Fully Quantum Measurements of the Electron Magnetic Moment

Brian Odom Research performed at Harvard University,

Gabrielse group Les Houches Physics with Trapped Charged Particles January 2012

New values for g and α

Funding: NSF

Newer values for g and α

Phys. Rev. Lett. 106, 080801 (2011)

The electron g-factor

Classical, non-relativistic Dirac equation as single- particle wave equation Quantum Electrodynamics (QED)

2.002 319 304g = ...

2g =

1g =

2e S

gm

µ

= −

B

Sgµ

µ≡

or, if you prefer …

g-factor from QED 2 3 4

1 2 3 41 ...2g C C C C non QEDα α α α

π π π π = + + + + + + −

Why measure the electron g-factor?

• Determination of α, using QED calculations

Why measure the electron g-factor?

• Determination of α, using QED calculations

• Precision test of QED

α-1137.03599 137.03600 137.03601

∆α / α (ppb)-100-50050100

muonium h.f. structure

electron g, UW 1987

quantum Hall effect

ac Josephson effect & γp,h

h / mn

h / mCs, optical trans- itions, mass ratios

electron g, Harvard 2006

h / mRb, mass ratios

Testing QED with measurements of α

?

Why measure the electron g-factor?

• Determination of α, using QED calculations

• Precision test of QED

• Probe for electron sub-structure (R < 10-18 m)

Why measure the electron g-factor?

• Determination of α, using QED calculations

• Precision test of QED

• Probe for electron sub-structure (R < 10-18 m)

• Precision test of Lorentz, CPT symmetry

Why measure the electron g-factor?

• Determination of α, using QED calculations

• Precision test of QED

• Probe for electron sub-structure (R < 10-18 m)

• Precision test of Lorentz, CPT symmetry

• Complement to the muon g-factor measurement

Why measure the electron g-factor?

• Determination of α, using QED calculations

• Precision test of QED

• Probe for electron sub-structure (R < 10-18 m)

• Precision test of Lorentz, CPT symmetry

• Complement to the muon g-factor measurement

• Prospects for improved proton to electron mass ratio

A single electron in a Penning trap

motion frequency hυ/kB damping axial 200 MHz 9.6 mK 1 Hz

cyclotron 149.0 GHz 7.2 K 0.02 Hz

spin 149.2 GHz 7.2 K 10-12 Hz

magnetron 130 kHz 6.4 µK 10-17 Hz

g-factor measurement

( )( ) ( )

2 1a z c 2

21 1c z c2 2

2=1

2 2g ω ω ω δ

ω δ ω ω δ− +

++ + +

g in free space:

g-2 in free space:

a

c

21 12 2g g ω

ω−= + = +

g-2 in a Penning trap:

[ Brown and Gabrielse. Rev. Mod. Phys. 58, 1 (1986) ] (3 orders of magnitude for free)

B s s

B B c

22Bg

Sω ωµ

µ µ ω

µµ

≡ = = =

Cylindrical Penning trap construction

Dilution refrigerator and magnet

A tabletop experiment … if you have a high ceiling

Experimental setup

B-fi

eld

shift

(ppb

)

-20

-10

0

10

20

time (hours)0 10 20 30 40 50 60

dew

ar te

mpe

ratu

re (C

)

19.0

19.5

20.0

20.5

21.0

21.5

B-fi

eld

shift

(ppb

)

-20

-10

0

10

20

time (hours)0 20 40 60

dew

ar te

mpe

ratu

re (C

)

19.0

19.5

20.0

20.5

21.0

21.5

B-field stability against room temperature

•Magnet with two broken shims •No temperature regulation

•Magnet with working shims •Shed temperature regulated

Friday, Saturday construction

< 1 ppb noise and drift at night

~ 0.1 K temperature regulation of dewar

The axial oscillator is coupled to a tuned-circuit amplifier

Response to a resonant rf-drive applied to an endcap

Detection of single electron axial motion

Feedback for self-excitation

B. D’Urso, R. Van Handel, B. Odom, D. Hanneke, and G. Gabrielse. Phys. Rev. Lett. 94, 113002 (2005)

Quantized cyclotron motion

Can we observe quantum jumps between the cyclotron states?

Cylindrical cavity suppresses decay

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

15

τ = 16 s

• In free space, cyclotron lifetime = 0.08 s

• In our cylindrical traps, we have

achieved up to a 16 s lifetime

[ Peil and Gabrielse. Phys. Rev. Lett. 83, 1287 (1999) ]

Magnetic transitions are detected by a shift in the axial frequency

4

0

1

2

3

z

“Magnetic bottle” couples magnetic and axial oscillators

Detection of magnetic transitions

2z 0 2B B B z= + 2 21

E s c 22 k z zU Bµ ++=

Sub-Kelvin cyclotron temperature… Thermal Jumps

[ Peil and Gabrielse. Phys. Rev. Lett. 83, 1287 (1999) ]

• Permits single-quantum cyclotron spectroscopy

Sub-Kelvin cyclotron temperature… Thermal Jumps

[ Peil and Gabrielse. Phys. Rev. Lett. 83, 1287 (1999) ]

• Permits single-quantum cyclotron spectroscopy

Relativistic Corrections

• Eliminates relativistic error from ωc uncertainty

Single quantum cyclotron spectroscopy

time (s)0 100 200 300

axia

l fre

quen

cy s

hift

(Hz)

-3

0

3

6

9

12

15

Procedure: 1. Turn FET amplifier off 2. Apply a microwave drive pulse of ~150 GHz 3. Turn FET amplifier on, check for axial frequency shift 4. Plot a histograms of excitations vs. frequency

frequency - υc (ppb)0 100 200 300

# of

cyc

lotr

on e

xcita

tions

Single quantum cyclotron spectroscopy

time (s)0 100 200 300

axia

l fre

quen

cy s

hift

(Hz)

-3

0

3

6

9

12

15

Procedure: 1. Turn FET amplifier off 2. Apply a microwave drive pulse of ~150 GHz 3. Turn FET amplifier on, check for axial frequency shift 4. Plot a histograms of excitations vs. frequency

Poor amp heat sinking, amp off during excitation Tz = 16 K

frequency - υc (ppb)0 100 200 300

# of

cyc

lotr

on e

xcita

tions

Single quantum cyclotron spectroscopy

time (s)0 100 200 300

axia

l fre

quen

cy s

hift

(Hz)

-3

0

3

6

9

12

15

Procedure: 1. Turn FET amplifier off 2. Apply a microwave drive pulse of ~150 GHz 3. Turn FET amplifier on, check for axial frequency shift 4. Plot a histograms of excitations vs. frequency

Good amp heat sinking, amp on during excitation Tz = 3.7 K

frequency - υc (ppb)0 100 200 300

# of

cyc

lotr

on e

xcita

tions

Single quantum cyclotron spectroscopy

time (s)0 100 200 300

axia

l fre

quen

cy s

hift

(Hz)

-3

0

3

6

9

12

15

Procedure: 1. Turn FET amplifier off 2. Apply a microwave drive pulse of ~150 GHz 3. Turn FET amplifier on, check for axial frequency shift 4. Plot a histograms of excitations vs. frequency

Good amp heat sinking, amp off during excitation Tz = 0.32 K

An unpleasant surprise:

Temperature-dependent B

•We observed a huge shift of B-field vs. trap temperature •Heat load changes are unavoidable as:

•Amplifier cycles on/off

•Anomaly drive is applied

•10 ppb / mK is far too much!

tem

pera

ture

(mK

)

707580859095

100105

time (hours)0 2 4 6 8 10

B fi

eld

shift

(ppb

)

-300-250-200-150-100

-500

Shift of -10 ppb / mK at 75 mK !!!

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

temperature-1 (Kelvin-1)

0 5 10 15-100

0

100

200

300

400

500

600

700

•Nuclear paramagnetism makes standard Penning trap materials (copper, MACOR) incompatible with a stable B-field below 1 K

Curie-law paramagnetism…OF OUR TRAP!

40 ppb / K-1

New silver trap

Prototype silver tripod

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

•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

Finally—narrow line shapes

frequency - 170 410 496.7 Hz-0.5 0.0 0.5 1.0 1.5

spin

flip

frac

tion

0.00

0.05

0.10

0.15

0.20

2 ppb

frequency - 170 410 496.7 Hz-2 0 2 4 6 8 10

spin

flip

frac

tion

0.00

0.05

0.10

0.15

0.20

10 ppb

Comparison of line shapes

U. Wash. anomaly line

Harvard anomaly line

U. Wash. Harvard

Tz (K) 6 0.6 0.1 υz (MHz) 60 200 0.09 B2 (T/m2) 150 1500 10

H

UW

∆∆

H

UW0.1∆

∆ =

[ Van Dyck et al. Phys. Rev. Lett. 59, 26 (1987) ]

Scatter in g-factor measurements

uWave power (a.u.)0 20 40 60 80

176

178

180

182

184

186

Harvard 2006 UW 1987

UW 1991

uWave power (a.u.)0 20 40 60 80

176

178

180

182

184

186

Cavity mode structure

• Parametric response of large e- cloud maps cavity mode structure [ Tan and Gabrielse. App. Phys. Lett. 55, 2144 (1989) ]

Cavity mode structure

• Parametric response of large e- cloud maps cavity mode structure

• Modes coupling to centered single e- cloud are easily identified [ Tan and Gabrielse. App. Phys. Lett. 55, 2144 (1989) ]

TE 1n1 TM 1n1

First observation of cavity shift of g

Final Error Budget

Measurement summary

Harvard g-factor measurement: • Fully quantum measurement eliminates relativistic shift

( 1 ppt per quantum level )

• Low temperature allows quantum spectroscopy and narrows lines

• Cylindrical trap allows first quantitative treatment of cavity shift

Results : g / 2 = 1.001 159 652 180 85 (76) (0.76 ppt) α = 137.035 999 710 (90) (32) 137.035 999 710 (96) (0.70 ppb)

-1

Measurement summary

Harvard g-factor measurement: • Fully quantum measurement eliminates relativistic shift

( 1 ppt per quantum level )

• Low temperature allows quantum spectroscopy and narrows lines

• Cylindrical trap allows first quantitative treatment of cavity shift

g / 2 = 1.001 159 652 180 85 (76) (0.76 ppt) α = 137.035 999 710 (90) (32) 137.035 999 710 (96) (0.70 ppb)

-1

Results :

New values for g and α

g / 2 = 1.001 159 652 180 85 (76) (0.76 ppt) α = 137.035 999 710 (90) (32) 137.035 999 710 (96) (0.70 ppb)

g-factor from QED 2 3 4

1 2 3 41 ...2g C C C C non QEDα α α α

π π π π = + + + + + + −

Harvard 2008 measurement

Harvard 2008 measurement

Harvard 2008 measurement

α, a wrinkle and a new measurement

Phys. Rev. Lett. 106, 080801 (2011)

QED…Still standing 57 years later