PHYSICAL REVIEW LETTERS week ending PRL 97, 030801 (2006) 21 JULY 2006 New Measurement of the Electron Magnetic Moment Using a One-Electron Quantum Cyclotron B. Odom,* D. Hanneke, B. D’Urso,† and G. Gabrielse‡ Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA (Received 17 May 2006; published 17 July 2006) A new measurement resolves cyclotron and spin levels for a single-electron quantum cyclotron to obtain an electron magnetic moment, given by g=2 1:001 159 652 180 85 760:76 ppt. The uncer- tainty is nearly 6 times lower than in the past, and g is shifted downward by 1.7 standard deviations. The new g, with a quantum electrodynamics (QED) calculation, determines the fine structure constant with a 0.7 ppb uncertainty—10 times smaller than for atom-recoil determinations. Remarkably, this 100 mK measurement probes for internal electron structure at 130 GeV. DOI: 10.1103/PhysRevLett.97.030801 PACS numbers: 06.20.Jr, 12.20.Fv, 13.40.Em, 14.60.Cd Measurements of the electron magnetic moment () for a 10 times more stringent test. Third, even though muon probe the electron’s interaction with the fluctuating vac- g values [13] have nearly 1000 times larger uncertainties uum of QED, and also probe for possible electron sub- compared to the electron g, heavy particles (possibly un- structure. As an eigenstate of spin S, the electron (charge known in the standard model) make a contribution that is ÿe and mass m) has / S, relatively much larger for the muon. However, the contri- e@ S bution is small compared to the calculated QED contribu- ÿg : (1) tion which depends on and must be subtracted out. The 2m @ electron g provides and a confidence-building test of the The constant g is a dimensionless measure of the moment, required QED. with the dimensions and approximate size given by the The electron g determines the spin frequency g @ s 2 c Bohr magneton, e = 2m. If the electron was a mechanical for a free electron in a magnetic field Bz^. To weakly system with an orbital angular momentum, then g would confine the electron, an electric quadrupole potential, V depend upon the relative distributions of the rotating 2z2 ÿ 2, is added, with xx^ yy^. Optimal biasing of charge and mass, with g 1 for identical distributions. the electrodes [Fig. 2(a)] of an orthogonalized cylindrical [Cyclotron motion of a charge in a magnetic field B,at Penning trap [9] minimizes an undesired z4 term. The frequency c eB= 2m, is one example.] A Dirac point electron-trap system has four eigenfrequencies. The spin particle has g 2. QED predicts that vacuum fluctuations and trap-modified cyclotron frequencies are approximately and polarization slightly increase this value. Electron sub- equal at 149 GHz. A harmonic axial oscillation structure [1] would make g deviate from the Dirac-QED s c along B is at z 200 MHz, and an orthogonal circular prediction (as quark-gluon substructure does for a proton). magnetron oscillation is at 134 kHz. The latter three Measurements of the electron g have a long history m frequencies are shifted by the unavoidable leading imper- [2,3], with a celebrated measurement [4] providing the fections of a real Penning trap—harmonic distortions of accepted value [5] since 1987. The new g has a 6 times the quadrupole potential, and a misalignment of the elec- smaller standard deviation and is shifted by 1.7 standard trode axis and B [14]. Silver trap electrodes were used after deviations [Fig. 1(a)]. A one-electron quantum cyclotron the nuclear paramagnetism of copper electrodes caused [6], cavity-inhibited spontaneous emission [7], a self- unacceptable temperature-dependent fluctuations in B excited oscillator (SEO) [8], and a cylindrical Penning near 100 mK. trap [9] contribute to the extremely small uncertainty. For The spin motion is undamped, being essentially un- the first time, spectroscopy is done with the lowest cyclo- coupled from its environment [15]. The cyclotron motion tron and spin levels of a single electron fully resolved via quantum nondemolition (QND) measurements [6], and a (a) ppt = 10-12 (b) ppb = 10-9 cavity shift of g is directly observed. 0510-15 -10 -5 0 5 10 What can be learned from the more accurate electron g? The first result beyond g itself is the fine structure constant, Harvard (2006) Rb (2006) Harvard g (2006) 2 Cs (2006) e = 4 @c, determined from g and QED with UW (1987) CODATA 2002 0 UW g (1987) 10 times smaller uncertainty compared to any other 180 185 190 7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0 -1 -6 method [10–12]. This fundamental measure of the strength (g / 2 - 1.001 159 652 000) / 10-12 (α - 137.035 990) / 10 of the electromagnetic interaction is a crucial ingredient in our system of fundamental constants [5]. Second, the most FIG. 1 (color). Measurements of the electron g (a). demanding test of QED continues to be a comparison of Determinations of [11,12], and the current CODATA value measured and calculated g, and the way is now prepared [5] (b). Measured g are converted to with current QED theory. 0031-9007=06=97(3)=030801(4) 030801-1 © 2006 The American Physical Society PHYSICAL REVIEW LETTERS week ending PRL 97, 030801 (2006) 21 JULY 2006 (a) trap cavity electron (b) n = 1 20 top endcap 10 (a) (b) n = 2 ν δ / 2 shift fc = c - 3 0 electrode z in ppb in quartz spacer ν - 3δ / 2 ν compensation c n = 0 0 2040600 204060 electrode n = 1 ν ν / 2 − ν time (s) time (s) ring electrode nickel rings a = g c c compensation ν δ / 2 0.5 cm c - electrode 0.20 (c) (d) bottom endcap n = 0 electrode 0.15 field emission point ms = -1/2 ms = 1/2 0.10 1 ppb 1 ppb FIG. 2. Cylindrical Penning trap cavity used to confine a single 0.05 0.00 electron and inhibit spontaneous emission (a), and the cyclotron quantum jump fraction and spin levels of an electron confined within it (b). -1 0 1 2 01 frequency - νa in Hz frequency - fc in kHz would damp in 0:1svia synchrotron radiation in free FIG. 3. Sample z shifts for a spin flip (a) and for a one- space. This spontaneous emission is greatly inhibited in the quantum cyclotron excitation (b). Quantum jump spectroscopy line shapes for anomaly (c) and cyclotron (d) transitions, with a trap cavity (to 6.7 or 1.4 s here) when B is tuned so c is far from resonance with cavity radiation modes [7,15]. maximum likelihood fit to the calculated line shapes (solid). The bands indicate 68% confidence limits for distributions of mea- Blackbody photons that would excite the cyclotron ground surements about the fit values. state are eliminated by cooling the trap and vacuum en- closure below 100 mK with a dilution refrigerator [6]. (Thermal radiation through the microwave inlet makes circuit that is amplified and fed back to drive the oscilla- <1 excitation=h.) The axial motion, damped by a resonant tion. QND couplings of spin and cyclotron energies to z circuit, cools below 0.3 K (from 5 K) when the axial [6] arise because saturated nickel rings [Fig. 2(a)] produce 2 2 detection amplifier is off for crucial periods. The magne- a small magnetic bottle, B 2 z ÿ =2z^ ÿ z^ 2 tron motion radius is minimized with axial sideband cool- with 2 1540 T=m . ing [15]. Anomaly transitions are induced by applying potentials For the first time, g is deduced from observed transitions oscillating at a to electrodes, to drive an off-resonance between only the lowest of the spin (ms 1=2) and axial motion through the bottle’s z gradient. The electron cyclotron n 0; 1; 2; ... energy levels [Fig. 2(b)], sees the oscillating magnetic field perpendicular to B as needed to flip its spin, with a gradient that allows a simul- g 1 1 1 2 E n; ms hcms n hc ÿ h n ms : taneous cyclotron transition. Cyclotron transitions are in- 2 2 2 2 duced by microwaves with a transverse electric field that (2) are injected into and filtered by the cavity. The electron samples the same magnetic gradient while and f eB= m a c The needed c 2 (for a free electron in a transitions are driven, because both drives are kept on, magnetic field) is related to the observable eigenfrequen- with one detuned slightly so that only the other causes cies by the Brown-Gabrielse invariance theorem [14], transitions. 2 2 2 2 c c z m ; (3) A measurement starts with the SEO turned on to verify that the electron is in the upper of the two stable ground which applies despite the mentioned imperfection shifts states, jn 0;ms 1=2i. Simultaneous c ÿ =2 and a of the three eigenfrequencies. The third term in Eq. (2), drives prepare this state as needed. The magnetron radius is the leading relativistic correction [15] with =c reduced with 1.5 s of strong sideband cooling [15]at 2 ÿ9 z hc= mc 10 , would add uncertainty to the measure- m, and the detection amplifier is turned off. After 1 s, ment if cyclotron energy levels were not resolved. either an fc drive, or a a drive, is on for 2 s. The detection The anomaly and spin-up cyclotron frequencies [a amplifier and the SEO are then switched on to check for a 173 MHz and fc in Fig.