wave: in its frame, the atom is submitted to an inertial force

Bloch oscillations in a periodic potential

LKB (1996) (E.P.)

per cycle

17/29

Atom in an accelerated lattice

1

& 2

Velocity of the lattice v=(1

-

2

)/2kLight shifts : Periodic potential

m1 2

2vr

Velocity distribution Wannier

function (center at v=0)

Wannier

function(center at 2Nvr

)

v

2

U0

vAcceleration

vt)cos(2k2

Ut)U(x, 0

18/29

Bloch oscillations and atomic interferometry

-10 -5 0 5 10-15 -10 -5 0 5 10 15

2 2 22

selectionF=2 → F=1

measurement F=1 → F=2

acceleration deceleration

detection

blow away beam

TRTR

high sensitivity of atomic interferometry+

high efficiency of Bloch oscillations

space

time

/2 /2/2/2

TR

TR

N Bloch oscillations

v0

v+2Nv0 r

v-2Nv0 r

19/29

Measurement of the recoil velocity

+

k1

k2

k2

k1

2 spectra

+k1

k1k2

k2

2 spectra

upwards acceleration

mes

sel

downwards acceleration sel

mes

Acceleration in both opposite directions

: )NN(2

VVv downup

downup

r

21

measselkkδδ

ΔV

We measure (Doppler effect)

:

mkv B

r

B21

downup

downmeassel

upmeassel

kkk)NN(2m

)V,VAvg(V 1,2 2,1with

(no contribution of g)

g

20/29

«

Atom

elevator

»

2 2 22

selectionF=2 → F=1

measurement F=1 → F=2

accelerationdeceleration

detection

blow

away beam

TRTR

acc dec

Bloch oscillations = high efficiency (99.95% per recoil) “increase”

the size of the vacuum chamber

more recoils transferred to the atoms higher accuracy on recoil determination

sel

acc

dec

acc

dec

mes

±300 ±300 -500 50021/29

Results with new vacuum chamber

170 measurements (14 hours)

Each measurement : 6μ10±9

(h/m) and 3 μ10±9

(α)

Relative uncertainty on h/m

: 4.4μ10±10 and 2.2μ10±10

on α22/29

Testing for correlations in measurements : T.Witt

BIPM : no correlations

Results : correlation ?

Metrologia

44 201 (2007) 23/29

Laser frequencies 0 1.3Beams alignment

- 3.3

3.3

Wave front curvature and Gouy phase - 25.1 3.0 2nd order Zeeman effect 4.0

3.0

Gravity gradient

-2.0

0.2 Light shift (one photon) 0 0.1Light shift (two photon) 0 0.1 Light shift (Bloch oscillations)

0 0.1

Index of refraction (cold atomic cloud and backgrd

vapor) 2.0Global systematic effects -26.4 5.9Statistical uncertainty 2.0Rydberg

constant and mass ratio

2.2

TOTAL UNCERTAINTY 6.6

Error budget

Source

(1/α) (×10-10) (×10-10)

24/29

Systematics

: beams alignment

/2)θ2k(1cosθ2kkk 221

Maximum angle estimated 40 µrad

from coupling between optical fibers 3.3μ10-10

25/29

Wavefront

curvature and Gouy

phase shift

Momentum of the photon ?

zp

where Φ

is the phase of the laser beam

p= ћk

holds

only

for a perfect

plane wave

For a Gaussian beam :

2

22

4

2

2 Rkr

w4r

w4

2k1k

zΦ

where w is

the

waist

of

the

beam, R the

wavefront

curavture

and

r the

distance from

the

propagation axes of

the

beam.

Beam diagnostics with a Shack-Hartmann analyser

w=3.6 mm R > 30 m

Largest systematic effect 25μ10-10

larger beam size

more light power

26/29

Determination of the fine structure constant

LKB-10 (Paris)

: α-1=137.035 999 037 (91) [6.6μ10-10] PRL 106,080801 (2011)

ae

=0.001 159 652 181 13 (84)

Harvard Uni-08

: ae

=0.001 159 652 180 73 (28) PRL 100,120801 (2008)

α-1=137.035 999 084 (51) [3.7μ10-10]

(-1

–

137.03) ×

105

599,8 599,85 599,9 599,95 600 600,05 600,1

h/m(Cs) 2002

ae

(UW) 1987

ae

(Harvard, 2006)

h/m(Rb) 2006

h/m(Rb) 2008

ae

(Harvard, 2008)

h/m(Rb) 2010

27/29

...hadronweak,,/mm,/mmaπαC

παC

παC

παCa τeμe

4

4

3

3

2

21e

Test of QED: electron

anomaly

14eee 1089)(40theoameasaaδ

First test of the QED at

the 10-9

level

First test of the muonicand hadronic

corrections

2008

2010

10-3

10-6

10-9

10-12

100

πα

2

πα

4

πα

3

πα

5

πα

μe

mm

τmme

a(weak)a(hadron)

176 180 184 188 192(ae

–

0.001 159 652 000)/10-12

UW 1987Harvard 2008

Rb 2010Rb 2010 –

only

electronic

QED contributions

28/29

Our team thanks you for your attention

29/29