Reported stored-lithium-ion Ives-Stilwell Doppler experiment at 0.338c (2014)

Quantities and apparatus notation. The laboratory speed parameter is $\beta =v\mathrm{/}c$, with $v$ the selected ion-class speed in an approximately inertial laboratory description; its quoted value is nominal. Frequencies ${\nu }_1,{\nu }_2,{\nu }_p,{\nu }_a>0$ are scalar estimates in hertz of the two rest transitions and the parallel/antiparallel laser readings, respectively; the ideal-model convention is $\nu =\omega \mathrm{/}(2\pi )$ and $\lambda =c\mathrm{/}\nu$ from Null wave covectors, frame frequency data and the observer frequency of an ideal plane wave, while the real apparatus has finite-width spectra. Here $\mathrm{M}\mathrm{H}\mathrm{z}={10}^6{\mathrm{s}}^{-1}$, $\mathrm{n}\mathrm{m}={10}^{-9}\mathrm{m}$, and $\mu \mathrm{r}\mathrm{a}\mathrm{d}={10}^{-6}$ radians. For a single-peaked nonnegative profile, its full width at half maximum (FWHM) is the separation of the two frequencies (or speeds) at half its peak height, when those crossings exist; the reported $\delta v\mathrm{/}v$ uses the speed width divided by the selected speed. Beam divergence and alignment are angles between ion trajectories and the specified beam axis. A Lambda system here means three source-labelled atomic levels, two lower and one common upper, with the two listed driven transitions; the symbols $3S_1$, $3P_2$ and $F$ distinguish the report's atomic levels, rather than supplying a quantum-state model. Metastable lifetime means the reported characteristic persistence time of the prepared level; no survival distribution is inferred. The Penning source, ESR (Experimental Storage Ring), electron cooling, lasers and optics below are the report's named preparation and control apparatus; their performance is reported empirically, not derived from SR. The energy label $\mathrm{M}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{u}$ is energy per atomic-mass unit: $1\mathrm{M}\mathrm{e}\mathrm{V}={10}^6$ electronvolts, with one electronvolt the energy acquired by elementary charge across one volt; it is not used here to deduce the nominal speed.

The experiment used ${}^7{\mathrm{L}\mathrm{i}}^{+}$ ions from a Penning ion gauge source, accelerated to $58.6\mathrm{M}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{u}$ and stored in the ESR at nominal $\beta =0.338$. Electron cooling reduced the reported longitudinal velocity spread to $\delta v\mathrm{/}v\approx 9\times {10}^{-6}$ (FWHM) and transverse divergence to less than $100\mu \mathrm{r}\mathrm{a}\mathrm{d}$. About ${10}^7$ to ${10}^8$ ions were stored, with an estimated metastable fraction below $1\mathrm{\%}$. The metastable $3S_1$ lifetime was $15$–$30\mathrm{s}$ in the ring, compared with $50\mathrm{s}$ in vacuum. The Lambda system used the transitions $3S_1(F=3\mathrm{/}2)\to 3P_2(F=5\mathrm{/}2)$ and $3S_1(F=5\mathrm{/}2)\to 3P_2(F=5\mathrm{/}2)$. Parallel and antiparallel beams near $386\mathrm{n}\mathrm{m}$ and $780\mathrm{n}\mathrm{m}$ addressed its two legs; an additional $780\mathrm{n}\mathrm{m}$ beam repumped ions into the system.

Count and fit objects. A spectrum in this account is a finite list of frequency settings $f_i\in \mathbb{R}$ with detected pulse counts $n_{i,h}\in {\mathbb{N}}_0$ in each switching configuration $h\in \{1,2,3,4\}$ and window. Fluorescence denotes the emitted light counted by the photomultipliers; a dark count is a registered pulse without applied excitation. Subtracted or averaged spectra are real-valued lists, not nonnegative integer counts or independent trials. A Lorentzian peak plus linear background means a fitted function $L:\mathbb{R}\to \mathbb{R}$ of the form $L(f)=A\mathrm{/}[1+4(f-f_0{)}^2\mathrm{/}{w}^2]+b_0+b_1(f-f_0)$ with $A>0$, $w>0$; $f_0,w$ have frequency units, and the peak term has half height at $f_0\pm w\mathrm{/}2$. Residuals are the differences of processed values and the fitted values at the recorded settings. A weighted average of fitted centers has form ${\sum }_i{w}_i{f}_{0,i}\mathrm{/}{\sum }_i{w}_i$ for nonnegative weights with positive sum; the report does not give the weights or their construction. Frequency locking means controlling a laser reading relative to a specified reference or offset; second-harmonic generation denotes conversion to twice the optical frequency. Optical-optical double resonance is the selected condition that both listed transitions are resonant for the same ion velocity class; it is an apparatus selection model, not a measured probability law.

The parallel beam was generated by doubling a $772\mathrm{n}\mathrm{m}$ Ti:Sa laser locked to an iodine marker previously calibrated with a frequency comb against a rubidium clock. The antiparallel diode laser was offset-locked to a rubidium reference, with a $600\mathrm{M}\mathrm{H}\mathrm{z}$ tuning range. Polarization-maintaining fibers transported the beams. Motorized stages controlled angles and positions to $20\mu \mathrm{r}\mathrm{a}\mathrm{d}$ and $50\mu \mathrm{m}$; scrapers established laser/ion overlap within $80\mu \mathrm{r}\mathrm{a}\mathrm{d}$. Photomultipliers counted fluorescence in four successive $100\mu \mathrm{s}$ configurations: both excitation beams, antiparallel only, parallel only, and neither. Dark counts were subtracted, then the separate-beam background was subtracted from the both-beam spectrum. Optical-optical double resonance selected ions for which both Lambda legs were resonant at the same velocity. To compensate metastable depletion, an $81$-point scan was paused after $69$ points for injection and cooling. Figure 2 shows an example averaged over $44$ scans, with $49\mathrm{M}\mathrm{H}\mathrm{z}$ laboratory linewidth and $35\mathrm{M}\mathrm{H}\mathrm{z}$ rest-frame linewidth. The authors fitted $40$ spectra with Lorentzians on linear backgrounds and formed a weighted average. Raw count tables, fit weights, a covariance matrix and a complete likelihood are not published in this report.

Table I gives the frequency estimates, in MHz, ${\nu }_1=546455143.0$, ${\nu }_2=546474960.7$, ${\nu }_a=384225534.98$ and ${\nu }_p=777210326.98$. The rest-frame inputs ${\nu }_1,{\nu }_2$ come from earlier measurements cited as references 21–22; those papers were not retrieved or read here. Equation (4) reports $ϵ=\sqrt{{\nu }_a{\nu }_p\mathrm{/}({\nu }_1{\nu }_2)}-1=(1.5\pm 2.3)\times {10}^{-9}$ at nominal $\beta =0.338$. These are estimates with the source-labelled $1\sigma$ uncertainty budget, not exact values or guaranteed bounds. The rounded Table I centers yield approximately $1.4744849565\times {10}^{-9}$, consistent with the reported rounded estimate, without reconstructing unrounded data.

The Gouy phase is the additional spatial phase of a focused beam relative to a plane-wave phase; for a $C^1$ phase correction $\psi :J\to \mathbb{R}$ on a proper-time interval $J\subseteq \mathbb{R}$, the phase-reading convention assigns a frequency correction $-(2\pi {)}^{-1}d\psi \mathrm{/}d\tau$ under the stipulated phase-reading interpretation. No beam profile $\psi$ is reconstructed here. The $1\sigma$ notation below is the source's label for an uncertainty scale; without its full estimation model it specifies neither a guaranteed interval nor a sampling distribution.

Table I labels its uncertainties $1\sigma$. In MHz, the aggregate entries are $0.43$ for each rest frequency, $1.6$ for the antiparallel frequency and $1.25$ for the parallel frequency. Antiparallel contributions are: weighted-fit statistical uncertainty $0.6$, rubidium reference $0.05$, diode stabilization $0.64$, background linearization $0.5$, Gouy phase $1.0$, angular alignment $0.6$, and ion divergence $0.4$. Parallel contributions are: iodine reference $0.06$, Ti:Sa stabilization $0.12$, Gouy phase $1.0$, alignment $0.6$, and divergence $0.4$. The inferred Gouy correction for each laser was $0.44\pm 1.0\mathrm{M}\mathrm{H}\mathrm{z}$, from experimentally validated numerical simulation using measured beam parameters. The authors reported no significant residual asymmetry; simulated background linearization was assigned $0.5\mathrm{M}\mathrm{H}\mathrm{z}$. The excess linewidth, about five times the expected width, was attributed mainly to drift-induced fluorescence. The $0.6\mathrm{M}\mathrm{H}\mathrm{z}$ statistical entry comes from the weighted analysis of forty fitted spectra, not forty independent raw clock readings. These reported statistical and systematic scales are preserved without inventing independence, a Gaussian total sampling law, fit weights, covariance or coverage.

The measured observable tests the combined Doppler/time-dilation, active atomic-transition and vacuum-wave invariance assumptions together with resonance selection, calibration, beam alignment and correction models. The source reports no resolved deviation at its uncertainty level and quotes $\mathrm{\mid }\alpha \mathrm{\mid }\le 2.0\times {10}^{-8}$ for $\alpha =ϵ\mathrm{/}{\beta }^2$. That quotation is recorded as the source's test-theory sensitivity statement, rather than a newly certified hard bound or confidence interval. RMS and SME parameter translations and future higher-boost prospects remain contextual research topics; they are not prerequisites or claims of the comparison theorem.