Angular Momentum Projection Research Articles

Multiplicity, Parity and Angular Momentum of a Cooper Pair in Unconventional Superconductors of D4h Symmetry: Sr2RuO4 and Fe-Pnictide Materials

Sr2RuO4 and Fe-pnictide superconductors belong to the same point group symmetry D4h. Many experimental data confirm odd pairs in Sr2RuO4 and even pairs in Fe-pnictides, but opposite conclusions also exist. Recent NMR results of Pustogow et al., which revealed even Cooper pairs in Sr2RuO4, require reconsideration of symmetry treatment of its SOP (superconducting order parameter). In the present work making use of the Mackey–Bradley theorem on symmetrized squares, a group theoretical investigation of possible pairing states in D4h symmetry is performed. It is obtained for I4/mmm , i.e., space group of Sr2RuO4, that triplet pairs with even spatial parts are possible in kz direction and in points M and Y. For the two latter cases pairing of equivalent electrons with nonzero total momentum is proposed. In P4/nmm space group of Fe- pnictides in point M, even and odd pairs are possible for singlet and triplet cases. It it shown that even and odd chiral states with angular momentum projection m=±1 have nodes in vertical planes, but Eg is nodal , whereas Eu is nodeless in the basal plane. It is also shown that the widely accepted assertion that the parity of angular momentum value is directly connected with the spatial parity of a pair is not valid in a space-group approach to the wavefunction of a Cooper pair.

Wave function of a photon produced in the resonant scattering of twisted light by relativistic ions

We present a theoretical investigation of the resonant elastic scattering of twisted light, carrying angular momentum, by partially stripped ions. Special emphasis is placed on a question of whether the scattered radiation is also twisted. In order to investigate such an "angular momentum transfer", we develop an approach that allows us to find a quantum state of the final photon without projecting it onto a detector state. A general expression for this so--called \textit{evolved} state of outgoing radiation is derived and it can be used to analyze the resonant scattering by any ion, independently of its shell structure. Here, we illustrate our approach with the strong electric dipole $n S_{0} \to n' P_{1} \to n S_{0}$ transitions, which play an important role for the Gamma Factory project at CERN. For the incident Bessel light, the scattered radiation is shown to be in a superposition of twisted modes with the projections of the total angular momentum $m_f = 0, \pm 1$. The larger values of $m_f$ can be efficiently generated by inducing transitions of higher multiplicity. This angular momentum transfer, together with a remarkable cross section that is many orders of magnitude larger than that of the backward Compton scattering, makes the resonant photon scattering an effective tool for the production of twisted x-- and even gamma--rays at the Gamma Factory facility.

Generation of hard twisted photons by charged particles in cholesteric liquid crystals

We study the radiation from charged particles crossing a cholesteric plate in the shortwave approximation when the wavelength of photons is much smaller than the pitch of the cholesteric helix, whereas the escaping angle of the photon and the anisotropy of the permittivity tensor can be arbitrary. The radiation of photons is treated in the framework of quantum electrodynamics with classical currents. The radiation of the plane-wave photons and the photons with definite projection of the angular momentum (the twisted photons) produced by charged particles crossing the cholesteric plate and moving rectilinearly and uniformly is considered. The explicit expressions for the average number of radiated photons and their spectra with respect to the energy and the projection of the angular momentum are obtained in this case. It turns out that in the paraxial approximation the projection of the orbital angular momentum, l, of radiated twisted photons is related to the harmonic number n∈Z as l=2n+1, i.e., the given system is a pure source of twisted photons as expected. It is shown that in the paraxial shortwave regime the main part of radiated photons is linearly polarized with l=±1 at the harmonics n={-1,0}. The applicability conditions of the approach developed are discussed. As the examples, we consider the production of 6.3eV twisted photons from uranium nuclei and the production of x-ray twisted photons from 120 MeV electrons.

Radiation of Twisted Photons in Elliptic Undulators

The explicit expressions for the average number of twisted photons radiated by a charged particle in an elliptical undulator in the classical approximation as well as in the approach accounting for the quantum recoil are obtained. It is shown that radiation emitted by a particle moving along an elliptical helix which evolves around the axis specifying the angular momentum of twisted photons obeys the selection rule: $m+n$ is an even number, where $m$ is a projection of the total angular momentum of a twisted photon and $n$ is the harmonic number of the undulator radiation. This selection rule is a generalization of the previously known selection rules for radiation of twisted photons by circular and planar undulators and it holds for both classical and quantum approaches. The class of trajectories of charged particles that produce the twisted photon radiation obeying the aforementioned selection rule is described.

Role of triaxiality in deformed halo nuclei

It is known that nuclear deformation plays an important role in inducing the halo structure in neutron-rich nuclei by mixing several angular momentum components. While previous theoretical studies on this problem in the literature assume axially symmetric deformation, we here consider non-axially symmetric deformations. With triaxial deformation, the $\Omega$ quantum number is admixed in a single-particle wave function, where $\Omega$ is the projection of the single-particle angular momentum on the symmetric axis, and the halo structure may arise even when it is absent with the axially symmetric deformation. In this way, the area of halo nuclei may be extended when triaxial deformation is considered. We demonstrate this idea using a deformed Woods-Saxon potential for nuclei with neutron number N=13 and 43.

On the rules for filling electron shells and the properties of the atomic Hamiltonian

The aim of this work is to determine the dependence of the total energy of the atomic system on the mutual orientation of the orbital angular moments of electrons. It is shown that the minimum value of the energy is attained for those configurations that are an eigenfunction of the orbital angular momentum operator . Thus, at least for individual atoms of the periodic table it is shown that Dirac’s postulate on the necessity of a wave function to be an eigenfunction of is a consequence that follows from the properties of the Hamiltonian, and it is not a complementary condition. Additionally, it is shown that the second Hund’s rule is incorrect with respect to the mutual orientation of the electron orbital angular momentum projections.

Rotating deformed halo nuclei and shape decoupling effects

To explore the rotational excitation of deformed halo nuclei, the angular momentum projection (AMP) has been implemented in the deformed relativistic Hartree-Bogoliubov theory in continuum (DRHBc), in which both the mean field and collective wave functions are expanded in terms of Dirac Woods-Saxon basis. The DRHBc + AMP approach self-consistently describes the coupling between single particle bound states and the continuum not only in the ground state but also in rotational states. The rotational modes of deformed halos in 42,44Mg are investigated by studying properties of rotational states such as the excitation energy, configuration, and density distribution. Our study demonstrates that the deformed halo structure persists from the ground state in the intrinsic frame to collective states. Especially, the typical behavior of shape decoupling effects in rotating deformed halo nuclei is revealed.

Axial shape asymmetry and high-spin states in nuclei with Z=100 suggested by the projected total energy surface approach

The axial-shape asymmetry of yrast states in $^{246--256}\mathrm{Fm}$ is studied by performing the projected total-energy surface (PTES) calculations, which consider the beyond-mean-field effects associated with the restoration of rotational symmetry and shape variation at the same time. The results show a large elongation deformation but also a considerable large triaxiality for their ground and high spin states, the triaxial deformation $\ensuremath{\gamma}\ensuremath{\approx}{11}^{\ensuremath{\circ}}$ in average. In comparison, the TRS calculations have also been performed for these nuclei, and the results show a well-established axial quadrupole shape in their ground states. The presence of the significant triaxial deformation can be attributed to the beyond-mean-field effects generated by the angular-momentum projection. The axial asymmetric shape for the yrast states of nuclei with $Z=100$, suggested by the present variation after projection (VAP) calculations, indicates that the triaxial degree of freedom may also play a significant role in other transfermium and even superheavy nuclei. The present PTES calculations have well reproduced the available experimental energies of the ground-band states and predict the rest yrast states up to spin 30 in each nucleus. The calculated yrast bands of $^{246--256}\mathrm{Fm}$ present the back bending phenomenon at about the state ${18}^{+}$, caused by the alignment excitations of the two quasiparticle neutrons of $\ensuremath{\nu}{j}_{15/2}[743]7/2$ or of $\ensuremath{\nu}{h}_{11/2}[761]1/2$. It is worth confirming the predicted band structures by the future spectroscopic experiments in the transfermium nuclei for the study of the single-particle structure in the superheavy mass region.

Dynamics of a spin qubit in an optical dipole trap

We present a theoretical investigation of coherent dynamics of a spin qubit encoded in hyperfine sublevels of an alkali-metal atom in a far off-resonant optical dipole trap. The qubit is prepared in the "clock transition" utilizing the Zeeman states with zero projection of the spin angular momentum. We focus on various dephasing processes such as the residual motion of the atom, fluctuations of the trapping field and its incoherent scattering and their effects on the qubit dynamics. We implement the most general fully-quantum treatment of the atomic motion, so our results remain valid in the limit of close-to-ground-state cooling with low number of vibrational excitations. We support our results by comparison with an experiment showing reasonable correspondence with no fitting parameters.

Universal stereodynamics of cold atom-molecule collisions in electric fields

We use numerically exact quantum dynamics calculations to demonstrate universal stereoselectivity of cold collisions of $^2\Pi$ molecules with $^1$S-state atoms in an external electric field. We show that cold collisions of OH molecules in their low-field-seeking $f$ states, whose dipole moments are oriented against the field direction, are much more likely to lead to inelastic scattering than those of molecules oriented along the field direction, causing nearly perfect steric asymmetry in the inelastic collision cross sections. The universal nature of this effect is due to the threshold suppression of inelastic scattering between the degenerate $\pm M$ Stark sublevels of the high-field-seeking $e$-state, where $M$ is the projection of the total angular momentum of the molecule on the field axis. Above the $\Lambda$-doublet threshold, the stereodynamics of inelastic atom-molecule collisions can be tuned via electric-field-induced resonances, which enable effective control of Ne + OH scattering over the range of collision energies achievable in current merged beam experiments.

Controlling rotation in the molecular frame with an optical centrifuge

We computationally demonstrate a new method for coherently controlling the rotation-axis direction in asymmetric top molecules with an optical centrifuge. Appropriately chosen electric-field strengths and the centrifuge's acceleration rate allow to generate a nearly arbitrary rotational wavepacket. For D$_2$S and 2H-imidazole (C$_3$H$_4$N$_2$) we created wavepackets at large values of the rotational quantum number $J$ with the desired projections of the total angular momentum onto two of the molecules' principal axes of inertia. One application of the new method is three-dimensional alignment with a molecular axis aligned along the laser's wave vector, which is important for the three-dimensional imaging of molecules yet not accessible in standard approaches. The simultaneous orientation of the angular momentum in the laboratory frame and in the molecular frame could also be used in robust control of scattering experiments.

A transverse energy flow at the tight focus of light with higher-order circular-azimuthal polarization

Tight focusing of light with mth-order circular-azimuthal polarization was investigated. This is a new type of inhomogeneous hybrid polarization that combines the properties of mth order cylindrical polarization and circular polarization. Using the Richards-Wolf formalism, we obtained analytical expressions in the focal spot for the projections of the electric and magnetic field, the intensity distribution, the projections of the Poynting vector, and the spin angular momentum. It was shown theoretically and numerically that at the focus, the intensity has 2(m+1) local maxima located on a circle centered on an on-axis intensity null. It was shown that 4m vortices of a transverse energy flow were produced at the focus, with their centers located between the local intensity maxima. It was also shown that in the focal plane, the transverse energy flow changes the direction of rotation 2(2m+1) times around the optical axis. It is interesting that the longitudinal projection of the spin angular momentum at the focus changes sign 4m times. In those areas of the focal plane where the transverse energy flow rotates counterclockwise, the longitudinal projection of the spin angular momentum is positive, and the polarization vector rotates counterclockwise in the focal plane. Conversely, if the energy flow rotates clockwise, the polarization vector rotates clockwise, and the longitudinal projection of the spin angular momentum is negative. Numerical simulations are in agreement with the theoretical investigation.

Elastic scattering of twisted neutrons by nuclei

We present a theoretical formalism for scattering of the twisted neutrons by nuclei in a kinematic regime where interference between Coulomb interaction and the strong interaction is essential. Twisted neutrons have definite quantized values of an angular momentum projection along the direction of propagation, and we show that it results in novel observable effects for the scattering cross section, spin asymmetries and polarization of the scattered neutrons. We demonstrate that additional capabilities provided by beam's orbital angular momentum enable new techniques for measuring both real and imaginary parts of the scattering amplitude. Several possible observables are considered, for which the targets may be either well-localized with respect to the spatial beam profile, or the scattering occurs incoherently on nuclei in a bulk target. The developed approach can be applied to other nuclear reactions with strongly interacting twisted particles.

Determination of quantum labels based on projections of the total angular momentum on the molecule-fixed axis

Molecular line lists, particularly those computed for high temperature applications, often have very few states assigned local quantum numbers, i.e rotational and vibrational quantum labels. These are often important components for accurately determining line shape parameters required for radiative transfer simulations. Through variational calculations, the projection of the total angular momentum onto the molecule fixed axis (k) is investigated in the Radau internal coordinate system to determine when it can be considered a good quantum number. In the Radau coordinate system, when the square of the th component of the wavefunction is greater than one half, then we can classify k as a good quantum number in accordance with the theorem of Hose and Taylor, a quantum analogue of the Kolmogorov–Arnold–Moser theorem. When the theorem is satisfied, it is shown that k can reliably be used to determine oblate and prolate quantum labels Ka and Kc. This approach is tested on the states of the water and ozone molecules.

Microscopic coupled-channel calculation of proton and alpha inelastic scattering to the $4^+_1$ and $4^+_2$ states of $^{24}\textrm{Mg}$

Abstract The triaxial and hexadecapole deformations of the $K^\pi=0^+$ and $K^\pi=2^+$ bands of $^{24}$Mg have been investigated by the inelastic scatterings of various probes, including electrons, protons, and alpha($\alpha$) particles, for a prolonged time. However, it has been challenging to explain the unique properties of the scatterings observed for the $4^+_1$ state through reaction calculations. This paper investigates the structure and transition properties of the $K^\pi=0^+$ and $K^\pi=2^+$ bands of $^{24}$Mg employing the microscopic structure and reaction calculations via inelastic proton and $\alpha$ scattering. In particular, the $E4$ transitions to the $4^+_1$ and $4^+_2$ states are reexamined. The structure of $^{24}$Mg was calculated employing the variation after the parity and total angular momentum projections in the framework of the antisymmetrized molecular dynamics (AMD). The inelastic proton and $\alpha$ reactions were calculated by the microscopic coupled-channel (MCC) approach by folding the Melbourne $g$-matrix $NN$ interaction with the AMD densities of $^{24}$Mg. Reasonable results were obtained on the properties of the structure, including the energy spectra and $E2$ and $E4$ transitions of the $K^\pi=0^+$ and $K^\pi=2^+$ bands owing to the enhanced collectivity of triaxial deformation. The MCC+AMD calculation successfully reproduced the angular distributions of the $4^+_1$ and $4^+_2$ cross sections of proton scattering at incident energies of $E_p=40$–100 MeV and $\alpha$ scattering at $E_\alpha=100$–400 MeV. This is the first microscopic calculation to describe the unique properties of the $0^+_1\to 4^+_1$ transition. In the inelastic scattering to the $4^+_1$ state, the dominant two-step process of the $0^+_1\to 2^+_1\to 4^+_1$ transitions and the deconstructive interference in the weak one-step process were essential.

Projection on particle number and angular momentum: Example of triaxial Bogoliubov quasiparticle states

Many quantal many-body methods that aim at the description of self-bound nuclear or mesoscopic electronic systems make use of auxiliary wave functions that break one or several of the symmetries of the Hamiltonian in order to include correlations associated with the geometrical arrangement of the system's constituents. Such reference states have been used already for a long time within self-consistent methods that are either based on effective valence-space Hamiltonians or energy density functionals, and they are presently also gaining popularity in the design of novel ab-initio methods. A fully quantal treatment of a self-bound many-body system, however, requires the restoration of the broken symmetries through the projection of the many-body wave functions of interest onto good quantum numbers. The goal of this work is three-fold. First, we want to give a general presentation of the formalism of the projection method starting from the underlying principles of group representation theory. Second, we want to investigate formal and practical aspects of the numerical implementation of particle-number and angular-momentum projection of Bogoliubov quasiparticle vacua, in particular with regard of obtaining accurate results at minimal computational cost. Third, we want to analyze the numerical, computational and physical consequences of intrinsic symmetries of the symmetry-breaking states when projecting them.

Superatomic Signature and Reactivity of Silver Clusters with Oxygen: Double Magic Ag 17 – with Geometric and Electronic Shell Closure

Understanding the stability and reactivity of silver clusters toward oxygen provides insights to design new materials of coinage metals with atomic precision. Herein, we report a systematic study o...

Optical pumping schemes for nuclear spin polarization of Ar35

Quantum-state preparation has a wide range of applications ranging from quantum optics to quantum metrology to fundamental physics precision measurements. In the context of nuclear $\ensuremath{\beta}$-decay correlation experiments such as the ones using argon, one needs to prepare a source of highly polarized atoms. Creating such a state-prepared sample amounts to generating a so-called stretched state, i.e., a state with the maximal projection of the total angular momentum along the quantization axis. This is typically achieved through optical pumping. Since this technique inherently depends on cycles of absorption followed by spontaneous emission, it may lead to undesirable heating and population loss. It is therefore crucial to devise polarization methods with both optimized efficiency and minimized drawbacks. We propose and compare various schemes, which we have been investigating numerically in the case of argon-35 atoms. We show that polarization degrees as high as $99.99%$ can be obtained within less than 140 $\ensuremath{\mu}\mathrm{s}$.

Radiation of twisted photons from charged particles moving in cholesterics

The radiation of twisted photons by charged particles traversing a cholesteric plate is studied in the framework of quantum electrodynamics in an anisotropic inhomogeneous dispersive medium. The complete set of solutions to the Maxwell equations in the cholesteric plate is constructed in the paraxial and small anisotropy approximations. The explicit expressions for the average numbers of plane-wave and twisted photons created by a charged point particle crossing the cholesteric plate are derived. The selection rules for the twisted photons radiated at the harmonic n ∈ ℤ are established. In the paraxial regime, the projection of the orbital angular momentum of a radiated twisted photon obeys the selection rule l = ± (2n + 1). In the approximation of a small anisotropy of the permittivity tensor, the selection rule becomes m = ± 2n, where m is the projection of the total angular momentum of a radiated twisted photon. The sign “±” in these selection rules is related to the choice of the forward or reflected waves in the cholesteric and is realized at the different energies of radiated photons. As the examples, the radiation of optical twisted photons by electrons with the Lorentz factors γ = 235 and γ = 500 and by uranium nuclei with γ = 2 are considered. It is shown that charged particles traversing normally a cholesteric plate can be used as a pure source of twisted photons. These results can be employed for elaboration of compact flexible sources of twisted photons, new particle detectors, and for diagnostics of the structure of liquid crystals.

A detailed study of nuclear structure of odd-mass Pm isotopes near N = 82 shell closure

A systematic study of the positive as well as negative parity yrast structure for the odd-mass 145−149Pm isotopes is carried out by using the angular momentum projection technique implemented in the projected shell model (PSM). By assuming an axial symmetry in the deformed basis, we have been able to calculate the yrast states up to a maximum spin of 47/2 for both positive and negative parity, and as a result, a good agreement is obtained with available experimental data for all isotopes. The intrinsic structure of positive and negative parity yrast states has been found to be built on πg7/2 $$ \otimes $$ νh11/2 and πh11/2 $$ \otimes $$ νh11/2 configurations, respectively. The phenomenon of back-bending has also been discussed in the present work. Reduced electric and magnetic transitions probabilities [B(E2) and B(M1)] are also calculated using the PSM wave functions. Since experimental data on B(E2) and B(M1) are not available, these results can serve as motivation for experimentalists to look for these data. However, the theoretical values of transition strength ratio B(M1)/B(E2) have been found to be matching excellently with the experimental ones.