Proton-to-electron Mass Ratio Research Articles

The cosmological constant and energy quantization in hydrogen atom in light of a doubly special relativity with a minimum speed

We show the existence of an invariant minimum speed in space–time as a new fundamental constant of nature by forming the basis of a doubly special relativity so-called Symmetrical Special Relativity (SSR). Such an observer-independent minimum speed emerges from the Dirac’s Large Number Hypothesis (LNH), which leads us to build SSR-theory. This allows us to understand that the hydrogen atom represents the most stable bound state in the universe as being a fundamental structure in space–time with two speed limits, i.e., the speed of light c and a minimum speed V. So the symmetry in space–time given by an invariant minimum speed, which has origin in the electrical and gravitational bound states in hydrogen atom is able to provide a deeper understanding of the proton-electron mass ratio. We investigate how the minimum speed plays a fundamental role in the quantization of energy in hydrogen atom. The minimum speed V is related to a preferred background frame (ultra-referential SV) represented by the surface of a dark cone forbidden to the world line of any particle. It is related to the vacuum energy that leads to the cosmological constant. An investigation of SSR is done and a hypothetical experiment of Bose–Einstein condensation is proposed to detect the minimum speed as a possible Lorentz violation at very low energies.

Author Correction: Proton–electron mass ratio by high-resolution optical spectroscopy of ion ensembles in the resolved-carrier regime

Author Correction: Proton–electron mass ratio by high-resolution optical spectroscopy of ion ensembles in the resolved-carrier regime

Precision measurement based on rovibrational spectrum of cold molecular hydrogen ion

A molecular hydrogen ion HD<sup>+</sup>, composed of a proton, a deuteron, and an electron, has a rich set of rovibrational transitions that can be theoretically calculated and experimentally measured precisely. Currently, the relative accuracy of the rovibrational transition frequencies of the HD<sup>+</sup> molecular ions has reached 10<sup>–12</sup>. By comparing experimental measurements with theoretical calculations of the HD<sup>+</sup> rovibrational spectrum, the precise determination of the proton-electron mass ratio, the testing of quantum electrodynamics(QED) theory, and the exploration of new physics beyond the standard model can be achieved. The experiment on HD<sup>+</sup> rovibrational spectrum has achieved the highest accuracy (20 ppt, 1 ppt = 10<sup>–12</sup>) in measuring proton-electron mass ratio. This ppaper comprehensively introduces the research status of HD<sup>+</sup> rovibrational spectroscopy, and details the experimental method of the high-precision rovibrational spectroscopic measurement based on the sympathetic cooling of HD<sup>+</sup> ions by laser-cooled Be<sup>+</sup> ions. In Section 2, the technologies of generating and trapping both Be<sup>+</sup> ions and HD<sup>+</sup> ions are introduced. Three methods of generating ions, including electron impact, laser ablation and photoionization, are also compared. In Section 3, we show the successful control of the kinetic energy of HD<sup>+</sup> molecular ions through the sympathetic cooling, and the importance of laser frequency stabilization for sympathetic cooling of HD<sup>+</sup> molecular ions. In Section 4, two methods of preparing internal states of HD<sup>+</sup> molecular ions, optical pumping and resonance enhanced threshold photoionization, are introduced. Both methods show the significant increase of population in the ground rovibrational state. In Section 5, we introduce two methods of determining the change in the number of HD<sup>+</sup> molecular ions, i.e. secular excitation and molecular dynamic simulation. Both methods combined with resonance enhanced multiphoton dissociation can detect the rovibrational transitions of HD<sup>+</sup> molecular ions. In Section 6, the experimental setup and process for the rovibrational spectrum of HD<sup>+</sup> molecular ions are given and the up-to-date results are shown. Finally, this paper summarizes the techniques used in HD<sup>+</sup> rovibrational spectroscopic measurements, and presents the prospects of potential spectroscopic technologies for further improving frequency measurement precision and developing the spectroscopic methods of different isotopic hydrogen molecular ions.

Extraction of spin-averaged rovibrational transition frequencies in HD+ for the determination of fundamental constants

We present a comprehensive analysis of all currently available high-accuracy frequency measurements of rotational and rovibrational transitions in the hydrogen molecular ion . Our analysis utilises the theoretically calculated hyperfine structure to extract the values of three spin-averaged transition frequencies through a global linear least-squares adjustment that takes into account theory-induced correlations between the different transitions. We subsequently use the three spin-averaged transition frequencies as input data in a second adjustment which employs precise theoretical expressions for the transition frequencies, written as a function of the proton, deuteron and electron relative atomic masses, the Rydberg constant, and the proton and deuteron charge radii. Our analysis shows that the data may significantly improve the value of the electron relative atomic mass and the proton-electron mass ratio, in particular if combined with recent high-precision measurements of particle atomic masses and mass ratios obtained from Penning traps.

Effect of correlated hyperfine theory errors in the determination of rotational and vibrational transition frequencies in HD+

We revisit the problem of separating the purely rovibrational and hyperfine contributions to observed rovibrational transition frequencies. The separation of hyperfine frequency shifts played an essential role in recent works in which the value of the proton-electron mass ratio was determined from Doppler-free rovibrational spectroscopy and high-precision theory of the molecular ion. Here, we present an alternative approach to removing theoretically computed hyperfine shifts from a limited set of observed rovibrational transition frequencies, based on a least-squares adjustment. The adjustment facilitates a quantitative study of the effect of correlations between the uncertainties of lower- and upper-state hyperfine energies on the derived value of the rovibrational transition frequency. We find that correlations may lead to significant shifts of the extracted rovibrational transition frequencies. In addition, we propose and demonstrate a method to quantify the additional shift and uncertainty, which is of importance to other applications of the spectroscopic data, such as the determination of fundamental physical constants.

Leading-order relativistic corrections to the g factor of H2+

Relativistic corrections of order ${\ensuremath{\alpha}}^{2}$ to the $g$ factor of ${\mathrm{H}}_{2}^{+}$ are calculated with a high accuracy of nine significant digits for a wide range of rovibrational states. The precision of previous calculations [R. A. Hegstrom, Phys. Rev. A 19, 17 (1979) is improved by about five orders of magnitude by performing nonadiabatic variational calculations and by including recoil corrections. These results allow for nondestructive identification of the internal state through the measurement of spin-flip transition frequencies, which is a crucial requirement for proposed spectroscopy experiments on ${\mathrm{H}}_{2}^{+}$ and its antimatter counterpart ${\overline{\mathrm{H}}}_{2}{}^{\ensuremath{-}}$ in Penning traps [E. G. Myers, Phys. Rev. A 98, 010101(R) (2018)]. Further, they pave the way towards precision calculations of the $g$ factor through the calculation of higher-order QED corrections and hence to an alternative precision route to obtaining the proton-electron mass ratio.

A study of submillimeter methanol absorption toward PKS 1830−211:

Context. Methanol is an important tracer to probe physical and chemical conditions in the interstellar medium of galaxies. Methanol is also the most sensitive target molecule for probing potential space-time variations of the proton-electron mass ratio, μ, a dimensionless constant of nature. Aims. We present an extensive study of the strongest submillimeter absorption lines of methanol (with rest frequencies between 300 and 520 GHz) in the z = 0.89 molecular absorber toward PKS 1830−211, the only high-redshift object in which methanol has been detected. Our goals are to constrain the excitation of the methanol lines and to investigate the cosmological invariance of μ based on their relative kinematics. Methods. We observed 14 transitions of methanol, five of the A-form and nine of the E-form, and three transitions of A-13CH3OH, with ALMA. We analyzed the line profiles with a Gaussian fitting and constructed a global line profile that is able to match all observations after allowing for variations of the source covering factor, line opacity scaling, and relative bulk velocity offsets. We explore methanol excitation by running the non local thermal equilibrium radiative transfer code RADEX on a grid of kinetic temperatures and H2 volume densities. Results. Methanol absorption is detected in only one of the two lines of sight (the southwest) to PKS 1830−211. There, the excitation analysis points to a cool (∼10 − 20 K) and dense (∼104 − 5 cm−3) methanol gas. Under these conditions, several methanol transitions become anti-inverted, with excitation temperatures below the temperature of the cosmic microwave background. In addition, we measure an abundance ratio A/E = 1.0 ± 0.1, an abundance ratio CH3OH/H2 ∼ 2 × 10−8, and a 12CH3OH/13CH3OH ratio 62 ± 3. Our analysis shows that the bulk velocities of the different transitions are primarily correlated with the observing epoch due to morphological changes in the background quasar’s emission. There is a weaker correlation between bulk velocities and the lower level energies of the transitions, which could be a signature of temperature-velocity gradients in the absorbing gas. As a result, we do not find evidence for variations of μ, and we estimate Δμ/μ=(−1.8 ± 1.2) × 10−7 at 1-σ from our multivariate linear regression. Conclusions. We set a robust upper limit |Δμ/μ| < 3.6 × 10−7 (3σ) for the invariance of μ at a look-back time of half the present age of the Universe. Our analysis highlights that systematics need to be carefully taken into account in future radio molecular absorption studies aimed at testing Δμ/μ below the 10−7 horizon.

Proton–electron mass ratio by high-resolution optical spectroscopy of ion ensembles in the resolved-carrier regime

Optical spectroscopy in the gas phase is a key tool to elucidate the structure of atoms and molecules and of their interaction with external fields. The line resolution is usually limited by a combination of first-order Doppler broadening due to particle thermal motion and of a short transit time through the excitation beam. For trapped particles, suitable laser cooling techniques can lead to strong confinement (Lamb-Dicke regime, LDR) and thus to optical spectroscopy free of these effects. For non-laser coolable spectroscopy ions, this has so far only been achieved when trapping one or two atomic ions, together with a single laser-coolable atomic ion [1,2]. Here we show that one-photon optical spectroscopy free of Doppler and transit broadening can also be obtained with more easily prepared ensembles of ions, if performed with mid-infrared radiation. We demonstrate the method on molecular ions. We trap approximately 100 molecular hydrogen ions (HD$^{+}$) within a Coulomb cluster of a few thousand laser-cooled atomic ions and perform laser spectroscopy of the fundamental vibrational transition. Transition frequencies were determined with lowest uncertainty of 3.3$\times$10$^{-12}$ fractionally. As an application, we determine the proton-electron mass ratio by matching a precise ab initio calculation with the measured vibrational frequency.

SI-traceable molecular transition frequency measurements at the 10−12 relative uncertainty level

Accurate and spectroscopic measurements of molecular transition frequencies are increasingly being employed in a variety of rigorous tests of physics, including the validity of quantum electrodynamics, the proton–electron mass ratio, and the dipole moment of the electron. Near-infrared molecular transitions may also underpin secondary frequency standards for length metrology and provide convenient wavelength standards for telecommunication and spectroscopy. To report progress in this field, we describe measurements of near-infrared Doppler-broadened carbon dioxide line positions. Our experiment uses cavity ring-down spectroscopy in the linear domain and provides an accurate International System of Units (SI)-traceable frequency axis via active locking of the probe laser and ring-down cavity to a Cs-clock-referenced optical frequency comb. The approach has several key attributes: the ability to rapidly scan the probe laser while maintaining a phase lock to the optical frequency comb, continuously tunable spectrum frequency axes that are not limited by the cavity mode spacing, and high signal-to-noise-ratio spectra resulting in stationary statistics amenable to long-term averaging. This performance is achieved by phase locking a single-mode probe laser to the optical frequency comb, coherent electro-optical phase modulation for rapid cavity mode-to-mode shifting of the probe laser, and sub-kilohertz-level frequency stabilization of the ring-down cavity relative to the probe laser. We report vacuum transition frequencies of five transitions in the ( 30012 ) ← ( 00001 ) vibrational band of 12 C 16 O 2 near 1.57 µm and achieve combined standard uncertainties as low as 212 Hz, corresponding to a relative combined standard uncertainty near 10 − 12 .

A very precise ratio

Metrology The value of the ratio of the masses of the proton and the electron has a bearing on the values of other physical constants. This ratio is known to a very high precision. Patra et al. improved this precision even further by measuring particular frequencies in the rovibrational spectrum of the hydrogen deuteride molecular ion (HD+) (see the Perspective by Hori). To reach this high precision, the researchers placed the HD+ molecules in an ion trap and surrounded them by beryllium ions. The cold beryllium ions then helped cool the HD+ molecules, making the HD+ spectral lines narrow enough that the proton-electron mass ratio could be extracted by comparison with theoretical predictions. Science , this issue p. [1238][1]; see also p. [1160][2] [1]: /lookup/doi/10.1126/science.aba0453 [2]: /lookup/doi/10.1126/science.abb9186

Long term measurement of theSr87clock frequency at the limit of primary Cs clocks

We report on a series of 42 measurements of the transition frequency of the 429 THz (5s2) S01–(5s5p) P03 line in Sr87 taken over three years from 2017 to 2019. They were performed at the Physikalisch-Technische Bundesanstalt (PTB) between the laboratory strontium lattice clock and the primary caesium fountain clocks CSF1 and CSF2. The length of each individual measurement run has been extended by use of a hydrogen maser as a flywheel to improve the statistical uncertainty given by the Cs clocks. We determine an averaged transition frequency of 429228004229873.00(0.07) Hz with 1.5×10−16 fractional uncertainty at the limit of the current realization of the unit hertz. Analysis of the data provides an improved limit on the coupling of the gravitational potential of the Sun to the proton-electron mass ratio μ and confirms the limits on the temporal drift of μ.Received 11 May 2020Accepted 20 July 2020DOI:https://doi.org/10.1103/PhysRevResearch.2.033242Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasAtomic spectraGravitationLaser spectroscopyAtomic, Molecular & OpticalGravitation, Cosmology & Astrophysics

Proton-electron mass ratio from laser spectroscopy of HD+ at the part-per-trillion level.

Recent mass measurements of light atomic nuclei in Penning traps have indicated possible inconsistencies in closely related physical constants such as the proton-electron and deuteron-proton mass ratios. These quantities also influence the predicted vibrational spectrum of the deuterated molecular hydrogen ion (HD+) in its electronic ground state. We used Doppler-free two-photon laser spectroscopy to measure the frequency of the v = 0→9 overtone transition (v, vibrational quantum number) of this spectrum with an uncertainty of 2.9 parts per trillion. By leveraging high-precision ab initio calculations, we converted our measurement to tight constraints on the proton-electron and deuteron-proton mass ratios, consistent with the most recent Penning trap determinations of these quantities. This results in a precision of 21 parts per trillion for the value of the proton-electron mass ratio.

On the proposal of an Eddington ratio of natural energies, ε

Eddington in 1923, first identified four dimensionless numbers, derived from combinations of the basic physical constants, which are known as the “Eddington constants.” In formulating these dimensionless numbers, Eddington, a leading physicist of his time, claimed that they are characteristic of the structure and dynamics of the Universe at large, on the microscopical scale, and at the macroscopical scale. These four dimensionless ratios are labeled here, and elsewhere, as α (also called the fine structure constant), and β (the electron-proton mass ratio), and γ, (the ratio of electrical-to-gravitational force of the proton on the electron), and δ (a ratio involving the cosmological constant and other constants). Here, in this communication, is defined a 5th Eddington ratio, labeled as e (a ratio of characteristic energies of diatomic and monatomic hydrogen). The uncanny fitting of these 5 fundamental ratios to simple formulas involving the mathematical constants e, the base of natural logarithms, π, the familiar circular constant, ϕ, the golden ratio and “2,” the only even prime number, is described to some degree along with a tabulation of characteristics of these 5 ratios.

Nonadiabatic QED Correction to the Dissociation Energy of the Hydrogen Molecule.

The quantum electrodynamic correction to the energy of the hydrogen molecule has been evaluated without expansion in the electron-proton mass ratio. The obtained results significantly improve the accuracy of theoretical predictions reaching the level of 1MHz for the dissociation energy, in very good agreement with the parallel measurement [Hölsch etal., Phys. Rev. Lett. 122, 103002 (2019)PRLTAO0031-900710.1103/PhysRevLett.122.103002]. Molecular hydrogen has thus become a cornerstone of ultraprecise quantum chemistry, which opens perspectives for determination of fundamental physical constants from its spectra.

A determination of fundamental constants using HD+ ion and atomic hydrogen spectroscopy data

A determination of fundamental constants using HD+ ion spectroscopy data is discussed from comparisons between precision measurements and accurate theoretical predictions by taking into account recent measurements and updated CODATA values of the fundamental constants. The deuteron-proton mass ratio is determined with an uncertainty of 10-9. The ratio between the HD+ reduced mass and the electron mass is determined with an uncertainty of 7.3 × 10-10. The Rydberg constant, the proton-electron mass ratio and the deuteron-electron mass ratio are consistently determined with 10-9 , 10-6 , and 10-6 level uncertainties from an adjustment of the (v,L)=(0,0)→(0,1) and (v,L)=(0,2)→(8,3) HD+ ion transitions and of the (n,l,j,f)=1S1/2f=1→2S1/2f=1 atomic hydrogen transition. The result of the adjustment provides a test of the consistency of the two-body and three-body quantum electrodynamics energy level calculations for the atomic hydrogen and the HD+ ion.

Role of Four Gravitational Constants in Nuclear Structure

This paper attempts to understand the role of the four gravitational constants in the nuclear structure whichhelps in understanding the nuclear elementary charge, the strong coupling constant, nuclear charge radii,nucleon magnetic moments, nuclear stability, nuclear binding energy and Neutron life time. The three assumed atomic gravitational constants help in understanding neutron-proton stability. Electromagnetic and nuclear gravitational constants play a role in understanding proton-electron mass ratio, Bohr radius and characteristic atomic radius. With reference to the weak gravitational constant, it is possible to predict the existence of a weakly interacting fermion of rest energy 585 GeV, called Higg’s fermion. Cosmological ‘dark matter’ research and observations can be carried out in this direction also.

Prospects for a polar-molecular-ion optical probe of varying proton-electron mass ratio

Molecules with deep vibrational potential wells provide optical intervals sensitive to variation in the proton-electron mass ratio ($\mu$). On one hand, polar molecules are of interest since optical state preparation techniques have been demonstrated for such species. On the other hand, it might be assumed that polar species are unfavorable candidates, because typical molecule-frame dipole moments reduce vibrational state lifetimes and cause large polarizabilities and associated Stark shifts. Here, we consider single-photon spectroscopy on a vibrational overtone transition of the polar species TeH$^+$, which is of practical interest because its diagonal Franck-Condon factors should allow rapid state preparation by optical pumping. We point out that all but the ground rotational state obtains a vanishing low-frequency scalar polarizability from coupling with adjacent rotational states, because of a fortuitous relationship between rigid rotor spacings and dipole matrix elements. We project that for good choices of spectroscopy states, demonstrated levels of field control should make possible uncertainties of order $1 \times 10^{-18}$, similar to those of leading atomic ion clocks. The moderately long lived vibrational states of TeH$^+$ make possible a frequency uncertainty approaching $1 \times 10^{-17}$ with one day of averaging for a single trapped ion. Observation over one year could probe for variation of $\mu$ with a sensitivity approaching the $1 \times 10^{-18}/\textrm{yr}$ level.

Discovery of OH Absorption from a Galaxy at z ∼ 0.05: Implications for Large Surveys with SKA Pathfinders

Abstract We present the first detection of OH absorption in diffuse gas at z > 0, along with another eight stringent limits on OH column densities for cold atomic gas in galaxies at 0 < z < 0.4. The absorbing gas detected toward Q0248+430 (z q = 1.313) originates from a tidal tail emanating from a highly star-forming galaxy G0248+430 (z g = 0.0519) at an impact parameter of 15 kpc. The measured column density is N(OH) = (6.3 ± 0.8) × 1013 cm−2, where and T ex are the covering factor and the excitation temperature of the absorbing gas, respectively. In our Galaxy, the column densities of OH in diffuse clouds are of the order of N(OH) ∼ 1013–14 cm−2. From the incidence (number per unit redshift; n 21) of H i 21 cm absorbers at 0.5 < z < 1 and assuming no redshift evolution, we estimate the incidence of OH absorbers (with log N(OH) > 13.6) to be n OH = at z ∼ 0.1. Based on this we expect to detect such OH absorbers from the MeerKAT Absorption Line Survey (MALS). Using H i 21 cm and OH 1667 MHz absorption lines detected toward Q0248+430, we estimate (ΔF/F) = (5.2 ± 4.5) × 10−6, where , α is the fine structure constant, μ is the electron–proton mass ratio, and g p is the proton gyromagnetic ratio. This corresponds to Δα/α(z = 0.0519) = (1.7 ± 1.4) × 10−6, which is among the stringent constraints on the fractional variation of α.

Toward a Determination of the Proton-Electron Mass Ratio from the Lamb-Dip Measurement of HD.

Precision spectroscopy of the hydrogen molecule is a test ground of quantum electrodynamics (QED), and it may serve for the determination of fundamental constants. Using a comb-locked cavity ring-down spectrometer, for the first time, we observed the Lamb-dip spectrum of the R(1) line in the overtone of hydrogen deuteride (HD). The line position was determined to be 217 105 182.79±0.03_{stat}±0.08_{syst} MHz (δν/ν=4×10^{-10}), which is the most accurate rovibrational transition ever measured in the ground electronic state of molecular hydrogen. Moreover, from calculations including QED effects up to the order m_{e}α^{6}, we obtained predictions for this R(1) line as well as for the HD dissociation energy, which are less accurate but signaling the importance of the complete treatment of nonadiabatic effects. Provided that the theoretical calculation reaches the same accuracy, the present measurement will lead to a determination of the proton-to-electron mass ratio with a precision of 1.3 parts per billion.

Fine-Structure Constant from Golden Ratio Geometry

After a brief review of the golden ratio in history and our previous exposition of the fine-structure constant and equations with the exponential function, the fine-structure constant is studied in the context of other research calculating the fine-structure constant from the golden ratio geometry of the hydrogen atom. This research is extended and the fine-structure constant is then calculated in powers of the golden ratio to an accuracy consistent with the most recent publications. The mathematical constants associated with the golden ratio are also involved in both the calculation of the fine-structure constant and the proton-electron mass ratio. These constants are included in symbolic geometry of historical relevance in the science of the ancients.