Principal Quantum Number Research Articles

Small molecule activation by well-defined compounds of heavy p-block elements.

The creative design and exploration of new bonding motifs and molecular architectures in main group chemistry have pushed the boundaries of reactivity in this field of research. In this context, the activation of small molecules represents a set of benchmark reactions and offers valuable opportunities for the development of innovative synthetic methods. In addition to significant progress by use of transition metal complexes and compounds of lighter p-block elements, compounds based on heavy p-block elements (with the principle quantum number n > 4) have seen major advances in recent years. Fundamental properties originating from their high atomic number (such as the size, energy, and polarizability of atomic orbitals) set them apart from more established species in small molecule activation. Challenges and opportunities arising from this scenario are analyzed and highlighted.

Electrostatic trapping vibrationally excited Rydberg NO molecules

Nitric oxide (NO) molecules travelling in pulsed supersonic beams have been laser photoexcited to long-lived Rydberg states in series converging to the lowest lying, , and first excited, , vibrational states in the ground electronic state of NO. After excitation, the molecules were decelerated in the travelling electric traps of a chip-based Rydberg–Stark decelerator. Deceleration to rest in the laboratory-fixed frame of reference allowed subsequent electrostatic trapping before in situ detection by pulsed electric field ionisation. The decay rates of the molecules from the traps were measured for states with principal quantum numbers between 32 and 48. Comparison of the corresponding trap decay time constants, ranging from 245 to s for states with , with those recorded for molecules in states with , allowed bounds to be placed on the vibrational autoionisation rates of the ℓ-mixed Stark states prepared in the experiments of Hz.

Generalized relativistic small‐core pseudopotentials accounting for quantum electrodynamic effects: Construction and pilot applications

AbstractA simple procedure to incorporate one‐loop quantum electrodynamic (QED) corrections into the generalized (Gatchina) nonlocal shape‐consistent relativistic pseudopotential model is described. The pseudopotentials for Lu, Tl, and Ra replacing only inner core shells (with principal quantum numbersn ≤ 3 for the two former elements andn ≤ 4 for the latter one) are derived from the solutions of reference atomic SCF problems with the Dirac–Coulomb–Breit Hamiltonian to which the model Lamb shift operator added. QED contributions to atomic valence excitation energies evaluated at the SCF level are demonstrated to exceed the errors introduced by the pseudopotential approximation itself by an order of magnitude. Pilot applications of the new model to calculations of excitation energies of two‐valence‐electron atomic systems using the intermediate‐Hamiltonian relativistic Fock space coupled cluster method reformulated here for incomplete main model spaces are reported. Implications for high‐accuracy molecular excited state calculations are discussed.

High-dimensional SO(4)-symmetric Rydberg manifolds for quantum simulation

We develop a toolbox for manipulating arrays of Rydberg atoms prepared in high-dimensional hydrogen-like manifolds in the regime of linear Stark and Zeeman effect. We exploit the SO(4) symmetry to characterize the action of static electric and magnetic fields as well as microwave and optical fields on the well-structured manifolds of states with principal quantum number n. This enables us to construct generalized large-spin Heisenberg models for which we develop state-preparation and readout schemes. Due to the available large internal Hilbert space, these models provide a natural framework for the quantum simulation of quantum field theories, which we illustrate for the case of the sine-Gordon and massive Schwinger models. Moreover, these high-dimensional manifolds also offer the opportunity to perform quantum information processing operations for qudit-based quantum computing, which we exemplify with an entangling gate and a state-transfer protocol for the states in the neighborhood of the circular Rydberg level.

AIHFLTF: Integrals in Laguerre function bases for electronic structure calculations in atoms

A program for calculating atomic integrals over Laguerre-type basis functions is provided. The Λ functions are used. They can generate complete orthonormal systems for bound states of atoms. The orthogonality property makes it possible to use large-scale basis sets without loss of accuracy. We have previously published the results of our nonrelativistic Hartree-Fock calculations for He–Og. Total energies were obtained to 30 significant figures for the atoms in the first to third periods. The integrals for this computation were calculated according to the analytical formulation given by Freund and Hill using a multiple-precision arithmetic package. This program for calculating integrals has been rewritten using Gauss-Laguerre quadrature. This version is twice as fast as the previous one while maintaining precision. Since quadruple precision arithmetic implemented in Fortran is used in this version, the program can be further accelerated by OpenMP parallelization. This version has been released. Program summaryProgram Title: AIHFLTFCPC Library link to program files:https://doi.org/10.17632/t4nxzbyssc.1Licensing provisions: MITProgramming language: Fortran 95Nature of problem: The basis set expansion method is widely used in studies of the electronic structure of atoms using the Hartree-Fock method or the post-HF method. The Λ functions which are able to generate complete orthonormal systems for bound states of atoms are used in the present paper. Because of orthogonality, the number of expansion terms can be increased without problems arising from linear dependence, enabling accurate electronic structure calculations. However, evaluation of 2-electron repulsion integrals demands much CPU-time. It is therefore vital in studies of the electronic structure of atoms to calculate 2-electron integrals efficiently.Solution method: Numerical integration based on Gauss-Laguerre quadrature is used to evaluate efficiently 2-electron repulsion integrals over the Λ functions. For this purpose, quadruple precision arithmetic is used, implemented in the Fortran programming language. Accuracy to 30 significant figures is confirmed for the integrals generated, even when the principal quantum number n is up to 200. The most time-consuming step in generating the integral is multiply-add arithmetic in which the values of the integrands at abscissae (zeros of a Laguerre polynomial) are multiplied by the weights. This step can be parallelized by OpenMP.

Dynamics of frustrated tunneling ionization driven by inhomogeneous laser fields

We theoretically investigated frustrated tunneling ionization (FTI) driven by spatially inhomogeneous strong laser fields induced by surface plasmon resonance within a bow-tie metal nanostructure. The results show that the FTI probability and the principal quantum number distribution exhibit similar oscillatory behavior as a function of the pulse duration. Our analysis reveals that the periodic defocusing and refocusing of the electron spatial distribution due to the inhomogeneous laser field is responsible for the oscillatory structures. In addition, the initial tunneling coordinates and the angular momentum distributions of the FTI events and theirs pulse duration dependence are also explored. Moreover, our results show that the frequency of the oscillatory structures depends sensitively on the electron quiver amplitude and the inhomogeneity strength. Thus, the electron quiver amplitude and the size of the gap between bow-tie nanostructure are useful and efficient knobs for controlling the yield and properties of exited Rydberg states.

Corresponding States for Volumes of Elemental Solids at Their Pressures of Polymorphic Transformations

Many non-molecular elemental solids exhibit common features in their structures over the range of 0 to 0.5 TPa that have been correlated with equivalent valence electron configurations. Here, it is shown that the pressures and volumes at polymorphic transitions obey corresponding states given by a single, empirical universal step-function Vtr/L = −0.0208(3) · Ptr + Ni, where Vtr is the atomic volume in Å3 at a given transformation pressure Ptr in GPa, and L is the principal quantum number. Ni assumes discrete values of approximately 20, 30, 40, etc. times the cube of the Bohr radius, thus separating all 113 examined polymorphic elements into five discrete sets. The separation into these sets is not along L. Instead, strongly contractive polymorphic transformations of a given elemental solid involve changes to different sets. The rule of corresponding states allows for predicting atomic volumes of elemental polymorphs of hitherto unknown structures and the transitions from molecular into non-molecular phases such as for hydrogen. Though not an equation of state, this relation establishes a basic principle ruling over a vast range of simple and complex solid structures that confirms that effective single-electron-based calculations are good approximations for these materials and pressures The relation between transformation pressures and volumes paves the way to a quantitative assessment of the state of very dense matter intermediate between the terrestrial pressure regime and stellar matter.

Noise self-canceling picoscale twisted interferometer.

We show a noise self-canceling real-time picometer scale interferometer by exploiting the unique spiral phase structure of twisted light. We use a single cylindrical interference-lens to implement the twisted interferometer and perform simultaneous measurement on N phase-orthogonal single-pixel intensity pairs chosen on the petal of the daisy-flower-like interference pattern. A cancellation of various noises by three orders of magnitude was achieved in our setup compared with a conventional single-pixel detection, enabling a sub-100 picometer resolution in measuring a non-repetitive intracavity dynamic event in real-time. Furthermore, the noise cancellation capability of the twisted interferometer scales up statistically for higher radial and azimuthal quantum numbers of the twisted light. The proposed scheme could find applications in precision metrology and in developing analogous ideas for twisted acoustic beam, electron beams, and matter waves.

Comment on: “On the interaction of an electron with a nonuniform electric field under the influence of a cut-off point induced by the spiral dislocation topology”

We analyze a recently proposed model for the interaction of an electron with a nonuniform electric field under the influence of a cut-off point induced by the topology of a spiral dislocation (Maia and Bakke, 2021). We show that the analytical eigenvalues derived by the authors are qualitatively and quantitatively incorrect. Besides, the infinite degeneracy of the energy levels and the limit on the possible values of the radial quantum number predicted by the authors are mere artifacts of the approximation used to solve the differential equation.

New acetylene line list near 3.8-µm - Part I

Two acetylene spectra were recorded between 2350 and 2750 cm−1 with Fourier Transform Spectrometer (FTS) using high optical path length. 14 bands were observed for the first time including one cold band. The positions and the intensities of the lines have been measured for 824 transitions of the 12C2H2 isotopologue. The band centers, the band strengths, the upper level spectroscopic constants and the Herman-Wallis parameters were retrieved for all bands. The effective dipole moment parameters of the ΔP = 4 region were refined, P being the polyad number defined as P = 5V1 + 3V2 + 5V3 + V4 + V5, where Vi are the vibrational normal mode principal quantum numbers, with i = 1 − 5. An updated line list for the ∆P = 4 series of transitions is proposed as supplementary data in order to improve the spectroscopic databases such as HITRAN or GEISA.

Quantum oscillations in the black hole horizon

By applying Rosen's quantization approach to the historical Oppenheimer and Snyder gravitational collapse and by setting the constraints for the formation of the Schwarzschild black hole (SBH), in a previous paper [1] two of the Authors (CC and FF) found the gravitational potential, the Schrodinger equation, the solution for the energy levels, the area quantum and the quantum representation of the ground state at the Planck scale of the SBH. Such results are consistent with previous ones in the literature. It was also shown that the traditional classical singularity in the core of the SBH is replaced by a quantum oscillator describing a non-singular two-particle system where the two components, named the "nucleus" and the "electron", strongly interact with each other through a quantum gravitational interaction. In agreement with the de Broglie hypothesis, the "electron" is interpreted in terms of the quantum oscillations of the BH horizon. In other words, the SBH should be the gravitational analogous of the hydrogen atom. In this paper, it is shown that these results allow us to compute the SBH entropy as a function of the BH principal quantum number in terms of Bekenstein-Hawking entropy and three sub-leading corrections. In addition, the coefficient of the formula of Bekenstein-Hawking entropy is reduced to a quarter of the traditional value. Then, it is shown that, by performing a correct rescaling of the energy levels, the semi-classical Bohr-like approach to BH quantum physics, previously developed by one of the Authors (CC), is consistent with the obtained results for large values of the BH principal quantum number. After this, Hawking radiation will be analysed by discussing its connection with the BH quantum structure. Finally, it is shown that the time evolution of the above mentioned system solves the BH information paradox.

Convergent close-coupling calculations of electron scattering on HeH+

We use the molecular convergent close-coupling (MCCC) method to perform calculations of 10--1000 eV electron scattering on the ground state of ${\mathrm{HeH}}^{+}$. Cross sections are presented for excitation of the $n=2\ensuremath{-}3$ singlet and triplet states (where $n$ is the united-atoms-limit principle quantum number), as well as ionization. We also present cross sections for ${\mathrm{He}}^{+}$ and ${\mathrm{H}}^{+}$ ion production following dissociative excitation and ionization. The ${\mathrm{He}}^{+}$ production cross section is compared with the measurements of Lecointre, Jureta, Urbain, and Defrance [J. Lecointre, J. J. Jureta, X. Urbain, and P. Defrance, J. Phys. B At. Mol. Opt. Phys. 47, 015203 (2014)]. We find that the MCCC results are up to 30% higher than experiment.

Decoherence of Higher Order Orbital Angular Momentum Entangled State in Non-Kolmogorov Turbulence

The decay of OAM entanglement in non-Kolmogorov turbulence has been numerically evaluated. In this work, we explore the evolution of OAM entanglement with higher-order OAM mode in the weak scintillation regime. In particular, the results of the numerical evaluation show that the OAM entanglement state with higher value of the azimuthal mode and larger radial quantum number survives over a longer distance. Meanwhile, the beam parameters and turbulence parameters usually have significant influences on OAM entanglement. In addition, it is demonstrated that the effect of turbulence on the OAM entanglement is the most serious when the generalized exponent is around 3.07.

Probing van der Waals interactions and detecting polar molecules by Förster-resonance energy transfer with Rydberg atoms at temperatures below 100 mK

Electric-field-controlled F\"orster resonance energy transfer (FRET) between Rydberg helium (He) atoms and ground-state ammonia (NH$_3$) molecules has been studied at translational temperatures below 100 mK. The experiments were performed in an intrabeam collision apparatus with pulsed supersonic beams of NH$_3$ seeded in He. A range of F\"orster resonances, between triplet Rydberg states in He with principal quantum numbers of 38, 39 and 40, and the inversion intervals in NH$_3$ were investigated. Resonance widths as low as $100\pm20$ MHz were observed for Rydberg-Rydberg transitions with electric dipole transition moments of 3270 D. These widths result from binary collisions at a mean center-of-mass speed of $19.3\pm2.6$ m/s. For transitions in which the initially prepared Rydberg states were strongly polarized, with large induced static electric dipole moments, van der Waals interactions between the collision partners increased the resonance widths to $\sim750$ MHz. From measurements of the rate of FRET for the narrowest resonances observed, a density of NH$_3$ of $(9.4\pm0.3)\times10^{9}$ cm$^{-3}$ in the upper ground-state inversion sublevel in the interaction region of the apparatus was determined non-destructively.

Spectroscopy of nanoantenna-covered Cu2O : Towards enhancing quadrupole transitions in Rydberg excitons

Plasmonic antennas channel incoming light into a confined near field that comes along with strong field gradients at the edges. Transverse field gradients are capable of driving quadrupole transitions in a more effective way than the weak longitudinal field gradient in dipolar light. Cuprous oxide Rydberg excitons with principal quantum number $n$ between 5 and 15 are several tens to a few hundreds of nanometers in diameter and can have different orbital angular momentum quantum numbers. Usually, only $P$- and $F$-excitons are electric dipole allowed, while $S$- and $D$-excitons remain dark. Here, we use 30-nm-wide and 60- to 110-nm-long plasmonic aluminum nanoantennas deposited on the cuprous oxide crystal surface to create transverse field gradients in similar spatial dimensions as the size of the Rydberg excitons. This way, light field and matter wave function overlap spatially and quadrupole transitions of $S$excitons are enhanced. The relative enhancement of the $S$-exciton oscillator strength with respect to the $P$-exciton oscillator accounts for a factor of up to 1.4.

An Analytical Formula for 1s 2s2p 3s3p3d Energy Shifts of Finite-Size Nuclei due to Fluctuating Vacuum Fields

In quantum electrodynamics, the vacuum is not empty but has fluctuating fields which cause an orbiting lepton (electron or muon) to deviate from its quantum orbit and as a result, weaken its coupling with the nucleus, thus causing small changes in lepton energy states. Therefore, in this study, the quantum electrodynamics theory is then applied to determine qualitatively the mean square positions of an oscillating lepton by taking into account the vacuum fields and nuclear size effects. The first-order time-independent perturbation theory as an approximation method is applied to solve for the small changes in the nuclear lepton interaction caused by the two effects. The general expression that can be applied to calculate the changes in lepton energy states is deduced and then applied to calculate numerically these effects on the energy states of single-electron and single muonic atoms. The results showed the dependence of these effects on the principal quantum number n, the orbital quantum number l, the azimuthal quantum number m and the proton number Z. A more significant effect on s (l = 0) states of the large atomic nucleus and muonic atoms is observed. The obtained energy level corrections can be considered as additional energy level corrections that have to be investigated for reanalyzing the existing measured nuclear structure effects and for comparison with future spectroscopic investigations. Moreover, the new proposed formula would be suitable for calculating the energy level corrections in both muonic and electronic atoms caused by the fluctuating vacuum fields.

On the Pasternack–Sternheimer theorem for bound states in hydrogenic atoms and ions derived by operator calculus

The Pasternack–Sternheimer theorem for bound states (Pasternack and Sternheimer 1962 J. Math. Phys. 3 1280) is obtained directly by the methods of operator calculus set out earlier (Hey 2006 J. Phys. B: At. Mol. Phys. 39 2641–64), on the basis of the factorisation technique of Infeld and Hull (Infeld and Hull 1951 Rev. Mod. Phys. 23 21–68). The present derivation, which complements the group theoretical treatments of Armstrong (Armstrong 1970 J. Phys. Colloq. 31 C4-17–23) and Cunningham (Cunningham 1972 J. Math. Phys. 13 33–9), not only elucidates the original result in terms of fundamental quantum mechanical theory, but also reveals some apparently new inter-connections between different radial matrix elements (for given n, diagonal and off-diagonal in ℓ, ℓ′) of hydrogenic atoms and ions. The key equation used to derive the theorem here is shown to follow identically in the non-relativistic limit from the treatment of the generalised Kepler problem by Crubellier and Feneuille (Crubellier and Feneuille 1971 J. Physique 32 405–11). This work is a continuation of studies employing operator methods to provide results of potential usefulness for spectroscopic studies of laboratory and astrophysical plasmas, in particular to transitions between states of high principal quantum number, as in the high-n radio recombination lines (Hey 2013 J. Phys. B: At. Mol. Opt. Phys. 46 175702; Peach 2014 Adv. Space Res. 54 1180-83).

Simulations for x-ray imaging of wave-packet dynamics

Previous work on imaging wave packet dynamics with x-ray scattering revealed that the scattering patterns deviate substantially from the notion of instantaneous momentum density of the wave packet. Here we show that scattering patterns can provide clear insights into the electron wave packet dynamics if the final state of the scattered electron and the scattered photon momentum are determined simultaneously. The scattering probability is shown to be proportional to the modulus square of the Fourier transform of the instantaneous spatial wave function weighted by the final state of the electron. Several cases for the choice of final state of the electron are explored. First, the case where the final state can be measured up to a given principal quantum number $n$ and orbital angular momentum $l$ are presented. Next, the case where the final states can only be determined up to a given energy is discussed. Finally, the case of an initial wave packet consisting of a large amount of a known stationary state and a small amount of an unknown stationary state is examined. The scattering profile is used to determine the properties of the unknown state in the wave packet.

Magic Numbers in Octahedral Ligated Metal-Chalcogenide Superatoms.

The attainment of the superatomic state offers a unifying framework for the periodic classification of atomic clusters. Metallic clusters attain the superatomic state via the confined nearly free electron gas model that leads to groupings of quantum states marked by radial and angular momentum quantum numbers. We examine ligated octahedral metal-chalcogenide clusters where the nearly free electron gas model is invalid; however, the high symmetry can also lead to the bunching of electronic states. For octahedral TM6E8L6 clusters (TM = transition metal; E = chalcogen; L = ligand), the electronic shells are filled for valence electron counts of 96, 100, and 114 electrons. These magic electron counts are marked by large highest occupied molecular orbital-lowest unoccupied molecular orbital (HOMO-LUMO) gaps, high ionization energies, and low electron affinity─all classic signatures of the superatomic state. We also find that clusters with electron counts differing from the magic counts show periodic patterns reminiscent of those observed in the periodic table of elements.

Self-Kerr Effect across the Yellow Rydberg Series of Excitons in Cu_{2}O.

We investigate the nonlinear refraction induced by Rydberg excitons in Cu_{2}O. Using a high-precision interferometry imaging technique that spatially resolves the nonlinear phase shift, we observe significant shifts at extremely low laser intensity near each exciton resonance. From this, we derive the nonlinear index n_{2}, present the n_{2} spectrum for principal quantum numbers n≥5, and report large n_{2} values of order 10^{-3} mm^{2}/mW. Moreover, we observe a rapid saturation of the Kerr nonlinearity and find that the saturation intensity I_{sat} decreases as n^{-7}. We explain this with the Rydberg blockade mechanism, whereby giant Rydberg interactions limit the exciton density, resulting in a maximum phase shift of 0.5rad in our setup.