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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
This is an Open Access article distributed under the terms of the Creative Commons Attribution License 2.0, whichpermits unrestricted use distribution, and reproduction in any medium, provided the original work is properly cited,
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 !