Stark Maps Research Articles

Cesium Rydberg-state ionization study by three-dimensional ion-electron correlation: Toward a monochromatic electron source

We study the excitation and ionization of cesium Rydberg states in an electric field ($\ensuremath{\approx}\phantom{\rule{0.16em}{0ex}}2200\phantom{\rule{4pt}{0ex}}\mathrm{V}/\mathrm{cm}$) near the classical field-ionization threshold ($\ensuremath{\approx}\ensuremath{-}280\phantom{\rule{4pt}{0ex}}{\mathrm{cm}}^{\ensuremath{-}1}$ binding energy) by three-dimensional (3D) ion-electron coincidence spectroscopy. Cesium atoms are produced by an effusive oven and excited with lasers to Stark-shifted Rydberg states or directly ionized. Using a double time-of-flight setup we record 3D ($X, Y$, time of flight) coincidence imaging of electrons and ions. Above-threshold photoionization creates broad images with poor electron-ion spatial correlation. Fast ionizing states produce very good correlations and the images reveal the electric-field map of the ionization region. Slow ionizing states show that the relatively high atomic velocity is detrimental to the correlations. Experimental data are accurately reproduced by detailed Monte Carlo excitation and ionization simulations based on Stark maps obtained with local-frame transformation (LFT) theory. Agreement between our spectroscopic experiment and LFT theory is very good, with better that hundreds of megahertz accuracy. But, on rare particular states, several gigahertz discrepancy is found. This study can be used to select appropriate states for the creation of ion and electron beams with high brightness, good correlation, and low energy dispersion.

ARC: An open-source library for calculating properties of alkali Rydberg atoms

We present an object-oriented Python library for the computation of properties of highly-excited Rydberg states of alkali atoms. These include single-body effects such as dipole matrix elements, excited-state lifetimes (radiative and black-body limited) and Stark maps of atoms in external electric fields, as well as two-atom interaction potentials accounting for dipole and quadrupole coupling effects valid at both long and short range for arbitrary placement of the atomic dipoles. The package is cross-referenced to precise measurements of atomic energy levels and features extensive documentation to facilitate rapid upgrade or expansion by users. This library has direct application in the field of quantum information and quantum optics which exploit the strong Rydberg dipolar interactions for two-qubit gates, robust atom-light interfaces and simulating quantum many-body physics, as well as the field of metrology using Rydberg atoms as precise microwave electrometers. Program summaryProgram Title: ARC: Alkali Rydberg CalculatorProgram Files doi:http://dx.doi.org/10.17632/hm5n8w628c.1Licensing provisions: BSD-3-ClauseProgramming language: Python 2.7 or 3.5, with C extensionExternal Routines: NumPy [1], SciPy [1], Matplotlib [2]Nature of problem: Calculating atomic properties of alkali atoms including lifetimes, energies, Stark shifts and dipole–dipole interaction strengths using matrix elements evaluated from radial wavefunctions.Solution method: Numerical integration of radial Schrödinger equation to obtain atomic wavefunctions, which are then used to evaluate dipole matrix elements. Properties are calculated using second order perturbation theory or exact diagonalisation of the interaction Hamiltonian, yielding results valid even at large external fields or small interatomic separation.Restrictions: External electric field fixed to be parallel to quantisation axis.Supplementary material: Detailed documentation (.html), and Jupyter notebook with examples and benchmarking runs (.html and .ipynb).[1] T.E. Oliphant, Comput. Sci. Eng. 9, 10 (2007). http://www.scipy.org/.[2] J.D. Hunter, Comput. Sci. Eng. 9, 90 (2007). http://matplotlib.org/.

Quantum chaos and statistical properties of energy levels for atom in parallel electric and magnetic fields

In this paper, the statistical properties of energy levels are studied numerically for atom in parallel electric and magnetic fields, which is an ideal system to examine the contributions of external fields and ionic core to quantum chaos. The Stark maps of diamagnetic spectra and nearest neighbor spacing (NNS) distributions are obtained by diagonalization method incorporating core effect. We identify obvious level anti-crossing and large value of [Formula: see text] for barium, indicating that core effect has predominant contribution to chaotic dynamics in barium. To study the core effect in detail, we sweep the quantum defect artificially and find that larger core effect will undoubtedly induce stronger chaotic dynamics.

Atom-interferometric measurement of Stark level splittings

Multiple adiabatic/diabatic passages through avoided crossings in the Stark map of cesium Rydberg atoms are employed as beam splitters and recombiners in an atom-interferometric measurement of energy-level splittings. We subject cold cesium atoms to laser-excitation, electric-field and detection sequences that constitute an (internal-state) atom interferometer. For the read-out of the interferometer we utilize state-dependent collisions, which selectively remove atoms of one kind from the detected signal. We investigate the dependence of the interferometric signal on timing and field parameters, and find good agreement with time-dependent quantum simulations of the interferometer. Fourier analysis of the interferometric signals yield coherence frequencies that agree with corresponding energy-level differences in calculated Stark maps. The method enables spectroscopy of states that are inaccessible to direct laser-spectroscopic observation, due to selection rules, and has applications in field metrology.

Multichannel quantum defect theory of strontium bound Rydberg states

Using the reactance matrix approach, we systematically develop new multichannel quantum defect theory (MQDT) models for the singlet and triplet S, P, D and F states of strontium below the first ionization limit, based on improved energy level measurements. The new models reveal additional insights into the character of doubly excited perturber states, and the improved energy level measurements for certain series allow fine structure to be resolved for those series’ perturbers. Comparison between the predictions of the new models and those of previous empirical and ab initio studies reveals good agreement with most series; however, some discrepancies are highlighted. Using the MQDT wave functions derived from our models we calculate other observables such as Landé -factors and radiative lifetimes. The analysis reveals the impact of perturbers on the Rydberg state properties of divalent atoms, highlighting the importance of including two-electron effects in the calculations of these properties. The work enables future investigations of properties such as Stark maps and long-range interactions of Rydberg states of strontium.

Core effect in the Stark spectrum of highly excited barium

We have studied the Stark spectrum of highly excited barium both experimentally and theoretically. The observed spectra are well explained by diagonalization of the Hamiltonian incorporating core effects. Core effects were studied by examining the Stark maps of barium as well as by comparing Stark spectra of hydrogen and barium directly. An anti-crossing is revealed by the Stark maps. The core-induced interaction between the neighboring channels of p state results in level anti-crossing, providing an efficient way of populating the neighboring states inaccessible at free field.

Stark spectrum of barium in highly excited Rydberg states

We present observations of Stark spectra of barium in highly excited Rydberg states in the energy region around n = 35. The one-photon excitation concerns the π transition. The observed Stark spectra at electric fields ranging from 0 to 60 V·cm−1 are well explained by the diagonalization of the Hamiltonian incorporating the core effects. From the Stark maps, the anti-crossings between energy levels are identified experimentally and theoretically. The time of flight spectra at the specified Stark states are recorded, where the deceleration and acceleration of barium atoms are observed. This is very consistent with the prediction derived from the Stark maps from the point of view of energy conservation.

Spectroscopy of a cold strontium Rydberg gas

We present a study of a cold strontium Rydberg gas. The narrowband laser excitation of Rydberg states in the range n = 20–80 from a 6 mK cloud of strontium atoms is detected using the spontaneous ionization of the Rydberg atoms. Using a high-resolution step-scanning technique, we perform detailed measurements of the Stark maps of selected Rydberg states. We find excellent agreement between the measured Stark maps and a numerical calculation based on an independent electron model. Finally we show that excitation of the second valence electron can be used to probe the dynamics of the Rydberg gas with nanosecond temporal resolution via autoionization.

Trapping cold molecular hydrogen

Translationally cold H(2) molecules excited to non-penetrating |M(J)| = 3 Rydberg states of principal quantum number in the range 21-37 have been decelerated and trapped using time-dependent inhomogeneous electric fields. The |M(J)| = 3 Rydberg states were prepared from the X (1)Σ(+)(u)(v = 0, J = 0) ground state using a resonant three-photon excitation sequence via the B (1)Σ(+)(u)(v = 3, J = 1) and I (1)Π(g) (v = 0, J = 2) intermediate states and circularly polarized laser radiation. The circular polarization of the vacuum ultraviolet radiation used for the B ← X transition was generated by resonance-enhanced four-wave mixing in xenon and the degree of circular polarization was determined to be 96%. To analyse the deceleration and trapping experiments, the Stark effect in Rydberg states of molecular hydrogen was calculated using a matrix diagonalization procedure similar to that presented by Yamakita et al., J. Chem. Phys., 2004, 121, 1419. Particular attention was given to the prediction of zero-field positions of low-l states and of avoided crossings between Rydberg-Stark states with different values of |M(J)|. The calculated Stark maps and probabilities for diabatic traversal of the avoided crossings were used as input to Monte-Carlo particle-trajectory simulations. These simulations provide a quantitatively satisfactory description of the experimental data and demonstrate that particle loss caused by adiabatic traversals of avoided crossings between adjacent |M(J)| = 3 Stark states of H(2) is small at principal quantum numbers beyond n = 25. The main source of trap losses was found to be from collisional processes. Predissociation following the absorption of blackbody radiation is estimated to be the second most important trap-loss mechanism at room temperature, and trap loss by spontaneous emission is negligible under our experimental conditions.

Study of Rydberg states of lithium in a static electric field using B-spline functions

The B-spline expansion technique and the method of model potential of atom have been applied to study the properties of Stark states of lithium. Our results for Stark maps, oscillator strength distributions, avoided crossing positions and widths for Li are in good agreement with other reported theoretical calculations and experimental measurement. It shows that the method we used can be applied to investigate the properties of alkali-metal atoms in external static electric field. Finally, for the first time, we obtained the evolution of the oscillator strength spectrum of Li with varying static electric field.

Molecular quantum defect orbital study of the Stark effect in theH3molecule

Stark maps and the distribution of oscillator strengths within a Stark manifold in the ${\mathrm{H}}_{3}$ molecule have been calculated by diagonalization of the Hamiltonian matrix using a molecular quantum defect orbital (MQDO) basis set. Our perturbative treatment, in which only experimental quantum defects have been used and no fitting parameters are required, explicitly accounts for the coupling between the rotation of the molecular core and the Rydberg-electron motion. A pure Hund's case (d) has been adopted for all the Rydberg states except for the $np$ series, for which Hund's case (b) is considered to be more realistic. A mixing of some of the $np$ Stark states with those belonging to other Rydberg series appears to occur. The presently achieved results conform very well with other reliable theoretical values, in the absence of experimental data. The adequacy of the MQDO procedure for accurately dealing with properties related to the Stark effect in molecules is suggested.

The Stark effect in atomic Rydberg states through a quantum defect approach

AbstractA basis set of quantum defect orbitals (QDOs) has been adopted for the diagonalization of the Hamiltonian matrix of nonhydrogenic atoms in the presence of an external electric field, so that the Stark structure of the Rydberg states has been possible to determine. The presently obtained Stark maps are in excellent agreement with those resulting from theory and experiment, as reported in the literature for a few representative atoms. The adequacy of the Stark quantum defect orbital (SQDO) procedure for accurately dealing with properties related to the Stark effect in atoms is suggested. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005

Stark maps and Rydberg transitions in the presence of an electric field for Li, Na, and K. A quantum defect orbital approach

The Stark structure of the Rydberg states of the lighter alkali atoms, Li, Na, and K, and the Stark oscillator strength distribution for Li have been calculated by diagonalization of the quantum defect orbital (QDO) Hamiltonian matrix. The presently obtained Stark maps are in excellent agreement with those resulting from theory and experiment, as reported in the literature by other authors. Good accord is also found for the presently calculated oscillator strength distribution, within a Stark manifold, with that of other theoretical approach and experimental measurements. The adequacy of the QDO procedure for accurately dealing with properties related to the Stark effect in atoms is suggested.

Stark structure of the Rydberg states of alkaline-earth atoms

The Stark effects of the Rydberg states in the alkaline-earth atoms are studied theoretically. Using a method similar to the treatment of alkali atoms, the properties of the Stark states of Mg, Ca, Sr and Ba atoms in the regions far away from the perturbers are investigated. The Stark maps for Mg (n = 16, M = 0), Ca (n = 10, M = 0), Sr (n = 12, M = 0) and Ba (n = 13, |M| = 0, 1) are presented. Topics such as the general methods of calculation, the treatment of fine structure and the structure of level anti-crossings are discussed. The comparison between the theoretical and experimental Stark maps is satisfactory.

Strontium spectra in crossed electric and magneticfields

We report new experiments on the spectrum of Sr atoms in crossed electric and magnetic fields. We apply an experimental geometry which enables the two circular polarizations to be detected and recorded separately, thereby considerably simplifying the observed structure. This experimental procedure allows Stark maps for the diamagnetic structure to be constructed. We compare the data for Sr with analogous data reported recently for Ba. We also present theoretical calculations of the crossed-field spectrum, which agree well with our observations.

Discrete-time picture for multiphoton microwave spectroscopy

The time evolution of atomic and molecular systems driven by periodic external fields is reformulated to combine Floquet's theorem with time-dependent representations of the atomic degrees of freedom. This approach increases the flexibility of Floquet calculations of multiphoton processes in strong radiation fields, particularly at low frequencies, by incorporating adiabatic states into the basis. The depopulation of Rydberg states of He and Na in strong microwave fields illustrates the utility of the method. In particular, a study of the depopulation of the 11s state of Na in an oscillatory field reveals a strong correspondence between dc-field Stark maps and quasienergy spectra and predicts a resonance feature associated with Landau-Zener transitions between dc Stark levels.

Atomic supersymmetry and the Stark effect.

A search is conducted for physical quantum-mechanical supersymmetries involving the hydrogen atom. In all coordinate systems admitting a direct separation of the Schr\"odinger equation, the structure of the separated equations is examined for possible supersymmetric extensions. In addition to the known supersymmetry involving the radial equation for spherical coordinates, we uncover a related supersymmetry involving the radial equation for conical coordinates and a pair of supersymmetries involving parabolic coordinates. The associated spectra and possible physical import of the latter are discussed. They connect certain eigenfunctions of the hydrogen and lithium atoms in the unbroken-symmetry limit. Following the established procedure for the case of spherical coordinates, the breaking of these parabolic supersymmetries is incorporated in a model constructed using notions of quantum-defect theory. The model yields analytical wave functions in parabolic coordinates for the valence electron of alkali-metal atoms, while correctly reproducing the eigenvalue spectra. These ideas are applied to the study of the Stark effect in alkali-metal atoms. Using supersymmetry-based quantum-defect eigenfunctions, we obtain Stark maps for lithium and sodium. The spherical case shows striking agreement with experiment.

Stark structure of the Rydberg states of alkali-metal atoms

The authors describe practical methods for calculating the Stark structure of Rydberg states of the alkali metals based on diagnolization of the energy matrix. A survey of Stark structures is presented for all of the alkali metals in the vicinity of $n=15$. Topics discussed include general methods for evaluating radial matrix elements, the treatment of fine structure, oscillator-strength distribution, scaling laws, the structure of a level anticrossing, and sources of error. Experimental Stark maps are compared with calculated results for lithium and cesium. Experimental studies of the oscillator-strength distribution within a Stark manifold and the structure of a level anticrossing are also presented.

Further observations on the site of bone prospective areas in the chick embryo wing bud.

The portion of the pre-axial region of the wing bud anterior to the cranial boundary of the skeletogenous territory of the stylo-zeugopod, according to Stark and Searls' recent maps, was surgically isolated in chicken embryos of the stages 18-25, and autoplastically grafted to the dorsal surface of the hind-limb bud or to the trunk. In other embryos of the same stages the cranial half or the cranial two thirds of the pre-axial region mentioned were isolated and heterotopically implanted. The experimental results consistently showed that the isolate contained a variously large portion of the presumptive radius and a significant part of the proximal half of the prospective humerus. In fact, the skeletal parts mentioned did not develop in situ in the donor wing while they differentiated, in general, in the implantation site. In this regard, the present research confirms data from previous experiments showing that Stark and Searls' maps do not offer an accurate representation of the position and the cranio-caudal width of the presumptive areas of the skeletal pieces of the wing. Our findings seem also to indicate that the major (or longitudinal) axis of the future radius is arranged along the proximodistal axis of the wing bud throughout the developmental stages mentioned; instead, the major axis of the future humerus undergoes a gradual shift, being first (stages 18-20) arranged nearly parallel to the bud base and getting more and more parallel to the proximo-distal axis of the wing bud through stages 21-24. Besides, between stage 18 and 25 the cranio-caudal width of the presumptive skeletogenous territory of the zeugopod, and to a lesser extent of the stylopod, seems to undergo a slight relative reduction with respect to the cranio-caudal thickness of the whole wing bud; this might depend on a process of aggregation of the prospective skeletogenous cells into gradually denser and more defined precartilaginous blastemes.