GabrielseOne-Electron Quantum Cyclotron:

A New Measurement of the

Electron Magnetic Moment

and the Fine Structure Constant

Gerald Gabrielse

Leverett Professor of Physics, Harvard University

2006 DAMOP Thesiswith Shannon Fogwell and Josh Dorr

Earlier contributions: David Hanneke

Brian Odom,

Brian D’Urso,

Steve Peil,

Dafna Enzer,

Kamal Abdullah

Ching-hua Tseng

Joseph Tan N$F0.1 µm

2ψ

20 years

7 theses

2006 DAMOP Thesis

Prize Winner

Gabrielse

Three Programs to Measure Electron g

U. Michigan

beam of electrons

spins precess

with respect to

U. Washington

one electron

Harvard

one electron

quantum jump

spectroscopy of

cylindrical

Penning trap

inhibit with respect to

cyclotron motion

keV 4.2 K

spectroscopy of

quantum levels

100 mK

inhibit

spontaneous

emission

measure

cavity shifts

self-excited

oscillatorCrane, Rich, …

Dehmelt,

Van Dyck

Gabrielse

References

2006: B. Odom, D. Hanneke, B. D’Urso, G. Gabrielse

PRL 97, 030801 (2006).

2008: D. Hanneke, S. Fogwell and G. Gabrielse

PRL 100, 120801 (2008).

Two recent review chapters in book Lepton Moments, edited by Two recent review chapters in book Lepton Moments, edited by

Roberts and Marciano, (World Scientific, Singapore, 2009)

G. Gabrielse, “Measuring the Electron Magnetic Moment”

G. Gabrielse, “Determining the Fine Structure Constant”

Gabrielse

New Measurement of Electron Magnetic Moment

B

Sgµ µ=

��

ℏ

magnetic

moment

spin

Bohr magneton2

e

m

ℏ

13

/ 2 1.001159 652 180 73

0.000 000 000 000 28 2.8 10

g

−

=

± ×

• First improved measurements (2006, 2008) since 1987

• 15 times smaller uncertainty

• 1.7 standard deviation shift

• 2500 times smaller uncertainty than muon g

D. Hanneke, S.

Fogwell and G.

Gabrielse

Gabrielse

New Determination of the Fine Structure Constant2

0

1

4

e

cα

πε=

ℏ

1

10

137.035 999 084 (12) (37) (33)

0.000 000 051 3.7 10

α −

−

=

± ×

• Strength of the electromagnetic interaction

• Important component of our system of

fundamental constants

• Increased importance for new mass standard

• First lower uncertainty since 1987 (2006, 2008)

• 20 times more accurate than atom-recoil methods

D. Hanneke, S. Fogwell and G. Gabrielse

Gabrielse

Key: Resolving One-Quantum Transitions

for One Trapped Electron

one-quantum

cyclotron

transition

spin flip

electron magnetic moment

in Bohr magnetons

Gabrielse

Twenty Years and 7 PhD Theses?

• One-electron quantum cyclotron

• Resolve lowest cyclotron states as well as spin

• Quantum jump spectroscopy of spin and cyclotron motions

Takes time to develop new ideas and methods needed to determine

g/2 to 2.8 parts in 1013 uncertainty – one PhD at a time

firs

t m

easu

rem

ent

wit

h

thes

e new

met

hods

• Quantum jump spectroscopy of spin and cyclotron motions

• Cavity-controlled spontaneous emission

• Radiation field controlled by cylindrical trap cavity

• Cooling away of blackbody photons

• Synchronized electrons identify cavity radiation modes

• Trap without nuclear paramagnetism

• One-particle self-excited oscillator firs

t m

easu

rem

ent

wit

h

thes

e new

met

hods

Gabrielse

Orbital Magnetic Moment

B

Lgµ µ=

��

ℏ

magnetic

moment

angular momentum

Bohr magneton2

e

m

ℏ

e.g. What is g for identical charge and mass distributions?

2( )2 2 2 2

e ev L e e LIA L

mv m m

v

ρµ πρ

πρ ρ= = = = =

ℏ

ℏ

Bµ1g =�

ρ

v ,e m

Gabrielse

Spin Magnetic Moment

B

Sgµ µ=

��

ℏ

magnetic

moment

angular momentum

Bohr magneton2

e

m

ℏ

1g = identical charge and mass distribution1g =

2g =

2.002 319 304 ...g =

identical charge and mass distribution

spin for simplest Dirac point particle

simplest Dirac spin, plus QED

(if electron g is different � electron has substructure)

Gabrielse

Why Measure the Electron Magnetic Moment?

1. Electron g - basic property of simplest of elementary particles

2. Determine fine structure constant – from measured g and QED

(May be even more important when we change mass standards)

3. Test QED – requires independent αααα

4. Test CPT – compare g for electron and positron ���� best lepton

testtest

5. Look for new physics beyond the standard model

• Is g given by Dirac + QED? If not ���� electron substructure

(new physics)

• Muon g search for new physics needs α α α α and test of QED

Gabrielse

Gabrielse

150 GHzc

f ≈

One Electron in a Magnetic Field

n = 1

n = 2

n = 3

n = 4

0.1

µ

2ψ

- - - - - - -Trap with

charges

Tesla6B ≈

n = 0

n = 1

0.1

µm

2ψ

µm7.2 kelvinchυ =

Need low

temperature

cyclotron motion

T << 7.2 K

- - - - - - -

Gabrielse

Electron Cyclotron Motion

Comes Into Thermal Equilibrium

T = 100 mK << 7.2 K � ground state always

Prob = 0.99999…

n = 4

hot

cavity

absorb

blackbody

photons

spontaneous

emission

cold

n = 0

n = 1

n = 2

n = 3

n = 4

electron “temperature” – describes its energy distribution

on average

Gabrielse

Electron in Cyclotron Ground State

0.23

0.11

0.03

QND Measurement of Cyclotron Energy vs. Time

9 x 10-39

On a short time scale

� in one Fock state or another

Averaged over hours

� in a thermal state

average number

of blackbody

photons in the

cavity

S. Peil and G. Gabrielse, Phys. Rev. Lett. 83, 1287 (1999).

Gabrielse

David Hanneke G.G.

Gabrielse

First Penning Trap Below 4 K ���� 70 mK

Need low

temperature

cyclotron motion

T << 7.2 K

Gabrielse

Gabrielse

Spin ���� Two Cyclotron Ladders of Energy Levels

1

2c

eB

mν

π=

n = 0

n = 1

n = 2

n = 3

n = 4

n = 1

n = 2

n = 3

n = 4

cν

cν

cν

cν

cν

cν

cν

cν

2s c

gν ν=

Cyclotron

frequency:

Spin

frequency:

2 mπn = 0

ms = -1/2 ms = 1/2

cν 2

s c

Gabrielse

Basic Idea of the Fully-Quantum Measurement

1

2c

eB

mν

π=

n = 0

n = 1

n = 2

n = 3

n = 4

n = 1

n = 2

n = 3

n = 4

cν

cν

cν

cν

cν

cν

cν

cν

2s c

gν ν=

Cyclotron

frequency:

Spin

frequency:

2c

mπn = 0

ms = -1/2 ms = 1/2

cν 2

s c

12

c

c

s s

c

g ν

ν

ν

ν

ν= = +

−Measure a ratio of frequencies: B in free

space

310−≈

• almost nothing can be measured better than a frequency

• the magnetic field cancels out (self-magnetometer)

Gabrielse

Special Relativity Shift the Energy Levels δδδδ

2 c

eB

mπν =

n = 0

n = 1

n = 2

n = 3

n = 4

n = 1

n = 2

n = 3

n = 4

3 / 2cν δ−

/ 2c

ν δ−

5 / 2c

ν δ−

7 / 2c

ν δ−5 / 2

cν δ−

3 / 2c

ν δ−

7 / 2c

ν δ−

9 / 2c

ν δ−

2s c

gν ν=

Cyclotron

frequency:

Spin

frequency:

mn = 0

ms = -1/2 ms = 1/2

/ 2c

ν δ− 2s c

9

210c

c

h

mc

νδ

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

Cylindrical Penning Trap2 2 2~ 2V z x y− −

• Electrostatic quadrupole potential � good near trap center

• Control the radiation field � inhibit spontaneous emission by 200x

(Invented for this purpose: G.G. and F. C. MacKintosh; Int. J. Mass Spec. Ion Proc. 57, 1 (1984)

Gabrielse

One Electron in a Penning Trap

• very small accelerator

• designer atom

12 kHz 200 MHzcool detect

153 GHzElectrostatic

quadrupole

potentialMagnetic field

need to

measure

for g/2

Gabrielse

Frequencies Shift

B in Free SpacePerfect Electrostatic

Quadrupole Trap

Imperfect Trap

• tilted B

• harmonic

distortions to V

'c cν ν< cν

'z cν ν= zνc

eB

mν =

2s c

gν ν=

'z cν ν=

m zν ν=

z

mν

2s c

gν ν=

2s c

gν ν=

Problem: not a measurable eigenfrequency in an

imperfect Penning trap

Solution: Brown-Gabrielse invariance theorem

2 2 2( ) ( ) ( )c c z mν ν ν ν= + +

2

s

c

g ν

ν=

Gabrielse

M

Measured sideband frequency – function of B tilt and harmonic distortion

( , , ) ( , , )ω θ φ ε θ ωω φ ε+ − ≈+

An Aside Arising from the Last of These Conferences

only an approx.

More detail: G. Gabrielse, Int. J. of Mass Spec. 279, 107 (2009)

( , , ) ( , , ) cω θ φ ε θ ωω φ ε+ − ≈+

c c cω ω θ ω ε∆ +∼ ~1%

2 29 1

4 2cω θ ε ω−

∆ −

≈

dimensional analysis :

using invariance theorem:

(and a bit of symmetry)

How bad is this common approximation?

only an approx.

much, much smaller

Gabrielse

Spectroscopy in an Imperfect Trap

• one electron in a Penning trap

• lowest cyclotron and spin states

( )

2

s c s c c a

c c c

v vg ν ν ν ν

ν ν ν

+ − += = =

2

2

( )

21

( )32

2 2

c c c

za

c

z

c

c

g

f

ν

ν

νδ

ν

ν −

≈ +

+ +

To deduce g � measure only three eigenfrequencies

of the imperfect trap

expansion for c z mv ν ν δ≫ ≫ ≫

GabrielseFeedback Cooling of an Oscillator

Electronic Amplifier Feedback: Strutt and Van der Ziel (1942)

Basic Ideas of Noiseless Feedback and Its Limitations: Kittel (1958)

Applications: Milatz, … (1953) -- electrometer

Dicke, … (1964) -- torsion balance

Forward, … (1979) -- gravity gradiometer

Ritter, … (1988) -- laboratory rotor

e

Dissipation : (1 ) Fluctuations: (1 )

Fluctuation-Dissipation Invariant: /

e e

e

g T T g

T const

Γ = Γ − = −

Γ =faster damping rate

� higher temperature

Ritter, … (1988) -- laboratory rotor

Cohadon, … (1999) -- vibration mode of a mirror

Proposal to apply Kittel ideas to ion in an rf trap

Dehmelt, Nagourney, … (1986) never realized

Proposal to “stochastically” cool antiprotons in trap

Beverini, … (1988) – stochastic cooling never realized

Rolston, Gabrielse (1988) – same as feedback cooling (same limitations)

Realization of feedback cooling with a trapped electron (also include noise)

D’Urso, Odom, Gabrielse, PRL (2003)

D’Urso, Van Handel, Odom, Hanneke, Gabrielse, PRL 94, 11302 (2005)

Gabrielse

QND Detection

of One-Quantum Transitions

n=0

cyclotron

ground

n=0

cyclotron

ground

n=1

cyclotron

excited

elec

tro

n s

elf-

exci

ted

osc

illa

tor

2 2

2

2 2

2

1

2z

B B z B zm zH ω µ∆ = −→ =�

12

( )cyclotron c

freq E hf n= = +n=1

n=0

ground

state

ground

state

excited

state

on

e-el

ectr

on

sel

f

time

Gabrielse

Quantum Non-demolition Measurement

B

QND

H = Hcyclotron + Haxial + Hcoupling

[ Hcyclotron, Hcoupling ] = 0

QND: Subsequent time evolution

of cyclotron motion is not

altered by additional

QND measurements

QND

condition

Gabrielse

One-Electron Self-Excited Oscillator

V(t)

measure voltage

I2R

damping

amplitude, φfeedback

dampingaxial motion

200 MHz

of

trapped

electron crucial to limit

the osc. amplitude

GabrielseObserve Tiny Shifts of the Frequency

of a One-Electron Self-Excited Oscillator

one quantum

cyclotron

excitation

spin flipspin flip

Unmistakable changes in the axial frequency

signal one quantum changes in cyclotron excitation and spin

B"Single-Particle Self-excited Oscillator"

B. D'Urso, R. Van Handel, B. Odom and G. Gabrielse

Phys. Rev. Lett. 94, 113002 (2005).

Gabrielse

Quantum Jump Spectroscopy

• one electron in a Penning trap

• lowest cyclotron and spin states

“In the dark” excitation

� turn off all detection

and cooling drives

during excitation

Gabrielse

Application of Cavity QED

nu

mb

er

of

n=

1 t

o n

=0 d

ecays

20

30

ax

ial fr

eq

uen

cy s

hif

t (H

z)

9

12

15ττττ = 16 s

excite,

measure time in excited state

Inhibited Spontaneous Emission

decay time (s)

0 10 20 30 40 50 60

nu

mb

er

of

n=

1 t

o n

=0 d

ecays

0

10

time (s)

0 100 200 300a

xia

l fr

eq

uen

cy s

hif

t (H

z)

-3

0

3

6

9

Gabrielse

Cavity-Inhibited Spontaneous Emission

Free Space

B = 5.3 T

Within

ms75

1=γ

1

Purcell

Kleppner

Gabrielse and Dehmelt

Within

Trap Cavity

frequency

1

16 secγ =

cν

cavity

modes

Inhibited

By 210!B = 5.3 T

Inhibition gives the

averaging time needed

to resolve a

one-quantum transition

Gabrielse

Big Challenge: Magnetic Field Stability

12

c c

s ag ω

ω ω

ω= = +

n = 0

n = 1n = 2n = 3

n = 0

n = 1n = 2

ms = -1/2ms = 1/2

But: problem when B

drifts during the

Magnetic field cancels out

drifts during the

measurement

Magnetic field take

~ month to stabilize

Gabrielse

Self-Shielding Solenoid Helps a Lot

Flux 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)

Gabrielse

Eliminate Nuclear Paramagnetism

Deadly nuclear magnetism of copper and other “friendly” materials

� Had to build new trap out of silver

� New vacuum enclosure out of titanium~ 1 year

setback

Gabrielse

Measurement Cycle

12

c c

s ag ω

ω ω

ω= = +

n = 0

n = 1n = 2n = 3

n = 0

n = 1n = 2

ms = -1/2ms = 1/2

simplified

1. Prepare n=0, m=1/2 � measure anomaly transition

2. Prepare n=0, m=1/2 � measure cyclotron transition

3. Measure relative magnetic field

3 hours

0.75 hour

Repeat during magnetically quiet times

Gabrielse

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 = 3n = 2

Precision:Sub-ppb line splitting (i.e. sub-ppb precision of a g-2 measurement)

is now “easy” after years of work

n = 0

n = 1n = 2n = 3

n = 0

n = 1n = 2

ms = -1/2ms = 1/2

line s: L. Brown

Gabrielse

Cavity Shifts of the Cyclotron Frequency

12 c c

s ag ω

ω ω

ω= = −

n = 0

n = 1n = 2n = 3

n = 0

n = 1n = 2

ms = -1/2ms = 1/2

Within a Trap Cavity

frequency

1

16 secγ =

cν

cavity

modes

spontaneous emission

inhibited by 210B = 5.3 T

cyclotron frequency

is shifted by interaction

with cavity modes

Gabrielse

Cavity modes and Magnetic Moment Error

use synchronization of electrons to get cavity modes

Operating between modes of cylindrical trap

where shift from two cavity modes

cancels approximately

first measured

cavity shift of g

Gabrielse

2008 Measurement

���� reduced cavity

shift correction

uncertainty

• Measuring lifetime and g

as a function of B

• Attempting cavity sideband

cooling

Gabrielse

Gabrielse

New: Careful Study of the Cavity Shifts

Gabrielse

Electron as Magnetometer ���� Lineshapes

Gabrielse

Shifts and Uncertainties g (in ppt = 10-12 )

cavity shifts not a problem lineshape broadening

Gabrielse

New Measurement of Electron Magnetic Moment

B

Sgµ µ=

��

ℏ

magnetic

moment

spin

Bohr magneton2

e

m

ℏ

13

/ 2 1.001159 652 180 73

0.000 000 000 000 28 2.8 10

g

−

=

± ×

• First improved measurements (2006, 2008) since 1987

• 15 times smaller uncertainty

• 1.7 standard deviation shift

• 2500 times smaller uncertainty than muon g

D. Hanneke, S.

Fogwell and G.

Gabrielse

Gabrielse

Gabrielse

Dirac + QED Relates Measured g and Measured αααα

42 3

2 3 41 .2

..1 C C C Cg

aα α α α

π ππδ

π

= + + + + +

B

Sgµ µ=

��

ℏ

2

0

1

4

e

cα

πε=

ℏ

fine structure

constantmagnetic

moment

QED Calculation

Measure

hadronic, weak

standard model

tiny

1. Use measured g and QED to extract fine structure constant

2. Wait for another accurate measurement of α α α α ���� Test QED

Kinoshita, Nio,

Remiddi, Laporta, etc.

Dirac

point

particle

GabrielseBasking in the Reflected Glow of Theorists

1

2

2

3

3

12

C

C

C

g α

π

α

π

α

π

= +

+

+

KinoshitaRemiddi G.G

2004

5

3

4

4

5

. ..

C

C

a

C

π

α

δ

π

α

π

+

+

+

+

Gabrielse“Simple” Analytic Expressions from QED

(where calculations are completed)

72 Feynman diagrams

7 Feynman diagrams

polylog:

871 Feynman diagrams

12672 Feynman diagrams

Gabrielse

42 3

2 3 41 .2

..1 C C C Cg

aα α α α

π ππδ

π

= + + + + +

experimental

uncertainty

Gabrielse

New Determination of the Fine Structure Constant2

0

1

4

e

cα

πε=

ℏ

1

10

137.035 999 084 (12) (37) (33)

0.000 000 051 3.7 10

α −

−

=

± ×

• Strength of the electromagnetic interaction

• Important component of our system of

fundamental constants

• Increased importance for new mass standard

• First lower uncertainty since 1987 (2006, 2008)

• 20 times more accurate than atom-recoil methods

D. Hanneke, S. Fogwell and G. Gabrielse

Gabrielse

Fine Structure Constant Uncertainties

Gabrielse

Next Most Accurate Way to Determine α (α (α (α (use Cs example)

Combination of measured Rydberg, mass ratios, and atom recoil

Chu, …Pritchard, …

Haensch, …

42

2 3 2

0 0

2

1 1

4 (4 ) 2

2

2

e

e

p

e m ceR

hc h c

R h

c m

MM fR h h

απε πε

α

∞

∞

≡ ← ≡

=

= ← =

• Now this method is >10 times less accurate

• We hope that will improve in the future � test QED

(Rb measurement is similar except get h/M[Rb] a bit differently)

Haensch, …

Tanner, …

Werthe, Quint, … (also Van Dyck)

Bir

aben

, …

2

2

1

2 12

2

1 12

22

( )

4( )

pCs recoil

Cs p e Cs D

recoil Cs C

D C e

MM fR h hc

c M M m M f

f M MR c

f M mα

∞

∞

= ← =

=

Gabrielse

Several Measurements Needed for the Atom-Recoil

Determinations of the Fine Structure Constant

Gabrielse

Earlier Measurements

(On Much Larger Uncertainty Scale)

Gabrielse

Test of QED

Most stringent test of QED: Comparing the measured electron g

to the g calculated from QED using

an independent α

12

exp| ( ) | 15 10eriment theoryg g gδ α −= − < ×

• The uncertainty does not comes from g and QED, + SM

• 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

In Progress

Better measurement of the electron magnetic moment

and the fine structure constant

Better comparison of the positron and electron magnetic

moments

Spinoffs

Observe a proton spin flip with one-proton

self-excited oscillator (close, see new PRL)

Attempt to make a one-electron qubit

(see new PRA)

Goldman talk this afternoon

Gabrielse

Close to observing a proton spin flip (we hope).

(Mainz talk by Ulmer is coming.)

Gabrielse

New Dilution Refrigerator and Suspension

support plates

dilution refrigerator

4.2 K superconducting solenoid

100 mK trap

4.2 K superconducting solenoid

old new

100 mK trap and suspended electron

moves with the 4.2 K superconducting solenoid

Gabrielse

New Superconducting Solenoid

Gabrielse

Solenoid in Place

Larger

• to allow positron access

• to let electron move with

the solenoid

One superconducting lead

was broken inside.

� We had to repair

Gabrielse

New Silver Trap

Room for positron entry path

Improved cavity mode spectrum (to allow cavity sideband cooling)

old new

Gabrielse

Conclusion

Took many years to develop the quantum methods to measure the

electron magnetic moment

� Most precise measurement of the electron magnetic moment

� Most precise determination of the fine structure constant

� Most stringent test of QED (higher order and precision)

Soon: Most precise test of CPT invariance with leptons

More precise electron magnetic moment

More precise determination of the fine structure constant

Spinoffs: proton magnetic moment

one-electron qubit