Symmetry Of Wave Function Research Articles

Particle on a Ring Spectroscopic Selection Rules Determined by Group Theory

Early introduction of symmetry concepts in the Physical Chemistry curriculum has been shown to help students understand the relative simplicity of spectroscopic selection rules. Most Quantum Mechanics and Spectroscopy textbooks begin with various one-dimensional problems. Application of symmetry arguments to the particle-in-a-box problem is presented in some books, but no sources have been found where symmetry arguments are used to determine the selection rules for particle-on-a-ring spectroscopic transitions. This hinders the early introduction of symmetry concepts. This article removes this hindrance by deriving the particle-on-a-ring rotational selection rules using group theory symmetry arguments. The D∞h character table is used to define rotational wave function symmetry, and the D∞h direct product table is used to determine the nonzero behavior of the transition dipole moment integral. A survey of the symmetry of allowed transitions leads to the well-known rotational selection rules of Δm = 0 for Ra...

Fundamental mechanisms of the dependence of spin states on exchange correlations in quantum dots: Feynman path integrals

The effect of permutation symmetry of nonrelativistic electron wave functions in quantum dots on the spatial correlations, energy, and distribution over the spin states is investigated. The quantitative analysis is based on ab initio calculations using the technique of Feynman path integrals of entangled quantum states in model spherically symmetrical quantum dots with the parabolic confining field. In the computer simulation, the quantum-mechanical indistinguishability and spin variable are exactly taken into account. It is demonstrated that the detailed description of the high-order spatial correlations plays a key role in the correct calculation of the equilibrium spin number and energy values.

Decoherence of nuclear spins in the frozen core of an electron spin

Hybrid qubit systems combining electronic spins with nearby (``proximate'') nuclear spin registers offer a promising avenue towards quantum information processing, with even multispin error-correction protocols recently demonstrated in diamond. However, for the important platform offered by spins of donor atoms in cryogenically cooled silicon, decoherence mechanisms of $^{29}\mathrm{Si}$ proximate nuclear spins are not yet well understood. The reason is partly because proximate spins lie within a so-called ``frozen core'' region where the donor electronic hyperfine interaction strongly suppresses nuclear dynamics. We investigate the decoherence of a central proximate nuclear qubit arising from quantum spin baths outside, as well as inside, the frozen core around the donor electron. We consider the effect of a very large nuclear spin bath comprising many $(\ensuremath{\gtrsim}{10}^{8})$ weakly contributing pairs outside the frozen core (the ``far bath''). We also propose that there may be an important contribution from a few (of order 100) symmetrically sited nuclear spin pairs (``equivalent pairs''), which were not previously considered because their effect is negligible outside the frozen core. If equivalent pairs represent a measurable source of decoherence, nuclear coherence decays could provide sensitive probes of the symmetries of electronic wave functions. For the phosphorus donor system, we obtain ${T}_{2n}$ values of order 1 second for both the far-bath and equivalent-pair models, confirming the suitability of proximate nuclei in silicon as very-long-lived spin qubits.

The Andreev reflection of zero line mode in graphene-superconductor hybrid junction

The zero line mode (ZLM) in two dimensional materials provides a quasi-one dimensional path for electronic transport. We report the theoretical investigation of the Andreev reflection of ZLM by using the staggered graphene-superconductor based models. For a two-terminal system in which the valley index is well preserved, when graphene is zigzag edged, the Andreev reflection coefficient can be either large or strongly suppressed depending on the symmetric properties of the transverse wave function in graphene ribbon. However, the Andreev reflection coefficient, independent of the staggering profile in the armchair edged model, is large due to the absence of wave function symmetry. When ZLM changes its direction in a vertical path, a perfect Andreev reflection could happen when the incident ZLM stems from a zigzag edged graphene ribbon. In a zigzag edged four-terminal hybrid model, the interference of reflected holes leads to perfect Andreev reflection with probability unity and the annihilation of the crossed Andreev reflection. For the armchair edged model, the interference effect disappears because the Andreev reflection from one of the paths is prohibited. The interference of Andreev reflections in four-terminal models is investigated by spacial local density of states in the central scattering region as well.

Vibrational State-Selective Resonant Two-Photon Photoelectron Spectroscopy of AuS(-) via a Spin-Forbidden Excited State.

Vibrational state-selective resonant two-photon photoelectron spectra have been obtained via a triplet intermediate state ((3)Σ(-)) of AuS(-) near its detachment threshold using high-resolution photoelectron imaging of cryogenically cooled AuS(-) anions. Four vibrational levels of the (3)Σ(-) excited state are observed to be below the detachment threshold. Resonant two-photon absorptions through these levels yield vibrational state-selective photoelectron spectra to the (2)Σ final state of neutral AuS with broad and drastically different Franck-Condon distributions, reflecting the symmetries of the vibrational wave functions of the (3)Σ(-) intermediate state. The (3)Σ(-) excited state is spin-forbidden from the (1)Σ(+) ground state of AuS(-) and is accessed due to strong relativistic effects. The nature of the (3)Σ(-) excited state is confirmed by angular distributions of the photoelectron images and quantum calculations.

Scattering of a particle with internal structure from a single slit: exact numerical solutions

Scattering of a quantum particle with internal structure is fundamentally different from that of a point particle and shows quantum effects such as the modification of transmission due to tunnelling and trapping of the particle. As in a preceding paper (Shore et al 2014 New J. Phys. 17 013046) we consider a model of a symmetric, rigid rotor travelling through an aperture in a thin but impenetrable screen which is perpendicular to both the direction of motion and the rotation axis. We determine the quantum mechanical properties of this two-dimensional geometrical model using a quasi one-dimensional scattering problem with unconventional boundaries. Our calculations rely on finding the Green's function, which has a direct connection to the scattering matrix. Evaluated on a discrete lattice the Hamiltonian is ‘dressed’ by a self-energy correction that takes into account the open boundary conditions in an exact way. We find that the passage through the aperture can be suppressed or enhanced as a result of the rotational motion. These effects manifest themselves through resonances in the transmission probability as a function of incident energy and symmetry of the incident wavefunction. We determine the density-of-states to reveal the mode structure of resonant states and to exhibit the lifetimes of temporary trapping within the aperture.

Recent progress in Rashba spin orbit coupling on metal surface

Spin-orbit coupling (SOC) is a bridge between the spin and orbital of an electron. Through SOC, spin of the electron can possibly be controlled throuth external electric fields. It is found that many novel physical phenomena in solids are related with SOC, for example, the magnetic anisotropy of magnetic materials, the spin Hall effect, and the topological insulator, etc. In the surface of solid or at the interface of heterostructure, Rashba SOC is induced by the structure inversion asymmetry. It was observed first in semiconductor heterostructure, which has an inversion asymmetric potential at the interface. Because Rashba SOC at the interface can be easily controlled through gate voltage, it is of great significance in the field of electric control of magnetism. Metal surface subsequent to semiconductor becomes another main stream with large Rashba SOC. In this paper, we review the recent progress in Rashba SOC in metal surfaces, including both the magnetic and nonmagnetic metal surfaces. We demonstrate the findings in Au(111), Bi(111), Gd(0001), etc., and discuss the possible factors that could influence Rashba SOC, including the surface potential gradient, atom number, the symmetry of the surface wavefunction, and the hybridization between the different orbitals in the surface states, etc. We also discuss the manipulation of Rashba SOC through electric field or surface decoration. In addition, on magnetic surface, there coexist Rashba SOC and magnetic exchange interaction, which provides the possibility of controlling magnetic properties by electric field through Rashba SOC. The angle-resolved photoemission spectroscopy and the first-principles calculations based on density functional theory are the two main methods to investigate the Rashba SOC. We review the results obtained by these two approaches and provide a thorough understanding of the Rashba SOC in metal surface.

Understanding the electric field control of the electronic and optical properties of strongly-coupled multi-layered quantum dot molecules.

Strongly-coupled quantum dot molecules (QDMs) are widely employed in the design of a variety of optoelectronic, photovoltaic, and quantum information devices. An efficient and optimized performance of these devices demands engineering of the electronic and optical properties of the underlying QDMs. The application of electric fields offers a way to realise such a control over the QDM characteristics for a desired device operation. We performed multi-million-atom atomistic tight-binding calculations to study the influence of electric fields on the electron and hole wave function confinements and symmetries, the ground-state transition energies, the band-gap wavelengths, and the optical transition modes. Electrical fields parallel (Ep) and anti-parallel (Ea) to the growth direction were investigated to provide a comprehensive guide for understanding the electric field effects. The strain-induced asymmetry of the hybridized electron states is found to be weak and can be balanced by applying a small Ea electric field, of the order of 1 kV cm(-1). The strong interdot couplings completely break down at large electric fields, leading to single QD states confined at the opposite edges of the QDM. This mimics a transformation from a type-I band structure to a type-II band structure for the QDMs, which is a critical requirement for the design of intermediate-band solar cells (IBSCs). The analysis of the field-dependent ground-state transition energies reveals that the QDM can be operated both as a high dipole moment device by applying large electric fields and as a high polarizability device under the application of small electric field magnitudes. The quantum confined Stark effect (QCSE) red shifts the band-gap wavelength to 1.3 μm at the 15 kV cm(-1) electric field; however the reduced electron-hole wave function overlaps lead to a decrease in the interband optical transition strengths by roughly three orders of magnitude. The study of the polarisation-resolved optical modes indicates the benefits of applying small electric fields, which leads to an isotropic polarisation response, a desirable property for semiconductor optical amplifiers (SOAs).

Orbital dependent interaction of quantum well states for catalytic water splitting

Orientational dependence of catalytic activity for water splitting reaction on a two-dimensional gold cluster supported on MgO/Ag(001) has been identified using first-principles calculations. Strong oscillations are found in water adsorption energy, the dissociation barrier, and the binding energy of the dissociated H atom, with two different orientational patterns. These two patterns correlate with the wavefunction symmetry of frontier orbitals of selected quantum well states (QWSs). This finding reveals a new aspect of orbital symmetry in catalytic reactions without involving changes in the shape or size of the atomic cluster, and is promising for potential applications in chemical reactions using the orbital degree of freedom of QWSs.

Effects of molecular symmetry on quantum reaction dynamics: novel aspects of photoinduced nonadiabatic dynamics.

Nonadiabatic coupling terms (NACTs) between different electronic states lead to fast radiationless decay in photoexcited molecules. Using molecular symmetry, i.e., symmetry with respect to permutation of identical nuclei and inversion of the molecule in space, the irreducible representations of the NACTs can be determined with a combination of molecular symmetry arguments and quantization rules. Here, we extend these symmetry rules for electronic states and coupling elements and demonstrate the importance of molecular symmetry for nonadiabatic nuclear dynamics. As an example, we consider the NACTs related to the torsion around the CN bond in C5H4NH. We present the results of quantum dynamical simulations of the photoinduced large amplitude torsion on three coupled electronic states and show how the interference between wavepackets leads to radiationless decay, which depends on the symmetry of the NACTs. Moreover, we show that the nuclear spin of the system determines the symmetry of the initial nuclear wave function and thus influences the torsional dynamics. This may open new possibilities for nuclear spin selective laser control of nuclear dynamics.

Full dimensional Franck-Condon factors for the acetylene à (1)Au-X̃ (1)Σ(g)(+) transition. II. Vibrational overlap factors for levels involving excitation in ungerade modes.

A full-dimensional Franck-Condon calculation has been applied to the à (1)Au-X̃ 1Σg+ transition in acetylene in the harmonic normal mode basis. Details of the calculation are discussed in Part I of this series. To our knowledge, this is the first full-dimensional Franck-Condon calculation on a tetra-atomic molecule undergoing a linear-to-bent geometry change. In the current work, the vibrational intensity factors for levels involving excitation in ungerade vibrational modes are evaluated. Because the Franck-Condon integral accumulates away from the linear geometry, we have been able to treat the out-of-plane component of trans bend (ν4('')) in the linear X̃ state in the rotational part of the problem, restoring the χ Euler angle and the a-axis Eckart conditions. A consequence of the Eckart conditions is that the out-of-plane component of ν4('') does not participate in the vibrational overlap integral. This affects the structure of the coordinate transformation and the symmetry of the vibrational wavefunctions used in the overlap integral, and results in propensity rules involving the bending modes of the X̃ state that were not previously understood. We explain the origin of some of the unexpected propensities observed in IR-UV laser-induced fluorescence spectra, and we calculate emission intensities from bending levels of the à state into bending levels of the X̃ state, using normal bending mode and local bending mode basis sets. Our calculations also reveal Franck-Condon propensities for the Cartesian components of the cis bend (ν5('')), and we predict that the best Ã-state vibrational levels for populating X̃-state levels with large amplitude bending motion localized in a single C-H bond (the acetylene↔vinylidene isomerization coordinate) involve a high degree of excitation in ν6(') (cis-bend). Mode ν4(') (torsion) populates levels with large amplitude counter-rotational motion of the two hydrogen atoms.

Novel Effect Induced by Pseudo-Jahn-Teller Interactions: Broken Cylindrical Symmetry in Linear Molecules.

It is shown that in linear molecules the pseudo-Jahn-Teller (PJT) interaction of a Σ or Π term with a Δ term induces a bending instability that is angular dependent, reducing the symmetry of the adiabatic potential energy surface from expected D∞h to D4h and C∞v to C2v or C4v. This spontaneously broken cylindrical symmetry (BCS) emerges from the solution of the vibronic coupling equations of the PJT effect (PJTE) problems (Σ+Δ)⊗w, (Π+Δ)⊗w, (Π+Σ+Δ)⊗w, and (Δ+Δ)⊗w, where w includes linear, quadratic, and fourth order vibronic coupling terms, and it is confirmed by ab initio calculations for a series of triatomic molecules with ground or excited Δ terms. The BCS is due to the angular symmetry of the electronic wave functions of the Δ term, ∼cos 2φ, and ∼sin 2φ, split by the fourth order vibronic coupling, which in overlap with the other symmetry wave functions of the Σ or Π term provides for the periodical symmetry of the added covalency that facilitates the bending. The mechanism of this PJT-induced BCS effect is discussed in detail; the numerical values of the vibronic coupling parameters for the molecules under consideration were estimated by means of combining separate ab initio calculations of some of them with a procedure fitting the analytical expressions to ab initio calculated energy profiles. It is also shown that the bending of linear molecules in Δ states, similar to Π states, is exclusively a PJT (not Renner-Teller) effect. The BCS revealed in this paper illustrates again the predicting power of the PJTE.

High performance current and spin diode of atomic carbon chain between transversely symmetric ribbon electrodes.

We demonstrate that giant current and high spin rectification ratios can be achieved in atomic carbon chain devices connected between two symmetric ferromagnetic zigzag-graphene-nanoribbon electrodes. The spin dependent transport simulation is carried out by density functional theory combined with the non-equilibrium Green's function method. It is found that the transverse symmetries of the electronic wave functions in the nanoribbons and the carbon chain are critical to the spin transport modes. In the parallel magnetization configuration of two electrodes, pure spin current is observed in both linear and nonlinear regions. However, in the antiparallel configuration, the spin-up (down) current is prohibited under the positive (negative) voltage bias, which results in a spin rectification ratio of order 104. When edge carbon atoms are substituted with boron atoms to suppress the edge magnetization in one of the electrodes, we obtain a diode with current rectification ratio over 106.

Giant Rectification Ratios of Azulene-like Dipole Molecular Junctions Induced by Chemical Doping in Armchair-Edged Graphene Nanoribbon Electrodes

Electron transport properties of an azulene-like dipole molecule anchored with carbon atomic chains sandwiched between two graphene nanoribbon (GNR) electrodes are theoretically investigated at the ab initio level. The molecular junctions are constructed with a strategy of modulating symmetry of Bloch wave functions. The chemical doping in an armchair-edged GNR is shown to play a significant role in determining the conductance behavior and rectifying performance of the molecular junctions. Giant rectification ratios up to 104 at low bias voltages are obtained for the molecular junctions with asymmetric arrangement of undoped zGNR and doped aGNR electrodes. The boron (aluminum) dopants in the aGNR electrode induce a better rectifying performance for the molecular junctions than the respective nitrogen (phosphorus) dopants. Moreover, the boron or nitrogen doping is more advantageous than the respective aluminum or phosphorus doping in view of improving rectifying behaviors of the molecular junctions. Taking...

Toward weak confinement regime in epitaxial nanostructures: Interdependence of spatial character of quantum confinement and wave function extension in large and elongated quantum dots

In this paper we present a comprehensive and detailed analysis of carrier/exciton wave function extension in large low-strain ${\mathrm{In}}_{0.3}{\mathrm{Ga}}_{0.7}$As quantum dots (QDs). They exhibit rather shallow confinement potential with electron/hole localization energy below 30 meV and confinement strength substantially weakened in comparison to typical epitaxial quasi-zero-dimensional semiconductor nanostructures. The aim of this study is to investigate the influence of different factors on the wave function (probability density distribution) for carriers or excitons in this regime, i.e., object shape anisotropy as well as strain, piezoelectricity, and Coulomb interactions, and to identify the physical mechanisms determining the properties of optical emission. To probe the wave function symmetry, polarization-resolved photoluminescence has been performed, and the spatial extensions of the corresponding probability densities have been verified in magneto-optical measurements. The observed diamagnetic coefficients in the range of (15--31) $\ensuremath{\mu}\mathrm{eV}/{\mathrm{T}}^{2}$ reflect large in-plane QD size. These studies also enable us to investigate the importance of light hole states admixture to the valence band ground state in such nanostructures, which can be addressed via the degree of linear polarization of emission as well as the exciton ${g}_{X}$ factor. The linear-polarization-resolved measurements revealed an exceptionally low exciton fine structure splitting of $5 \ensuremath{\mu}\mathrm{eV}$ on average as well as a low emission polarization degree of \ensuremath{-}0.05, with the polarization perpendicular to the QD elongation direction dominating. The increased light hole contribution to the lowest energy hole level is reflected in the decreased exciton ${g}_{X}$ factor (in the range of 0--1) and is consistent with the results of the eight-band k\ifmmode\cdot\else\textperiodcentered\fi{}p modelling. Based on the temperature dependence of the diamagnetic coefficient, the problem of individual QD uniformity has additionally been discussed. To evaluate the impact of the confinment potential and the structure geometry on the optical properties of the QDs, a comparison between the investigated dots and InAs/InGaAlAs/InP quantum dashes exhibiting a much deeper confining potential is presented.

Electronic structure of low-pressure and high-pressure phases of silicon disulfide

Self-consistent density functional theory calculations of band spectra, densities of states as well as the spatial distribution of valence electron charge density were carried out for the low-pressure α-phase and the high-pressure β-phase of SiS2. Group-theoretical analysis performed for both phases enabled the symmetry of wave functions in a number of high-symmetry points of the Brillouin zone as well as the structure of valence band representations to be found. Based on the calculations of the band structure, the orthorhombic α-phase of SiS2 was determined to be an indirect-gap semiconductor with the band gap Egi = 2.44 eV (T1 → X8 transition), while the β-phase was shown to be direct gap with Egd = 2.95 eV (Г3 → Г2 transition). The calculated energy distribution of the total density of states in the valence band of α-SiS2 qualitatively and quantitatively correlates with the main experimental features of the X-ray photoelectron spectrum.

Exchange symmetry, fluctuation-compressibility relation, and thermodynamic potentials of quantum liquids.

Liquid helium does not obey the Gibbs fluctuation-compressibility relation, which was noted more than six decades ago. However, still missing is a clear explanation of the reason for the deviation or the correct fluctuation-compressibility relation for the quantum liquid. Here we present the fluctuation-compressibility relation valid for any grand canonical system. Our result shows that the deviation from the Gibbs formula arises from a nonextensive part of thermodynamic potentials. The particle-exchange symmetry of many-body wave function of a strongly degenerate quantum gas is related to the thermodynamic extensivity of the system; a Bose gas does not always obey the Gibbs formula, while a Fermi gas does. Our fluctuation-compressibility relation works for classical systems as well as quantum systems. This work demonstrates that the application range of the Gibbs-Boltzmann statistical thermodynamics can be extended to encompass nonextensive open systems without introducing any postulate other than the principle of equal a priori probability.

Magnetic exchange interaction in topological insulators

The effective Hamiltonian that describes magnetic exchange interaction in a topological insulator is calculated, projecting bulk contact $s$-$d$ interaction onto surface electron states. This model consistently describes the indirect exchange between magnetic atoms mediated either by massless Dirac fermions in thick films or by massive excitations in a thin slab with tunneling between opposite surfaces. It is shown that an effective $s$-$d$ exchange is sensitive to the symmetry of surface wave functions and depends on the position of an impurity spin relative to slab surfaces. Under a vertical bias the indirect exchange between localized spins is switchable between opposite surfaces, and the oscillating range function reveals zero magnetic field beating as a signature of Rashba spin splitting.

Abnormal superfluid fraction of harmonically trapped few-fermion systems.

Superfluidity is a fascinating phenomenon that, at the macroscopic scale, leads to dissipationless flow and the emergence of vortices. While these macroscopic manifestations of superfluidity are well described by theories that have their origin in Landau's two-fluid model, our microscopic understanding of superfluidity is far from complete. Using analytical and numerical ab initio approaches, this Letter determines the superfluid fraction and local superfluid density of small harmonically trapped two-component Fermi gases as a function of the interaction strength and temperature. At low temperature, we find that the superfluid fraction is, in certain regions of the parameter space, negative. This counterintuitive finding is traced back to the symmetry of the system's ground state wave function, which gives rise to a diverging quantum moment of inertia I(q). Analogous abnormal behavior of I(q) has been observed in even-odd nuclei at low temperature. Our predictions can be tested in modern cold atom experiments.

Terahertz emission from ac Stark-split asymmetric intersubband transitions

Transitions between the two states of an AC Stark-split doublet are forbidden in centro-symmetric systems, and thus almost impossible to observe in experiments performed with atomic clouds. However, electrons trapped in nanoscopic heterostructures can behave as artificial atoms, with the advantage that the wavefunction symmetry can be broken by using asymmetric confining potentials. Here we develop the many-body theory describing the intra-doublet emission of a resonantly pumped intersubband transition in a doped asymmetric quantum well, showing that in such a system the intra-doublet emission can be orders of magnitude higher than in previously studied systems. This emission channel, which lies in the terahertz range, and whose frequency depends upon the pump power, opens the way to the realization of a new class of monolithic and tunable terahertz emitters.