State Multipoles Research Articles

Multipoles from Majorana constellations

Majorana stars, the 2S spin coherent states that are orthogonal to a spin-S state, offer an elegant method to visualize quantum states, disclosing their intrinsic symmetries. These states are naturally described by the corresponding multipoles. These quantities can be experimentally determined and allow for an SU(2)-invariant analysis. We investigate the relationship between Majorana constellations and state multipoles, thus providing insights into the underlying symmetries of the system. We illustrate our approach with some relevant and informative examples. Published by the American Physical Society 2024

Performing competitive analysis in QFD studies using state multipole moments and bootstrap sampling

ABSTRACTQFD often makes use of Likert scales to convert opinions into data for analysis. Likert scales are prone to bias in relation to their labeling, interpretation, and distributional assumptions. Analysis is often based on means, without regard to variation or distribution shape, making inferential statistical procedures unreliable. We present a method for extracting increased information from Likert data using State Multipole Moments, and use bootstrap sampling to enable the moments to be statistically compared. We demonstrate application using a published QFD competitor analysis, showing how this method can enable practitioners to perform statistical comparisons resulting in improved decision making.

Angular momentum polarization of the molecules in the Li + HF reaction

A theoretical analysis of the orientation and alignment of the rotational angular momenta of the reactants and products of the Li + HF(v r = 0, j r = 3) → LiF(v, j) + H reaction at a collision energy of E coll = 0.317 eV is performed. The polarization of the angular momentum of the molecules involved in the reaction is based on the technique of spherical tensor operators (state multipoles). Quantum-mechanical calculations of the S-matrix of the reaction are carried out using the wave packet method. In particular, the influence of the orientation of the angular momentum of the HF reactant on the differential cross section is examined. It is shown that the contribution to the differential cross section comes only from be reactants with an angular momentum perpendicular to the reaction plane. In addition, the angular dependence of the orientation and alignment of the angular momenta of the reaction products are examined. It is shown that, for an isotropic distribution of reactant molecules, the orientation of the angular momenta of the products differs from zero only in the direction perpendicular to the reaction plane. Different experimental geometries, based on radiation enhanced multiphoton ionization, are proposed to detect the predicted effects.

The Analysis of Likert Scales Using State Multipoles

Though ubiquitous, Likert scaling’s traditional mode of analysis is often unable to uncover all of the valid information in a data set. Here, the authors discuss a solution to this problem based on methodology developed by quantum physicists: the state multipole method. The authors demonstrate the relative ease and value of this method by examining college students’ endorsement of one possible cause of prejudice: segregation. Though the mean level of students’ endorsement did not differ among ethnic groups, an examination of state multipoles showed that African Americans had a level of polarization in their endorsement that was not reflected by Hispanics or European Americans. This result could not have been obtained with the traditional approach and demonstrates the new method’s utility for social science research.

Vector correlations in the F + HD reaction

A theoretical study of the orientation of product rotational angular momenta for two chemical reaction channels: F + HD(ν r = 0, j r = 0) → HF(ν, j) + D and F + HD(ν r = 0, j r = 0) → DF(ν, j) + H at a E coll = 78.54 meV collision energy was performed. Angular momentum orientation was described on the basis of irreducible tensor operators (state multipoles) expressed through anisotropy transfer coefficients, which contained quantum-mechanical scattering T matrices determined on the basis of exact solutions to quantum scattering equations obtained using the hyperquantization algorithm. The possibility of the existence of substantial orientation of the angular momentum of reaction products j in the direction perpendicular to the scattering plane was demonstrated. The dependences of differential reaction cross sections and state multi-poles on the ν and j quantum numbers were calculated and analyzed. A experimental scheme based on the multiphoton ionization method was suggested. The scheme can be used to detect predicted reaction product angular momentum orientation.

Vector Correlation Analysis for Inelastic and Reactive Collisions between Partners Possessing Spin and Orbital Angular Momentum

A general reactive collision of the type A + B --> C + D is considered where both the collision partners (A and B) or the products (C and D) may possess internal, i.e., spin, orbital or rotational, angular momenta. Compact expressions are derived using a rigorous quantum mechanical analysis for the angular momentum anisotropy of either of the products (C or D) arising from an initially polarized distribution of the reactant angular momentum. The angular momentum distribution of the product is expressed in terms of canonical spherical tensors multiplied by anisotropy-transforming coefficients c(K(i)q(k))(K)(K(r),L). These coefficients act as transformation coefficients between the angular momentum anisotropy of the reactants and that of the product. They are independent of scattering angle but depend on the details of the scattering dynamics. The relationship between the coefficients c(K(i)q(k))(K)(K(r),L) and the body-fixed scattering S matrix is given and the methodology for the quantum mechanical calculation of the anisotropy-transforming coefficients is clearly laid out. The anisotropy-transforming coefficients are amenable to direct experimental measurement in a similar manner to vector correlation and alignment parameters in photodissociation processes. A key aspect of the theory is the use of projections of both reactant and product angular momenta onto the product recoil vector direction. An important new conservation rule is revealed through the analysis, namely that if the state multipole for reactant angular momentum distribution has a projection q(k) onto the product recoil vector the state multipoles for the product angular momentum distribution all have this same projection. Expressions are also presented for the distribution of the product angular momentum when its components are evaluated relative to the space-fixed Z-axis. Notes with detailed derivations of all the formulas are available as Supporting Information.

Tensor cross sections and the collisional evolution of state multipoles: OH(XΠ2)–Ar

By means of a kinetic analysis, we show that the overall rate constant for the collisional loss of orientation or alignment of a rotational level is the sum of the rate constant for elastic depolarization and the sum of the rate constants for all rotationally inelastic transitions out of the level under consideration. An expression for the depolarization cross section is derived in terms of tensor cross sections, and the relationship of depolarization to m-resolved transitions is discussed. We use this formalism in simulations, based on high-quality ab initio potential energy surfaces, of the depolarization of the open-shell molecule OH(X (2)Pi) through collisions with Ar. Good agreement is seen with the results of the two-color polarization spectroscopy experiments of Paterson et al. [J. Chem. Phys. 129, 074304 (2008)]. In addition, we show that the major contribution to elastic collisional depolarization occurs not from weak, glancing collisions but from encounters which probe the inner wall of the potential energy surface.

Fluorescence of polarized atoms excited by polarized electrons

A general expression for the cross section describing properties of the fluorescence radiation from polarized atoms excited by polarized electrons is obtained in the distorted wave (DW) approximation. The expression is presented in the form of multiple expansion over the state multipoles of all particles taking part in the two-step process. It is used to derive the alignment parameters of excited atom and angular distribution of emitted fluorescence. The polarization degree of radiation from the lowest autoionizing states np5(n+1)s22P3/2 following the excitation of non-polarized Na and K atoms by non-polarized electrons is calculated.

Anisotropy-dependent circular polarization spectroscopy

Experimental and theoretical study of atomic collisions with neutral atoms, molecules, and ions is a key to understand the energy and charge transfer processes, depolarization cross sections, dynamical evolution of the state multipoles, and the accurate description of the intermediate levels of the collision complex 1‐5. Although the study of collision dynamics is not new, significant advances in techniques for laser cooling and trapping of atoms, molecules and ions has enormously increased the interest in this area of research 6,7. For example, collisions involving cold Rydberg atoms 8‐10 has been a great interest in many fields of research due to the interesting properties of these atoms in astrophysical and laboratory plasmas. Also, high-resolution photoassociation spectroscopy of ultracold molecules 11‐16 has become an important subject to accurately determine the potential curves of these molecules in atomic and molecular physics. The measurement of collisional depolarization cross section between an alkali metal and rare gas atom provides valuable information about the interaction between the two species such as relaxation rates of the electronic moments. Experimental investigations of collisional depolarization in 6p 2 P 3/2 level cesium atoms, up to now, were carried out using incoherent light sources as optical excitation 17,18. Guiry et al. 17 has experimentally studied the disorienta

The parity-adapted basis set in the formulation of the photofragment angular momentum polarization problem: The role of the Coriolis interaction

We present a theoretical framework for calculating the recoil-angle dependence of the photofragment angular momentum polarization taking into account both radial and Coriolis nonadiabatic interactions in the diatomic/linear photodissociating molecules. The parity-adapted representation of the total molecular wave function has been used throughout the paper. The obtained full quantum-mechanical expressions for the photofragment state multipoles have been simplified by using the semiclassical approximation in the high-J limit and then analyzed for the cases of direct photodissociation and slow predissociation in terms of the anisotropy parameters. In both cases, each anisotropy parameter can be presented as a linear combination of the generalized dynamical functions fK(q,q',q,q') of the rank K representing contribution from different dissociation mechanisms including possible radial and Coriolis nonadiabatic transitions, coherent effects, and the rotation of the recoil axis. In the absence of the Coriolis interactions, the obtained results are equivalent to the earlier published ones. The angle-recoil dependence of the photofragment state multipoles for an arbitrary photolysis reaction is derived. As shown, the polarization of the photofragments in the photolysis of a diatomic or a polyatomic molecule can be described in terms of the anisotropy parameters irrespective of the photodissociation mechanism.

Photofragment angular momentum distribution beyond the axial recoil approximation: Predissociation

We present the quantum mechanical expressions for the angular momentum distribution of the photofragments produced in slow predissociation. The paper is based on our recent theoretical treatment [J. Chem. Phys. 123, 034307 (2005)] of the recoil angle dependence of the photofragment multipole moments which explicitly treat the role of molecular axis rotation on the electronic angular momentum polarization of the fragments. The electronic wave function of the molecule was used in the adiabatic body frame representation. The rigorous expressions for the fragment state multipoles which have been explicitly derived from the scattering wave function formalism have been used for the case of slow predissociation where a molecule lives in the excited quasibound state much longer than a rotation period. Possible radial nonadiabatic interactions were taken into consideration. The optical excitation of a single rotational branch and the broadband incoherent excitation of all possible rotational branches have been analyzed in detail. The angular momentum polarization of the photofragments has been treated in the high-J limit. The polarization of the photofragment angular momenta predicted by the theory depends on photodissociation mechanism and can in many cases be significant.

Determination of ionization dynamical parameters by time-resolved photoelectron imaging

A theoretical formalism is developed for photoelectron angular distributions from an aligned ensemble of molecules observed by a time-resolved photoelectron imaging experiment (TRPEI) in the gas phase. Rotational motion is treated quantum mechanically, and photo-ionization in the perturbative regime is considered. We introduce state multipoles of the ionization laser field and express the photoelectron anisotropy parameters by a double expansion with molecular axis alignment parameters and state multipoles of the field. In order to analyse the feasibility of the full determination of the dynamical parameters and examine multiple solutions for the quadratic equations of the dynamical parameters, we introduce G and H matrices by decoupling the geometrical factors and the already known dynamical factors from the unknown dynamical factors. The usefulness of multichannel quantum defect theory in the analysis of TRPEI is described.

Electronic angular momentum polarizations of photofragments: A case study of ICN photodissociation from a perpendicular transition

A quantum treatment on ICN photodissociation from an initial perpendicular transition (Omega'=+/-1<--Omega"=0) to the asymptote CN(|Sigma+,J'M'N'1/2>)+I(2P3/2) is presented. Density matrices of both photofragments are derived and explicit expressions of the state multipoles in terms of the angular momentum coupling coefficients and the rotation-bending factors have been obtained. To perceive the physical origin of electronic angular momentum polarizations of the iodine photofragments, a correlation scheme which considers the magnetic dipolar and the electrostatic dipole-quadrupole interactions between I and CN cofragments is proposed. For ICN precursors in the vibrational ground state or in the equally populated l-type split levels, the alignment parameters of the iodine photofragments in the molecular frame can be calculated according to this long-range interaction model. For the perpendicular transition |1Pi1><--|1Sigma0+>, its alignment parameters of I(2P3/2) from the incoherent and coherent transitions to the |Omega'=1> and |Omega'=-1> components are rho(0)2(1Pi1)=0.756 and rho2(2)(1Pi1)=-0.656, respectively. For the perpendicular transition to |3Pi1>, rho(0)2(3Pi1)=-0.878 and rho2(2)(3Pi1)=0.328 are from the incoherent transition, whereas rho(0)2(3Pi1)=0.122 and rho2(2)(3Pi1)=0.328 are from the coherent transition. To analyze the photoion images of iodine photofragments, angular distributions of I+ from the 2+1 resonance-enhanced multiphoton ionization detection scheme are derived.

Quantum Theory of ICN Photodissociation: Density Matrices of Photofragments from a Parallel Transition

AbstractA quantum treatment on ICN photodissociation from an initial parallel transition (Ω′ = 0 ← Ω″ = 0) to 1 the asymptote CN(|)+ I(2P1/2) is presented. Density matrices of both photofragments are derived, and explicit expressions of the state multipoles in terms of the angular momentum coupling coefficients and the rotation‐bending factors have been obtained. The present theoretical framework provides a foundation to study photofragments with non‐null electronic and/or spin angular momenta. To investigate the angular momentum polarizations phenomena, these density matrices can play a prime role in laser‐based detections of state‐selected photofragments.

Photofragment angular momentum distribution beyond the axial recoil approximation: The role of molecular axis rotation

We present the quantum-mechanical expressions for the recoil angle dependence of the photofragment multipole moments which explicitly treat the role of molecular axis rotation on the electronic angular momentum polarization of the fragments. The paper generalizes the result of Siebbeles et al. [J. Chem. Phys. 100, 3610 (1994)] to the case of dissociation of rotating molecules. The electronic wave function of the molecule was used in the adiabatic body-frame representation. The obtained rigorous expressions for the fragment state multipoles have been explicitly derived from the scattering wave-function formalism and then simplified using the quasiclassical approximation in the high-J limit. Possible radial and Coriolis nonadiabatic interactions have been taken into consideration. It is shown that the rotation of the molecular axis is described by a number of rotation factors which depend on the rank of the incident-photon polarization matrix, on the dissociation mechanism, and on the classical angle of rotation of the molecular axis gamma.

Relaxation of atomic state multipoles by radiation re-absorption: Neon [formula omitted] atoms in a discharge plasma

In a positive column of a glow discharge in the magnetic field of 36.4 G, a linearly polarized laser pulse or a circularly polarized laser pulse has produced polarized neon atoms (alignment or orientation) in the 2 p 2 (Paschen notation) level from the 1 s 3 level. The subsequent fluorescence to the 1 s 2 level was observed with its polarized components resolved. Depopulation, disorientation and disalignment rates of the 2 p 2 atom were measured and their discharge current dependences were examined for a discharge current from 0.4 to 2.0 mA. The degrees of radiation re-absorption, or the optical thickness, of the transition lines from the 2 p 2 level to the 1 s 2 - 1 s 5 levels were measured as functions of the discharge current. A Monte Carlo simulation was performed by which the depopulation, disorientation and disalignment rates by the radiation re-absorption for these transitions were determined. The calculated rates were compared with the observed ones and found to reproduce the their discharge current dependences. D’Yankonov and Perel's analytical expression for these rates was quantified from comparison with the Monte Carlo results.

Study on the orientation parameter of excited sodium scattered by electrons

The orientation parameter is used to describe the collision dynamic process and the structure informations of excited atomic state in electron scattering. In density matrix theory, the state multipoles express the structure of atom assembles impacted by particles, and the state multipole of K=1 expresses the orientation vector. In the sysmetry scattering plane system, the orientation vector is denoted by the orientation parameter L⊥, which is calculated from scattering amplititutes, C-G coefficient and 6-j symbol. The orientation parameters of sodium S→P excited by low energy electron are calculated in this paper, and the results are consistent with that obtained from electron-photon coincidence scattering experiment from 2eV—100eV incident energy. The characteristics of orientation angular-distribution vs incident energy are also discussed.

Orbital alignment in N2O photodissociation. I. Determination of all even rank anisotropy parameters.

We present a general method for determination of the photofragment K=4 state multipoles in an ion imaging experiment. These multipoles are important for determining the full density matrix for any photofragment with j(a)> or =2. They are expressed in terms of laboratory frame anisotropy parameters that have distinct physical origins and possess characteristic angular distributions. The explicit expression for the (2+1) resonant multiphoton ionization absorption signal for the case of arbitrarily polarized probe light is derived and a procedure for isolation of the rank-4 state multipoles from all others is shown. This treatment is applied to the case of O((1)D) produced in the 193 nm photodissociation of N2O. The results show nonzero values for all K=4 anisotropy parameters, indicating the complexity of the photodissociation dynamics in this system.

Cascade effects in the excitation ofnp5(n+1)pstates of krypton and xenon atoms by polarized electrons

The integrated Stokes parameters of the decay photons from the excited np{sup 5}(n+1)p[5/2]{sub 3}, np{sup 5}(n+1)p{sup '}[3/2]{sub 2}, and np{sup 5}(n+1)p{sup '}[3/2]{sub 1} (n=5 and 6) states of Kr and Xe, respectively, are measured from near threshold to 100 eV using incident polarized electrons. The effects of the cascade photons on the Stokes parameters have been investigated using a pulsed beam of polarized electrons. The cascade effects for the final excited states are more significant for the linear polarization P{sub 1} parameter than for the circular polarization P{sub 3} parameter and they do not always have depolarization effects. The integrated state multipoles, determined from the integrated Stokes parameters corrected for cascade effects, are closer to, but not in agreement with, the values from semirelativistic R matrix and relativistic distorted-wave approximation methods.

Using ion imaging to analyze higher-order moments of the angular distributions of photofragments

We investigate the effect of the hexadecapole (K=4) polarization moment on the spatial distributions of angular momenta for atoms produced during the photodissociation of diatomic or triatomic molecules by polarized radiation. We derive general expressions for the angular distributions of the atomic density matrix for K = 2, 4 and expressions for the corresponding anisotropy parameters that contain all information on the photodissociation dynamics. We show that these anisotropy parameters can be experimentally determined by using ion imaging. We consider oxygen atoms in the 1 D 2 state aligned with respect to the orbital angular momentum as an example and provide ion images of the signals that correspond to the population of the atomic magnetic sublevels ¦m¦ = 0, 1, 2. We show the contributions from the second-and fourth-rank state multipoles to the angular distributions of the atomic density matrix to be comparable in magnitude and significantly different in form.