3 lectures : 24, 25, 26 April 2012

Lucile JULIEN

QED 2012, Cargèse

Experimental tests of QEDin bound and isolated systems

QED & Quantum Vaccum, Low Energy Frontier, 03003 (2012) DOI: 10.1051/iesc/2012qed03003 © Owned by the authors, published by EDP Sciences, 2012

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Article published online by EDP Sciences and available at http://www.iesc-proceedings.org or http://dx.doi.org/10.1051/iesc/2012qed03003

Experimental tests of QED in bound systems

Already discussed :

• Spectroscopy of hydrogen and helium atomsincluding experimental methods

In this third lecture :

• Pure leptonic atomic systems : positronium and muonium

• Other exotic atoms : muonic helium and antiprotonic helium• Muonic hydrogen (proton radius puzzle)• He+ ion (electronic and muonic)

• Highly charged ions : H-like, Li-like and He-like

• g – 2 mesurements : ions, electron, muon

• determinations of the fine structure constant

The electron magnetic moment

The magnetic moment of a particle with charge q and mass mis related to its angular momentum by the g factor J

m

qg=

2μ

For an electron, one introduces the Bohr magnetone

B m

e

2

−=μ

and then Jg Bμμ =

In a magnetic field, the coupling energy of a given magnetic moment

with the field is given by so that :B.μ−

- the magnetic moment precessesaround the field at the Larmor frequency

- energy levels in an atomic bound system are shifted (Zeeman effect) by a quantity which is simply related to the g factor and to the various quantum numbers of the level

Dirac’s theory predicted 2=g for a free electron (pure spin momentum)

m

qBgL 2

−=ω

Why to measure the anomalous magnetic moment of electron ?

In 1947, the first precise measurement of the hyperfine structure in hydrogen and deuterium showed a discrepancy betweenmeasured and calculated values derived from Dirac equation

This result and the 2S hydrogen Lamb shift, also measured in 1947, gave the first experimental evidence of deviations from the Dirac theory

For a bound electron in the ground state of an H-like atom, the predictions are :

•••• ( ) ( ) ( ) +−−=−+= ...12

1

3

112121

3

2 422ααα ZZZg D Dirac

Extremely precise measurements of g - 2 can be performed in ion trapsand then provide stringent tests of QED

( ) ( ) ( ) ( ) 64242 107053.171...2

1

4

1

12

1

3

11

)(−×−=+++−−=

−

−

N

e

e

e

m

mZZZZ

g

Hgα

π

αααα

•••• If radiative corrections and recoil effect are taken into account :

...)2

1(2)1(2 ++=+=−

π

αZag ee

and :

g factor measurements in light H-like ions

Experiments performed at Mainz

A single ion is stored in the strongmagnetic field (∼ 4 T) of a Penning trap

cyclotron frequencyof the trapped ion : B

M

Q

ionc =ω

e

Q

M

mg

ion

e

c

LJ ⋅⋅=

ω

ω2

external input

Larmor frequencyof the electron

The Larmor frequency is measuredthrough the spin-flip rate of the electronas a function of a driving microwave field

(see later)

width ∼∼∼∼ 30 mHz

12C5+

g factor measurements in light H-like ions

Results : • gJ (12C5+ ) = 2.001 041 596 4 (8) (6) (44)stat syst

the contribution of the electron’s atomic mass dominates the error budget

These results are in very good agreement with theoretical calculations which takeinto account one-loop all orders in Z terms for the bound-state QED contributions

• gJ (16O7+ ) = 2.000 047 025 4 (15) (44)

They can be used to derive a value of the electron mass

more precise than the CODATA valueme = 0.000 548 579 909 6 (4) u

• gJ (28Si13+ ) = 1.995 348 958 7 (5) (3) (8)

in excellent agreement with the theoretical value

gJ (28Si13+ ) = 1.995 348 958 0 (17)

including QED contributions up to the two-loop level in (Z )2 and (Z )4

g factor measurements in heavy highly charged ions

Experiment planned at the HITRAP facility , GSI (Darmstadt)

in a series of H-like elements : Pb81+, Bi82+, U91+

Ions injected and confined in a cryogenic Penning trapwhere a constant homogeneous magnetic field is applied

A double laser-microwave resonance technique will be usedto measure the gF factor of a given hyperfine sublevel

on the ppb level

The combined measurement of hfstransition frequency and gF factor

will allow the simultaneousdetermination of electronic gJ

and nuclear gI factors

simplest caseI = 1/2( )

Novel technique of deceleration, trapping and cooling of HCI

Magnetic moment of free leptonic particles

electron, positron, muon

• is the most stringent test of QED (*)

• is the most stringent test of CPT invariance with leptons(comparaison electron - positron)

(*) if is known

• is a test for physics beyond the Standard Model (e.g. electron substructure) (*)

and, if one is confident in QED calculations,

• gives the most precise determination of if there is no physics beyondthe Standard Model

Jm

qg=

2μ

The g-value is a dimensionless measure of the magnetic moment

Its experimental determination

Measurements of the electron g-factor

For a free electron the g-factor is simply deduced from the ratio between

the Larmor frequency and the cyclotron frequencye

L m

qBg

2−=ω

ec m

qB=ω

2

2−= eg

eaand its anomaly is defined as

In 1947, the first determination of ae was performed by Kusch and Foley.

Their result was ae = 0.00119 (5) in agreemeent with the prediction of Schwinger :

00116.02

=π

α

For a review of early measurements, see :

Measurements of the electron g-factor

• Direct observation of the electron spin precession in a given magnetic field(Michigan)

100 keV electrons (100 ns pulses)are polarized and trappedin a 1 kG magnetic field

during a predetermined time T

The spin rotation during this time is recorded versus T

First experimental method :

The same method is easily extended to the g-factor of positron

3 ppm accuracy on ae

Measurements of the electron g-factor

Second experimental method :

• Study of transitions induced by a RF field in a Penning trapin a given magnetic field (Washington, Mainz, Stanford, Harvard)

The energy levels of one electronin a magnetic field are given by :

( ) Lscs mnmnE ωω ++=2

1,

wherec

a

c

L g

ω

ω

ω

ω+== 1

2

and cLa ωωω −=aω is the anomaly frequency

Rabi-Landau levels

1=n

0=n

1=n

0=n

2=n

2

1−=sm

2

1+=sm

cω

Lω

aω

directly related to ae

c

aea

ω

ω=

hyperbolic Penning trap

In a Penning trap

A quadrupolar electric potential is appliedwhich confine the electron along the z axis

( ) +−=

22

1,,

2222 yx

zmzyxV zeω

the transverse confinement is obtainedby the application of the magnetic field

z

B

e

The electron movement is the sum of :- a cyclotron rotation at a slightly modified frequency- an oscillation along the z axis- a slow rotation at the magnetron frequency

An electron suspended in a Penning trap is a homemade atom called geonium

Pionneer work in Washington

- Penning trap at 4K- single electron strored

Measurement of the cyclotron frequencyand of the anomaly frequency

Detection of spin-flip through the induced shift of the axial frequency

Results :

and 1210)1.25.0(1/ −×±+=+− ee

gg

)40(200652159001.12/ =−eg

4 x 10-11

and the most precise determination of the electron g-factor

• Cylindrical Penning trap invented to form a microwave cavitythat could inhibit spontaneous emission (by a factor of up to 250)

narrowed linewith

• Trap cavity cooled to 100 mKthe electron cyclotron motion is its ground state

• Careful control and probe of radiation field and magnetic field in the trap cavity

The one quantum change in cyclotron motion is resolved

The Harvard experiment

Electron’s lowest cyclotron and spin levels

Special relativity makes the transition cyclotron frequency decrease by a very small shift

( )smn ++− 1δ

where 92

102/

−≈=mc

c

c

ω

πω

δ

not negligible at this level of precision !

• 5.4 T stabilized magnetic field• very stable trapping potential provided by a charged capacitor

Numerical values : GHz1502/ ≈πωc

MHz2002/ ≈πωz

kHz1332/ ≈πωm

MHz1742/ ≈πωa

(measured)

Quantum nondemolition (QND) detection

The one-quantum sensitivity is obtained through the coupling of the cyclotron motionto the thermally driven axial motion at 200 MHz frequency

where electronic detection is efficient

This coupling is obtained with a magnetic bottle gradient of B (Ni rings)

here 4 Hz in 200 MHz ( )sBz mn +=Δ δωThe shift is

spin flip anomaly transition one-quantum cyclotron excitationinduced by applying an oscillating potential to

the electrodes

induced by microwavesinjected into the cavity

Phys. Rev. Lett. 97, 030801 (2006)

Quantum-jump spectroscopy

cyclotrontransitions

anomalytransitions

measuring the quantum jumpsper attempt to drive them

as a function of drive frequency

Result :

in 2006

in 2008

)76(85180652159001.12/ =g

)28(73180652159001.12/ =g

7.6 x 10-13

2.8 x 10-13

electron anomaly : discussion

• Since the experimental uncertainty is less than 1% of

the coefficient is needed to match the precision of theory with experiment

124

1029 −×≈π

α

( )81A

• Taking into account the presence of the muon and tau particles, the QED contribution to the electron g - 2 can be written :

( ) ( ) ( )τμτμ mmmmAmmAmmAAa eeeee , 3221 +++=

where ( ) ( ) ( ) ( ) ...4

83

62

42 +++=π

α

π

α

π

α

π

αiiiii AAAAA and ( )

2

121 =A

The last result obtained in Harvard is :

1210)28.0(73.1806521591 −×=ea 2.4 x 10-10

But the comparison of theory with measured electron anomaly needs alsoa value of obtained by an independent measurement

• In addition, the total non QED (hadronic) contribution to ae is 1.72(2) x 10-12

electron anomaly and fine structure constant

On another hand, the last measurement of the electron g-factor, combined with recent calculations of A1

(8) cefficient and estimation of A1(10) one

gives the most precise determination of the fine structure constant

)39()33(084999035.1371 =−αexp. theor.

3.7 x 10-10

The other determinations of the fine structure constant are discussed in the following

Measurements of the muon g-factor

• First experiment at the Columbia-Nevis Cyclotron : the direct observation of the muon magnetic moment precessionin a magnetic field provided the indication that the muon behavelike a heavy electron with a g-factor given by

)8(00122.12

=g in agreement with predictions of QED

Columbia

• During the following years, experiments were performed at CERN, either at the synchrocyclotron or at the proton synchrotron :

leading to increasing accuracy

• Experiment in Brookhaven by V. Hughes and collaboratorsStudy of the spin precession relative to the momentum in a given magnetic field

Measurements of the muon g-factor

In both CERN and Brookhaven experiments, relativistic muons are stored in circularorbits in a uniform magnetic field and are detected through their decay electrons

E821 collaboration

Columbia

310)63(91920165.1 −×=μa

The anomaly frequencyis observed as an oscillation

of the detected electrons

The magnetic field is deducedfrom proton NMR

Result :

muon anomaly : discussion

• Due to the mass dependence, the muon g - 2 is 4 x 104 times more sensitive to hadronic and electroweak effects than the electron g - 2

The hadronic contribution to aμ is 60 ppmThis result allows to check the contribution of virtual hadrons to the muon anomaly

The average current experimental result is :

1110)63(080592161 −×=μa 0.5 ppm

• Including all contributions, the theoretical value of aμ in the Standard Model is :

1110)62(791591161 −×=μa

which differs from the experimental value by 289 (86) x 10-11 3.4

Both theory and experiment must be improved to know if it is or not an indicator of physics beyond the standard model

The various ways to determine

We have seen that a determination of can be deduced :- from the hyperfine structure of muonium with an accuracy of 5.8 x 10-8

- from the fine structure of helium with an uncertainty of 3.1 x 10-8

limited by uncalculated high-order QED terms- from the electron g - 2 magnetic moment anomaly with an uncertainty of 3.7 x 10-10

partly due to both experiment and calculations

The two most precise methods only are discussed in the following

Other independent measurements are strongly needed to test QED

Gyromagnetic ratio of a shielded proton (H2O) in high field 3.7 x 10-8

Josephson effect 3.1 x 10-7

Quantum Hall effect 1.8 x 10-8

Diffraction of neutrons 2.4 x 10-8

Recoil effect (measurement of the h/M ratio) 6.6 x 10-10

Other methods which have been used :

The Quantum Hall effect

Fixed current I in a quantum Hall effect deviceat low temperature RH

Plateaus are observed when the resistanceis recorded versus the magnetic field

2)(

ei

h

i

RiR K

H == where i is integer

0.5 mIts measurement by comparison to a known resistance is doneby means of a calculable capacitor

and is limited to 1.8 x 10-8

(NIST, NML, NPL, BNM)

as the deduced value of

α

μ

20

2

c

e

hRK ==The Von Klitzing constant RK is simply related to

c0μ( - impedance of vacuum - is exact in SI)

The constant can also be deduced from recoil

Rydberg constant in terms of energy : 22

2

1αcmRch e=∞

Measurement of h/M allows to determine

M

h

m

M

c

R

e

××= ∞22αRelative uncertainty on each term :

- Rydberg constant : 5 x 10-12

- atom-to-electron mass ratio : 4 x 10-10

The recoil velocity gives the ratio h/M

The recoil velocity can be measured very precisely as a frequency shift

Mvr

k=Mk rv

Experiments have been performed : - in Stanford (Cs atoms)- in Paris (Rb atoms)

Ultracold Rb atoms at 4 μK

The Paris experiment

Final uncertainty : σ (vr) σ (v)

• Selection of an initial subrecoil velocity class• Coherent acceleration : N Bloch oscillations

momentum transfer of• Measurement of the final velocity distribution

kN2

Principle :

N × 2 kMOT

+optical

molasses

Subrecoil velocity selectionStimulated Raman transitionin the ground state (ππππ pulse)between the two hyperfine levels

Contra-propagating laser beamsDoppler shift velocity selection

Mk2222 k1111

( non selected atoms left in F=2 level are blown out )

5P3/2

5S1/2

F=2F=1

~ 1000 GHz

Velocity selective stimulated Raman transition from F = 1 to F = 2

Frequency scan of one beam through the profile

velocity distribution measurementafter the coherent acceleration

(momentum transfer)

5S1/2

F=2F=1

Experimental sequenceN

2/(N

1+N

2)

~ vr /50

15000

vHz1 r

v ↔≈σ

The F = 2 relative population is recordedversus the difference between the selection

and the detection laser frequencies

for the Raman beams

Signal

F=2 F=2F=1(acceleration)

Experimental arrangementVertical geommetry

Pump

Rubidium vapor

MOT

Detection area

Raman and Blochlaser beams

The populations in F=2 and F=1 levels after the velocity detectionare successively measured by fluorescence using a time of flight technique

Bloch Raman

Mgcoherent

acceleration

velocity selectionand detection

To cancel the effect of gravity,a differential measurement is used

(up and down accelerations)

Improvement of the velocity selection

In our earlier experiment (π−π configuration), two π Raman pulses were used - to select a subrecoil velocity distribution- and to measure the final velocity distribution

In our present experiment ({π/2, π/2}− {π/2, π/2} configuration), - the first pair of π/2 pulses

(frequency 1)selects a velocity pattern

velocity

1/TR

Ramsey patternTR : time between the two pulses

- the second pair of π/2 pulses (frequency 2) selects another velocity pattern

Fit of the central partprecise determination

of the velocity

Frequency δ = δ2 - δ1

When the detection frequency 2is swept, the signal obtained is

the convolution of two Ramsey patterns

Interferometric method

Atom interferometryThe manipulation of ultracold atoms with laser beams

has opened the way to atom interferometry

π

|a

|b

|a

|b

With a resonant laser light, a π pulse acts as a mirror and a π/2 pulse as a beam splitter

π/2

|a

|b

|a

time

momentum

With 3 or 4 pulses, one realizes an interferometer

Ramsey-Bordé

π π/2π/2 π/2π/2 π/2 π/2

N= 30 determination of αwith an accuracy of 7.7 x 10-9

|a> |b>

π/2 π/2

π/2

π/2

T

T

π

(2vr)

π

(2vr)

The Stanford experiment

The dephasing between the two pathsis proportional to the recoil velocity

The two pairs of π/2 pulses are usedto separate and recombine

the two paths of the interferometer

N π pulses are applied to increasethe number of transferred recoils

Atomic fountain of cold Cs atoms

Stimulated Raman transitionin a given hyperfine level

Mk2222 k1111

2vr

Coherent acceleration of the atoms (I)

Linear frequency sweep of the frequency of one beam :

Momentum

Ene

rgy

hνννν1111

hνννν2

0

rvk2

rvk10

k2−

rvk6

k6k4k2

k2

Momentum transfer :

per cycle

t∝−= 21 ννδ

( )δ

F=1

νννν1111 νννν2

5P3/240 GHz

Bloch oscillations in the fundamental energy band

In the laboratory frame : accelerated optical lattice

In the accelerated frame at

aMF −=

( )tk

a 21 ννπ −= : standing wave but inertial force

Coherent acceleration of the atoms (II)

Principle :

First measurement :

up to 1000 recoils

Transfer efficiency of 99.95% per B.O.

Experimental sequence

Raman beams(interferometric method)

+ Bloch oscillations

Signal

Four spectra recorded in 5 min.6 ppb accuracy on h/M

upwards and downwards accelerationsto cancel the effect of gravity

exchange between Raman beamsto cancel systematic shifts

Recent measurements performed in Paris

4.6 x 10-9

6.6 x 10-10

1370 spectra have been recorded

-1 = 137.035 999 037 (91) Result :

137.035 990 137.036 000 137.036 010

h/m (neutron)

α-1

quantum Hall effectSolid state

physics

ΓΓΓΓ’p,h-90

hfs muonium

QEDg – 2 of the electron (UW)

g 2 of the electron (Harvard)

h/m (Cs)

h / mh/m (Rb)

2006

2008

He fine structure

2010

Comparaison between various measurements of

The mesurement of in different domains of physicsis a test of the consistency of theory

CODATAdeduced from all experiments

deduced from experimentsexcept g - 2

(α-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

The comparison between recent h/M and g-2 determinations of gives a test of QED at a level better than 10-9

Fine structure constant : discussion

The value of deduced from ae is the most accurate one but is fully dependent on QED calculations

It has been shifted by 4.7 x 10-9 in 2007 after a correction in calculations

Electron anomaly and recoil effect : conclusion

First test of the QED at the 10-9 level

(up to 4-loop terms)

The accuracy is nowat the level of muonic

and hadronic corrections

( ) ...,,/,/4

4

3

3

2

21 +++++= hadronweakmmmmaCCCCa eee τμπ

α

π

α

π

α

π

α

( ) ( ) ( ) 1410)8940( −×±−=−= theoameasaa eeeδfrom h/M g - 2

In red : theoretical uncertainties

General conclusion

The spectroscopy of atomic systems and free particles at low energyprovides very accurate tests of QED

A lot of experiments give results in good agreement with calculated predictions

The most accurate test is the comparison between determinations of the fine structure constant derived from electron g - 2 and h/M measurement

Previous discrepancies are now solved …- fine structure of helium- orthopositronium lifetime

… but others are not solved or recently appeared !- g-2 anomaly of muon > 3 - Lamb shift of muonic hydrogen (proton size puzzle) ~ 5

The work must continue for both theoreticians and experimentalists

Bibliography

• " Quantum electrodynamics " Ed. T. Kinoshita, Advanced Series on Directions in High Energy Physics- Vol.7, World Scientific (1990)

• " Introduction to the Physics of Highly Charged Ions " Ed. H.F. Beyer and V.P. Shevelko,Series in Atomic and Molecular Physics,Institute of Physics (2003)

• " The Hydrogen Atom " Ed. G.F. Bassani, M. Inguscio and T.W. HänschSpringer Verlag (1989)

• " Lepton Dipole Moments " Ed. B.L. Roberts and W.J. Marciano Advanced Series on Directions in High Energy Physics- Vol.20, World Scientific (2009)

Many thanks to all my colleagues for their help, comments and suggestions ...

... and thank you for your attention !