We have observed the cold collision frequency shift of the 1S-2S transition in trapped spin-polarized atomic hydrogen. We find $Δν_{1S-2S} = -3.8(8)\times 10^{-10} n Hz cm^3$, where $n$ is the sample density. From this we derive the 1S-2S s-wave triplet scattering length, $a_{1S-2S}=-1.4(3)$ nm, which is in fair agreement with a recent calculation. The shift provides a valuable probe of the distribution of densities in a trapped sample.
We study the proton-hydrogen collision in the energy range from 0 to 5 K where the hyperfine structure of the hydrogen atom is important. The proper multichannel treatment of the hyperfine structure is found to be crucial at cold temperatures compared to the elastic approximation used for higher temperatures. Both elastic and hyperfine-changing inelastic processes are investigated using the newly developed multichannel quantum-defect theory (MQDT) as well as the traditional coupled-channel numerical method. Cross section results from the two methods coincide well throughout the energy range. T…
Laser spectroscopy studies are being prepared to measure the 1s ground state hyperfine splitting in trapped cold highly charged ions. The purpose of such experiments is to test quantum electrodynamics in the strong electric field regime. These experiments form part of the HITRAP project at GSI. A brief review of the planned experiments is presented.
We have studied hydrogen-hydrogen-alkali three-body systems in the adiabatic hyperspherical representation. For the spin-stretched case, there exists a single $X$H molecular state when $X$ is one of the bosonic alkali atoms: $^7$Li, $^{23}$Na, $^{39}$K, $^{87}$Rb and $^{133}$Cs. As a result, the {\em only} recombination process is the one that leads to formation of $X$H molecules, H+H+$X$$\rightarrow$$X$H+H, and such molecules will be stable against vibrational relaxation. We have calculated the collision rates for recombination and collision induced dissociation as well as the elastic cross-s…
My previous calculations of the Lamb shift in muonic hydrogen are reviewed and compared with other work. In addition, numerical results for muonic deuterium are presented.
Theoretical calculations of the Lamb shift provide the basis required for the determination of the Rydberg constant from spectroscopic measurements in hydrogen. The recent high-precision determination of the proton charge radius drastically reduced the uncertainty in the hydrogen Lamb shift originating from the proton size. As a result, the dominant theoretical uncertainty now comes from the two- and three-loop QED effects, which calls for further advances in their calculations. We review the present status of theoretical calculations of the Lamb shift in hydrogen and light hydrogen-like ions …
The Lamb shift in muonic hydrogen continues to be a subject of experimental and theoretical investigation. Here my older work on the subject is updated to provide a complementary calculation of the energies of the 2p-2s transitions in muonic hydrogen.
Higher order $(α/π)^2 (Z α)^6$ logarithmic corrections to the hydrogen Lamb shift are calculated. The results obtained show the two-loop contribution has a very peculiar behavior, and significantly alter the theoretical predictions for low lying S-states.
In this work we report the results concerning calculations for quantum-mechanical rotational transitions in molecular hydrogen, H$_2$, induced by an ultra-cold ground state anti-hydrogen atom $\bar{H}_{1s}$. The calculations are accomplished using a non-reactive close-coupling quantum-mechanical approach. The H$_2$ molecule is treated as a rigid rotor. The total elastic scattering cross section $σ_{el}(ε)$ at energy $ε$, state-resolved rotational transition cross sections $σ_{jj'}(ε)$ between states $j$ and $j'$ and corresponding thermal rate coefficients $k_{jj'}(T)$ are computed in the tempe…
We present a detailed study about ion-atom-atom three-body recombination in hydrogen and deuterium plasmas based on classical trajectory calculations in hyperspherical coordinates. Our results, due to the predominant role of the long-range charged-induced dipole interaction, indicate that H$_2^+$ and D$_2^+$ are the main reaction products in the case of hydrogen and deuterium plasmas, respectively. Besides, we find a more steep energy-dependent reaction rate when the collision energy surpasses the dissociation energy of the molecular ion, thus entering a new dynamical regime dominated by short…
Accurate calculations of the nuclear recoil effect to the Lamb shift of hydrogen-like atoms are presented. Numerical results are reported for the $ns$ states with $n \leq 5$ and for the $2p_{1/2}$ and $2p_{3/2}$ states. The calculations are performed to the first order in the electron-nucleus mass ratio and to all orders in the nuclear binding strength parameter $Zα$ (where $Z$ is the nuclear charge number and $α$ is the fine structure constant). The obtained results provide accurate predictions for the higher-order remainder beyond the known $Zα$-expansion terms. In the case of hydrogen, the …
The 7 standard deviations between the proton rms charge radius from muonic hydrogen and the CODATA-10 value from hydrogen spectroscopy and electron-scattering has caused considerable discussions. Here, we review the theory of the 2S-2P Lamb shift and 2S hyperfine splitting in muonic hydrogen combining the published contributions and theoretical approaches. The prediction of these quantities is necessary for the determination of both proton charge and Zemach radii from the two 2S-2P transition frequencies measured in muonic hydrogen.
In the comment we discuss an additional term to the $α^3$ correction to the Lamb shift, which was missed in the calculation by Kinoshita and Nio (Phys. Rev. Lett. 82, 3240 (1999)).
Frequency shifts from background gas collisions currently contribute significantly to the inaccuracy of atomic clocks. Because nearly all collisions with room-temperature background gases that transfer momentum eject the cold atoms from the clock, the interference between the scattered and unscattered waves in the forward direction dominates these frequency shifts. We show they are ~10 times smaller than in room-temperature clocks and that van der Waals interactions produce the cold-atom background-gas shift. General considerations allow the loss of Ramsey fringe amplitude to bound this freque…
A Lamb-shift polarimeter, which has been built for a fast determination of the polarization of protons and deuterons of an atomic-beam source and which is frequently used in the ANKE experiment at COSY-Jülich, is shown to be an excellent device for atomic-spectroscopy measurements of metastable hydrogen isotopes. It is demonstrated that magnetic and electric dipole transitions in hydrogen can be measured as a function of the external magnetic field, giving access to the full Breit-Rabi diagram for the $2^2S_{1/2}$ and the $2^2P_{1/2}$ states. This will allow the study of hyperfine structure, $…
We investigate black-body induced energy shifts for low-lying levels of atomic systems, with a special emphasis on transitions used in current and planned high-precision experiments on atomic hydrogen and ionized helium. Fine-structure and Lamb-shift induced black-body shifts are found to increase with the square of the nuclear charge number, whereas black-body shifts due to virtual transitions decrease with increasing nuclear charge as the fourth power of the nuclear charge. We also investigate the decay width acquired by the ground state of atomic hydrogen, due to interaction with black-body…
We report high-precision calculations of the nuclear recoil effect to the Lamb shift of hydrogen-like atoms to the first order in the electron-nucleus mass ratio and to all orders in the nuclear binding strength parameter $Zα$. The results are in excellent agreement with the known terms of the $Zα$ expansion and allow an accurate identification of the nonperturbative higher-order remainder. For hydrogen, the higher-order remainder was found to be much larger than anticipated. This result resolves the long-standing disagreement between the numerical all-order and the analytical $Zα$-expansion a…
The largest contributions to the $n=2$ Lamb-shift, fine structure interval and $2s$ hyperfine structure of muonic hydrogen are calculated by exact numerical evaluations of the Dirac equation, rather than by a perturbation expansion in powers of $1/c$, in the framework of non-relativistic quantum electrodynamics. Previous calculations and the validity of the perturbation expansion for light elements are confirmed. The dependence of the various effects on the nuclear size and model are studied
Calculation of higher-order two-loop corrections is now a limiting factor in development of the bound state QED theory of the Lamb shift in the hydrogen atom and in precision determination of the Rydberg constant. Progress in the study of light hydrogen-like ions of helium and nitrogen can be helpful to investigate these uncalculated terms experimentally. To do that it is necessary to develop a theory of such ions. We present here a theoretical calculation for low energy levels of helium and nitrogen ions.
We calculate the DC Stark effect for three molecular hydrogen ions in the non-relativistic approximation. The effect is calculated both in dependence on the rovibrational state and in dependence on the hyperfine state. We discuss special cases and approximations. We also calculate the AC polarisabilities for several rovibrational levels, and therefrom evaluate accurately the black-body radiation shift, including the effects of excited electronic states. The results enable the detailed evaluation of certain systematic shifts of the transitions frequencies for the purpose of ultra-high-precision…
We use the dressed atom formalism to calculate the frequency shift in a hydrogen maser induced by applied radiation near the Zeeman frequency, and find excellent agreement with a previous calculation made in the bare atom basis. The maser oscillates on the Delta_F = 1, Delta_m_F = 0 hyperfine transition, while the applied field is swept through the F = 1, Delta_m_F = \pm 1 Zeeman resonance. We determine the effect of the applied field on the Zeeman levels using the dressed atom picture, and then calculate the maser frequency shift by coupling the dressed states to the microwave cavity. Qualita…
We present here a new method based on a transfer of population by adiabatic passage that allows to prepare cold atomic samples with a well defined ratio of atomic density and atom number. This method is used to perform a measurement of the cold collision frequency shift in a laser cooled cesium clock at the percent level, which makes the evaluation of the cesium fountains accuracy at the $10^{-16}$ level realistic. With an improved set-up, the adiabatic passage would allow measurements of atom number-dependent phase shifts at the $10^{-3}$ level in high precision experiments.
The correction to the hydrogen Lamb shift due to the proton electric and magnetic polarizabilities is expressed analytically through their static values, which are known from experiment. The numerical value of the correction to the hydrogen 1S state is $ - 71 \pm 11 \pm 7$ Hz. Correction to the H-D 1S-2S -- isotope shift due to the proton and neutron polarizabilities is estimated as $ 53 \pm 9 \pm 11$ Hz.
Quantum electrodynamics in very strong Coulomb fields is one scope which has not yet been tested experimentally with suffcient accuracy to really determine whether the perturbative approach is valid. One sensitive test is the determination of the 1s Lamb Shift in highly-charged very heavy ions. The 1s Lamb Shift of hydrogen-like lead (Pb81+) and gold (Au78+) has been determined using the novel detector concept of silicon microcalorimeters for the detection of hard X-rays. The results of (260 +- 22) eV for lead and (208 +- 13) eV for gold are within error bars in good agreement with theoretical…
The two-loop self-energy correction to the Lamb shift of hydrogen-like ions is calculated for the $1s$, $2s$, and $2p_{1/2}$ states and nuclear charge numbers $Z = 30$-$100$. The calculation is performed to all orders in the nuclear binding strength parameter $Zα$. As compared to previous calculations of this correction, numerical accuracy is improved by an order of magnitude and the region of the nuclear charges is extended. An analysis of the $Z$-dependence of the obtained results demonstrates their consistency with the known $Zα$-expansion coefficients.
We evaluate a number of systematic effects that are important for an experimental microwave measurement of the $n$$=$$2$ S-to-P intervals in atomic hydrogen. The analysis is important for both re-evaluating the best existing measurement [Lundeen and Pipkin, PRL 46, 232 (1981)] of the 2S$_{1/2}$-to-2P$_{1/2}$ Lamb shift, and for a new measurement that is ongoing in our laboratory. This work is part of a larger program to understand the several-standard-deviation discrepancies between various methods for determining the proton charge radius.
We report on a theoretical analysis of the phase-shift in compact atomic clocks working either with cold or thermal atoms. It is well known that in a microwave cavity with electromagnetic losses, a traveling wave adds to the standing wave of a given resonant mode. We calculate the spatial varying phase related to this travelling wave for two geometries of interest in clock applications, the cylindrical cavity and the spherical cavity. Due to their motion, the atoms probe different regions of the cavity and then experience different phases of the interrogating microwave field. We show that this…
The magic wavelength for an optical lattice for hydrogen atoms that cancels the lowest order AC Stark shift of the 1S-2S transition is calculated to be 513 nm. The magnitude of AC Stark shift $ΔE=-1.19$ kHz/(10kW/cm$^2$) and the slope $dΔE/dν= -27.7$ Hz/(GHz $\cdot$ 10 kW/cm$^2$) at the magic wavelength suggests that a stable and narrow linewidth trapping laser is necessary to achieve a deep enough optical lattice to confine hydrogen atoms in a way that gives a small enough light shift for the precision spectroscopy of the 1S-2S transition.
We calculate the cross-damping frequency shift of a laser-induced two-photon transition monitored through decay fluorescence, by adapting the analogy with Raman scattering developed by Amaro et al. [P. Amaro et al., PRA 92, 022514 (2015)]. We apply this method to estimate the frequency shift of the 1S-3S transition in hydrogen and deuterium. Taking into account our experimental conditions, we find a frequency shift of less than 1 kHz, that is smaller than our current statistical uncertainty.
The correction to the hydrogen Lamb shift due to the proton electric and magnetic polarizabilities is expressed analytically through their static values, which are known from experiment. The numerical value of the correction is $ - 71 \pm 11 \pm 7$ Hz.
We investigate collisional shifts of spectral lines involving excited hydrogenic states, where van der Waals coefficients have recently been shown to have large numerical values when expressed in atomic units. Particular emphasis is laid on the recent hydrogen 2S-4P experiment (and an ongoing 2S-6P experiment) in Garching, but numerical input data are provided for other transitions (e.g., involving S states), as well. We show that the frequency shifts can be described, to sufficient accuracy, in the impact approximation. The pressure related effects were separated into two parts, (i) related t…
From the comparison between the experimental and theoretical results for the energy difference between 2s1/2 and 2p1/2 states in hydrogen and hydrogen-like ions with different values of Z, we estimate bounds for the presence of an extra fractional dimension. These bounds are higher when we work with muonic hydrogen and muonic deuterium, which open the possibility that future measurements of the Lamb shift in muonic atoms with one electron and higher values of Z may show evidence of the extra dimension.
The radiative correction to the nuclear charge radius contribution to the Lamb shift of order $α(Zα)^5m_r^3<r^2>$ is calculated. In view of the recent high precision experimental data, this theoretical correction produces a significant contribution to the hydrogen-deuterium isotopic shift.
Recently, we studied the magic wavelength for the atomic hydrogen 1S-2S transition [A.K., Phys. Rev. A 92, 042507 (2015)]. An explicit summation over virtual atomic states of the discrete part of the hydrogen spectrum was performed to evaluate the atomic polarizability. In this paper, we supplement the contribution of the continuum part of the spectrum and add the reduced-mass correction. The magic wavelength, at which the lowest-order ac Stark shifts of the 1S and 2S states are equal, is found to be equal to 514.6 nm. The ac Stark shift at the magic wavelength is -221.6 Hz / (kW/cm^2), and th…
Recently the muonic hydrogen lamb shift has been measured with unprecedented accuracy, allowing for a precise determination of the proton radius. This determination is 5 sigma away from the previous CODATA value obtained from (mainly) the hydrogen lamb shift and the electron-proton scattering. Within an effective field theory formalism, I will define the proton radius and briefly review some aspects of the theoretical prediction for the muonic hydrogen lamb shift, studying both the pure QED-like computation and the hadronic effects.
We develop an exact sum rule that relates the spectral shift of a trapped gas undergoing cold collisions to measurable quantities of the system. The method demonstrates the dependence of the cold collision frequency shift on the quantum degeneracy of the gas and facilitates extracting scattering lengths from the data. We apply the method to analyzing spectral data for magnetically trapped hydrogen atoms and determine the value of the $1S-2S$ scattering length.
The Lorentz oscillator system is studied to interpret the spectral lines of hydrogen atoms. The dielectric constant of this system is analyzed, which takes into account the electrical polarization of hydrogen atoms. This dielectric constant gives the red shift of the spectral line and the appearance of the optical spectrum dip. This dip is on the blue side of the spectral position of the shifted line. The value of this red shift and the width of this dip strongly depend on the hydrogen atom concentration and the spectral position of the not shifted line. This red shift increases with an increa…
When the hydrogen atom moves, the proton current generates a magnetic field which interacts with the hydrogen electron. A simple analyze shows that this interaction between the hydrogen momentum and the electron is of order of $α^3\frac{v}{c}m$, where $α$ is the fine structure constant, $v$ is the atom velocity, $c$ is the speed of light, and $m$ is the electron mass. Using the Bethe-Salpeter equation, the two velocity-dependent operators of this order are derived for the hydrogen velocity $\frac{v}{c}<<α$. As well known, the degeneracy of the energy levels with the same principal quantum numb…
A system of Lorentz oscillators is considered to interpret of spectral lines of hydrogen atoms. The dielectric permittivity of this system, which takes into account its electric polarization, is considered. The substance is examined in the gas state or in the form of small dust of matter. It is shown that the considering of the electric polarization results in a redshift of spectral lines of the substance and the appearance of dip of curve of spectrum. This dip takes place in the blue side with respect to the spectral position of the already shifted line. The magnitude of the red shift of spec…
We consider the application of a basic principle of quantum theory, the tracelessness of a certain class of hamiltonians, to the precision spectroscopy of the molecular hydrogen ions. We show that it is possible to obtain the spin-averaged transition frequencies between states from a simple weighted sum of experimentally accessible spin-dependent transition frequencies. We discuss the cases ${\rm H}_{2}^{+}$ and ${\rm HD}^{+}$, which are distinct in the multipole character of their rovibrational transitions. Inclusion of additional frequencies permits canceling also the electric quadrupole shi…
The hadronic shift in pionic hydrogen has been redetermined to be $ε_{1s}=7.086\,\pm\,0.007(stat)\,\pm\,0.006(sys)$\,eV by X-ray spectroscopy of ground state transitions applying various energy calibration schemes. The experiment was performed at the high-intensity low-energy pion beam of the Paul Scherrer Institut by using the cyclotron trap and an ultimate-resolution Bragg spectrometer with bent crystals.
We report direct, time-resolved observations of single-photon superradiance in a highly extended, elliptical sample of cold 87Rb atoms. The observed rapid decay rate is accompanied by its counterpart, the cooperative Lamb shift. The rate of the strongly directional decay, and the associated shift, scale linearly with the number of atoms and show a strong dependence on the sample shape, demonstrating the collective nature of the observed quantities.
Recently, p-wave cold collisions were shown to dominate the density-dependent shift of the clock transition frequency in a 171Yb optical lattice clock. Here we demonstrate that by operating such a system at the proper excitation fraction, the cold collision shift is canceled below the 5x10^{-18} fractional frequency level. We report inelastic two-body loss rates for 3P0-3P0 and 1S0-3P0 scattering. We also measure interaction shifts in an unpolarized atomic sample. Collision measurements for this spin-1/2 171Yb system are relevant for high performance optical clocks as well as strongly-interact…
Recently a high-precision measurement of the Lamb shift in muonic hydrogen has been performed. An accurate value of the proton charge radius can be extracted from this datum with a high accuracy. To do that a sufficient accuracy should be achieved also on the theoretical side, including an appropriate treatment of higher-order proton-structure effects. Here we consider a higher-order contribution of the finite size of the proton to the Lamb shift in muonic hydrogen. Only model-dependent results for this correction have been known up to date. Meantime, the involved models are not consistent eit…
The Lamb shift measurement and theory are now both a dynamically developing field and we give a review of the current data. Critical comparison of theory and experiment can be done using a value of the proton charge radius and we pay attention to several results of its determination.
The Lamb shift in muonic hydrogen was measured some time ago to a high accuracy. The theoretical prediction of this value is very sensitive to the proton-finite-size effects. The proton radius extracted from muonic hydrogen is in contradiction with the results extracted from elastic electron-proton scattering. That creates a certain problem for the interpretation of the results from the muonic hydrogen Lamb shift. For the latter we need also to take into account the two-photon-exchange contribution with the proton finite size involved. The only way to describe it relies on the data from the sc…
We demonstrate that criticisms concerning our work nucl-th/9704043, made in the recent preprint hep-ph/9903352, are unfounded. The improvements over our results claimed in hep-ph/9903352 are in fact spurious, being based mainly on misunderstandings and on dubious theoretical models.
GRASIAN: Shaping and characterization of the cold hydrogen and deuterium beams for the forthcoming first demonstration of gravitational quantum states of atoms
Stefan Meyer Institute for Subatomic Physics, Austrian Academy of Sciences（奥地利科学院斯特凡·迈耶亚原子物理研究所）；Institute for Particle Physics and Astrophysics, eth（苏黎世联邦理工学院粒子物理与天体物理研究所）；Department of Physics and Astronomy, University of Turku（图尔库大学物理与天文系）；Laboratoire Kastler Brossel, Sorbonne Université, CNRS, ENS-PSL Université, Collège de France（索邦大学卡斯特尔-布罗塞尔实验室（CNRS、巴黎高等师范学院-PSL大学、法兰西公学院））；Institut Max von Laue - Paul Langevin（马克斯·冯·劳厄-保罗·朗之万研究所）
The radiative ortho-para transition in the molecular hydrogen is studied. This highly forbidden transition is very sensitive to relativistic and subtle nonadiabatic effects. Our result for the transition rate in the ground vibrational level $Γ(J=1\to J=0) = 6.20(62)\cdot 10^{-14} \iyr$ is significantly lower in comparison to all the previous approximate calculations. Experimental detection of such a weak line by observation of, for example, the cold interstellar molecular hydrogen is at present unlikely.
Contributions to the energy levels in light muonic atoms and, in particular, to the Lamb shift fall into a few well-distinguished classes. The related diagrams are calculated using different approaches. In particular, there is a specific kind of non-relativistic contributions. Here we consider such corrections to the Lamb shift in order $α^5m_μ$. These contributions are due to free vacuum polarization loops as well as to various effects of light-by-light scattering. The closed loop in the related diagrams is an electronic one, which allows a non-relativistic consideration of the muon. Both kin…
We show that the resonance shifts in the fluorescence of a cold gas of rubidium atoms substantially differ from those of thermal atomic ensembles that obey the standard continuous medium electrodynamics. The analysis is based on large-scale microscopic numerical simulations and experimental measurements of the resonance shifts in a steady-state response in light propagation.
The recent conundrum with the proton charge radius inspires reconsideration of the corrections that enter into determinations of the proton size. We study the two-photon proton-structure corrections, with special consideration of the non-pole subtraction term in the dispersion relation, and using fits to modern data to evaluate the energy contributions. We find that individual contributions change more than the total, and present results with error estimates.
We comprehensively analyse the theoretical prediction for the Lamb shift in muonic hydrogen, and the associated determination of the proton radius. We use effective field theories. This allows us to relate the proton radius with well-defined objects in quantum field theory, eliminating unnecessary model dependence. The use of effective field theories also helps us to organize the computation so that we can clearly state the parametric accuracy of the result. In this paper we review all (and check several of) the contributions to the energy shift of order $α^5$, as well as those that scale like…
We discuss the feasibility to realize a space cold atom clock with counter-propagating cold atoms in microgravity. The design of the space clock is based on atomic beam clock with a Ramsey cavity, except a magneto-optical trap (MOT) is placed at each side. Cold atoms are launched from MOTs at both side of the clock simultaneously and move at counter-direction towards each other. The velocity of launched atoms is precisely controlled to the Ramsauer-Townsend resonance so that no additional collision frequency shift is taken place. Such a configuration can efficiently cancel the frequency shift …
