The measurement of the anomalous electron magnetic moment $g-2$ through quantum transitions of a single trapped electron is the most stringent test of quantum field theory. These experiments are now so precise that they must account for the effects of the cavity containing the electron. Classical calculations of this "cavity shift" must subtract the electron's divergent self-field, and thus require knowledge of the exact Green's function for the cavity's electromagnetic field. We perform the first fully quantum calculation of the cavity shift in a closed cavity, which instead involves subtracting linearly divergent cavity mode sums and integrals. Using contour integration methods, we find perfect agreement with existing classical results for both spherical and cylindrical cavities, justifying their current use. Moreover, our mode-based results can be naturally generalized to account for systematic effects, necessary to push future measurements to the next order of magnitude in precision.
The measurement of the anomalous electron magnetic moment g−2 through quantum transitions of a single trapped electron is the most stringent test of quantum field theory. These experiments are now so precise that they must account for the effects of the cavity containing the electron. Classical calculations of this "cavity shift" must subtract the electron's divergent self-field, and thus require knowledge of the exact Green's function for the cavity's electromagnetic field. We perform the first fully quantum calculation of the cavity shift in a closed cavity, which instead involves subtracting linearly divergent cavity mode sums and integrals. Using contour integration methods, we find perfect agreement with existing classical results for both spherical and cylindrical cavities, justifying their current use. Moreover, our mode-based results can be naturally generalized to account for systematic effects, necessary to push future measurements to the next order of magnitude in precision.
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.
A cylindrical Penning trap is of such high quality that a single electron is observed with good signal-to-noise. The trap is also a good approximation to an ideal cylindrical microwave cavity. This is documented in situ at 4 K with a new probe utilizing trapped electrons. An extraordinary control of the cavity-electron interaction results and opens the way to a new generation of electron magnetic moment measurements, not limited by either cavity frequency shifts or damping widths, and to sideband cooling of an electron motion to very low (mK) temperatures.
The energy-level shift of a ground-state atom in front of a nondispersive dielectric half-space is calculated by quantizing the electric field by means of a normal-mode expansion and applying second-order perturbation theory to the electric-dipole Hamiltonian $\mathit{\ensuremath{\mu}}\ensuremath{\cdot}\mathbf{E}.$ It is shown that the contributions to this shift coming from traveling and from evanescent waves can be combined into a single expression which lends itself readily to asymptotic analysis for large atom-surface separations, while in the opposite asymptotic regime when the atom is close to the surface the combined expression is less convenient. Employing a Green's-function formalism instead of the normal-mode expansion leads directly to the combined formula, and in that case it is advantageous to be able to apply the same transformation backwards and split the energy shift into a sum of distinct contributions corresponding to different physical processes. The analysis serves to shed light on common sources of error in the literature and paves the way for the study of more complicated models in cavity quantum electrodynamics.
In 1965, the Nobel Prize was awarded for the modern formulation of Quantum Electrodynamics (QED) to R.P. Feynman for his diagrammatic method, J. Schwinger for the operator method, and S.-I. Tomonaga for his relativistic derivation. Subsequently, P. Kusch performed a precise measurement of the electron's anomalous magnetic moment, providing a critical test validating QED's computational methods, which proved to be extremely accurate. Nevertheless, QED is based on a physically unrealistic assumption—mathematically circumvented—that the electron is a dimensionless point particle endowed with physical properties "ex abrupto." This work, developed within a deterministic and non-local dBBZ (modified de Broglie–Bohm) theory, models the electron as an entangled and distributed structure, as proposed in previous studies. It enables the calculation of the electron's anomalous magnetic moment as a consequence of its intrinsic structure, without employing QED techniques. The method involves calculating, for each orbital, the anomalous moment modified to account for the influence of existing magnetic fields. Subsequently, entanglement is imposed on the sum of these moments, and from the total orbital anomalous moment thus obtained, the theoretical magnetic anomaly is derived and compared with the corresponding experimental value, yielding relative errors on the order of 10^-10. This procedure, which allows for greater theoretical precision , need a similar , albeit more complex, calculation of the muon's and tau anomalous magnetic moment. This is because a specific parameter, termed the "source parameter," selected within an allowable range requires , at least , dual verification to be adopted with high precision. The theoretical determination of the muon's anomalous magnetic moment is also presented in a subsequent document, employing analogous computational procedures.
The interaction of an atomic system with an unquantized electromagnetic field is studied by means of the Heisenberg-operator equations of motion. The electromagnetic fields created by the atom are taken into account by assuming that the charge and probability current densities are the actual charge and current source terms in Maxwell's equation. When Ehrenfest's theorem is written to order (1/c)2, the equations of motion are found to have a constant of motion that can be interpreted as stating that the sum of the atomic energy, energy of interaction, and energy stored in the electromagnetic field is constant. This constant of integration can be expressed as an expectation of a new semiclassical Hamiltonian that now includes the effects of the atomic self-fields to order 1/c2. This Hamiltonian is related to the classical Darwin Lagrangian. Since the new approximate Hamiltonian is the sum of an atomic and a field Hamiltonian, it provides a formulation of semiclassical radiation theory that is formally close to the usual formulation of quantum electrodynamics. A Schr\"odinger equation can be derived by applying the variational principle to the expectation of the new Hamiltonian. The result is a nonlinear integro-differential equation, in \ensuremath{\Psi}, which is somewhat similar to Hartree's self-consistent equation for a multielectron atom.When the Heisenberg equations are written to include the next-higher-order terms, 1/c3, it is found that the total energy of the atom and the electromagnetic field in the vicinity of the atom decreases at a rate that is given by the Larmor power formula (2e2/3c3)(d〈v〉/dt)2, where 〈v〉 is the quantum-mechanical expectation of the electron's velocity operator. This approximate formulation of semiclassical radiation theory is applied to semiclassical calculations of atomic radiative shifts. It is then shown that, if some retardation effects are included to all orders of 1/c in the vector potential, the semiclassical analysis provides a formula for radiative energy-level shifts that are time dependent and that contain coefficients similar to the starting point of the Bethe calculation of the Lamb shift. The incorporation of mass renormalization in semiclassical theory is then discussed.
The study of the appearance of quantum effects in an essentially classical experiment---involving the passage of an electron through a resonant cavity---as the frequency of the electromagnetic field increases, was begun in a previous paper, in which the electron velocity was examined. In the present paper, an analysis is made of the electric field in the same experiment. Both the expectation value of the field, and the expectation value of the square of the deviation from the expectation value (the dispersion) of the field are calculated to the lowest order in the electron charge. The expectation value of the field is seen to differ from the classical value by a small amount, the difference being due to the wave properties of the electron. The dispersion in the field is shown to consist of three parts: one due to the quantum-mechanical properties of the field alone, the second due to the wave properties of the electron, and the third due to the interaction between the quantum-mechanical properties of the field and the classical properties of the electron. The orders of magnitude of the three terms are shown to decrease in the order in which they are enumerated. A calculation is then made of the effect on the field of a number of electrons passing through the cavity, the interaction between the electrons being neglected. The number of electrons required in a "bunch" to produce a field comparable to the dispersion in the field is calculated, and, for a typical set of experimental conditions, it is seen that the highest frequency at which this requirement can be met is of the order of 3\ifmmode\times\else\texttimes\fi{}1012/sec.