Regulated soft coherent photon-count prediction — Examples

Use the one-charge prescribed current of Regulated soft coherent photon-count prediction with final velocity $v=\frac{1}{2}$ along $\widehat{\mathbf{z}}$. Choose any upper photon-energy cutoff $\mathrm{\Lambda }>0$ in joules and set $\lambda =\mathrm{\Lambda }\mathrm{/}\mathrm{e}$, where $\mathrm{e}$ is Euler's number. The angular coefficient is $C=\frac{\alpha }{\pi }(2\log 3-2)>0.$ The exact cutoff coherent state therefore has mean photon count $C$, vacuum probability $e^{-C}$, and mean photon energy $C\mathrm{\Lambda }(1-{\mathrm{e}}^{-1})$ joules. Each halving of $\lambda$ increases the mean count by $C\log 2$; the mean energy remains bounded and tends to $C\mathrm{\Lambda }$ as $\lambda \downarrow 0$. These are model predictions for the prepared free state.

The primary report is X. Fan, T. G. Myers, B. A. D. Sukra, and G. Gabrielse, “Measurement of the Electron Magnetic Moment,” arXiv:2209.13084v2 (7 December 2022), published as Physical Review Letters 130, 071801 (2023). It reports a one-electron quantum cyclotron in a cylindrical silver-electrode Penning trap. A magnetic field near $5.3\mathrm{T}$ and an electrostatic quadrupole potential confine the electron; the source describes a nominal $50\mathrm{m}\mathrm{K}$ trap on a $4.2\mathrm{K}$ solenoid and a cavity cooled below $100\mathrm{m}\mathrm{K}$. The axial detection amplifier heats the axial motion to a source-reported $T_z\approx 0.5\mathrm{K}$; this is distinct from the nominal trap temperature.

The source defines the magnetic moment ratio by $-\mu \mathrm{/}{\mu }_B=g\mathrm{/}2$, with ${\mu }_B=e\mathrm{\hslash }\mathrm{/}(2m)$. Here $e>0$ is the elementary-charge magnitude in coulombs, $\mathrm{\hslash }$ is the reduced Planck constant in joule-seconds, and $m>0$ is the electron rest mass in kilograms; accordingly ${\mu }_B$ has SI unit $\mathrm{A}{\mathrm{m}}^2$ (equivalently $\mathrm{J}\mathrm{/}\mathrm{T}$). The electron charge is $-e$. This item denotes the reported dimensionless electron gyromagnetic ratio as $g_s\mathrm{/}2$, distinguishing it from the QFT coupling $g_{\mathrm{e}\mathrm{m}}$ in Example observable types and scoped meanings. The exact finite-regulator, formal-coefficient, and source-reported estimate types are those in Example observable types and scoped meanings.

The ideal linear model Ideal linear Penning mode-frequency invariant gives a conditional sum-of-squares identity for its complete stable mode spectrum. That lemma is an ideal model supplier. The reported apparatus analysis additionally applies relativistic-frequency and cavity corrections and is not reconstructed from the ideal identity alone.

The authors infer $g\mathrm{/}2$ from spin and cyclotron frequencies, using the anomaly frequency ${\nu }_a={\nu }_s-{\nu }_c$ and the ratio $g\mathrm{/}2=1+{\nu }_a\mathrm{/}{\nu }_c$. The report gives trap-modified frequencies near ${\bar{\nu }}_c=149\mathrm{G}\mathrm{H}\mathrm{z}$, ${\bar{\nu }}_a=173\mathrm{M}\mathrm{H}\mathrm{z}$, and ${\bar{\nu }}_z=114\mathrm{M}\mathrm{H}\mathrm{z}$ at the stated field. It uses the Brown–Gabrielse invariance relation to obtain the free cyclotron frequency, then a corrected ratio involving fitted frequencies, the relativistic shift, and an added cavity shift.

Cyclotron and anomaly quantum-jump trials are alternated. Each begins by preparing the spin-up cyclotron ground state and applying one second of magnetron sideband cooling. Cyclotron transitions are driven for five seconds with excitation kept below 20 percent to limit saturation; anomaly transitions are driven for 30 seconds. Axial-frequency shifts detect the spin and cyclotron jumps. The authors fit measured line shapes to infer the relevant frequencies and correct magnetic-field drift with a quadratic fit. Their cyclotron line-shape fit includes Gaussian broadening to accommodate a reported mismatch between the predicted and observed cyclotron line shape.

The cavity correction uses measured mode frequencies and quality factors. The report describes three consistent measurement methods and a renormalized calculation that subtracts ideal contributions for 72 modes and adds back contributions using measured mode frequencies and quality factors. The leading cavity-shift uncertainty is attributed to field modifications from cavity imperfections and misalignment.

The authors report the weighted average of eleven cavity-corrected determinations: $-\frac{\mu }{{\mu }_B}=\frac{g}{2}=1.00115965218059(13)\mathrm{.}$ They state that the parenthetical digits give a $1\sigma$ uncertainty of $1.3\times {10}^{-13}$ in this dimensionless ratio. The reported consistency across the eleven field settings is a source observation; this item does not turn it into a proof of QED or of the apparatus model.

The paper's uncertainty table lists, in units of ${10}^{-13}$, statistical $0.29$, cyclotron broadening $0.94$, cavity correction $0.90$, nuclear paramagnetism $0.12$, anomaly power shift $0.10$, magnetic-field drift $0.09$, and total $1.3$. The authors treat cavity uncertainties as correlated for nearby field settings and do not recommend averaging this determination with their 2008 result because correlations between similar methods are difficult to determine. This record preserves the authors' stated $1\sigma$ uncertainty without treating it as a posterior probability or asserting Gaussian, independent, or frequentist-coverage assumptions. The raw trial record, full covariance matrix, and weights are not reconstructed here.

This item records the source-reported estimate and its stated apparatus, analysis, and uncertainty qualifications. The paper separately compares the measurement with a Standard Model prediction that uses QED coefficients and independent inputs; that calculation and comparison are outside this record. The estimate is not an exact constant, and agreement with a theory is not a deductive proof of that theory.

Let the source-reported dimensionless ratio be denoted $\widehat{g_s\mathrm{/}2}:=-\widehat{\mu }\mathrm{/}{\mu }_B$, where ${\mu }_B=e\mathrm{\hslash }\mathrm{/}(2m)$ is the Bohr magneton, with $e>0$ the elementary-charge magnitude in coulombs, $\mathrm{\hslash }$ the reduced Planck constant in joule-seconds, and $m>0$ the electron rest mass in kilograms, as specified in Fan2022 reported electron magnetic-moment determination. The hat marks an empirical estimate, not an exact constant. Define the derived dimensionless anomaly estimate ${\widehat{a}}_e:=\widehat{g_s\mathrm{/}2}-1$. Under the empirical premise from Fan2022 reported electron magnetic-moment determination, whose reported central value is $\widehat{g_s\mathrm{/}2}=1.00115965218059$, exact subtraction gives ${\widehat{a}}_e=\mathrm{0.00115965218059.}$ The source-stated $1\sigma$ uncertainty remains $1.3\times {10}^{-13}$, and all apparatus, correction, weighting, correlation, and source-data qualifications of the experiment remain in force. This is only an affine transformation of a reported estimate; it supplies no comparison with a theoretical prediction.