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We investigate the long-range, two-body interactions between rubidium and potassium atoms in highly excited ($n=70$) Rydberg states. After establishing properly symmetrized asymptotic basis states, we diagonalize an interaction Hamiltonian consisting of the standard Coulombic potential expansion and atomic fine structure to calculate electronic potential energy curves. We find that when both atoms are excited to either the $70s$ state or the $70p$ state, both the $\Omega=0+$ symmetry interactions and the $\Omega=0-$ symmetry interactions demonstrate a deep potential well capable of supporting many bound levels; the size of the corresponding dimer states are on the order of 2.25 $\mu$m. We estabish $n$-scaling relations for the equilibrium separation $R_e$ and the dissociation energy $D_e$ and find these relations to be consistent with similar calculations involving the homonuclear interactions between rubidium and cesium. We discuss the specific effects of $\ell$-mixing and the exact composition of the calculated potential well \textit{via} the expansion coefficients of the asymptotic basis states. Finally, we apply a Landau-Zener treatment to show that the dimer states are stable with respect to predissociation., Comment: 8 pages, 4 figures, 4 tables

Neutral atoms are promising for large-scale quantum computing, but accurate neutral-atom entanglement depends on large Rydberg interactions which strongly limit the interatomic distances. Via a phase accumulation in detuned Rabi cycles enabled by a Rydberg interaction of similar magnitude to the Rydberg Rabi frequency, we study a controlled-phase gate with an arbitrary phase and extend it to the controlled-NOT gate. The gates need only three steps for coupling one Rydberg state, depend on easily accessible van der Waals interaction that naturally arises between distant atoms, and have no rotation error in the weak interaction regime. Importantly, they can work with very weak interactions so that well-separated qubits can be entangled. The gates are sensitive to the irremovable fluctuation of Rydberg interactions, but can still have a fidelity over 98\% with realistic position fluctuation of qubits separated over 20~$\mu$m., Comment: 7 pages, 3 figures

Photodissociation dynamics of the jet-cooled vinoxy radical (CH2CHO) via the B2A″ state was studied in the near-ultraviolet (near-UV) region of 308-328 nm using high-n Rydberg H atom time-of-flight (HRTOF) and resonance-enhanced multiphoton ionization (REMPI) techniques. The vinoxy radical beam was produced by 193 nm photolysis of ethyl vinyl ether followed by supersonic expansion. The H + CH2CO product channel was observed directly in the H atom TOF spectra. The H atom photofragment yield (PFY) spectra were obtained by integrating the H atom TOF spectra and measuring the H atom REMPI signals, and showed several vibronic bands of the B2A″ state. The translational energy distributions of the H + CH2CO products, P(ET)'s, were obtained at several vibronic transitions. The P(ET) distributions were broad, peaking at a low energy of ∼3500 cm-1. The product translational energy release was moderate; the average translational energy release in the maximum available energy, ⟨fT⟩, was in the range of 0.24-0.27. The product angular distributions in this wavelength region were slightly anisotropic, with the β parameter in the range of 0.10-0.24. The near-UV photodissociation mechanism of the H + CH2CO product channel of the vinoxy radical is consistent with unimolecular dissociation on the electronic ground state (X2A″) following internal conversion from the B2A″ state to the A2A' state and then to the X2A″ state (although unimolecular dissociation from the first excited A2A' may also contribute).

Intermolecular interactions, stereodynamics, and coupled potential energy surfaces (PESs) all play a significant role in determining the outcomes of molecular collisions. A detailed knowledge of such processes is often essential for a proper interpretation of spectroscopic observations. For example, nitric oxide (NO), an important radical in combustion and atmospheric chemistry, is commonly quantified using laser-induced fluorescence on the A2Σ+ ← X2Π transition band. However, the electronic quenching of NO (A2Σ+) with other molecular species provides alternative nonradiative pathways that compete with fluorescence. While the cross sections and rate constants of NO (A2Σ+) electronic quenching have been experimentally measured for a number of important molecular collision partners, the underlying photochemical mechanisms responsible for the electronic quenching are not well understood. In this paper, we describe the development of high-quality PESs that provide new physical insights into the intermolecular interactions and conical intersections that facilitate the branching between the electronic quenching and scattering of NO (A2Σ+) with H2, N2, and CO. The PESs are calculated at the EOM-EA-CCSD/d-aug-cc-pVTZ//EOM-EA-CCSD/aug-cc-pVDZ level of theory, an approach that ensures a balanced treatment of the valence and Rydberg electronic states and an accurate description of the open-shell character of NO. Our PESs show that H2 is incapable of electronically quenching NO (A2Σ+) at low collision energies; instead, the two molecules will likely undergo scattering. The PESs of NO (A2Σ+) with N2 and CO are highly anisotropic and demonstrate evidence of electron transfer from NO (A2Σ+) into the lowest unoccupied molecular orbital of the collision partner, that is, the harpoon mechanism. In the case of ON + CO, the PES becomes strongly attractive at longer intermolecular distances and funnels population to a conical intersection between NO (A2Σ+) + CO and NO (X2Π) + CO. In contrast, for ON + N2, the conical intersection is preceded by an ∼0.40 eV barrier. Overall, our work shines new light into the impact of coupled PESs on the nonadiabatic dynamics of open-shell systems.

Exciton polaron is a hypothetical many-body quasiparticle that involves an exciton dressed with a polarized electron-hole cloud in the Fermi sea. It has been evoked to explain the excitonic spectra of charged monolayer transition metal dichalcogenides, but the studies were limited to the ground state. Here we measure the reflection and photoluminescence of monolayer MoSe2 and WSe2 gating devices encapsulated by boron nitride. We observe gate-tunable exciton polarons associated with the 1 s–3 s exciton Rydberg states. The ground and excited exciton polarons exhibit comparable energy redshift (15~30 meV) from their respective bare excitons. The robust excited states contradict the trion picture because the trions are expected to dissociate in the excited states. When the Fermi sea expands, we observe increasingly severe suppression and steep energy shift from low to high exciton-polaron Rydberg states. Their gate-dependent energy shifts go beyond the trion description but match our exciton-polaron theory. Our experiment and theory demonstrate the exciton-polaron nature of both the ground and excited excitonic states in charged monolayer MoSe2 and WSe2. An exciton polaron is a quasiparticle composed of an exciton dressed with an electron-hole cloud, and this concept has been used to explain the ground excitonic states in charged monolayer transition metal dichalcogenides. Here the authors present experimental and theoretical evidence of exciton-polaron Rydberg states in monolayer MoSe2 and WSe2.

Two-photon absorption (2PA) spectra of liquid cyclohexane and hexanes are reported for the energy range 6.4-8.5 eV (177-145 nm), providing detailed information about their electronic structures in bulk liquid. Using a broadband pump-probe fashion, we measured the continuous 2PA spectra by simultaneous absorption of a 266 nm (4.6 eV) pump photon and one UV-vis probe photon from the white-light continuum (1.8-3.9 eV). Theoretical one-photon absorption (1PA) and 2PA cross sections of isolated gas phase molecules are computed by the equation of motion coupled-cluster method with single and double substitutions (EOM-CCSD) to substantiate the assignment of the experimental spectra, and the natural transition orbital (NTO) analysis provides visualization of the participating orbitals in a transition. Our analysis suggests that upon solvation transitions at the lowest excitation energy involving promotion of electron to the 3s Rydberg orbitals are blue-shifted (∼0.55 eV for cyclohexane and ∼0.18 eV for hexanes) to a greater extent as compared to those involving other Rydberg orbitals, which is similar to the behavior observed for water and alcohols. All other transitions experience negligible (cyclohexane) or minor red-shift by ∼0.15-0.2 eV (hexane) upon solvation. In both alkanes, the spectra are entirely dominated by Rydberg transitions: the most intense bands in 1PA and 2PA spectra are due to the excitation of electrons to the Rydberg "p" and "d" type orbitals, respectively, although one transition terminating in the 3s Rydberg has significant 2PA strength. This work demonstrates that the gas phase electronic transition properties in alkanes are not significantly altered upon solvation. In addition, electronic structure calculations using an isolated-molecule framework appear to provide a reasonable starting point for a semiquantitative picture for spectral assignment and also to analyze the solvatochromic shifts for liquid phase absorption spectra.

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

We report in this paper energy positions of Rydberg series from metastable state of Ca3+ ion.. Calculations are performed up to n = 20 using the Screening Constant by Unit Nuclear Charge (SCUNC). The present results compared wellwith the experimental data of Ghassan A Alna’washi which are the only available values.The accurate data presented in this work may be a useful guideline for future experimental and other theoretical studies.

We propose a method to continuously frequency shift a target laser that is frequency stabilized by a reference laser, which is several hundreds of nanometers detuned. We demonstrate the technique using the $\mathrm {5S_{1/2}} \rightarrow \mathrm {5P_{3/2}} \rightarrow \mathrm {29D_{5/2}}$ Rydberg transition in 87Rb vapor and lock the 482 nm target laser to the 780 nm reference laser using the cascaded electromagnetically induced transparency signal. The stabilized frequency of the target laser can be shifted by about 1.6 GHz by phase modulating the reference laser using a waveguide-type electro-optical modulator. This simple method for stable frequency shifting can be used in atomic or molecular physics experiments that require a laser frequency scanning range on the order of several GHz.

With the redefinition of the International System of Units (SI) that occurred in November 2018, there has been much attention on the development of atom-based sensors for metrology applications that are directly linked to the SI. In particular, great progress has been made in using Rydberg atom-based techniques for electric field (E-field) metrology. These Rydberg atombased E-field sensors have made it possible to develop atom-based receivers, which potentially have many benefits over conventional technologies in detecting and receiving modulated signals.

The near-edge X-ray absorption fine structure (NEXAFS) spectra of indole, 2,3-dihydro-7-azaindole, and 3-formylindole in the gas phase have been measured at the carbon and nitrogen K-edges. The spectral features have been interpreted based on density functional theory (DFT) calculations within the transition potential (TP) scheme, which is accurate enough for a general description of the measured C 1s NEXAFS spectra as well as for the assignment of the most relevant features. For the nitrogen K-edge, the agreement between experimental data and theoretical spectra calculated with TP-DFT was not quite satisfactory. This discrepancy was mainly attributed to the many-body effects associated with the excitation of the core electron, which are better described using the time-dependent density functional theory (TDDFT) with the range separated hybrid functional CAM-B3LYP. An assignment of the measured N 1s NEXAFS spectral features has been proposed together with a complete description of the observed resonances. Intense transitions from core levels to unoccupied antibonding π* states as well as several transitions with mixed-valence/Rydberg or pure Rydberg character have been observed in the C and N K-edge spectra of all investigated indoles.

Entangled photon pairs are a fundamental component for testing the foundations of quantum mechanics, and for modern quantum technologies such as teleportation and secured communication. Current state-of-the-art sources are based on nonlinear processes that are limited in their efficiency and wavelength tunability. This motivates the exploration of physical mechanisms for entangled photon generation, with a special interest in mechanisms that can be heralded, preferably at telecommunications wavelengths. Here we present a mechanism for the generation of heralded entangled photons from Rydberg atom cavity quantum electrodynamics (cavity QED). We propose a scheme to demonstrate the mechanism and quantify its expected performance. The heralding of the process enables non-destructive detection of the photon pairs. The entangled photons are produced by exciting a rubidium atom to a Rydberg state, from where the atom decays via two-photon emission (TPE). A Rydberg blockade helps to excite a single Rydberg excitation while the input light field is more efficiently collectively absorbed by all the atoms. The TPE rate is significantly enhanced by a designed photonic cavity, whose many resonances also translate into high-dimensional entanglement. The resulting high-dimensionally entangled photons are entangled in more than one degree of freedom: in all of their spectral components, in addition to the polarization—forming a hyper-entangled state, which is particularly interesting in high information capacity quantum communication. We characterize the photon comb states by analyzing the Hong-Ou-Mandel interference and propose proof-of-concept experiments.

Condensation of hydrogen Rydberg atoms (highly electronically excited) into the lowest energy state of condensed hydrogen i.e. the ultra-dense hydrogen phase, H(0), has gained increased attention not only from the fundamental aspects but also from the applied point of view. The physical properties of ultra-dense hydrogen H(0) were recently reviewed (Physica Scripta 2019 https://doi.org/10.1088/1402-4896/ab1276 ), summarizing the results reported in 50 publications during the last ten years. The main application of H(0) so far is as the fuel and working medium in nuclear particle generators and nuclear fusion reactors which are under commercial development. The first fusion process showing sustained operation above break-even was published in 2015 (AIP Advances) and used ultra-dense deuterium D(0) as fuel. The first generator giving a high-intensity muon flux intended for muon-catalyzed fusion reactors was patented in 2017, using H(0) as the working medium. Here, we first focus on the different nuclear processes using hydrogen isotopes for energy generation, and then on the detailed processes of formation of H(0). The production of H(0) employs heterogeneous catalysts which are active in hydrogen transfer reactions. Iron oxide-based, alkali promoted catalysts function well, but also platinum group metals and carbon surfaces are active in this process. The clusters of highly excited Rydberg hydrogen atoms H(l) are formed upon interaction with alkali Rydberg matter. The final conversion step from ordinary hydrogen Rydberg matter H(l) to H(0) is spontaneous and does not require a solid surface. It is concluded that the exact choice of catalyst is not very important. It is also concluded that the crucial feature of the catalyst is to provide excited alkali atoms at a sufficiently high surface density and in this way enabling formation and desorption of H(0) clusters. Finally, the relation to industrial catalytic processes which use H(0) formation catalysts is described and some important consequences like the muon and neutron radiation from H(0) are discussed.

We have used 308 nm photolysis of acetaldehyde to measure a photoionization spectrum of the formyl (HCO) radical between 8 and 11.5 eV using an 11 meV FWHM photoionization energy resolution. We have confirmed that the formyl radical is the carrier of the spectrum by generating an identical spectrum of the HCO product in the Cl + H2CO reaction. The spectrum of HCO and its deuterated isotopologue (DCO) have several resolved autoionizing resonances above the Franck-Condon envelope, which we assign to autoionization after initial excitation into neutral 3sσ and 3p Rydberg states converging to the first triplet excited state of HCO+(a 3A'). The quantum defects for these states are δ3sσ = 1.06 ± 0.02 and δ3p = 0.821 ± 0.019. We report absolute photoionization cross-section measurements of σHCOPI(9.907 eV) = 4.5 ± 0.9 Mb, σHCOPI(10.007 eV) = 4.8 ± 1.0 Mb, σHCOPI(10.107 eV) = 6.0 ± 1.2 Mb, σHCOPI(10.107 eV) = 5.7 ± 1.2 Mb, and σHCOPI(10.304 eV) = 10.6 ± 2.2 Mb relative to the photoionization cross section of the methyl radical. The absolute cross-section measurements are a factor of ∼1.5 larger than those determined in past studies, although the presence of strong autoionizing features supports a dependence on photoionization energy resolution. We propose that the semiempirical model of Xu and Pratt for estimation of free radical photoionization cross sections is more accurate when applied with a reference species containing the same atoms as the free radical rather than isoelectronic species with different atoms.

According to the existing paradigm, helium atoms and helium-like ions (hereafter, heliumic systems) in a relatively weak external static electric field do not exhibit the linear Stark effect—in distinction to hydrogen atoms and hydrogen-like ions. In the present paper we consider the classical dynamics of a muonic-electronic heliumic system in Rydberg states–starting from the concept from our previous paper. We show that there are two states of the system where the averaged electric dipole moment is non-zero. Consequently, in these states the heliumic system should exhibit the linear Stark effect even in a vanishingly small electric field, which is a counter-intuitive result. We also demonstrate the possibility of controlling the overall precession of the electronic orbit by an external electric field. In particular, we show the existence of a critical value of the external electric field that would “kill” the precession and make the electronic orbit stationary. This is another counter-intuitive result. We calculate analytically the value of the critical field and show that it is typically smaller or even much smaller than 1 V/cm.

We have theoretically investigated the modulation structures of the excited-state population of an argon atom exposed to a linearly polarized 800-nm laser pulse in the tunneling region. The numerical result identifies a nearly opposite phase modulation structure between the low-lying n = 4–7 states and over all higher $$n>7$$ states. This phase-dependent alternating population phenomenon is found to be related to the periodic channel-closing effect that results from ac-Stark-shifted multiphoton resonances. Moreover, the angular momentum states attributed to different parities have a similar alternating effect, which confirms a multiphoton process in the tunneling region. We introduce a mechanism of two-photon $$\Lambda $$ -type Raman transitions involving Rydberg and continuum states with a simultaneous redistribution of states to account successfully for this observed out-of-phase angular momentum state population.

Band-edge excitons of few-layer nickel phosphorous trisulfide (NiPS3) are characterized via micro-thermal-modulated reflectance (μTR) measurements from 10 to 300 K. Prominent μTR features of the A exciton series and B are simultaneously detected near the band edge of NiPS3. The A exciton series contains two sharp A1 and A2 levels and one threshold-energy-related transition (direct gap, E∞), which are simultaneously detected at the lower energy side of NiPS3. In addition, one broadened B feature is present at the higher energy side of few-layer NiPS3. The A series excitons may correlate with majorly d-to-d transition in the Rydberg series with threshold energy of E∞ ≅ 1.511 eV at 10 K. The binding energy of A1 is about 36 meV, and the transition energy is A1 ≅ 1.366 eV at 300 K. The transition energy of B measured by μTR is about 1.894 eV at 10 K. The excitonic series A may directly transit from the top of valence band to the conduction band of NiPS3, while the B feature might originate from the spin-split-off valence band to the conduction band edge. The direct optical gap of NiPS3 is ~1.402 eV at 300 K, which is confirmed by μTR and transmittance experiments.

A cold Rydberg gas, with its atoms prepared initially all in the excited state |n0> , with n0 »1, contains an excessive amount of energy, and presumably is to relax by the Penning-type molecular auto-ionization (MAI), in which a portion of excess energy of one atom is given to another near-by atom and ionizing it. Its complementary process, the resonant energy transfer (RET), is discussed, in which the excess energy of one atom is used on another to form a hyper-excited atomic state |na> with na»n0. This process is always present, provided certain resonance energy conditions are satisfied. In this report, the n0 and density dependences of the RET rates are studied in detail, employing a simple model: 1) at low densities, the RET is mediated by the dipole-dipole coupling Vdd and its rates are generally much smaller than that of MAI, especially for small n0. But 2) as the density increases, our model shows that the rates become of comparable magnitude or even larger than the MAI rates. The Vdd is no longer adequate. We, then construct a semi-empirical potential to describe the RET process. 3) At high densities, we show that the atomic orbital of |na> overlaps with that of neighboring atoms, and the electron-electron potential becomes prominent, resulting in much higher rates.

We report in this paper energy positions of the 2P°_2s22p2(1D)nd 2P, 2P°_2s22p2(1D)nd 2S, 2P°_2s22p2(1D)ns 2D, 2P°_2s22p2(1S)nd 2D, and 2P°_2s22p3(3P)np 2D Rydberg series in the photoionization spectra originating from 2P° metastable state of O+ ions. Calculations are performed up to n = 30 using the Modified Orbital Atomic Theory (MAOT). The present results are compared to the experimental data of Aguilar which are the only available values. The accurate data presented in this work may be a useful guideline for future experimental and other theoretical studies.

Multiple Rydberg series converging to the O2+c4Σ-u state, accessed by 20-25 eV extreme ultraviolet (XUV) light, serve as important model systems for the competition between nuclear dissociation and electronic autoionization. The dynamics of the lowest member of these series, the 3sσg state around 21 eV, has been challenging to study owing to its ultra-short lifetime (

We report a comprehensive study on the electronic excited states of tetrafluoro-1,4-benzoquinone, through high-resolution vacuum ultraviolet photoabsorption spectroscopy and time-dependent density functional theory calculations performed within the nuclear ensemble approach. Absolute cross section values were experimentally determined in the 3.8-10.8 eV energy range. The present experimental results represent the highest resolution data yet reported for this molecule and reveal previously unresolved spectral structures. The interpretation of the results was made in close comparison with the available data forpara-benzoquinone [Joneset al.,J. Chem. Phys., 2017,146, 184303]. While the dominant absorption features for both molecules arise from analogous π* ← π transitions, some remarkable differences have been identified. The perfluoro effect manifests in different ways: shifts in band positions and cross sections, appearance of features associated with excitations to σ CF* orbitals, and spectrum broadening by quenching of either vibrational or Rydberg progressions. The level of agreement between experiment and theory is very satisfactory, yet that required the inclusion of nuclear quantum effects in the calculations. We have also discussed the role of temperature on the absorption spectrum, as well as the involvement of core-excited resonances in promoting dissociative electron attachment reactions in the 3-5 eV range.

The fully state-selected reactions between H2+ molecules in the X+ 2Σg+(v+ = 0, N+ = 0) state and HD molecules in the X 1Σg+(v = 0, J = 0) state forming H3+ + D and H2D+ + H have been studied at collision energies Ecoll between 0 and kB·30 K with a resolution of about 75 mK at the lowest energies. H2 molecules in a supersonic beam were prepared in Rydberg-Stark states with principal quantum number n = 27 and merged with a supersonic beam of ground-state HD molecules using a curved surface-electrode Rydberg-Stark decelerator and deflector. The reaction between H2+ and HD was studied within the orbit of the Rydberg electron to avoid heating of the ions by stray electric fields. The reaction was observed for well-defined and adjustable time intervals, called reaction-observation windows, between two electric-field pulses. The first pulse swept all ions away from the reaction volume and its falling edge defined the beginning of the reaction-observation window. The second pulse extracted the product ions toward a charged-particle detector located at the end of a time-of-flight tube and its rising edge defined the end of the reaction-observation window. Monitoring and analysing the time-of-flight distributions of the H3+ and H2D+ products in dependence of the duration of the reaction-observation window enabled us to obtain information on the kinetic-energy distribution of the product ions and determine branching ratios of the H3+ + D and H2D+ + H reaction channels. The mean product-kinetic-energy release is 0.46(5) eV, representing 27(3)% of the available energy, and the H3+ + D product branching ratio is 0.225(20). The relative reaction rates correspond closely to Langevin capture rates down to the lowest energies probed experimentally (≈kB·50 mK).

In many areas of physics and chemistry, the Rydberg constant is a fundamental physical constant that plays an important role. It comes into play as an indispensable physical constant in basic equations for describing natural phenomena. The Rydberg constant appears in the formula for calculating the wavelengths in the line spectrum emitted from the hydrogen atom. However, this Rydberg wavelength formula is a nonrelativistic formula derived at the level of classical quantum theory. In this paper, the Rydberg formula is rewritten as a wavelength formula taking into account the theory of relativity. When this is done, we come to an unexpected conclusion. What we try to determine by measuring spectra wavelengths is not actually the value of the Rydberg constant R∞ but the value Rn,m of Formula (18). R∞ came into common use in the world of nonrelativistic classical quantum theory. If the theory of relativity is taken into account, R∞ can no longer be regarded as a physical constant. That is, we have continued to conduct experiments to this day in an attempt to determine the value of a physical constant, the Rydberg constant, which does not exist in the natural world.

The energy dependence of the rates of the reactions between He+ and ammonia (NY3, Y = {H,D}), forming NY2+, Y and He as well as NY+, Y2 and He, and the corresponding product branching ratios have been measured at low collision energies Ecoll between 0 and kB·40 K using a recently developed merged-beam technique [Allmendinger et al., ChemPhysChem, 2016, 17, 3596]. To avoid heating of the ions by stray electric fields, the reactions are observed within the large orbit of a highly excited Rydberg electron. A beam of He Rydberg atoms was merged with a supersonic beam of ammonia using a curved surface-electrode Rydberg–Stark deflector, which is also used for adjusting the final velocity of the He Rydberg atoms, and thus the collision energy. A collision-energy resolution of about 200 mK was reached at the lowest Ecoll values. The reaction rate coefficients exhibit a sharp increase at collision energies below ∼kB·5 K and pronounced deviations from Langevin-capture behaviour. The experimental results are interpreted in terms of an adiabatic capture model describing the rotational-state-dependent orientation of the ammonia molecules by the electric field of the He+ atom. The model faithfully describes the experimental observations and enables the identification of three classes of |JKMp〉 rotational states of the ammonia molecules showing different low-energy capture behaviour: (A) high-field-seeking states with |KM| ≥ 1 correlating to the lower component of the umbrella-motion tunnelling doublet at low fields. These states undergo a negative linear Stark shift, which leads to strongly enhanced rate coefficients; (B) high-field-seeking states subject to a quadratic Stark shift at low fields and which exhibit only weak rate enhancements; and (C) low-field-seeking states with |KM| ≥ 1. These states exhibit a positive Stark shift at low fields, which completely suppresses the reactions at low collision energies. Marked differences in the low-energy reactivity of NH3 and ND3—the rate enhancements in ND3 are more pronounced than in NH3—are quantitatively explained by the model. They result from the reduced magnitudes of the tunnelling splitting and rotational intervals in ND3 and the different occupations of the rotational levels in the supersonic beam caused by the different nuclear-spin statistical weights. Thermal capture rate constants are derived from the model for the temperature range between 0 and 10 K relevant for astrochemistry. Comparison of the calculated thermal capture rate coefficients with the absolute reaction rates measured above 27 K by Marquette et al. (Chem. Phys. Lett., 1985, 122, 431) suggests that only 40% of the close collisions are reactive.

In the literature, there were studies of Rydberg states of hydrogenic atoms/ions in a high-frequency laser field. It was shown that the motion of the Rydberg electron is analogous to the motion of a satellite around an oblate planet (for a linearly polarized laser field) or around a (fictitious) prolate planet (for a circularly polarized laser field): it exhibits two kinds of precession – one of them is the precession within the orbital plane and another one is the precession of the orbital plane. In this study, we study a helium atom or a helium-like ion with one of the two electrons in a Rydberg state, the system being under a high-frequency laser field. For obtaining analytical results, we use the generalized method of the effective potentials. We find two primary effects of the high-frequency laser field on circular Rydberg states. The first effect is the precession of the orbital plane of the Rydberg electron. We calculate analytically the precession frequency and show that it differs from the case of a hydrogenic atom/ion. In the radiation spectrum, this precession would manifest as satellites separated from the spectral line at the Kepler frequency by multiples of the precession frequency. The second effect is a shift of the energy of the Rydberg electron, also calculated analytically. We find that the absolute value of the shift increases monotonically as the unperturbed binding energy of the Rydberg electron increases. We also find that the shift has a nonmonotonic dependence on the nuclear charge Z: as Z increases, the absolute value of the shift first increases, then reaches a maximum, and then decreases. The nonmonotonic dependence of the laser field-caused energy shift on the nuclear charge is a counterintuitive result.

14 pags., 10 figs., 2 tabs., The nonadiabatic photodissociation dynamics of the CH$_3$ (and CD$_3$) radical from the 3p$_z$ and 3s Rydberg states is investigated by applying a one-dimensional (1D) wave packet model that uses recently calculated ab initio 1D electronic potential-energy curves and nonadiabatic couplings. Calculated predissociation lifetimes are found to be too long as compared to the experimental ones. The 1D dynamical model, however, is able to predict qualitatively and explain the fragmentation mechanisms that produce the hydrogen-fragment translational energy distributions (TED) measured experimentally for the ground vibrational resonance of the 3p$_z$ and 3s Rydberg states (CH$_3$($v=0$, 3p$_z$) and CH$_3$($v=0$, 3s)). The CH$_3$($v=0$, 3p$_z$) TED found experimentally displays a rather large energy spreading, while the experimental CH$_3$($v=0$, 3s) TED is remarkably more localized in energy. The present model also predicts a widely spread CH$_3$($v=0$, 3p$_z$) TED, produced by a complex dissociation mechanism which involves predissociation to one dissociative valence state through a nonadiabatic coupling, as well as transfer of population to a second valence state through three conical intersections. Also in agreement with experiment, the model predicts a rather localized CH$_3$($v=0$, 3s) TED because the conical intersections no longer operate in this photodissociation process, and predissociation occurs only into a single valence state. Another complex dissociation mechanism is predicted by the model for initial CH$_3$($v>0$, 3s) and CD$_3$($v>0$, 3s) resonances. In this case the mechanism is gradually activated, as vibrational excitation increases, by the interplay between the two nonadiabatic couplings connecting the 3s and 3p$_{x,y}$ Rydberg states with the dissociative $^2$A$_1$ valence state, and produces complex TEDs with signals from several resonances of both 3s and 3p$_{x,y}$. Thus the present 1D quantum model reveals a rich photodissociation dynamics of methyl, where a variety of complex fragmentation mechanisms is favored by the presence of different nonadiabatic couplings between the electronic states involved., This work was funded by the Ministerio de Ciencia e Innovacion (MICINN, Spain), Grant No. PGC2018-096444-B-100. The Centro de Supercomputacion de Galicia (CESGA, Spain) is acknowledged for the use of its resources.

The high-resolution absorption spectrum of 3-methoxyacrylonitrile (3MAN) was measured between 5.27 and 12.59 eV using a synchrotron-based Fourier-transform spectrometer. It was related to an absolute absorption cross-section scale. Complementary calculations at the DFT-MRCI/aug-cc-pVTZ level of theory document the vertical transition energies and oscillator strengths toward the first 19 states of both the E and Z geometrical isomers of 3MAN. Comparisons with the experimental absorption spectrum reveal the similarities and differences between 3MAN, a bifunctional molecule, with acrylonitrile and methylvinylether, where only one functional group is present. As in acrylonitrile, several broad valence transitions were observed up to the ionization limit. They are likely associated with the extended π-system induced by the nitrile group but might also involve σσ* transitions close to the ionization limit. As in methylvinylether, Rydberg series converging to the ionization limit are absent. This is attributed to a difference in neutral and cationic geometry due to a 60° rotation of the methyl group.

In our previous papers (Can. J. Phys. 91 (2013) 715; 92 (2014) 1405), we studied Rydberg states of systems consisting of a nucleus of charge Z, a muon, and an electron, both the muon and electron being in circular states. The studies of such quasimolecules μZe were motivated by numerous applications of muonic atoms and molecules, where one of the electrons is substituted by the heavier lepton μ–. We demonstrated that the muonic motion can represent a rapid subsystem, while the electronic motion can represent a slow subsystem. We showed that the spectral lines emitted by the muon in such systems experience a red shift compared to the corresponding spectral lines that would have been emitted by the muon in a muonic hydrogenic atom/ion. In the present paper, we also consider Rydberg states of quasimolecules μZe with Z > 1 (i.e., Rydberg states of muonic–electronic helium and helium-like ions). However, our current approach has important distinctions from our previous papers. The systems considered here are truly stable and the electron orbit is generally elliptical (although the relatively small influence of the electron on the muon is neglected). In our previous papers, the influence of the electron on the muon was taken into account; however, in the rotating frame used in our previous papers, the motion of the muon was only metastable (not truly stable), and furthermore, only circular orbits of the electron were considered in our previous paper. In the present paper, we show that the effective potential energy of the Rydberg electron is mathematically equivalent to the potential energy of a satellite moving around an oblate planet. Based on this, we demonstrate that the unperturbed orbital plane of the Rydberg electron undergoes simultaneously two different precessions: precession within the orbital plane and precession of the orbital plane around the axis of the muonic circular orbit. We provide analytical expressions for the frequencies of both precessions. The shape of the elliptical orbit of the Rydberg electron is not affected by the perturbation, which is the manifestation of the (approximate) conservation of the square of the angular momentum of the Rydberg electron. This means that the above physical systems have a higher than geometrical symmetry (also known as a hidden symmetry) which is a counterintuitive result of general physical interest. We note that the above problem of the motion of the Rydberg electron in muonic–electronic helium atoms or helium-like ions is mathematically equivalent to another problem from atomic physics: a hydrogen Rydberg atom in a linearly-polarized electric field of a high-frequency laser radiation.

Highly sensitive phase- and frequency-resolved detection of microwave electric fields is of central importance in a wide range of fields, including cosmology1,2, meteorology3, communication4 and microwave quantum technology5. Atom-based electrometers6,7 promise traceable standards for microwave electrometry, but their best sensitivity is currently limited to a few μV cm−1 Hz−1/2 (refs. 8,9) and they only yield information about the field amplitude and polarization10. Here, we demonstrate a conceptually new microwave electric field sensor—the Rydberg-atom superheterodyne receiver (superhet). The sensitivity of this technique scales favourably, achieving even 55 nV cm−1 Hz−1/2 with a modest set-up. The minimum detectable field of 780 pV cm−1 is three orders of magnitude smaller than what can be reached by existing atomic electrometers. The Rydberg-atom superhet allows SI-traceable measurements, reaching uncertainty levels of 10−8 V cm−1 when measuring a sub-μV cm−1 field, which has been inaccessible so far with atomic sensors. Our method also enables phase and frequency detection. In sensing Doppler frequencies, sub-μHz precision is reached for fields of a few hundred nV cm−1. This work is a first step towards realizing electromagnetic-wave quantum sensors with quantum projection noise-limited sensitivity. Such a device will impact diverse areas like radio astronomy, radar technology and metrology. The Rydberg-atom superhet, based on microwave-dressed Rydberg atoms and a tailored electromagnetically induced transparency spectrum, allows SI-traceable measurements of microwave electric fields with unprecedented sensitivity.

Trapped neutral atoms have become a prominent platform for quantum science, where entanglement fidelity records have been set using highly-excited Rydberg states. However, controlled two-qubit entanglement generation has so far been limited to alkali species, leaving the exploitation of more complex electronic structures as an open frontier that could lead to improved fidelities and fundamentally different applications such as quantum-enhanced optical clocks. Here we demonstrate a novel approach utilizing the two-valence electron structure of individual alkaline-earth Rydberg atoms. We find fidelities for Rydberg state detection, single-atom Rabi operations, and two-atom entanglement surpassing previously published values. Our results pave the way for novel applications, including programmable quantum metrology and hybrid atom-ion systems, and set the stage for alkaline-earth based quantum computing architectures., ISM and JPC contributed equally to this work

Quantum beats in fluorescence decay from Zeeman-split magnetic sublevels have been measured for helium Rydberg states excited by synchrotron radiation. The Zeeman quantum beats observed in this prototypical case were fitted with an equation from a theoretical formulation. It is proposed that Zeeman quantum beat measurement can be a useful way to simply evaluate the polarization characteristics of extreme ultraviolet light.

Cuprous oxide (Cu2O) is a semiconductor with large exciton binding energy and significant technological importance in applications such as photovoltaics and solar water splitting. It is also a superior material system for quantum optics that enabled the observation of intriguing phenomena, such as Rydberg excitons as solid-state analogue to highly-excited atomic states. Previous experiments related to excitonic properties focused on natural bulk crystals due to major difficulties in growing high-quality synthetic samples. Here, the growth of Cu2O microcrystals with excellent optical material quality and very low point defect levels is presented. A scalable thermal oxidation process is used that is ideally suited for integration on silicon, demonstrated by on-chip waveguide-coupled Cu2O microcrystals. Moreover, Rydberg excitons in site-controlled Cu2O microstructures are shown, relevant for applications in quantum photonics. This work paves the way for the wide-spread use of Cu2O in optoelectronics and for the development of novel device technologies. Cu2O is of great interest for its excitonic properties, yet challenges in its fabrication means that most experiments focus on naturally occurring samples. Here, scalable thermal oxidation is reported for the growth of Cu2O with low-defect content, allowing the observation of Rydberg excitons.

Quantum gates and entanglement based on dipole-dipole interactions of neutral Rydberg atoms are relevant to both fundamental physics and quantum information science. The precision and robustness of the Rydberg-mediated entanglement protocols are the key factors limiting their applicability in experiments and near-future industry. There are various methods for generating entangling gates by exploring the Rydberg interactions of neutral atoms, each equipped with its own strengths and weaknesses. The basics and tricks in these protocols are reviewed, with specific attention paid to the achievable fidelity and the robustness to the technical issues and detrimental innate factors., Comment: 57 pages, 10 figures

We report photoelectron circular dichroism of S-(+)-fenchone enantiomers recorded with state-state vibrational level resolution using picosecond laser (2 + 1) resonance enhanced multiphoton ionization via 3s and 3p Rydberg intermediate states. The 3p state decays to the 3s state on a picosecond time scale so that, above the 3p Rydberg excitation threshold, ionization of vibrationally hot 3s states competes with direct 3p-1 ionization. Complex vibronic dynamics of the 3p → 3s internal conversion weaken the Rydberg Δv = 0 propensity rule in both the 3p-1 and 3s-1 ionization channels. Large variations of the forward-backward chiral asymmetry factors are observed between the Δv = 0 and Δv > 0 vibrational transitions, including dramatic swings from up to ±17%. Such changes of sign indicate complete reversal of the preferred direction for photoelectron emission in the laboratory frame, associated with vibrational motion. These asymmetry switches easily exceed the amplitude and frequency of such vibrationally induced flips previously observed in single photon ionization.

Extended Hubbard models have proven to bear novel quantum states, but their experimental realization remains challenging. In this work we propose to use bosonic quantum gases dressed with molecular bound states in Rydberg interaction potentials for the observation of these quantum states. We study the molecular Rabi coupling with respect to principal quantum number and trapping frequency of the ground state atoms for various molecular potentials of Rubidium and Potassium, and the hereby resulting dressed interaction strength. Additionally, we propose a two-color excitation scheme which significantly increases the dressed interaction and cancels AC Stark shifts limiting the atomic motion in the itinerant regime. We study the various equilibrium phases of the corresponding extended Bose-Hubbard model by means of the Cluster Gutzwiller approach and perform time evolution simulations via the Lindblad master equation. We find a supersolid phase by slowly ramping the molecular Rabi coupling of an initially prepared superfluid and discuss the role of dissipation., Comment: 11 pages, 7 figures

We review classical studies of the oscillatory-precessional motion of an electron in the field of an electric dipole (the latter representing the polar molecule) with or without external magnetic or electric fields. The focus is on the most recent studies. In one study (at zero external field), it was shown that, generally, the oscillations being in the meridional direction and the precession being along parallels of latitude can take place on the same time scale—contrary to the statement from the previous literature. In another study, it was shown that a magnetic field enables new ranges of the bound oscillatory-precessional motion of the Rydberg electron and that in one of the new ranges, the period of the θ-oscillations has the non-monotonic dependence on primary parameter of the system. This is a counterintuitive result. In yet another study, it was shown that under the electric field there are two equilibrium circular states of a positive energy and one equilibrium state of a negative energy. The existence of the equilibrium state of the negative energy is a counterintuitive result since at the absence of the field, the bound state was possible only for the zero energy. Thus, it is a counterintuitive result that in this case the electric field can play the role of a stabilizing factor.

Hyperentanglement~(HE), the simultaneous entanglement between two particles in more than one degrees of freedom, is relevant to both fundamental physics and quantum technology. Previous study on HE has been focusing on photons. Here, we study HE in individual neutral atoms. In most alkaline-earth-like atoms with two valence electrons and a nonzero nuclear spin, there are two stable electronic states, the ground state and the long-lived clock state, which can define an electronic qubit. Meanwhile, their nuclear spin states can define a nuclear qubit. By the Rydberg blockade effect, we show that the controlled-Z~(C$_{\text{Z}}$) operation can be generated in the electronic qubits of two nearby atoms, and simultaneously in their nuclear qubits as well, leading to a C$_{\text{Z}}\otimes$C$_{\text{Z}}$ operation which is capable to induce HE. The possibility to induce HE in individual neutral atoms offers new opportunities to study quantum science and technology based on neutral atoms., 19 pages, 10 figures

Electronic states and their dynamics are of critical importance for electronic and optoelectronic applications. Here, we probe various relevant electronic states in monolayer MoS2, such as multiple excitonic Rydberg states and free-particle energy bands, with a high relative contrast of up to >200 via broadband (from ~1.79 to 3.10 eV) static third-harmonic spectroscopy, which is further supported by theoretical calculations. Moreover, we introduce transient third-harmonic spectroscopy to demonstrate that third-harmonic generation can be all-optically modulated with a modulation depth exceeding ~94% at ~2.18 eV, providing direct evidence of dominant carrier relaxation processes, associated with carrier-exciton and carrier-phonon interactions. Our results indicate that static and transient third-harmonic spectroscopies are not only promising techniques for the characterization of monolayer semiconductors and their heterostructures, but also a potential platform for disruptive photonic and optoelectronic applications, including all-optical modulation and imaging., 17 pages, 4 figures

The authors developed a highly sensitive method based on a quantum-control double-resonance-spectroscopy scheme to study electronically excited states of molecular ions that do not dissociate. The method is used to study the $3d\phantom{\rule{0}{0ex}}\ensuremath{\delta}$ Rydberg state of MgAr${}^{+}$, and the results contribute to the first extensive characterization of the Rydberg states of a molecular cation.

Dissipative systems often exhibit wavelength-dependent loss rates. One prominent example is Rydberg polaritons formed by electromagnetically induced transparency, which have long been a leading candidate for studying the physics of interacting photons and also hold promise as a platform for quantum information. In this system, dissipation is in the form of quantum diffusion, i.e., proportional to ${k}^{2}$ ($k$ being the wavevector) and vanishing at long wavelengths as $k\ensuremath{\rightarrow}0$. Here, we show that one-dimensional condensates subject to this type of loss are unstable to long-wavelength density fluctuations in an unusual manner: after a prolonged period in which the condensate appears to relax to a uniform state, local depleted regions quickly form and spread ballistically throughout the system. We connect this behavior to the leading-order equation for the nearly uniform condensate---a dispersive analog to the Kardar-Parisi-Zhang equation---which develops singularities in finite time. Furthermore, we show that the wavefronts of the depleted regions are described by purely dissipative solitons within a pair of hydrodynamic equations, with no counterpart in lossless condensates. We close by discussing conditions under which such singularities and the resulting solitons can be physically realized.

Quantum sensors offer the capability to reach unprecedented precision by operating at the standard quantum limit (SQL) or beyond by using quantum entanglement. But an emerging class of quantum sensors that use Rydberg electromagnetically induced transparency (EIT) to detect rf electric fields have yet to reach the SQL. In this work we prove that this discrepancy is due to fundamental limitations in the EIT probing mechanism. We derive the optimum sensitivity of a three-level quantum sensor based on EIT, or more generally coherent spectroscopy, and compare this to the SQL. We apply a minimal set of assumptions, while allowing strong probing fields, thermal broadening, and large optical depth. We derive the optimal laser intensities and optical depth, providing specific guidelines for sensitive operation under common experimental conditions. Clear boundaries of performance are established, revealing that ladder-EIT cannot achieve the SQL due to unavoidable absorption loss. The results may be applied to any EIT-based quantum sensor, but we particularly emphasize our results' importance to the growing field of Rydberg quantum sensing.

We investigate the indistinguishability of single photons retrieved from collective Rydberg excitations in cold atomic ensembles. The Rydberg spin waves are created either by off resonant two-photon excitation to the Rydberg state or by Rydberg electromagnetically induced transparency. To assess the indistinguishability of the generated single photons, we perform Hong-Ou-Mandel experiments between the single photons and weak coherent states of light. We analyze the indistinguishability of the single photons as a function of the detection window and for photons generated by off-resonant excitation we infer high value of indistinguishability going from 89% for the full waveform to 98% for small detection windows. In the same way, we also investigate for the first time the indistinguishability of single photons generated by Rydberg EIT, showing values lower than those corresponding to single photons generated by off-resonant excitation. These results are relevant for the use of Rydberg atoms as quantum network nodes., Comment: 8 pages, 8 figures. Submitted on November 9

Electric Rydberg-atom interferometry has been performed with helium atoms in coherent superpositions of the $1s56s\phantom{\rule{0.16em}{0ex}}^{3}S_{1}$ and $1s57s\phantom{\rule{0.16em}{0ex}}^{3}S_{1}$ Rydberg levels. The experiments were carried out in a longitudinal geometry with the atoms traveling at 2000 m/s in pulsed supersonic beams. After laser photoexcitation from the $1s2s\phantom{\rule{0.16em}{0ex}}^{3}S_{1}$ metastable level, coherent manipulation of the population of the Rydberg states was achieved using sequences of microwave pulses. The difference in the static electric dipole polarizabilities of the $1s56s\phantom{\rule{0.16em}{0ex}}^{3}S_{1}$ and $1s57s\phantom{\rule{0.16em}{0ex}}^{3}S_{1}$ levels allowed superpositions of external momentum states to be generated when inhomogeneous electric fields were used to exerted forces on the atoms prepared in superpositions of these internal states. Interference fringes, with contributions arising from the spatial separation of these Rydberg-atom wave packets in the direction of propagation of the atomic beam, were identified through changes in the internal-state populations as the magnitudes and durations of the time-dependent electric-field gradient pulses were adjusted. The maximal displacement of the atomic wave packets for which interference fringes were observed was $\ensuremath{\sim}0.75\phantom{\rule{0.28em}{0ex}}\mathrm{nm}$, limited by the longitudinal velocity spread in the atomic beam and the characteristics of the inhomogeneous electric-field distribution in the apparatus. The experimental data are in good quantitative agreement with the results of numerical calculations of the time evolution of the atomic states under the experimental conditions.

We present a general approach to converge excited state solutions to any quantum chemistry orbital optimization process, without the risk of variational collapse. The resulting Square Gradient Minimization (SGM) approach only requires analytic energy/Lagrangian orbital gradients and merely costs 3 times as much as ground state orbital optimization (per iteration), when implemented via a finite difference approach. SGM is applied to both single determinant $\Delta$SCF and spin-purified Restricted Open-Shell Kohn-Sham (ROKS) approaches to study the accuracy of orbital optimized DFT excited states. It is found that SGM can converge challenging states where the Maximum Overlap Method (MOM) or analogues either collapse to the ground state or fail to converge. We also report that $\Delta$SCF/ROKS predict highly accurate excitation energies for doubly excited states (which are inaccessible via TDDFT). Singly excited states obtained via ROKS are also found to be quite accurate, especially for Rydberg states that frustrate (semi)local TDDFT. Our results suggest that orbital optimized excited state DFT methods can be used to push past the limitations of TDDFT to doubly excited, charge-transfer or Rydberg states, making them a useful tool for the practical quantum chemist's toolbox for studying excited states in large systems.

The dipole blockade effect at laser excitation of mesoscopic ensembles of Rydberg atoms lies in the fact that the excitation of one atom to a Rydberg state blocks the excitation of other atoms due to the shift in the collective energy levels of interacting Rydberg atoms. It is used to obtain the entangled qubit states based on single neutral atoms in optical traps. In this paper, we present our experimental results on the observation of the dipole blockade for mesoscopic ensembles of 1-5 atoms when they are detected by the selective field ionization method. We investigated the spectra of the three-photon laser excitation $ 5S_{1/2} \to 5P_{3/2} \to 6S_{1/2} \to nP_{3/2} $ of cold Rb Rydberg atoms in a magneto-optical trap. We have found that for mesoscopic ensembles this method allows only a partial dipole blockage to be observed. This is most likely related to the presence of parasitic electric fields reducing the interaction energy of Rydberg atoms, the decrease in the probability of detecting high states, and the strong angular dependence of the interaction energy of Rydberg atoms in a single interaction volume., Comment: 16 pages, 7 figures

The relationship E = −K holds between the energy E and kinetic energy K of the electron constituting a hydrogen atom. If the kinetic energy of the electron is determined based on that relationship, then the energy levels of the hydrogen atom are also determined. In classical quantum theory, there is a formula called the Rydberg formula for calculating the wavelength of a photon emitted by an electron. In this paper, in contrast, the formula for the wavelength of a photon is derived from the relativistic energy levels of a hydrogen atom derived by the author. The results show that, although the Rydberg constant is classically a physical constant, it cannot be regarded as a fundamental physical constant if the theory of relativity is taken into account.

The response of a molecule to photoexcitation is governed by the coupling of its electronic states. However, if the energetic spacing between the electronically excited states at the Franck-Condon window becomes sufficiently small, it is infeasible to selectively excite and monitor individual states with conventional time-resolved spectroscopy, preventing insight into the energy transfer and relaxation dynamics of the molecule. Here, we demonstrate how the combination of time-resolved spectroscopy and extensive surface hopping dynamics simulations with a global fit approach on individually excited ensembles overcomes this limitation and resolves the dynamics in the n3p Rydberg states in acetone. Photoelectron transients of the three closely spaced states n3px, n3py, and n3pz are used to validate the theoretical results, which in turn allow retrieving a comprehensive kinetic model describing the mutual interactions of these states for the first time.

The photoionization microscopy of the Rydberg Rb atom exposed to a continuous infrared radiation laser field is investigated based on the semiclassical open orbit theory. In contrast to the photoionization of the Rydberg hydrogen atom, the ionic core-scattering effect plays an important role in the photoionization of the Rb atom. Due to the core-scattering effect and the laser field, the electron trajectories become chaotic. A huge number of ionization trajectories from the ionic source to the detector plane appear, which makes the oscillatory pattern in the electron probability distribution become much more complicated. The ρ–θ curve on the detector plane exhibits a self-similar fractal structure for the ionization trajectories of the Rydberg Rb atom in the laser field. Due to constructive and destructive quantum interference of different electron trajectories, a series of concentric rings appear in the photoionization microscopy interference patterns on the detector plane. The electron probability density distributions on the detector are found to be changed sensitively with the scaled electron energy and the laser wavelength. Even as the detector plane is located at a macroscopic distance from the photoionization source, the photoionization microscopy interference patterns can be observed clearly. These calculations may provide a valuable contribution to the actual experimental study of the photoionization microscopy of non-hydrogenic Rydberg atom in the laser field.