Zero-point confinement in a suitable trap is briefly discussed as a quantum-mechanical equivalent of the classical single particle at rest in free space. So far, such confinement has been realized only for the 150 GHz cyclotron motion in geonium, a single electron permanently confined in a Penning trap. The most important result of mono-ion spectroscopy is a new, 10 000 times smaller, radius for the electron. This result was obtained by analyzing our g-factor data, g/2 = 1.001 159 652 193 (4), on the basis of a near-Dirac particle model. RF spectroscopy in geonium relies on the Continuous Stem Gerlach Effect in which a spin flip is detected as a small change in the ~ 60 MHz axial oscillation frequency of the electron in the trap. The Kaufmann Effect or relativistic mass shift may become a superior alternative : operating a geonium apparatus as a frequency selective mini-synchro-cyclotron has produced an easily detectable shift in the axial frequency. Like in Habann's 1926 split-anode magnetron, energy is quickly transferred from the oscillating electron to a resonant circuit, but by us for the purpose of detection, damping and cooling of the oscillatory motion. Conversely, localizing the electron to ~ 60 μm in the node of a standing wave in the trap cavity, where this cavity looks like a short, when it is resonant with the cyclotron motion, has made it possible to decrease the natural line width ten-fold. By driving the axial frequency not on resonance but on a side band higher by the ~ 12 kHz magnetron frequency, it has been possible to force the magnetron motion to absorb the excess in the photon energy and thereby shrink the magnetron radius to ~ 15 μm. By an analogous laser spectroscopic procedure the oscillation amplitude of a Ba+ ion in a different trap has been reduced to ~ 120 nm or less. Like in Nuclear Magnetic Resonance, confinement is now much smaller than the wave length and no motional side bands appear in the optical spectrum. Such a mono-ion oscillator, but using Tl+ or a similar ion, may make an optical frequency standard with a 1 000-day reproducibility of 10 -18 possible and thereby allow one to watch the expansion of the universe from one's laboratory.
A single electron in a quantum cyclotron yields new measurements of the electron magnetic moment, given by g/2 = 1.00115965218073(28)[0.28 ppt], and the fine structure constant, α −1 = 137.035999084(51)[0.37 ppb], both significantly improved from prior results. The static magnetic and electric fields of a Penning trap confine the electron, and a 100 mK dilution refrigerator cools its cyclotron motion to the quantum-mechanical ground state. A quantum nondemolition measurement allows resolution of single cyclotron jumps and spin flips by coupling the cyclotron and spin energies to the frequency of the axial motion, which is self-excited and detected with a cryogenic amplifier. The trap electrodes form a high-Q microwave resonator near the cyclotron frequency; coupling between the cyclotron motion and cavity modes can inhibit spontaneous emission by over 100 timesthe free-space rate and shiftthe cyclotron frequency, a systematic effect that dominated the uncertainties of previous g-value measurements. A cylindrical trap geometry creates cavity modes with analytically calculable couplingsto cyclotron motion. Two independent methods use the cyclotron damping rate of an electron plasma or of the single electron itself as probes of the cavity mode structure and allow the identification ofthe modes by their geometries and couplings, the quantification of an offset between the mode and electrostatic centers, and the reduction of the cavity shift uncertainty to sub-dominant levels. Measuring g at four magnetic fields with cavity shifts spanning thirty times the final g-value uncertainty provides a check on the calculated cavity shifts. Magnetic field fluctuations limit the measurement of g by adding a noise-model dependence to the extraction of the cyclotron and anomaly frequencies from their resonance lines; the relative agreement of two line-splitting methods quantifies a lineshape model uncertainty. New techniques promise to increase field stability, narrow the resonance lines, and accelerate the measurement cycle. The measured g allows tests for physics beyond the Standard Model through searches for its temporal variation and comparisons with a “theoretical” g-value calculated from quantum electrodynamics and an independently measured fine structure constant.
One-electron
3+1 and
2+1 Dirac equations are used to calculate the motion of a relativistic electron in a vacuum in the presence of an external magnetic field. First, calculations are carried on an operator level and exact analytical results are obtained for the electron trajectories which contain both intraband frequency components, identified as the cyclotron motion, as well as interband frequency components, identified as the trembling motion, Zitterbewegung (ZB). Next, time-dependent Heisenberg operators are used for the same problem to compute average values of electron position and velocity employing Gaussian wave packets. It is shown that the presence of a magnetic field and the resulting quantization of the energy spectrum has pronounced effects on the electron ZB: it introduces intraband frequency components into the motion, influences all the frequencies, and makes the motion stationary (not decaying in time) in case of the
2+1 Dirac equation. Finally, simulations of the
2+1 Dirac equation and the resulting electron ZB in the presence of a magnetic field are proposed and described employing trapped ions and laser excitations. Using simulation parameters achieved in recent experiments of Gerritsma and coworkers, we show that the effects of the simulated magnetic field on ZB are considerable and can certainly be observed.