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Measurement of the Electron Magnetic Moment and the Fine Structure Constant

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Measurement of the Electron Magnetic Moment and the Fine Structure Constant PHYSICAL REVIEW LETTERS week ending PRL 100, 120801 (2008) 28 MARCH 2008 New Measurement of the Electron Magnetic Moment and the Fine Structure Constant D. Hanneke, S. Fogwell, and G. Gabrielse* Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA (Received 4 January 2008; published 26 March 2008) A measurement using a one-electron quantum cyclotron gives the electron magnetic moment in Bohr magnetons, g=2 1:001 159 652 180 73 28 [0.28 ppt], with an uncertainty 2.7 and 15 times smaller than for previous measurements in 2006 and 1987. The electron is used as a magnetometer to allow line shape statistics to accumulate, and its spontaneous emission rate determines the correction for its interaction with a cylindrical trap cavity. The new measurement and QED theory determine the fine structure constant, with ÿ1 137:035 999 084 51 [0.37 ppb], and an uncertainty 20 times smaller than for any independent determination of . DOI: 10.1103/PhysRevLett.100.120801 PACS numbers: 06.20.Jr, 12.20.Fv, 13.40.Em, 14.60.Cd The electron magnetic moment is one of the few order QED calculation (underway [7]) than by the mea- measurable properties of one of the simplest of elementary surement uncertainty in g=2. The accuracy of the new g particles—revealing its interaction with the fluctuating sets the stage for an improved CPT test with leptons. It also QED vacuum, and probing for size or composite structure will allow an improved test of QED, and will be part of the not yet detected. What can be accurately measured is g=2, discovery of low-mass dark-matter particles or the elimi- the magnitude of scaled by the Bohr magneton, B nation of this possibility [8], when a better independent e@= 2m. For an eigenstate of spin S, measurement of becomes available. Figure 3 represents the lowest cyclotron and spin energy g S levels for an electron weakly confined in a vertical mag- ÿ B @ ; (1) 2 =2 netic field Bz^ and an electrostatic quadrupole potential. with g=2 1 for a point electron in a renormalizable Dirac The latter is produced by biasing the trap electrodes of description. QED predicts that vacuum fluctuations and Fig. 2. The measured cyclotron frequency fc 149 GHz polarization slightly increase this value. Physics beyond (blue in Fig. 3) and the measured anomaly frequency a the standard model of particle physics could make g=2 173 MHz (red in Fig. 3) mostly determine g=2 [2] 2 deviate from the Dirac/QED prediction (as internal quark- g a ÿ z= 2fc gcav ’ 1 ; (2) gluon substructure does for a proton). 2 f 3=2 2= 2f 2 The 1987 measurement that provided the accepted g=2 c z c for nearly 20 years [1] was superceded in 2006 by a with only small adjustments for the measured axial fre- measurement that used a one-electron quantum cyclotron quency z 200 MHz, the relativistic shift =c 2 ÿ9 [2]. Key elements were quantum-jump spectroscopy and hc= mc 10 , and the cavity shift gcav=2. The latter quantum nondemolition (QND) measurements of the low- is the fractional shift of the cyclotron frequency caused by est cyclotron and spin levels [3], a cylindrical Penning trap the interaction with radiation modes of the trap cavity. The cavity [4] (Fig. 2), inhibited spontaneous emission [5], and Brown-Gabrielse invariance theorem [9] has been used to a one-particle self-excited oscillator (SEO) [6]. This Letter eliminate the effect of both quadratic distortions to the reports an improved measurement that has a 2.7 and electrostatic potential, and misalignments of the trap elec- 15 times lower uncertainty than the 2006 and 1987 mea- trode axis with B. Small terms of higher order in z=fc are surements, respectively, and confirms a 1.8 standard devia- neglected. tion shift of the 1987 value [Fig. 1(a)]. The interaction of the electron and its surrounding trap cavity is probed by measuring g=2 and the electron’s spontaneous emission rate as a function of magnetic field, thereby determining the corrections needed for good agreement between mea- surements at different fields. The electron is also used as its own magnetometer to accumulate quantum-jump line shape statistics over days, making it possible to compare methods for extracting the resonance frequencies. The new measurement and recently updated QED theory [7] determine with an uncertainty 20 times smaller than does any independent method [Fig. 1(b)]. The uncertainty FIG. 1. Most accurate measurements of the electron g=2 (a), in is now limited a bit more by the need for a higher- and most accurate determinations of (b). 0031-9007=08=100(12)=120801(4) 120801-1 © 2008 The American Physical Society PHYSICAL REVIEW LETTERS week ending PRL 100, 120801 (2008) 28 MARCH 2008 trap cavity electron top endcap citation, cavity-inhibited spontaneous emission provides electrode quartz spacer the time needed to turn on the electronic amplification compensation and feedback, so the SEO can reach an oscillation ampli- electrode tude at which the shift can be detected [6]. An anomaly nickel rings ring electrode transition is followed by a spontaneous decay to the spin- 0.5 cm compensation down ground state, jn 0;ms ÿ1=2i, and the QND bottom endcap electrode detection reveals the lowered spin energy. electrode field emission The expected line shapes arise from the thermal-axial microwave inlet point motion of the electron through the magnetic bottle gra- dient. The axial motion is cooled by a resonant circuit in FIG. 2 (color). Cylindrical Penning trap cavity used to confine about 0.2 s to as low as T 230 mK (from 5 K) when the a single electron and inhibit spontaneous emission. z detection amplifier is off. For the cyclotron motion these fluctuations are slow enough that the line shape is essen- Quantum-jump spectroscopy determines fc and a.For tially a Boltzmann distribution with a width proportional to each of many trials the system is prepared in the spin-up Tz [11]. For the anomaly resonance, the fluctuations are ground state, jn 0;ms 1=2i, after which the prepara- effectively more rapid, leading to a resonance shifted in tion drives and detection amplifier are turned off for 1 s. proportion to Tz. Either a cyclotron drive at a frequency near to f ,oran c We use the weighted average of a and fc from the line anomaly drive at frequency near a, is then applied for 2 s. shapes (indicated by the abscissa origins in Fig. 4) in The amplifier and a feedback system are turned on to Eq. (2) to determine g=2. With saturation effects avoided, provide QND detection of either a one-quantum cyclotron these pertain to the magnetic field averaged over the ther- excitation or a spin flip. Cavity-inhibited spontaneous mal motion. It is crucial that any additional fluctuations in emission makes the cyclotron excitation persist long B that are symmetric about a central value will broaden enough to allow such detection. Figure 4 shows the fraction such line shapes without changing the mean frequency. of the trials for which excitations were detected. To test this weighted mean method we compare maxi- The cyclotron drive is microwave radiation injected into mum likelihood fits to line shape models (Fig. 4). The data the trap cavity through a cold attenuator to keep black body fit well to a convolution (solid curve) of a Gaussian reso- photons from entering the trap. The anomaly drive is an lution function (solid inset curve) and a thermal-axial- oscillatory potential applied to electrodes at frequencies motion line shape [11] (dashed curve). The broadening near a to drive off-resonant axial motion through the magnetic bottle gradient from two nickel rings (Fig. 2). The electron, radially distributed as a cyclotron eigenstate, 0.2 147.5 GHz 147.5 GHz sees an oscillating magnetic field perpendicular to B as 0.1 needed to flip its spin, with a gradient that allows a simul- taneous cyclotron transition [10]. To ensure that the elec- 0.0 0.2 149.2 GHz 149.2 GHz tron samples the same magnetic variations while a and fc transitions are driven, both drives are kept on with one 0.1 detuned slightly so that only the other causes transitions. Low drive strengths keep transition probabilities below 0.0 20% to avoid saturation effects. 0.2 150.3 GHz 150.3 GHz QND detection of one-quantum changes in the cyclotron 0.1 and spin energies takes place because the magnetic bottle shifts the oscillation frequency of the self-excited axial excitation fraction 0.0 oscillation as z 4 n ms Hz. After a cyclotron ex- 0.2 151.3 GHz 151.3 GHz n = 2 0.1 νc - 5δ/2 0.0 νa n = 1 -5 0 5 10 -5 0 5 n = 2 ( ν - f ) / ppb ( ν - ν ) / ppb ν δ c a ν - 3δ/2 fc = c - 3 /2 c n = 0 n = 1 FIG. 4. Quantum-jump spectroscopy line shapes for cyclotron νa = gνc / 2 - νc νc - δ/2 (left) and anomaly (right) transitions, with maximum likelihood n = 0 fits to broadened line shape models (solid), and inset resolution ms = -1/2 ms = 1/2 functions. Vertical lines show the 1- uncertainties for extracted resonance frequencies. Corresponding unbroadened line shapes FIG. 3 (color online). Electron’s lowest cyclotron and spin are dashed. Gray bands indicate 68% confidence limits for levels. distributions about broadened fits. 120801-2 PHYSICAL REVIEW LETTERS week ending PRL 100, 120801 (2008) 28 MARCH 2008 TABLE I.
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