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

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

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.

Gabrielse

Magnetic Moments, Motivation and Results

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

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)

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

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

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).

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.

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.

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

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)

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)

GabrielseFamous Earlier MeasurementsNo Longer Fit on the Same Scale

ten timeslarger scale

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

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.

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

Gabrielse

The New Measurement

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

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

Gabrielse

First Penning Trap Below 4 K 70 mK

Cyclotron motion comes into thermal equilibrium with its container0.070 7.2 /cK K kω=

GabrielseParticle “Thermometer”

70 mK, lowest storage energyfor any charged

elementary particles

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

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

GabrielseWe Use Only the Lowest Cyclotron States

1 5 0 G H zcυ ≈Tesla6B ≈

n = 0n = 1n = 2n = 3n = 4

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:

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)

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

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)

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

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

=

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

Gabrielse

Detecting One Quantum Cyclotron Transitions

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= +

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

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

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

Gabrielse

Cylindrical Penning Trap

Electrostatic quadrupole potential good enough near trap center

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

Gabrielse

Better Detection Amplifier

Gabrielse

Detecting a High Axial Frequency

60 MHz 200 MHz

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).

Gabrielse

Use Digital Signal Processor DSP

• Real time fourier transforms• Use to adjust gain so oscillation stays the same

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)

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

Gabrielse

Quantum Jump Spectroscopy

• one electron in a Penning trap• lowest cyclotron and spin states

Gabrielse

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

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

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

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

Gabrielse

Some Challenges

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

GabrielseMagnetic Field – Takes Months to Settleafter a Change

Two months is a long time to wait for the magnetic field to settle.

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)

GabrielseA tabletop experiment …

if you have a high ceiling

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

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

Gabrielse

New Silver Trap

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

Gabrielse

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

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

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

Gabrielse

Cavity Shifts

Uncertainty completely limited nowby how well we understand the cavity shifts

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

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

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

Gabrielse

Summary of Uncertainties for g (in ppt = 10-12)

Test ofcavityshift

understandingMeasurement

of g-value

Gabrielse

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

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)

Gabrielse

Summary and Conclusion

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

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).

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)

Gabrielse

Stay Tuned.

We Have Some More Ideas for Doing Better

0.1 µm

2ψ