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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