Motional Stark effect (MSE) spectroscopy represents a unique diagnostic tool capable of determining the magnitude of the magnetic field and its direction in the core of fusion plasmas. The primary excitation channel for fast hydrogen atoms in injected neutral beams, with energy in the range of 25-1000 keV, is due to collisions with protons and impurity ions (e.g., He$^{2+}$ and heavier impurities). As a result of such excitation, at the particle density of 10$^{13}$-10$^{14}$ cm$^{-3}$, the line intensities of the Stark multiplets do not follow statistical expectations (i.e., the populations of fine-structure levels within the same principal quantum number $n$ are not proportional to their statistical weights). Hence, any realistic modeling of MSE spectra has to include the relevant collisional atomic data. In this paper we provide a general expression for the excitation cross sections in parabolic states within $n$=3 for an arbitrary orientation between the direction of the motion-induced electric field and the proton-atom collisional axis. The calculations make use of the density matrix obtained with the atomic orbital close coupling method and the method can be applied to other collisional systems (e.g., He$^{2+}$, Be$^{4+}$, C$^{6+}$, etc.). The resulting cross sections are given as simple fits that can be directly applied to spectral modeling. For illustration we note that the asymmetry detected in the first classical cathode ray experiments between the red- and blue-shifted spectral components can be quantitatively studied using the proposed approach.
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 discuss the opportunities of diagnosing the parameters of hypothetical "wild cables" in fusion plasmas via analyzing the Stark broadening of hydrogen spectral lines. The "wild cables" concept have suggested the observed long-lived skeletal structures in tokamaks and Z-pinches to be caused by a microdust-assembled skeleton. We present the results of calculating the widths of vacuum channels produced by the pressure of on-skeleton high-frequency (HF) electromagnetic waves of the TEM type, which thus might protect the skeletons from ambient high-temperature plasma. The values of amplitude of s…
A classical model of the hydrogen atom in a static electric field is studied, basing upon the work [ Hooker A. et al, {\it Phys. Rev. A}, 55 (1997) 4609 ]. In that work the electrons are supposed to move along Kepler orbits around the nucleus, while interacting with the external field. That classical model reproduces very well the true energy shift of the linear Stark effect. The agreement with the second order effect is poor. It is shown here that the results for the quadratic Stark effect may be considerably improved if the electrons are still allowed to move along classical trajectories, bu…
The second-order Stark effect for a planar Dirac one-electron atom in the ground state is analyzed within the framework of the Rayleigh-Schrödinger perturbation theory, with the use of the Sturmian series expansion of the generalized Dirac-Coulomb Green function. A closed-form analytical expression for the static dipole polarizability of that system is found. The formula involves a generalized hypergeometric function ${}_{3}F_{2}$ with the unit argument. Numerical values of the polarizabilities for relativistic planar hydrogenic atoms with atomic numbers $1\leqslant Z\leqslant68$ are provided …
An approximation-free, numerically efficient algorithm is presented for the Hamiltonian eigen-states of the Stark-Hydrogen problem describing a quantum particle exposed to the central Coulomb force and a homogeneous external field. As an example of application in a state-expansion with continuous energy, we calculate the time-dependent wavefunction of an electron tunneling from a hydrogen atom suddenly exposed to an external electric field.
The Kepler-Coulomb problem is solved in parabolic coordinates and the Larmor radiation problem is analyzed to complement a previous study performed for the usual representation in spherical polar coordinates. A comparison with quantum spontaneous decay rates shows that for azimuthal quantum number $m = 0$ states only transitions to nearby $n-Δn$ principal quantum number states are described properly by the Wentzel-Kramers-Brillouin quantized classical motions, but that for $m > 0$ reasonable results emerge for many values of $Δn$. A simple approximate expression for the lifetime of $m \ne 0$ s…
We compare the derivation of the dynamic Stark shift of hydrogenic energy levels in a classical framework with an adiabatically damped laser-atom interaction, which is equivalent to the Gell-Mann-Low-Sucher formula, and a treatment based on time-independent perturbation theory, with a second-quantized laser-atom dipole interaction Hamiltonian. Our analysis applies to a laser that excites a two-photon transition in atomic hydrogen or in a hydrogenlike ion with low nuclear charge number. Our comparisons serve to demonstrate why the dynamic Stark shift may be interpreted as a stimulated radiative…
The linear Stark effect for the first excited state of the hydrogen atom shows that, in the unperturbed states, the atom has a permanent electric dipole moment (EDM) of magnitude 3eao (ao is Bohr radius). The EDM is not induce by the external field but is inherent behavior of the atom. But the calculation of quantum mechanics tells us that unperturbed states of hydrogen atom have no EDM! In the effect, two of four states have no energy shift. What are the EDM of the hydrogen atoms in the two states? Quantum mechanics can not answer the problem. The statement that the EDM of the two states is p…
Numerical ab initio variational calculations of the transition probabilities and ac Stark shifts in two-photon transitions of antiprotonic helium atoms driven by two counter-propagating laser beams are presented. We found that sub-Doppler spectroscopy is in principle possible by exciting transitions of the type (n,L)->(n-2,L-2) between antiprotonic states of principal and angular momentum quantum numbers n~L-1~35, first by using highly monochromatic, nanosecond laser beams of intensities 10^4-10^5 W/cm^2, and then by tuning the virtual intermediate state close (e.g., within 10-20 GHz) to the r…
In this paper we study the ionization rate and the Stark shift of a one-dimensional model of the H$_{2}^{+}$ ion. The finding of these two quantities is reduced to the solutions of a complex eigenvalue problem. We solve this problem both numerically and analytically. In the latter case we consider the regime of small external electrostatic fields and small internuclear distances. We find an excellent agreement between the ionization rate computed with the two approaches, even when the approximate result is pushed beyond its expected validity. The ionization rate is very sensitive to small chan…
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…
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.
Stark effect of hydrogenic ions as a test of the indefinite-metric formalism of the eight-component relativistic wave equation for spin-$\frac{1}{2}$ particles
Center for Theoretical Physics, Faculty of Science, Parahyangan Catholic University（帕拉希扬天主教大学理学院理论物理中心）
该论文通过计算氢离子 Stark 位移验证了八分量相对论波动方程的不定度规形式体系，证明其与 Dirac 结果一致，并揭示了自旋-场耦合和负范数态的必要性。
We extend the recently developed technique of opto-optical modulation (OOM) to probe state-resolved ac-Stark-induced phase variations of a coherently excited ensemble of helium atoms. In a joint experimental and theoretical study, we find that the spatial redirection of the resonant emission from the OOM process is different for the low-lying 1s2p state as compared with the higher-lying Rydberg states, and that this redirection can be controlled through the spatial characteristics of the infrared (IR) probe beam. In particular, we observe that the intensity dependence of the IR-induced Stark p…
Plasma Screening Effects in Stark Broadening: A Fully Relativistic Close-Coupling Approach
College of Science, Xi’an University of Science and Technology（西安科技大学理学院）；National Key Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics（计算物理国家重点实验室，应用物理与计算数学研究所）；HEDPS, Center for Applied Physics and Technology, Peking University（高能密度物理研究中心，应用物理与技术中心，北京大学）；College of Science, Xi’an University of Architecture and Technology（西安建筑科技大学理学院）
We report on the appearance of a structure at low energies in the photo-electron momentum distribution of the hydrogen atom exposed to two-color counter-rotating bi-circular laser fields. These structures, which arise due to AC-Stark shifted resonances, break the three-fold symmetry, typical for the $ω-2ω$ bi-circular fields. We discuss the physical origin of this structure in terms of partial-wave interference between direct ionization channels and a resonant pathway, that passes through the AC-Stark shifted state and show how the underlying Rydberg state population depend on the field streng…
Radiative and collisional constants of excited atoms contain the matrix elements of the dipole transitions and when they are blocked one can expect occurring a number of interesting phenomena in radiation-collisional kinetics. In recent astrophysical studies of IR emission spectra it was revealed a gap in the radiation emitted by Rydberg atoms ($RA$) with values of the principal quantum number of $n\approx10$. Under the presence of external electric fields a rearrangement of $RA$ emission spectra is possible to associate with manifestations of the Stark effect. The threshold for electric field…
In this paper it is shown that if one accept assumption of de Broglie that "unitary wave-particle" exists simultaneously and this coexistence is real, then one can find the mean life time of the hydrogen atom of Bohr (intensities).
Precision spectroscopy of the fine and hyperfine structures of high molecular Rydberg-Stark states: Metrology of molecular hydrogen ions
ETH Zürich（苏黎世联邦理工学院）；Vrije Universiteit Amsterdam（阿姆斯特丹自由大学）；Université Paris-Saclay（巴黎萨克雷大学）；Centre National de la Recherche Scientifique（法国国家科学研究中心）
We investigate AC Stark-shifted resonances in argon with ultrashort near-infrared pulses. Using 30 fs pulses we observe periodic enhancements of the excitation yield in the intensity regions corresponding to the absorption of 13 and 14 photons. By reducing the pulse duration to 6 fs with only a few optical cycles, we also demonstrate that the enhancements are significantly reduced beyond what is measurable in the experiment. Comparing these to numerical predictions, which are in quantitative agreement with experimental results, we find that even though the quantum-state distribution can be bro…
We present new calculations of unified line profiles for hydrogen perturbed by collisions with protons. We report on new calculations of the potential energies and dipole moments which allow the evaluation of profiles for the lines of the Lyman series up to Lyman$δ$ and the Balmer series up to Balmer10. Unified calculations only existed for the lines Lyman$α$ to Lyman$γ$ and Balmer$α$ including the H$_2^+$ quasi-molecule. These data are available as online material accompanying this paper and should be included in atmosphere models, in place of the Stark effect of protons, since the quasi-mole…
Photoelectron momentum distributions from strong-field ionization of carbonyl sulfide with 800 nm central-wavelength laser pulses at various peak intensities from $4.6$ to $13\times10^{13}$ W/cm$^2$ were recorded and analyzed regarding resonant Rydberg states and photoelectron orbital angular momentum. The evaluation of the differentials of the momentum distributions with respect to the peak intensity highly suppressed the impact of focal volume averaging and allowed for the unambiguous recognition of Freeman resonances. As a result, previously made assignments of photoelectron lines could be …
We investigate resonant strong field ionization of atomic hydrogen with respect to the 1s-2p-transition. By "strong" we understand that Rabi-periods are executed on a femtosecond time scale. Ionization and AC Stark shifts modify the bound state dynamics severely, leading to nonperturbative signatures in the photoelectron spectra. We introduce an analytical model, capable of predicting qualitative features in the photoelectron spectra such as the positions of the Autler-Townes peaks for modest field strengths. Ab initio solutions of the time-dependent Schroedinger equation show a pronounced shi…
We consider the resummation of the perturbation series describing the energy displacement of a hydrogenic bound state in an electric field (known as the Stark effect or the LoSurdo-Stark effect), which constitutes a divergent formal power series in the electric field strength. The perturbation series exhibits a rich singularity structure in the Borel plane. Resummation methods are presented which appear to lead to consistent results even in problematic cases where isolated singularities or branch cuts are present on the positive and negative real axis in the Borel plane. Two resummation prescr…
Several gage-equivalent forms (including some novel ones) of the Schroedinger equation for a hydrogenlike atom in a time-dependent electric field of a laser pulse are presented. These forms allow to develop a perturbation theory for both small and rather large intensities of the electromagnetic field.
Free-free transitions in laser-assisted electron-hydrogen scattering in a bichromatic field of frequencies $ω$ and $2ω$ are studied at moderate intensities for fast projectiles. A hybrid approach is used, in which the field-projectile interaction is described exactly but the field-target one is described by second order perturbation theory; the projectile-target interaction is treated in the first Born approximation. The adopted description of the target enables a consistent study of the leading process to each of the considered sidebands. Numerical results are presented for the angular distri…
Two-photon free-free transitions in elastic laser-assisted electron-hydrogen collisions are studied in the domain of high scattering energies and low or moderate field intensities, in the third order of perturbation theory, taking into account all the involved Feynman diagrams. Based on the analytical expressions of the transition amplitudes, the differential cross sections for two-photon absorption/emission are computed at impact energy $E_i=100$ eV. The effect of field polarizations on the angular distribution and on the frequency dependence of the differential cross section is analyzed.
We calculate the two-photon 1S-2S spectrum of an atomic hydrogen Bose-Einstein condensate in the regime where the cold collision frequency shift dominates the lineshape. WKB and static phase approximations are made to find the intensities for transitions from the condensate to motional eigenstates for 2S atoms. The excited state wave functions are found using a mean field potential which includes the effects of collisions with condensate atoms. Results agree well with experimental data. This formalism can be used to find condensate spectra for a wide range of excitation schemes.
We present the first experimental data on strong-field ionization of atomic hydrogen by few-cycle laser pulses. We obtain quantitative agreement at the 10% level between the data and an {\it ab initio} simulation over a wide range of laser intensities and electron energies.
The tunneling ionization of a hydrogen atom excited in the presence of a static electric field is investigated for the case where, before being extracted, the electron has an orbital angular momentum L perpendicular to the field E. The escaping electron has a nonzero mean transverse velocity <vT> in the direction of E cross <L>. This asymmetry is similar to the Collins effect in the fragmentation into hadrons of a transversely polarized quark. In addition, the linear Stark effect make <L> and <vT> oscillate in time. The degree of asymmetry is calculated at leading order in E for an initial sta…
We present a theoretical study of the ionization dynamics of a hydrogen atom exposed to attosecond laser pulses in the extreme ultra violet region at very high intensities. The pulses are such that the electron is expected to reach relativistic velocities, thus necessitating a fully relativistic treatment. We solve the time dependent Dirac equation and compare its predictions with those of the corresponding non-relativistic Schr{ö}dinger equation. We find that as the electron is expected to reach about 20% of the speed of light, relativistic corrections introduces a finite yet small decrease i…
The broadening of lines by Stark effect is an important tool for inferring electron density and temperature in plasmas. Stark-effect calculations often rely on atomic data (transition rates, energy levels,...) not always exhaustive and/or valid for isolated atoms. We present a recent development in the detailed opacity code SCO-RCG for K-shell spectroscopy (hydrogen- and helium-like ions). This approach is adapted from the work of Gilles and Peyrusse. Neglecting non-diagonal terms in dipolar and collision operators, the line profile is expressed as a sum of Voigt functions associated to the St…
The electron detachment from the hydrogen negative ion in strong fields is studied using the two-electron and different single-electron models within the quasistatic approximation. A special attention is payed to over-the-barrier regime where the Stark saddle is suppressed below the lowest energy level. It is demonstrated that the single-electron description of the lowest state of ion, that is a good approximation for weak fields, fails in this and partially in the tunneling regime. The exact lowest state energies and detachment rates for the ion at different strengths of the applied field are…
In this paper it is shown that if one accept assumption of de Broglie that "unitary wave-particle" exists simultaneously and this coexistence is real, then one can find the mean life time of the hydrogen atom of Bohr (intensities). Something more, the acceptance of de Broglie's ideas show that a single excited hydrogen atom decays at exactly predictable moment (after excitation). The natural width of excited hydrogen atoms are found using the Bohr's model of this atom and de Broglie's ideas. The mean life time of the excited states is a characteristic only of a statistical ensemble of many ato…
We discuss here the effect of the photon helicity in laser induced and inverse bremstrahlung for {\it high energy scattering} of electrons by hydrogen atoms. We demonstrate that it is possible to find EDAD for high scattering energies of the electrons if the{\it \ dressing of the atomic target} by the laser field is taken into account. We consider higher optical frequencies of the laser field and we restrict ourselves to the use of moderate field intensities.
In astrophysical regimes where the collisional excitation of hydrogen atoms is relevant, the cross sections for the interactions of hydrogen atoms with electrons and protons are necessary for calculating line profiles and intensities. In particular, at relative velocities exceeding ~1000 km/s, collisional excitation by protons dominates over that by electrons. Surprisingly, the hydrogen-proton cross sections at these velocities do not exist for atomic levels of n >= 4, forcing researchers to utilize extrapolation via inaccurate scaling laws. In this study, we present a faster and improved algo…
The broadening of lines by Stark effect is widely used for inferring electron density and temperature in plasmas. Stark-effect calculations often rely on atomic data (transition rates, energy levels,...) not always exhaustive and/or valid only for isolated atoms. In this work, we first present a recent development in the detailed opacity code SCO-RCG for K-shell spectroscopy. The approach is adapted from the work of Gilles and Peyrusse. Neglecting non-diagonal terms in dipolar and collision operators, the line profile is expressed as a sum of Voigt functions associated to the Stark components.…
Splitting the energy levels of a hydrogen-like atom by the electric field nonuniform at the atomic scale is studied. This situation is important for the multi-level treatment of the phenomenon of Rydberg blockade [Yu.V. Dumin, J. Phys. B, v.47, p.175502 (2014)]. An explicit formula for the energy levels is derived. A typical value of the energy shift by the electric field gradient turns out to be proportional to the 4th power of the principal quantum number (i.e., the square of atomic size), as would be expected from a qualitative consideration. Finally, the fine spatial structure of the Rydbe…
A new simulation box setup is introduced for the precise description of the wavepacket evolution of two electronic systems in intense laser pulses. In this box, the regions of the hydrogen molecule H$_{2} $, and singly and doubly ionized species, H$_{2}^+ $ and H$_{2}^{+2} $, are well recognized and their time-dependent populations are calculated at different laser field intensities. In addition, some new regions are introduced and characterized as quasi-double ionization and their time-dependencies on the laser field intensity are calculated and analyzed. The adopted simulation box setup is s…
We discuss the use of a region of uniform and constant magnetic field in order to implement a two-state atomic polarizer for an H(2S) beam. We have observed that a device with such field configuration is capable of achieving an efficient polarization for a wide range of magnetic field intensities and atomic velocities. In addition, we establish a criterion that must be met to confirm a successful polarization. That is possible due to a specific beating pattern for the Lyman-$α$ radiation expected for the outgoing two-state atomic beam.
The linear Stark effect shows that the first excited state of hydrogen atom has large permanent electric dipole moment (EDM), d(H)=3eao (ao is Bohr radius). Using special capacitors our experiments discovered that the ground state K, Rb or Cs atom is polar atom with a large EDM of the order of eao as hydrogen atom of excited state. Their capacitance(C) at different voltage (V) was measured. The C-V curve shows that the saturation polarization of K, Rb or Cs vapor has be observed when the field E more than ten to the fifth power V/m. When the saturation polarization appeared, nearly all K, Rb o…
Supersonic beams of hydrogen atoms, prepared selectively in Rydberg-Stark states of principal quantum number $n$ in the range between 25 and 35, have been deflected by 90$^\circ$, decelerated and loaded into off-axis electric traps at initial densities of $\approx 10^6$ atoms/cm$^{-3}$ and translational temperatures of 150 mK. The ability to confine the atoms spatially was exploited to study their decay by radiative and collisional processes. The evolution of the population of trapped atoms was measured for several milliseconds in dependence of the principal quantum number of the initially pre…
We present ionization probability and line shape calculations for the two-step 3-photon ionization process, $1S \stackrel{2\hbar ω}{\longrightarrow}2S \stackrel{\hbar ω}{\longrightarrow}εP $, of the ground state of hydrogenic atoms in a non-monochromatic laser field with a time--dependent amplitude. Within the framework of a three--level model, the {\it AC Stark} shifts and non-zero ionization rates of all states involved were taken into account together with spatial and temporal inhomogeneities of the laser signal. In contrast with the usual perturbative technique, the time evolution of the a…
We present the results of the light-shift calculations in the ns-states of hydrogenicsystems in the presence of a weak laser field. This is done by means of analytical calculations of the scalar dynamic polarizabilities (DP) of the corresponding levels. In contrast to the usual technique, 1.e. by solving certain inhomogeneous differential equation. As a result, we obtained for the DP a closed analytical expression valid for all principal quantum numbers, n. Within a framework of such an approach we succeeded both in retrieving the well known results and finding several new ones, as well.Our in…
The effect of the relative phase between the components of a bichromatic field of frequencies $ω$ and $3ω$ is discussed in the case of free-free transitions in laser-assisted electron-hydrogen scattering. For fast projectile and low field intensities, the role of target dressing is pointed out.
We show that combination of a linearly polarized resonant microwave field and a parallel static electric field may be used to create a non-dispersive electronic wavepacket in Rydberg atoms. The static electric field allows for manipulation of the shape of the elliptical trajectory the wavepacket is propagating on. Exact quantum numerical calculations for realistic experimental parameters show that the wavepacket evolving on a linear orbit can be very easily prepared in a laboratory either by a direct optical excitation or by preparing an atom in an extremal Stark state and then slowly switchin…
The formation and various disintegration modes of $2s$ states for muonic hydrogen and deuterium atoms at kinetic energies both above and below the $2s-2p$ threshold are studied. The cross sections of the collision-induced radiative quenching, elastic scattering, and Coulomb deexcitation of $(μ^- p)_{2s}$ and $(μ^- d)_{2s}$ atoms in collisions with ordinary hydrogen and deuterium atoms at collision energies below the $2s-2p$ threshold are calculated in the framework of the close-coupling approach. The basis set includes all open and closed channels corresponding to exotic-atom states with princ…
We assess the partition function and ionization degree of magnetized hydrogen atoms at thermodynamic equilibrium for a wide range of field intensities, $B\approx 10^5$-$10^{12}$~G. Evaluations include fitting formulae for an arbitrary number of binding energies, the coupling between the internal atomic structure and the center-of-mass motion across the magnetic field, and the formation of the so-called decentered states (bound states with the electron shifted from the Coulomb well). Non-ideal gas effects are treated within the occupational probability method. We also present general mathematic…
When superimposing the potentials of external fields on the Coulomb potential of the hydrogen atom a saddle point appears, which is called the Stark saddle point. For energies slightly above the saddle point energy one can find classical orbits, which are located in the vicinity of this point. We follow those so-called quasi-Penning orbits to high energies and field strengths observing structural changes and uncovering their bifurcation behavior. By plotting the stability behavior of those orbits against energy and field strength the appearance of a stability apex is reported. A cusp bifurcati…
