Fractatomic Physics: Atomic Stability and Rydberg States in Fractal Spaces

With the use of second-order perturbation theory in the long-range interatomic interaction for the degenerate states of two Rydberg atoms we have obtained a general formula for the dependence of atomic interaction energy on the interatomic distance R in the presence of the Förster resonance. Inside of the ‘Förster sphere’ (R < RF) this dependence transforms to the formula for electric dipole interaction energy ΔEd − d = C3/R3 and for R > RF it transforms to the formula for the van der Waals interaction energy ΔEVdW = −C6/R6. The van der Waals constant C6 is represented as an expansion in terms of irreducible components which define the dependence on the interatomic axis orientation relative to the quantisation axis of projections M of the total angular momentum J. The numerical values of the irreducible components of tensor C6 were calculated for rubidium atoms in the same Rydberg states |nlJM〉 with large quantum numbers n. We present the calculated resonance interaction energy of two rubidium atoms in the states |43D5/2M〉, whose total energy exceeds by only 8 MHz the total energy of one of the atoms in the state |45P3/2M〉 and of the other in the state |41F7/2M〉.

On the basis of analytical expressions in the Fues model potential approach for the second-order Stark effect on single-electron Rydberg states in atoms and ions, general equations are derived for coefficients of polynomials in powers of the principal quantum number n in asymptotic presentations of static scalar and tensor dipole polarizabilities. The power dependence for polarizabilities of isolated Rydberg states |nl〉 at n ≫ 1 scales as n7, in contrast with that of polarizabilities for hydrogenic states, degenerate in the orbital quantum number l ⩽ n − 1, which scales as n6. This difference is demonstrated analytically in the asymptotic dependencies of the second-order matrix elements, determining the Stark shifts of the isolated and degenerate states. Analytical equations for polynomial coefficients use the data on quantum defects determined from the level energies of corresponding series of states. Numerical values of coefficients are presented for S-, P- and D-series of Rydberg states in neutral atoms of alkali-metal elements and helium in comparison with existing data of the literature. The analytical equations are also used for determining numerical values of coefficients in asymptotic polynomials for polarizabilities of Rydberg states in positive ions of alkaline-earth-metal elements.

There are many experimental and theoretical investigations devoted to the multiphoton ionization (MPI) and above threshold ionization (ATI) of atoms in a laser field. Most of them are carried out for the ionization of atoms in ground states. But atoms in highly excited states with large orbital moments behave different from atoms in ground states when exposed to a laser field. Investigations of stability of atoms in Rydberg states in strong laser fields theoretically are carried out using methods of classical mechanics by Vos and Gavrila (1992) and Shepelansky. For not very intensive laser fields perturbation theory may be used. But up to now using perturbation theory are investigated only ionization from ground states and excited states with main quantum number n not higher than ten. The simplest is to investigate hydrogen atom as for it analytical expressions for transition matrix elements can be written. Investigating behavior of Rydberg states one have to get asymptotic expressions for them in the case of large main quantum number n and orbital momentum otherwise they diverge.

The high-precision determination of microwave radiation parameters may be based on measurements of the spectral characteristics of radiation transitions between the Rydberg states of atoms. Frequencies and matrix elements are calculated for dipole transitions from even-parity nS1/2 and nD5/2 to odd-parity n′P3/2 and n′F7/2 (where n′ = n, n ± 1, n ± 2) for the Rydberg states of alkali-metal atoms. The matrix elements determine the splitting of Rydberg-state energy levels in the field of a resonance microwave (μw) radiation, which results in the splitting of the resonance in electromagnetic induced transparency (EIT). Numerical computations based on the single-electron quantum defect method (QDM) and the Fues’ model potential (FMP) approach with the use of the most reliable data of the current literature on quantum defect values were performed for the 2S, 2P, 2D and 2F series of the Rydberg states of Li, Na, K, Rb and Cs atoms. The calculated data were approximated by quadratic polynomials of the principal quantum number. The polynomial coefficients were determined with the use of a standard curve-fitting interpolation polynomial procedure for numerically presented functions. The approximation equations may be used for the accurate evaluation of the frequencies and matrix elements of μw transitions in wide ranges of the Rydberg-state quantum numbers n &gt;&gt; 1.

Purpose: This work aims at investigating the zinc atoms in the triplet preionization – Rydberg states. The energy levels of atoms having two electrons outside the closed shell were studied mainly by the optical spectroscopy methods. However, just using the microwave spectroscopy to measure the frequency of transitions between the two Rydberg states allows to increase the accuracy of measurements in two or more orders of magnitude. Disign/methodology/approach:A line of three dye lasers is used to excite the zinc atoms into the triplet Rydberg states with a predetermined set of quantum numbers. The radiation of the first two of them is transformed into the second harmonic in nonlinear crystals. Dye lasers are excited by the radiation of the second harmonic of one YAG: ND3+ laser. All three radiations are reduced to the zone of interaction with the laser and the microwave radiation, which is located between the plates of the ionization cell, where the pulsed electric field is created. The excited Rydberg atoms are recorded with the field ionization procedure. The beam of neutral atoms is created by an effusion cell under the vacuum conditions, the residual pressure does not exceed 10-5 mm Hg. A pulsed electric field of some certain intensity results inionization of atoms excited by microwave radiation and in acceleration of electrons, which have appeared in the direction of the secondary electron multiplier, though being insufficient for ionization of atoms excited only by the laser radiation and which are initial for interaction with microwaves. By scanning the microwave radiation frequency with the given step and measuring the signal intensity of the secondary electron multiplier, the excitation spectrum of the atoms under study can be obtained. Findings: Using the created laser-microwave spectrometer, the frequencies of the F→D, F→F and F→G transitions between the triplet Rydberg states of zinc atoms were measured. From the analysis made of the transition frequencies, the quantum defect decomposition constants were obtained by the Ritz formula for the D, F, and G states of zinc atoms. Conclusions: The frequencies of the F→D, F→F and F→G transitions between the triplet Rydberg states of zinc atoms were measured that allowed obtaining the quantum defect decomposition constants according to the Ritz formula for the D, F and G states of zinc atoms, that in turn had allowed to calculate the energy of these terms and the transition frequencies at least in two orders of magnitude more accurately as against the similar measurements made by the optical spectroscopy. Key words: zinc atom, triplet states of atoms, Rydberg states, laser excitation, microwave radiation

Beams of helium atoms in Rydberg states with principal quantum number $n=52$,\nand traveling with an initial speed of 1950 m/s, have been accelerated,\ndecelerated and guided while confined in moving electric traps generated above\na curved, surface-based electrical transmission line with a segmented center\nconductor. Experiments have been performed with atoms guided at constant speed,\nand with accelerations exceeding $10^7$ m/s$^2$. In each case the manipulated\natoms were detected by spatially resolved, pulsed electric field ionization.\nThe effects of tangential and centripetal accelerations on the effective\ntrapping potentials experienced by the atoms in the decelerator have been\nstudied, with the resulting observations highlighting contributions from the\ndensity of excited Rydberg atoms to the acceleration, deceleration and guiding\nefficiencies in the experiments.\n

We examine the possibilities of refining an asymptotic description and quantitative calculations of the effects induced by thermal blackbody radiation (BBR) of the environment on the Rydberg states of atoms. Numerical values are calculated and asymptotic expressions are proposed for simplified estimates of natural lifetimes and threshold photoionisation cross sections for Rydberg states of rubidium and caesium atoms with large values of the principal quantum number, n ⩾ 20, and small orbital momenta, l = 0, 1, 2, 3. Based on analytical expressions, we present numerical estimates for the contributions of photoionisation probabilities to the BBR-induced broadening of the Rydberg energy level, as well as the contributions of continuum integrals to thermally induced shifts in the Rydberg-state energy levels.

The Stark effect in molecular Rydberg states is qualitatively different from the Stark effect in atomic Rydberg states because of the anisotropy of the ion core and the existence of rotational and vibrational degrees of freedom. These uniquely molecular features cause the electric-field-induced decoupling of the Rydberg electron from the body frame to proceed in several stages in a molecule. Because the transition dipole moment among the same-n* Rydberg states is much larger than the permanent dipole moment of the ion core, the decoupling of the Rydberg electron from the ion core proceeds gradually. In the first stage, analyzed in detail in this paper, l and N are mixed by the external electric field, while N+ is conserved. In the further stages, as the external electric field increases, N+, n*, and v+ are expected to undergo mixing. We have characterized these stages in n*=13, v+=1 states of CaF. The large permanent dipole moment of CaF+ makes CaF qualitatively different from the other molecules in which the Stark effect in Rydberg states has been described (H2, Na2, Li2, NO, and H3) and makes it an ideal testbed for documenting the competition between the external and CaF+ dipole electric fields. We use the weak-field Stark effect to gain access to the lowest-N rotational levels of f, g, and h states and to assign their actual or nominal N+ quantum number. Lowest-N rotational levels provide information needed to disentangle the short-range and long-range interactions between the Rydberg electron and the ion core. We diagonalize an effective Hamiltonian matrix to determine the l-characters of the 3 < or = l < or = 5 core-nonpenetrating 2Sigma+ states and to characterize their mixing with the core-penetrating states. We conclude that the mixing of the l=4, N-N+=-4(g(-4)) state with lower-l 2Sigma+ states is stronger than documented in our previous multichannel quantum defect theory and long-range fits to zero-field spectra.

Uncontrollable sporadic distortions of global positioning system (GPS) satellite signals, caused by phase and group delays in the propagation of electromagnetic radiation through a medium, take place during periods of high solar activity and formation of geomagnetic disturbances in the Earth’s ionosphere. Determining ways of ensuring sustainability of GPS systems is a fundamental scientific and technical challenge.Above-background incoherent ultra-high frequency (UHF) radiation is formed at altitudes of the E and D layers of the Earth’s ionosphere. Wavelengths of this radiation correspond to a range from 1 dm to 1 mm. This emission is caused by transitions between Rydberg states of atoms and molecules, which are excited by electrons in plasma and are surrounded by a neutral particle environment. Reliable information about UHF radiation flux power in this wavelength range is not currently available. The answer to this question depends entirely on the knowledge of impact and radiation quenching of Rydberg state dynamics and the kinetics of their location in a lower ionosphere, i.e., on the quantum optical properties of a perturbed environment. Analysis of existing experimental data has shown that UHF radiation is formed in the atmospheric layer located at altitudes of 60–110 km. A physical mechanism of satellite signal delay is caused by cascade resonance re-emissions of electromagnetic waves in the decimeter range while passing through this layer over a set of Rydberg states. The most promising approach to studies of medium optical quantum properties can be a simultaneous analysis of background additional noise and GPS signal propagation time delay, which determines a positioning error. Using standard methods of noise measurement, one cannot detect physical and chemical processes responsible for noise formation and errors affecting positioning. Therefore, the problem can be solved if the level of a background noise is considered as a noise of the measured GPS signal because propagation delays of the latter are caused by one of the most important atmospheric collisional process, i.e., the orbital degeneracy of highly excited states. For this purpose, it is advisable to use the S/N ratio where a signal corresponds to a level of a signal obtained by the GPS receiver and a noise corresponds to a GPS signal fluctuation.In this chapter, a current state theory is examined, and manners of its further development are discussed. These are associated with the progress of theoretical methods for describing medium neutral particle impact effects on the dynamics of collision and radiation quenching focusing primarily on elementary processes involving molecules of nitrogen and oxygen. It has been shown that preliminary calculations of nonadiabatic transition dynamics between potential energy surfaces (PES) of Rydberg complexes, construction of appropriate electronic wave functions, calculations of allowed transition dipole moments, and determination of emission line shapes are necessary for the quantitative estimation of the influence of excited particles on a spectrum of incoherent UHF radiation of the atmosphere. These results should be included in the total kinetic scheme, which establishes the dependence of UHF radiation on temperature and density of the lower ionosphere. Then satellite-monitoring data of infrared (IR) radiation, which accompanies UHF radiation, can be directly used for the detection of Rydberg states and diagnostics of plasma parameters.KeywordsHighly excited states of atoms and moleculesD and Е layers of lower Earth’s ionosphereNeutral particles of mediumRydberg complexes l mixingNonequilibrium two-temperature plasmaUHF microwave radiation

A three-level atomic medium can be made transparent to a resonant probe field in the presence of a strong control field acting on an adjacent atomic transition to a long-lived state, which can be represented by a highly excited Rydberg state. The long-range interactions between the Rydberg state atoms then translate into strong, non-local, dispersive or absorptive interactions between the probe photons, which can be used to achieve deterministic quantum logic gates and single photon sources. Here we show that long-range dipole–dipole exchange interaction with one or more spins—two-level systems represented by atoms in suitable Rydberg states—can play the role of control field for the optically dense medium of atoms. This induces transparency of the medium for a number of probe photons np not exceeding the number of spins ns, while all the excess photons are resonantly absorbed upon propagation. In the most practical case of a single spin atom prepared in the Rydberg state, the medium is thus transparent only to a single input probe photon. For larger number of spins ns, all np ≤ ns photon components of the probe field would experience transparency but with an np-dependent group velocity.

This chapter introduces to the theory of atomic population kinetics and radiative properties of atomic and ionic bound–bound transitions. Particular attention is devoted to the general problems related to an extremely large number of kinetic equations describing populations of Rydberg and autoionization atomic states in plasmas. A new method of reduced kinetics for autoionizing states, the virtual contour shape kinetic theory (VCSKT), is described in details. The method is based on a probability method for LTE- and non-LTE-level populations that allows effective level reduction while preserving all detailed atomic transitions. The representation employs effective relaxation constants that have analytical solutions. The comparison with detailed level-by-level calculations demonstrates high accuracy and large efficiency of the VCSKT. In order to solve many states’ kinetic problems for Rydberg atomic states, the quasi-classical representation of the system of kinetic equations is proposed. In particular, the two-dimensional radiative cascades between Rydberg atomic states are described by a purely classical motion of atomic electrons in a Coulomb field that lose energy and orbital momentum. The general collisional-radiative model for large principal quantum numbers is reduced to an effective diffusion in two-dimensional energy and orbital momentum space. The results of these new kinetic models are compared with standard collisional-radiative kinetics demonstrating an important reduction of computer times, the possibility to obtain scaling relations and to independently study the precision of complex quantum calculations for these many level kinetic problems.

This book is devoted to the modern methods of calculating the energy eigenvalues of Rydberg atoms A** and molecules XY** perturbed by neutral particles of a medium and to the results of studying the interaction processes with them. Numerous applications in plasma chemistry, aeronomy, and astrophysics have contributed to conducting this study. These methods are based on the use of integral variant of the theory utilizing Green’s function approach. Because of the closeness to the continuum boundary, these energies cannot be properly described in terms of the standard quantum chemistry. When the radius of electronic cloud of excited states is large enough (i.e., $R{_{{\rm{c}}\;}} = 2\,{n^2} \gg \hskip-1.2pt1$ , where n is the principal quantum number), they cannot be regarded as isolated even in the case of a rarefied gas. The spectral distortion is the strongest when the number N of perturbing neutral particles falling into this region exceeds unity. This chapter is divided into four main parts. In the first part, the generalized method of finite-radius potential (FRP) is discussed. This method self-consistently takes into account the short- and long-range interactions in the two-center system under consideration. It adequately describes the scattering of a weakly bound electron by the ion core and a perturbing atom with nonzero angular momenta l and L with respect to these centers, thereby allowing the theory to be extended to the intermediate (on the order of and less than electron wavelength $\lambda \propto n$ ) interatomic distances $R$ . As an application of the theory, the detailed analysis is performed for the behavior of the potential energy surfaces (PESs) of a system composed of a highly excited atom $A^{\ast\ast}\;(n, \, l)$ and a neutral atom В with the filled electronic shell. It is demonstrated that the inclusion of nonzero momentum $L$ for the ${e^{-} } - B$ scattering results in the additional splitting of the PES into the separate groups of interacting terms classified by the projection m of electronic angular momentum l on the quasimolecular axis. At distances $R \gg n$ , the FRP method exactly transforms to the zero-radius pseudopotential (ZRP) model and, correspondingly, to the asymptotic theory in which the PESs acquire a simple analytic form. It turns out that, at large values $n \gg 1$ , the ZRP method is valid up to the distances $R\sim \, n.$ In the second part, the specific features of the diabatic and adiabatic PESs are discussed by taking into account the dissociative, covalent, and ion configurations. The potentialities and disadvantages of the existing ab initio approaches are analyzed. The matching method is suggested, which allows a unique self-consistent picture devoid of the above-mentioned disadvantages to be obtained for the terms. As an illustration, the potential curves are calculated for the $n l\, \left( { {}^{2s + 1}\Lambda } \right)$ states of the ${\hbox{Na}}^{\ast\ast} + {\hbox{He}}$ quasimolecule ( $n, \, \,l\,$ , and $\Lambda$ are the principal quantum number, angular momentum, and its projection on the molecular axis, respectively, and S is the spin of the system), and a detailed comparison with the computational results of other authors is carried out. In the third part, the possible applications of the theory to the shock ionization, excitation, and quenching processes are discussed for the Rydberg states (RSs). Among these are also the simplest dissociation, exchange, and charge exchange reactions. They can be schematically represented as 1.1a $X{Y^{\ast\ast}} + M \to X{Y^{\ast\ast}} + M,$ 1.1b $X{Y^{\ast\ast}} + M\;\; \to X + {Y^\ast} + M,$ 1.1c $X{Y^{\ast\ast}} + M\;\; \to XM + {Y^\ast},$ 1.1d $X{Y^{\ast\ast}} + M\;\; \to X{Y^{+} } + {e^{-} } + M,$ 1.1e $X{Y^{\ast\ast}} + M\;\; \to X{Y^{+} } + {M^{-} }$ The interaction of ХY** with a neutral particle M includes the interactions with both ion and a weakly bound electron. The former is characterized by small impact parameters, whereas the latter has large impact parameters. As a result, the total scattering cross-sections can appreciably exceed the gas-kinetic values. The material is presented in terms of the PES of the $XY^{\ast\ast} + M$ system followed by the description of the dynamics of processes (1.1a–e) within the framework of the integral variant of the multichannel quantum defect (MQD) theory using the renormalized Lippmann–Shwinger equation technique. Such a formulation of the MQD theory allows one to obtain a convenient representation for Green’s function of a highly excited molecule, which opens up wide possibilities for various applications. In the fourth part, the many-center perturbation of the atomic Rydberg states is analyzed for the situation wherein two (or more) perturbing neutral centers fall inside the electronic cloud. The behavior of Rydberg atom in a dense medium is considered with allowance for the influence of finite number N of the neutral particles chaotically distributed in its volume. The stochastic approach is proposed for the solution to this problem.

Rydberg states of alkali-metal atoms are highly sensitive to electromagnetic radiation in the GHz-to-THz regime because their transitions have large electric dipole moments. Consequently, environmental blackbody radiation (BBR) can couple Rydberg states together at µs timescales. Here, we track the BBR-induced transfer of a prepared Rydberg state to its neighbors and use the evolution of these state populations to characterize the BBR field at the relevant wavelengths, primarily at 130 GHz. We use selective field ionization readout of Rydberg states with principal quantum number n∼30 in Rb85 and substantiate our ionization signal with a theoretical model. With this detection method, we measure the associated blackbody-radiation-induced time dynamics of these states, reproduce the results with a simple semiclassical population transfer model, and demonstrate that this measurement is temperature sensitive with a statistical sensitivity to the fractional temperature uncertainty of 0.09 Hz−1/2, corresponding to 26 K Hz−1/2 at room temperature. This represents a calibration-free SI-traceable temperature measurement, for which we calculate a systematic fractional temperature uncertainty of 0.006, corresponding to 2 K at room temperature when used as a primary temperature standard.

Using the numerical techniques that have been successful in elucidating the low-lying part of the level scheme of atoms in magnetic fields pertaining to white dwarf stars and neutron stars, we have computed the energies and wave functions of ``circular'' Rydberg states (i.e., with \ensuremath{\Vert}m\ensuremath{\Vert}=n-1) with \ensuremath{\Vert}m\ensuremath{\Vert}=24--35 for laboratory fields up to several tens of teslas. We can, therefore, follow a number of physical properties, such as transition frequencies and radiative decay times, of circular states as functions of the magnetic field strength and make predictions that should be verifiable experimentally.

We have developed an optical frequency synthesizer for the precision spectroscopy of highly excited Rydberg states of Rb atoms. This synthesizer can generate a widely tunable 480 nm laser light with an optical power of 150 mW and an absolute frequency uncertainty of less than 100 kHz using a high-repetition-rate (325 MHz) Er fiber-based optical frequency comb and a tunable frequency-doubled diode laser at 960 nm. We demonstrate the precision two-photon spectroscopy of the Rydberg states of 87Rb atoms by observing the electromagnetically induced transparency in a vapor cell, and measure the absolute transition frequencies of 87Rb to nD (n = 53–92) and nS (n = 60–90) Rydberg states with an uncertainty of less than 250 kHz. It is the first direct frequency measurements of these transitions using an optical frequency comb.