New Measurement of the Electron Magnetic Moment and the Fine Structure Constant
Gabrielse New Measurement of the Electron Magnetic Moment...
Transcript of Gabrielse New Measurement of the Electron Magnetic Moment...
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GabrielseNew Measurement
of the Electron Magnetic Momentand the Fine Structure Constant
Gerald GabrielseLeverett Professor of Physics
Harvard University
Almost finished student: David HannekeEarlier contributions: Brian Odom,
Brian D’Urso, Steve Peil, Dafna Enzer, Kamal AbdullahChing-hua TsengJoseph Tan
N$F 0.1 µm
2ψ
(poster session)
20 years6.5 theses
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Gabrielse
Why Does it take Twenty Years and 6.5 Theses?
• One-electron quantum cyclotron• Resolve lowest cyclotron and spin states• Quantum jump spectroscopy• Cavity-controlled spontaneous emission• Radiation field controlled by cylindrical trap cavity• Cooling away of blackbody photons• Synchronized electrons probe cavity radiation modes• Elimination of nuclear paramagnetism• One-particle self-excited oscillator
Explanation 1: We do experiments much too slowly
Explanation 2: Takes time to develop new methods for measurement with 7.6 parts in 1013 uncertainty
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Gabrielse
New g and αGerald Gabrielse
Leverett Professor of PhysicsHarvard University
N$F
Magnetic moments, motivation and resultsBefore this measurement – situation unchanged since 1987
Quantum Cyclotron and Many New Methods electron gi.e. Why it took ~20 years to measure g to 7.7 parts in 1013
Determining the Fine Structure Constant Spin-off measurements• million-fold improved antiproton magnetic moment• best lepton CPT test (comparing g for electron and positron)• direct measurement of the proton-to-electron mass ratio
0.1 µm
2ψ
N. Guise (poster)Also Quint, et al.
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Gabrielse
Magnetic Moments, Motivation and Results
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Gabrielse
Magnetic Moments
e.g. What is g for identical charge and mass distributions?
2( )2 2 2 2e ev L e e LIA L
mv m mv
ρµ πρπρ ρ
= = = = =
Bµ
BLgµ µ=magnetic
momentangular momentum
Bohr magneton2em
1g =
ρ
v ,e m
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Gabrielse
Magnetic Moments
BJgµ µ=magnetic
momentangular momentum
Bohr magneton2em
1g =
2g =
2.002 319 304 ...g =
cyclotron motion, identical charge and mass distribution
spin for simplest Dirac particle
simplest Dirac spin, plus QED
(if electron g is different the electron must have substructure)
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Gabrielse
Previous Status: Electron and Muon Moments
( ) : / 2 1.001159 652 188 4 0.000 000 000 004 3 4.3( ) : / 2 1.001159 652 187 9 0.000 000 000 004
( 821) : / 2 1.001165 920 3 0.000 000 000 8 800( 821) : / 2 1.001165 921 4 0.000 000 000 8 80
3 4.3Muon E g p
Electron UW g pptPositron UW g ppt
ptMuon E g
µ
µ
+
−
= ±= ±
= ±
= ± 0 ppt
Electron magnetic moment• test QED (with another measured fine structure constant)• measure fine structure constant (with QED theory)• best lepton CPT test by comparing electron and positron
Muon magnetic moment• look for unexpected new physics in expected forms
Muon more sensitive to “unexpected” new physicsElectron more accurate by 200
UW, 1987
Magnetic Moment
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Gabrielse
Previous Status: Electron and Muon Moments
( ) : / 2 1.001159 652 188 4 0.000 000 000 004 3 4.3( ) : / 2 1.001159 652 187 9 0.000 000 000 004
( 821) : / 2 1.001165 920 3 0.000 000 000 8 800( 821) : / 2 1.001165 921 4 0.000 000 000 8 80
3 4.3Muon E g p
Electron UW g pptPositron UW g ppt
ptMuon E g
µ
µ
+
−
= ±= ±
= ±
= ± 0 ppt
Electron magnetic moment• test QED (with another measured fine structure constant)• measure fine structure constant (with QED theory)• best lepton CPT test by comparing electron and positron
Muon magnetic moment• look for unexpected new physics in expected forms
Muon more sensitive to “unexpected” new physicsElectron more accurate by 200 ( 1000)
UW, 1987 E821, 2004
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Gabrielse
New Measurement of Electron Magnetic Moment
BSgµ µ=magnetic
momentspin
Bohr magneton2em
13
/ 2 1.001 159 652 180 850.000 000 000 000 76 7.6 10
g−
=
± ×
• First improved measurement since 1987• Nearly six times smaller uncertainty• 1.7 standard deviation shift• Likely more accuracy coming• 1000 times smaller uncertainty than muon g
B. Odom, D. Hanneke, B. D’Urson and G. Gabrielse,Phys. Rev. Lett. (in press).
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Gabrielse
Why Measure the Electron Magnetic Moment?
1. The electron g-value is a basic property of the simplest ofelementary particles
2. Use measured g and QED to extract fine structure constant(Will be even more important when we change mass standards)
3. Wait for another accurate measurement of α test QED4. Best lepton CPT test compare g for electron and positron5. Look for new physics
• Is g given by Dirac + QED? If not electron substructure• Enable the muon g search for new physics
- provide α- test of QED
Needed so the large QED term can be subtracted to see if there is new physics – heavy particles, etc.
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Gabrielse
QED Relates Measured g and Measured α42 3
2 3 41 .2
..1 C C C Cg aα α α απ ππ
δπ
= + + + + +
QED CalculationMeasure
weak/strong
Sensitivity to other physics (weak, strong, new) is low
1. Use measured g and QED to extract fine structure constant2. Wait for another accurate measurement of α Test QED
Kinoshita, Nio,Remiddi, Laporta, etc.
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Gabrielse
KinoshitaRemiddi Gabrielse
After dinner at the Gabrielse apartment in St. Genis in 2004
4
2
2
3
3
4
1
.
2
..
1 C
C
C
C
g
a
απ
απ
απ
δ
απ
= +
+
+
+
+
Basking in the reflected glow of theorists
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Gabrielse
New Determination of the Fine Structure Constant2
0
14
ec
απε
=• Strength of the electromagnetic interaction• Important component of our system of
fundamental constants• Increased importance for new mass standard
1
10
137.035 999 7100.000 000 096 7.0 10
α −
−
=
± ×
• First lower uncertainty since 1987
• Ten times more accurate thanatom-recoil methods
G. Gabrielse, D. Hanneke, T. Kinoshita, M. Nio, B. Odom,Phys. Rev. Lett. (in press)
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Gabrielse
42
2 3 20 0
2
22
1
2 122
1 12
1 14 (4 ) 22
2 2( )
4( )
e
e
pCs recoil
Cs p e Cs D
recoil Cs C
D C e
e m ce Rhc h c
R hc m
MM fR h h cc M M m M f
f M MR cf M m
απε πε
α
α
∞
∞
∞
∞
≡ ← ≡
=
= ← =
=
Next Most Accurate Way to Determine α (use Cs example)
• Now this method is 10 times less accurate• We hope that will improve in the future test QED
Combination of measured Rydberg, mass ratios, and atom recoil
(Rb measurement is similar except get h/M[Rb] a bit differently)
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GabrielseFamous Earlier MeasurementsNo Longer Fit on the Same Scale
ten timeslarger scale
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Gabrielse
Test of QED
Most stringent test of QED: Comparing the measured electron gto the g calculated from QED usingan independent α
1215 10gδ −< ×
• None of the uncertainty comes from g and QED• All uncertainty comes from α[Rb] and α[Cs]• With a better independent α could do a ten times better test
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GabrielseFrom Freeman Dyson – One Inventor of QED
Dear Jerry,
... I love your way of doing experiments, and I am happy to congratulate you for this latest triumph. Thank you for sending the two papers.
Your statement, that QED is tested far more stringently than its inventors could ever have envisioned, is correct. As one of the inventors, I remember that we thought of QED in 1949 as a temporary and jerry-built structure, with mathematical inconsistencies and renormalized infinities swept under the rug. We did not expect it to last more than ten years before some more solidly built theory would replace it. We expected and hoped that some new experiments would reveal discrepancies that would point the way to a better theory. And now, 57 years have gone by and that ramshackle structure still stands. The theorists …have kept pace with your experiments, pushing their calculations to higher accuracy than we ever imagined. And you still did not find the discrepancy that we hoped for. To me it remains perpetually amazing that Nature dances to the tune that we scribbled so carelessly 57 years ago. And it is amazing that you can measure her dance to one part per trillion and find her still following our beat.
With congratulations and good wishes for more such beautiful experiments, yours ever, Freeman.
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Gabrielse
Test for Electron Substructure
2* 130 // 2
mm GeV cgδ
> =
Brodsky and Drell, 1980
2* 600 // 2
mm GeV cgδ
> =
limited by the independent αvalues
if our g uncertaintywas the only limit
* 10.3m TeV>
(from any difference between experiment and theory g)
LEP contact interaction limit
Not bad for an experiment done at 100 mK, but LEP does better
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Gabrielse
The New Measurement
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Gabrielse
The 1987 measurement is already very accurate124.3 4.3 10ppt −= ×
How Does One Measure g More Accurately ?
• One-electron quantum cyclotron• Resolve lowest cyclotron and spin states• Quantum jump spectroscopy• Cavity-controlled spontaneous emission• Radiation field controlled by cylindrical trap cavity• Cooling away of blackbody photons• Synchronized electrons probe cavity radiation modes• Elimination of nuclear paramagnetism• One-particle self-excited oscillator
NewMethods
Give introduction to some new and novel methods
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GabrielseQuantum Cyclotron
1 5 0 G H zcυ ≈
Tesla6B ≈
one electron quantum
average quantum number < 1, etc. quantum
n = 0n = 1n = 2n = 3n = 4
7.2 kelvincω =
To realize a quantum cyclotron: • need cyclotron temperature << 7.2 kelvin 100 mK apparatus• need sensitivity to detect a one quantum excitation
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Gabrielse
First Penning Trap Below 4 K 70 mK
Cyclotron motion comes into thermal equilibrium with its container0.070 7.2 /cK K kω=
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GabrielseParticle “Thermometer”
70 mK, lowest storage energyfor any charged
elementary particles
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Gabrielse
Lowest Energy Cyclotron Eigenstates States
n=0 n=1 n=2
n=3 n=4 n=5
(ignore magnetron motion in a trap)
0.1 µm
2ψ
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GabrielseFor Fun: Coherent State
0ψ =
0.1 µm
1ψ =
Coherent state with 1n =
'/ 20
,!
c
in in tnn
nee e nn
βωψ =∞ −−
== ∑
a α α α=
Fock statesdo not oscillate
0.1 µm
Eigenfunction of the lowering operator:
n=0 n = 1
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GabrielseWe Use Only the Lowest Cyclotron States
1 5 0 G H zcυ ≈Tesla6B ≈
n = 0n = 1n = 2n = 3n = 4
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Gabrielse
Spin Two Cyclotron Ladders of Energy Levels
ceBm
ν =n = 0n = 1n = 2n = 3n = 4
n = 0n = 1n = 2n = 3n = 4
ms = -1/2 ms = 1/2
cν
cν
cνcν
cν
cν
cνcν
2s cgν ν=
Cyclotron frequency:
Spin frequency:
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Gabrielse
Basic Idea of the Fully-Quantum Measurement
ceBm
ν =n = 0n = 1n = 2n = 3n = 4
n = 0n = 1n = 2n = 3n = 4
ms = -1/2 ms = 1/2
cν
cν
cνcν
cν
cν
cνcν
2s cgν ν=
Cyclotron frequency:
Spin frequency:
12
c
c
s s
c
g νν
ννν
= = +−Measure a ratio of frequencies: B in free
space310−∼
• almost nothing can be measured better than a frequency• the magnetic field cancels out (self-magnetometer)
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Gabrielse
Modifications to the Basic Idea
Special relativity shifts the levels
We add an electrostatic quadrupole potential V ~ 2z2 –x2-y2
to weakly confine the electron for measurementshifts frequencies
The electrostatic quadrupole potential is imperfectadditional frequency shifts
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Gabrielse
Modification 1: Special Relativity Shift δ
ceBm
ν =n = 0n = 1n = 2n = 3n = 4
n = 0n = 1n = 2n = 3n = 4
ms = -1/2 ms = 1/2
3 / 2cν δ−/ 2cν δ−
5 / 2cν δ−7 / 2cν δ−
5 / 2cν δ−3 / 2cν δ−
7 / 2cν δ−9 / 2cν δ−
2s cgν ν=
Cyclotron frequency:
Spin frequency:
92 10c
c
hmcνδ
ν−= ≈Not a huge relativistic shift,
but important at our accuracy
Solution: Simply correct for δ if we fully resolve the levels
(superposition of cyclotron levels would be a big problem)
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Gabrielse
Modification 2: Add Electrostatic Quadrupole2 2 22V z x y− −∼
• Electrostatic quadrupole potential good near trap center• Control the radiation field inhibit spontaneous emission by 200x
CylindricalPenning
Trap
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Gabrielse
Frequencies Shift
B in Free SpacePerfect Electrostatic
Quadrupole Trap
Imperfect Trap• tilted B• harmonic distortions to V
2s cgν ν=
'c cν ν< cν
'z cν ν
m zν νzν
mν
ceBm
ν =
2s cgν ν=
2s cgν ν=
Problem: not a measurable eigenfrequency in animperfect Penning trap
Solution: Brown-Gabrielse invariance theorem2 2 2( ) ( ) ( )c c z mν ν ν ν= + +
2s
c
g νν
=
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Gabrielse
Spectroscopy in an Imperfect Trap
• one electron in a Penning trap• lowest cyclotron and spin states
2
2
( )2
( )21
( )322 2
s c s c c a
c c c
za
c
z
cc
v vg
g
f
ν ν ν νν ν ν
νν
νδν
ν
+ − += = =
−≈ +
+ +
To deduce g measure only three eigenfrequenciesof the imperfect trap
expansion for c z mv ν ν δ
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Gabrielse
Detecting One Quantum Cyclotron Transitions
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Gabrielse
Detecting the Cyclotron Motion
cyclotronfrequency
νC = 150 GHz too high todetect directly
axialfrequency
νZ = 200 MHz relativelyeasy to detect
Couple the axial frequency νZ to the cyclotron energy.
Small measurable shift in νZindicates a change in cyclotron energy.B
nickelrings
2z 0 2B B B z= +
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Gabrielse
Couple Axial Motion and Cyclotron Motion
2 22 ˆ[( / 2) ]B B z z zρ ρ∆ = − −
change in µchanges effective ωz
Add a “magnetic bottle” to uniform B
n=0n=1n=2n=3
B2 22
212 zm zH B zµω −=
spin flipis also a change in µ
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Gabrielse
Quantum Non-demolition Measurement
B
QND
H = Hcyclotron + Haxial + Hcoupling
[ Hcyclotron, Hcoupling ] = 0
QND: Subsequent time evolutionof cyclotron motion is notaltered by additionalQND measurements
QNDcondition
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Gabrielse
An Electron in a Penning Trap
• very small accelerator• designer atom
12 kHz
153 GHz
200 MHz
Electrostaticquadrupolepotential
Magnetic field
cool detect
need tomeasurefor g/2
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Gabrielse
Cylindrical Penning Trap
Electrostatic quadrupole potential good enough near trap center
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Gabrielse
•The axial oscillator is coupled to a tuned-circuit amplifier
Signal Out
−20 −10 0 10 20
0.0
0.4
ampl
itude
(a. u
.)
−20 −10 0 10 20−0.5
0.0
0.5
frequency − νz (Hz)
•Axial motion is driven to increase signal
Detection of single electron axial motion
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Gabrielse
Better Detection Amplifier
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Gabrielse
Detecting a High Axial Frequency
60 MHz 200 MHz
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Gabrielse
First One-Particle Self-Excited Oscillator
Feedback eliminates damping
Oscillation amplitude must be kept fixedMethod 1: comparatorMethod 2: DSP (digital signal processor)
"Single-Particle Self-excited Oscillator"B. D'Urso, R. Van Handel, B. Odom and G. GabrielsePhys. Rev. Lett. 94, 113002 (2005).
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Gabrielse
Use Digital Signal Processor DSP
• Real time fourier transforms• Use to adjust gain so oscillation stays the same
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GabrielseCyclotron Quantum Jumps to Observe Axial Self-Excited Oscillator
• Use positive feedback to eliminate the damping
• Use comparator to make constant drive amplitude
Measure very large axial oscillations
(again using cyclotronquantum jump spectroscopy)
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GabrielseObserve Tiny Shifts of the Frequencyof the Self-Excited Oscillator
one quantumcyclotronexcitation
spin flip
Tiny, but unmistakable, changes in the axial frequencysignal one quantum changes in cyclotron excitation and spin
How?B
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Gabrielse
Quantum Jump Spectroscopy
• one electron in a Penning trap• lowest cyclotron and spin states
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Gabrielse
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Gabrielse
One-Electron in a Microwave Cavity
B n=0n=1n=2n=3
one-electron • cyclotron oscillator• within a cavity• QND measurement of the
cyclotron energycyclotron
energytime
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GabrielseCool to Eliminate Blackbody Photons
• one electron• Fock states of a cyclotron oscillators• due to blackbody photons
0.23
0.11
0.03
9 x 10-39
On a short time scalein one Fock state or another
Averaged over hoursin a thermal state
average numberof blackbody
photons in the cavity
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GabrielseControl Inhibition of Spontaneous Emissionwithin a Cavity
decay time (s)0 10 20 30 40 50 60
num
ber o
f n=1
to n
=0 d
ecay
s
0
10
20
30
time (s)0 100 200 300
axia
l fre
quen
cy s
hift
(Hz)
-3
0
3
6
9
12
15τ = 16 s
• In free space, cyclotron lifetime = 0.1 s
• In our cylindrical trap 16 s lifetime
How long to make a quantumjump down
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Gabrielse
Spontaneous Emission is Inhibited
Free Space
B = 5.3 T
WithinTrap Cavity
frequency
ms751
=γ
116 sec
γ =
cν
cavitymodes
InhibitedBy 210!
B = 5.3 T
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Gabrielse
Some Challenges
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Gabrielse
Big Challenge: Magnetic Field Stability
• large magnetic field (5 Tesla) from solenoid
• nuclear magnetism of trap apparatus
12 c c
s ag ωω ω
ω= = +
relatively insensitive to BExperimenter’s“definition”of the g value
n = 0n = 1n = 2n = 3
n = 0n = 1n = 2
ms = -1/2ms = 1/2
But: problem when Bdrifts during the measurement
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GabrielseMagnetic Field – Takes Months to Settleafter a Change
Two months is a long time to wait for the magnetic field to settle.
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Gabrielse
Self-Shielding Solenoid Helps a LotFlux conservation Field conservation
Reduces field fluctuations by about a factor > 150
“Self-shielding Superconducting Solenoid Systems”,G. Gabrielse and J. Tan, J. Appl. Phys. 63, 5143 (1988)
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GabrielseA tabletop experiment …
if you have a high ceiling
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Gabrielse
Eliminate Nuclear Paramagnetism
One Year Setback
Deadly nuclear magnetism of copper and other “friendly” materials
Had to build new trap out of silverNew vacuum enclosure out of titanium…
~ 1 year
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Gabrielse
temperature (Kelvin)
0.0 0.5 1.0 1.5 2.0
mag
netic
fiel
d sh
ift (p
pb)
-100
0
100
200
300
400
500
600
700
copper trapsilver trap
temperature-1 (Kelvin-1)
0 5 10 15-100
0
100
200
300
400
500
600
700
copper trapsilver trap
0.0 0.5 1.0 1.5 2.0
expa
nded
200
x
-10
0
10
20
30
• The new silver trap decreases T-dependence of the field by ~ 400• With the silver trap, sub-ppb field stability is “easily” achieved
Silver trap improvement
40 ppb / K-1
0.1 ppb / K-1
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Gabrielse
New Silver Trap
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Gabrielse
frequency - υc (ppb)0 100 200 300
# of
cyc
lotr
on e
xcita
tions
“In the Dark” Excitation Narrower Lines
time (s)0 100 200 300
axia
l fre
quen
cy s
hift
(Hz)
-3
0
3
6
9
12
15
1. Turn FET amplifier off 2. Apply a microwave drive pulse of ~150 GH
(i.e. measure “in the dark”)3. Turn FET amplifier on and check for axial frequency shift4. Plot a histograms of excitations vs. frequency
Good amp heat sinking, amp off during excitationTz = 0.32 K
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Gabrielse
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Gabrielse
Quantum Jump Spectroscopy
Lower temperature so there are no blackbody photons in the cavity
Introduce microwave photons intothe trap cavity – near resonance with the cyclotron frequency
cyclotronenergy
time
count the quantum jumps/timefor a particular drive frequency
quantum jumprate
drive frequency
look for resonance
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Gabrielse
Measurement Cycle
12 c c
s ag ωω ω
ω= = +
n = 0n = 1n = 2n = 3
n = 0n = 1n = 2
ms = -1/2ms = 1/2
1. Prepare n=0, m=1/2 measure anomaly transition2. Prepare n=0, m=1/2 measure cyclotron transition
3. Measure relative magnetic field
3 hours
0.75 hour
Repeat during magnetically quiet times
simplified
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Gabrielse
Precision:Sub-ppb line splitting (i.e. sub-ppb precision of a g-2 measurement) is now “easy” after years of work
It all comes together:• Low temperature, and high frequency make narrow line shapes• A highly stable field allows us to map these lines
Measured Line Shapes for g-value Measurement
cyclotron anomaly
n = 0n = 1n = 2n = 3
n = 0n = 1n = 2
ms = -1/2ms = 1/2
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Gabrielse
Cavity Shifts
Uncertainty completely limited nowby how well we understand the cavity shifts
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Gabrielse
Cavity Shifts of the Cyclotron Frequency
12 c c
s ag ωω ω
ω= = −
n = 0n = 1n = 2n = 3
n = 0n = 1n = 2
ms = -1/2ms = 1/2
Within a Trap Cavity
frequency
116 sec
γ =
cν
cavitymodes
spontaneous emissioninhibited by 210
B = 5.3 T
cyclotron frequencyis shifted by interactionwith cavity modes
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Gabrielse
Cylindrical Penning Trap
Good approximation to a cylindrical microwave cavity• Good knowledge of the field within• Challenge is to make a sufficiently good electrostatic quadrupole
Holes and slits to make a trap• Must still measure resonant EM modes of the cavity
have donethis
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Gabrielse
Cavity modes and Magnetic Moment Error
Operating between modes of cylindrical trap where shift from two cavity modes cancels approximately
first measured cavity shift of g
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Gabrielse
Summary of Uncertainties for g (in ppt = 10-12)
Test ofcavityshift
understandingMeasurement
of g-value
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Gabrielse
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Gabrielse
Spinoff Measurements Planned
• Use self-excited antiproton oscillator to measure the antiproton magnetic moment million-fold improvement?
• Compare positron and electron g-values to make best testof CPT for leptons
• Measure the proton-to-electron mass ration directly
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Gabrielse
Emboldened by the Great Signal-to-Noise
Make a one proton (antiproton) self-excited oscillatortry to detect a proton (and antiproton) spin flip
measure proton spin frequencywe already accurately measure antiproton cyclotron frequenciesget antiproton g value
• Hard: nuclear magneton is 500 times smaller• Experiment underway Harvard
also Mainz and GSI (without SEO)(build upon bound electron g values)
(Improve by factor of a million or more)
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Gabrielse
Summary and Conclusion
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Gabrielse
How Does One Measure g to 7.6 Parts in 1013?
• One-electron quantum cyclotron• Resolve lowest cyclotron and spin states• Quantum jump spectroscopy• Cavity-controlled spontaneous emission• Radiation field controlled by cylindrical trap cavity• Cooling away of blackbody photons• Synchronized electrons probe cavity radiation modes• Elimination of nuclear paramagnetism• One-particle self-excited oscillator
NewMethods
Summary
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Gabrielse
New Measurement of Electron Magnetic Moment
BSgµ µ=magnetic
momentspin
Bohr magneton2em
13
/ 2 1.001 159 652 180 850.000 000 000 000 76 7.6 10
g−
=
± ×
• First improved measurement since 1987• Nearly six times smaller uncertainty• 1.7 standard deviation shift• Likely more accuracy coming• 1000 times smaller uncertainty than muon g
B. Odom, D. Hanneke, B. D’Urson and G. Gabrielse,Phys. Rev. Lett. (in press).
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Gabrielse
New Determination of the Fine Structure Constant2
0
14
ec
απε
=• Strength of the electromagnetic interaction• Important component of our system of
fundamental constants• Increased importance for new mass standard
1
10
137.035 999 7100.000 000 096 7.0 10
α −
−
=
± ×
• First lower uncertainty since 1987
• Ten times more accurate thanatom-recoil methods
G. Gabrielse, D. Hanneke, T. Kinoshita, M. Nio, B. Odom,Phys. Rev. Lett. (in press)
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Gabrielse
Stay Tuned.
We Have Some More Ideas for Doing Better
0.1 µm
2ψ