Multichannel Quantum Defect Theory Research Articles

Engineering Rydberg-pair interactions in divalent atoms with hyperfine-split ionization thresholds

Quantum information processing with neutral atoms relies on Rydberg excitation for entanglement generation. While the use of heavy divalent or open-shell elements, such as strontium or ytterbium, has benefits due to their optically active core and a variety of possible qubit encodings, their Rydberg structure is generally complex. For some isotopes in particular, hyperfine interactions are relevant even for highly excited electronic states. We employ multichannel quantum defect theory to infer the Rydberg structure of isotopes with nonzero nuclear spin and perform nonperturbative Rydberg-pair interaction calculations. We find that due to the high level density and sensitivities to external fields, experimental parameters must be precisely controlled. Specifically in Sr87, we study an intrinsic Förster resonance, unique to divalent atoms with hyperfine-split thresholds, which simultaneously provides line stability with respect to external field fluctuations and enhanced long-range interactions. Additionally, we provide parameters for pair states that can be effectively described by single-channel Rydberg series. The explored pair states provide exciting opportunities for applications in the blockade regime, as well as for more exotic long-range interactions such as largely flat, distance-independent potentials. Published by the American Physical Society 2024

Addendum: Multichannel quantum defect theory of strontium bound Rydberg states (2014 J. Phys. B: At. Mol. Opt. Phys. 47 155001)

Abstract Newly calculated multichannel quantum defect theory parameters and channel fractions are presented for the singlet and triplet S, P and D series and singlet F series of strontium. These results correct those reported in Vaillant et al (2014 J. Phys. B: At. Mol. Opt. Phys. 47 155001).

An energy-modified quantum defect method for the analysis of Rydberg spectra: Application to 2-butyne.

The high resolution Rydberg absorption spectrum of 2-butyne C4H6 recorded previously at the SOLEIL synchrotron facility has been interpreted using multichannel quantum defect theory (MQDT). The calculations are based on the continuum scattering calculations of Xu et al., J. Chem. Phys. 136, 154303 (2012) and of Jacovella et al., J. Phys. Chem. A 119, 12339 (2015) pertaining to the dipole-allowed excited state symmetries in absorption from the ground state. In contrast to the traditional approach of calculating low-lying electronic states first and then attempting to extend the calculations to ever higher energy, here the analysis proceeds through the extension of these detailed calculations of the electronic continuum scattering down into the discrete region of the spectrum. The continuum reaction matrices and dipole transition moments are adapted to the discrete Rydberg region via the use of an energy-modified formulation of MQDT theory and associated energy dependences of the quantum defects. The analysis reproduces more than 40 Rydberg states from n ≈ 10 down to the 3d and 4s levels with an rms error of better than 20 cm-1. These belong to five Rydberg series with three different molecular symmetries. While the approach predicts many additional series, most of these are calculated and observed to carry only little oscillator strength. The analysis shows that the Rydberg spectrum is dominated by the excitation of an e″ symmetry electron of fδ and gπ type, in line with what previous studies of the above-threshold shape resonance of 2-butyne have shown. The present study is intended to serve as an example showing how first principles continuum calculations may be useful for the interpretation of highly bound discrete states in a range that poses problems for the standard abinitio techniques. The quantitative treatment of the dipole absorption cross sections is deferred to a future paper.

Multichannel Quantum Defect Theory with a Frame Transformation for Ultracold Atom-Molecule Collisions in Magnetic Fields.

We extend the powerful formalism of multichannel quantum defect theory combined with a frame transformation to ultracold atom-molecule collisions in magnetic fields. By solving the coupled-channel equations with hyperfine and Zeeman interactions omitted at short range, the extended theory enables a drastically simplified description of the intricate quantum dynamics of ultracold molecular collisions in terms of a small number of short-range parameters. We apply the formalism to ultracold Mg+NH collisions in a magnetic field, achieving a 10^{4}-fold reduction in computational effort.

Dissociative recombination of NS+ in collision with slow electrons

Cross sections and rate coefficients for the dissociative recombination (DR) of the NS+ ion induced by collisions with low-energy electrons are reported for temperatures between 10 and 1000 K, relevant to a large range of interstellar cloud temperatures. Uncertainties are discussed for these rates. Comparisons are made with DR rates for the isovalent NO+ molecular ion which are found to be much faster. The present findings lead to a moderate dissociative reaction rate coefficient, smaller by a factor of 2 than the current estimates reported in the different kinetic databases for a temperature of 10 K. We consider that our rate coefficients obtained through multichannel quantum defect theory for NS+ are likely to be better than those displayed in the different kinetic databases.

Mqdtfit: A collection of Python functions for empirical multichannel quantum defect calculations

The Python functions distributed with this article can be used for calculating the parameters of multichannel quantum defect theory models describing excited bound states of complex atoms. These parameters are obtained by fitting a model to experimental data provided by the user. The two main formulations of the theory are supported, namely the one in which the parameters of the model are a set of eigen channel quantum defects and a transformation matrix, and the one where these parameters are the elements of a reactance matrix. The distribution includes programs for calculating theoretical energy levels, calculating mixing coefficients and channel fractions and producing Lu-Fano plots. Program summaryProgram Title: mqdtfitCPC Library link to program files:https://doi.org/10.17632/nzgygzd96c.1Developer's repository link:https://github.com/durham-qlm/mqdtfitLicensing provisions: BSD 3-clause.Programming language: Python 3.Nature of problem: Multichannel quantum defect theory aims at giving a unified description of the excited states of multielectron systems whereby the properties of entire series can be predicted from models involving a relatively small number of parameters. These parameters are obtained by fitting theory to experimental data, in the empirical version of the theory. The present programs specifically concern the application of this theory to the bound states of complex atomic systems, such as divalent atoms, for which spectroscopic series are often perturbed by isolated states of a different symmetry, making a multichannel description necessary.Solution method: The functions forming this library are grouped and linked to each other into a single Python module. They can be used to calculate the parameters of a given multichannel quantum defect theory model by fitting the model to experimental data provided by the user. These parameters are either a set of eigen channel quantum defects and a transformation matrix, in the eigenchannel parametrization of the theory, or the elements of an orthogonal matrix, in the reactance matrix parametrization of the theory. Given these parameters, this software can also be used to calculate theoretical energy levels, calculate coefficients describing the mixing between the channels considered and produce Lu-Fano plots.

Reactive collisions between electrons and BeH+ above dissociation threshold.

Our previous studies of dissociative recombination and vibrational excitation/de-excitation of the BeH+ ion, based on the multichannel quantum defect theory, are extended to collision energies above the dissociation threshold, taking into account the vibrational continua of the BeH+ ion and, consequently, its dissociative excitation. We have also significantly increased the number of dissociative states of 2Π, 2Σ+ and 2Δ symmetry included in our cross section calculations, generating the most excited ones by using appropriate scaling laws. Our results are suitable for modeling the kinetics of BeH+ in edge fusion plasmas for collision energies up to 12 eV.

Bound and autoionizing potential energy curves in the CH molecule

This article presents a method of computing bound state potential curves and autoionizing curves using fixed-nuclei R-matrix data extracted from the Quantemol-N software suite. It is a method based on two related multichannel quantum-defect theory approaches. One is applying bound-state boundary conditions to closed-channel asymptotic solution matrices, and the other is searching for resonance positions via eigenphase shift analysis. We apply the method to the CH molecule to produce dense potential-curve datasets presented as graphs and supplied as tables in the publication supplement.

On single particle states and resonances in multichannel reactions

"The Multichannel Quantum Defect Theory and the Reduced R-Matrix are formally related and physically equivalent; both theories describe not only the internal dynamics but also the interactions in space of eliminated channels. One proves the Multichannel Quantum Defect Theory is Reduced Collision Matrix describing effect of eliminated channel on observed ones. The multichannel resonances originating in bound or quasistationary single particle states are described in terms of Reduced Collision Matrix. The single particle states are defined by Bound- or Quasistationary- State equation in the eliminated channel, relating channel logarithmic derivative to R-Matrix."

Dissociative recombination of N_2hbox {H}^+: a revisited study

Dissociative recombination of N_2H^+ is explored in a two-step theoretical study. In a first step, a diatomic (1D) rough model with a frozen NN bond and frozen angles is adopted, in the framework of the multichannel quantum defect theory (MQDT). The importance of the indirect mechanism and of the bending mode is revealed, in spite of the disagreement between our cross section and the experimental one. In the second step, we use our recently elaborated 3D approach based on the normal mode approximation combined with R-matrix theory and MQDT. This approach results in satisfactory agreement with storage-ring measurements, significantly better at very low energy than the former calculations.

Population of ground and lowest excited states of sulfur via the dissociative recombination of SH+ in the diffuse interstellar medium

ABSTRACT Our previous study on dissociative recombination of ground state SH+ through 2Π states of SH is extended by taking into account the contribution of 4Π states recently explored by quantum chemistry methods. Multichannel quantum defect theory is employed for the computation of cross sections and rate coefficients for dissociative recombination, but also for vibrational excitation. Furthermore, we produce the atomic yields resulting from recombination, quantifying the generation of sulfur atoms in their ground (3P) and lowest excited (1D) states, respectively.

Coherent Control of Ultracold Molecular Collisions: The Role of Resonances.

We consider the coherent control of ultracold molecule-molecule scattering, impacted by a dense set of rovibrational resonances. To characterize the resonance spectrum, a rudimentary model based on multichannel quantum defect theory has been used to study the control of the scattering cross section and the reaction rate. Complete control around resonance energies is shown to be possible, but thermal averaging over a large number of resonances significantly reduces the extent of control of reaction rates related to the random distribution of optimal control parameters between resonances. We show that measuring the extent of coherent control could be used to extract meaningful information about the relative contribution of direct scattering versus collision complex formation, as well as about the statistical regime.

Isolated-core quadrupole excitation of highly excited autoionizing Rydberg states

The structure and photoexcitation dynamics of high lying doubly excited states of the strontium atom with high angular momenta are studied in the vicinity of the Sr$^+(N=5)$ threshold. The spectra recorded using resonant multiphoton isolated core excitation are analyzed with calculations based on configuration interaction with exterior complex scaling, which treats the correlated motion of the two valence electrons of Sr from first principles. The results are rationalized with a model based on multichannel quantum-defect theory and transition dipole moments calculated with a perturbative treatment of electron correlations. Together, both approaches reveal that most of the lines observed in the spectra arise from the interaction of a single optically active state, coupled to the initial state by an electric-dipole transition, with entire doubly-excited Rydberg series. The long-range electron correlations responsible for this interaction unexpectedly vanish for identical values of the initial and final principal quantum numbers, a fact related to the quasi hydrogenic nature of the high-$l$ Rydberg electron. This special situation, and in particular the vanishing interaction, leads to the surprising observation of an electric \emph{quadrupole} isolated-core excitation with a similar intensity as the neighboring electric dipole transitions.

The fine and hyperfine structures of the low-n 3Π u Rydberg states of H2 revisited

The experimental fine and hyperfine structures of low Rydberg states with 2–4 measured in the 1970's for various vibration-rotation levels by Lichten and collaborators, and by Miller and Freund, are reanalysed in the framework of molecular frame-transformation multichannel quantum defect theory MQDT. Instead of the level by level treatment in terms of an effective Hamiltonian as was carried out at the time, a global theory is applied to all levels at once. It is found that the essence of the observed level patterns is well represented by MQDT in terms of a substantially reduced number of dynamical parameters – quantum defects – although the experimental accuracy of the experiments is not yet quite reached. Possible reasons for the remaining discrepancies are given. Since quantum defect theory MQDT, a formalism derived from scattering theory, is still rather little used in the domain of modern ultra-high precision spectroscopy, the occasion is taken here to review some of its key elements.

Electron collisions with ArH+ molecular ions: highly excited vibrational states and dissociative excitation

A theoretical investigation of dissociative recombination and dissociative excitation processes involving electron collisions with the argonium ion (ArH+) at energies up to 7 eV is presented. Curves and couplings obtained using R-matrix calculations are used to provide the input for molecular dynamics calculations based on the multichannel quantum defect theory. A full set of vibrationally resolved cross sections and rate coefficients is presented for the kinetic modeling of argon-containing non-equilibrium plasma.

Differential Measurement of Electron Ejection after Two-Photon Two-Electron Excitation of Helium.

We report the measurement of the photoelectron angular distribution of two-photon single-ionization near the 2p^{2} ^{1}D^{e} double-excitation resonance in helium, benchmarking the fundamental nonlinear interaction of two photons with two correlated electrons. This observation is enabled by the unique combination of intense extreme ultraviolet pulses, delivered at the high-repetition-rate free-electron laser in Hamburg (FLASH), ionizing a jet of cryogenically cooled helium atoms in a reaction microscope. The spectral structure of the intense self-amplified spontaneous emission free-electron laser pulses has been resolved on a single-shot level to allow for post selection of pulses, leading to an enhanced spectral resolution, and introducing a new experimental method. The measured angular distribution is directly compared to state-of-the-art theory based on multichannel quantum defect theory and the streamlined R-matrix method. These results and experimental methodology open a promising route for exploring fundamental interactions of few photons with few electrons in general.

Analyzing the Rydberg-based optical-metastable-ground architecture for Yb171 nuclear spins

Neutral alkaline earth(-like) atoms have recently been employed in atomic arrays with individual readout, control, and high-fidelity Rydberg-mediated entanglement. This emerging platform offers a wide range of new quantum science applications that leverage the unique properties of such atoms: ultra-narrow optical "clock" transitions and isolated nuclear spins. Specifically, these properties offer an optical qubit ("o") as well as ground ("g") and metastable ("m") nuclear spin qubits, all within a single atom. We consider experimentally realistic control of this "omg" architecture and its coupling to Rydberg states for entanglement generation, focusing specifically on ytterbium-171 ($^{171}\text{Yb}$) with nuclear spin $I = 1/2$. We analyze the $S$-series Rydberg states of $^{171}\text{Yb}$, described by the three spin-$1/2$ constituents (two electrons and the nucleus). We confirm that the $F = 3/2$ manifold -- a unique spin configuration -- is well suited for entangling nuclear spin qubits. Further, we analyze the $F = 1/2$ series -- described by two overlapping spin configurations -- using a multichannel quantum defect theory. We study the multilevel dynamics of the nuclear spin states when driving the clock or Rydberg transition with Rabi frequency $\Omega_\text{c} = 2 \pi \times 200$ kHz or $\Omega_\text{R} = 2 \pi \times 6$ MHz, respectively, finding that a modest magnetic field ($\approx200\,\text{G}$) and feasible laser polarization intensity purity ($\lesssim0.99$) are sufficient for gate fidelities exceeding 0.99.

Coherent Light Shift on Alkaline-Earth Rydberg Atoms from Isolated Core Excitation without Autoionization

New experimental quantum simulation platforms have recently been implemented with divalent atoms trapped in optical tweezer arrays, with promising performance. The second valence electron also brings about new prospects through the so-called isolated core excitation (ICE). However, autoionization presents a strong limitation to this use. In this study, we propose and demonstrate a new approach to applying a sizable light shift to a Rydberg state with close-to-resonant ICE while avoiding autoionization. In particular, we investigate the ICE of ytterbium atoms in 1S0 Rydberg states. Spectroscopic studies of the induced autoionization and the light shift imparted to the Rydberg states are well accounted for with multichannel quantum defect theory. Such control over the inner electron without disturbing the Rydberg electron brings about a new tool for the targeted coherent manipulation of Rydberg states in quantum simulation or quantum computing experiments performed with alkaline-earth atoms.Received 1 November 2021Revised 15 February 2022Accepted 24 March 2022DOI:https://doi.org/10.1103/PRXQuantum.3.020327Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasAtom & ion trapping & guidingAutoionization & Auger processesPhysical SystemsRydberg atoms & moleculesAtomic, Molecular & OpticalQuantum Information

Spectroscopy and scattering matrices with nitrogen atom: Rydberg states and optical oscillator strengths

The scattering matrices of e + N+ with Jπ = 1.5+ in discrete energy regions are calculated using the eigenchannel R-matrix method. We obtain good parameters of multichannel quantum defect theory (MQDT) that vary smoothly as the function of the energy resulting from the analytical continuation property of the scattering matrices. By employing the MQDT, all discrete energy levels for N could be calculated accurately without missing anyone. The MQDT parameters (i.e., scattering matrices) can be calibrated with the available precise spectroscopy values. In this work, the optical oscillator strengths for the transition between the ground state and Rydberg series are obtained, which provide rich data for the diagnostic analysis of plasma.

Spectroscopy of Rydberg states in erbium using electromagnetically induced transparency

We present a study of the Rydberg spectrum in $^{166}\mathrm{Er}$ for series connected to the $4{f}^{12}(^{3}H_{6})6s$, ${J}_{c}=13/2$, and ${J}_{c}=11/2$ ionic core states using an all-optical detection based on electromagnetically induced transparency in an effusive atomic beam. Identifying approximately 550 individual states, we find good agreement with a multichannel quantum defect theory (MQDT) which allows assignment of most states to $ns$ or $nd$ Rydberg series. We provide an improved accuracy for the lowest two ionization thresholds to ${E}_{\mathrm{IP},{J}_{c}=13/2}=49260.750(1)\phantom{\rule{0.28em}{0ex}}{\mathrm{cm}}^{\ensuremath{-}1}$ and ${E}_{\mathrm{IP},{J}_{c}=11/2}=49701.184(1)\phantom{\rule{0.28em}{0ex}}{\mathrm{cm}}^{\ensuremath{-}1}$ as well as the corresponding quantum defects for all observed series. We identify Rydberg states in five different isotopes, and states between the two lowest ionization thresholds. Our results open the way for future applications of Rydberg states for quantum simulation using erbium and exploiting its special open-shell structure.