Angular Momentum Eigenstates Research Articles

Control of Photon Polarization in GaAs/AlAs Single Quantum Dot Emission

Polarization encoded single photons and photon pairs are expected to play a major role in future quantum communication schemes [1]. Therefore, control of polarization properties of single quantum dot emission is of a great value. We study magnetic field induced transition between anisotropy controlled and Zeeman controlled emission from individual GaAs/AlAs quantum dot (QD). We demonstrate the utility of these studies, involving polarized photon correlation measurements, for determination of the anisotropic exchange splitting of excitonic states in quantum dots. The sample was mounted directly on a microscope objective [2] and placed inside a pumped helium cryostat. Continuous wave excitation was performed with use of tunable Ti:Al2O3 laser at wavelength of 718 nm assuring excitation below the energy gap of the barrier material. The typical power of excitation beam was 100μW over the spot of a diameter of about 1μm. Temperature was kept constant at 1.8K. Photoluminescence was excited and collected through the same microscope objective. Correlation measurements were performed in a Hanbury-Brown and Twiss setup. The signal after polarization and spectral filtering was detected using two avalanche photodiodes. Polarization resolved crosscorrelations between biexciton (XX) and exciton (X) photons emitted subsequently in a radiative cascade were measured in magnetic field ranging from 0 to 0.5 T, applied in Faraday configuration. Polarization correlation of photon pairs in parallel linear polarizations appeared as a bunching peak in histograms of correlated counts in the absence of magnetic field. No polarization correlation in circular basis was observed at B = 0T. This is expected for QDs exhibiting in-plane anisotropy, since intermediate X state of the cascade is split by energy of electron-hole exchange interaction into two components emitting in orthogonal linear polarizations. Application of magnetic field transforms Anisotropic Exchange Splitting (AES) into Zeeman splitting of the X level, reducing thus the influence of the anisotropy. In a sufficiently high magnetic field pure exciton eigenstates of angular momentum M = +/1 are observed [3]. Expected conversion of linearly polarized excitonic states to circularly polarized ones is demonstrated by increase of the degree of circular polarization correlation with increasing magnetic field. We apply a simple model proposed by Besombes et al. [4] to describe the progressive change of the polarization correlation with increasing field. The model allows calculating of the degree of polarization correlation as a function of Zeeman splitting to AES ratio. By fitting the model to the experimental data, we were able to determine the anisotropic exchange splitting AES value of order of tens of μeV. This method of the AES determination can be used even in cases when direct measurement by polarization resolved microphotoluminescence is not sensitive enough.

Electron correlations in charge coupled vertically stacked quantum rings

We study spatial correlations for two electrostatically coupled electrons in a pair of vertically stacked quantum rings with configuration-interaction approach. We demonstrate that for distant quantum rings, the correlations undergo abrupt oscillations with the external magnetic field, which are strong for odd multiples of the flux quantum for which the electrons distinctly avoid being localized one above the other and are negligible for other values of the flux. The oscillations of the correlation strength vanish in the single-ring limit for which the electrons form a Wigner molecule. The wave function of the Hartree approximation is frustrated between reproducing the exact angular momentum eigenstate or the correlation between the electrons. We show that for distant rings the mean-field solution breaks the rotational symmetry of the system in a reentrant manner near odd multiples of half of the flux quantum to account for the correlation appearing in the exact solution.

Density functionals and transition-metal atoms

Density-functional calculations on transition-metal atoms are problematic due to the numerous possible ways, having inequivalent densities, of occupying the d orbitals. The problem is compounded by the issue of real orbitals versus complex orbitals. In this work we systematize the application of density-functional theories to transition-metal atoms using a current-density-dependent functional. For all the single-determinantal angular momentum eigenstates of ground-state terms, we obtain near degeneracy for the energies as we should. Also, we find a simple rule for occupying the real d orbitals that reproduces the energies of the (complex) angular momentum eigenstate results. Thus the long-standing confusion over how to compute transition-metal atom reference energies is resolved.

ENTANGLED STATE REPRESENTATION FOR A PARTICLE ON A CIRCLE STUDIED IN TERMS OF BOSONIC OPERATOR REALIZATION OF ANGULAR MOMENTUM

By introducing the bosonic operator realization of angular momentum, we establish the entangled state representation for describing quantum mechanics of a particle on a circle. The phase operator, the angular momentum eigenstates, the lowering and ascending operators for angular momentum are all well expressed in the bosonic realization with the aid of appropriate entangled states, i.e. we establish a new formalism for the quantum mechanics of a particle on a circle.

Angular momenta and eigenvectors of the weakly coupledT ⊗t Jahn-Teller system

The eigenvalues of the weakly coupled T ⊗ t Jahn-Teller problem are known for several decades, and the same holds also true for the eigenstates. These, however, are only given in the traditional position representation, which proves inconvenient if one attempts to extend the weak-coupling treatment into the region of stronger coupling. Here the solution of the T ⊗ t eigenvalue problem at weak coupling is derived in terms of creation and annihilation operators. This reformulation of the problem is nontrivial, since the algebraic form of the oscillator eigenvectors, being simultaneous angular-momentum eigenstates, has been worked out only recently and is probably still widely unknown. The electronic and oscillator eigenstates are then coupled to form eigenvectors of the total angular momentum. Finally, in preparation for an extension of the weak-coupling treatment, the action of the boson creation and annihilation operators on the oscillator eigenvectors is calculated, thus completing the algebraic approach to the weakly coupled T ⊗ t system.

Spin‐adapted states: A basis for quantum dot structure calculation

AbstractThe exact diagonalization method using a spin‐adapted basis is employed to calculate the electronic structure of a multi‐electron quantum dot. By isolating spin and orbital angular momentum eigenstates, we have significantly reduced the size of the matrices required in comparison with the standard configuration interaction method. A novel approach to the simplification of the interaction integrals that arise in the calculation is also presented, which allows exact evaluation of the Hamiltonian matrix required in the calculations. This Mathematica package permits accurate calculation of energy levels and wave functions for both ground and excited states of multiple electrons confined in a circular quantum dot. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006

Coherent states with elliptical polarization

Coherent states of the two-dimensional harmonic oscillator are constructed as superpositions of energy and angular momentum eigenstates. It is shown that these states are Gaussian wave-packets moving along a classical trajectory, with a well-defined elliptical polarization. They are coherent correlated states with respect to the usual Cartesian position and momentum operators. A set of creation and annihilation operators is defined in polar coordinates, and it is shown that these same states are precisely coherent states with respect to such operators.

Quantum behavior of a charged particle in an axial magnetic field

We study some aspects of the quantum theory of a charged particle moving in a time-independent, unidirectional magnetic field. When the field is uniform, we make a few clarifying remarks on the use of angular momentum eigenstates and momentum eigenstates with the diamagnetism of a free electron gas as an example. When the field is nonuniform but weakly varying, we discuss both perturbative and nonperturbative methods for studying a quantum mechanical system. As an application, we derive the quantized energy levels of a charged particle in a Helmholtz coil, which go over to the usual Landau levels in the limit of a uniform field.

Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon.

We propose interferometric methods capable of measuring either the total angular momentum, or simultaneously measuring the spin and orbital angular momentum of single photons. This development enables the measurement of any angular momentum eigenstate of a single photon. The work allows the investigation of single-photon two-qubit entangled states and has implications for high density information transfer.

Quantum spiral bandwidth of entangled two-photon states

We put forward the concept of quantum spiral bandwidth of the spatial mode function of the two-photon entangled state in spontaneous parametric downconversion. We obtain the bandwidth using the eigenstates of the orbital angular momentum of the biphoton states, and reveal its dependence with the length of the down converting crystal and waist of the pump beam. The connection between the quantum spiral bandwidth and the entropy of entanglement of the quantum state is discussed.

A non-commutativen-particle 3D Wigner quantum oscillator

An n-particle 3-dimensional Wigner quantum oscillator model is constructed explicitly. It is non-canonical in that the usual coordinate and linear momentum commutation relations are abandoned in favour of Wigner's suggestion that Hamilton's equations and the Heisenberg equations are identical as operator equations. The construction is based on the use of Fock states corresponding to a family of irreducible representations of the Lie superalgebra sl(1|3n) indexed by an A-superstatistics parameter p. These representations are typical for p\geq 3n but atypical for p<3n. The branching rules for the restriction from sl(1|3n) to gl(1) \oplus so(3) \oplus sl(n) are used to enumerate energy and angular momentum eigenstates. These are constructed explicitly and tabulated for n\leq 2. It is shown that measurements of the coordinates of the individual particles gives rise to a set of discrete values defining nests in the 3-dimensional configuration space. The fact that the underlying geometry is non-commutative is shown to have a significant impact on measurements of particle separation. In the atypical case, exclusion phenomena are identified that are entirely due to the effect of A-superstatistics. The energy spectrum and associated degeneracies are calculated for an infinite-dimensional realisation of the Wigner quantum oscillator model obtained by summing over all p. The results are compared with those applying to the analogous canonical quantum oscillator.

Merons in a finite quantum Hall system

We study quasiparticle states in a finite $\ensuremath{\nu}=1$ spin (as an additional degree of freedom) quantum Hall system. The skyrmion (topological) quasiparticle states are well established in the infinite system, but in finite systems, in the effective mean-field pictures that we have, as a rule, they are not eigenstates of hamiltonian and some other generators of the system symmetries. We present a class of states, which are eigenstates of certain combinations of the generators and are similar in structure to the orbital angular momentum eigenstates of the Laughlin quasihole in the same system. We refer to the states as states of a meron. They have an upper limit on the mean orbital angular momentum, and the limit defines an effective edge, at a distance less than the system radius, in an effective quasiparticle description of the meron. Remarkably, the meron edge (as the usual quasihole edge) is characterized by a power-law decay of the single-particle, static correlator defined between positional coherent states of the meron.

Quantum multiple scattering: Eigenmode expansion and its applications to proximity resonance

We show that for a general system of N s-wave point scatterers, there are always N eigenmodes. These eigenmodes or eigenchannels play the same role as spherical harmonics for a spherically symmetric target---they give a phase shift only. In other words, the T matrix of the system is of rank N, and the eigenmodes are eigenvectors corresponding to nonzero eigenvalues of the T matrix. The eigenmode expansion approach can give insight to the total scattering cross section; the position, width, and superradiant or subradiant nature of resonance peaks; the unsymmetric Fano line shape of sharp proximity resonance peaks based on the high-energy tail of a broadband; and other properties. Off-resonant eigenmodes for identical proximate scatterers are approximately angular-momentum eigenstates.

On uncertainty relations in noncommutative quantum mechanics

We discuss the uncertainty relations in quantum mechanics on noncommutative plane. In particular, we show that, for a given state, at most one out of three basic nontrivial uncertainty relations can be saturated. We consider also in some detail the case of angular momentum eigenstates.

A Galërkin approach to electronic near-degeneracies in molecular systems

We consider superposition states of various numbers of eigenstates in order to study vibrating quantum billiards semiquantally. We discuss the relationship between Galërkin methods, inertial manifolds, and partial differential equations such as nonlinear Schrödinger equations and Schrödinger equations with time-dependent boundary conditions. We then use a Galërkin approach to study vibrating quantum billiards. We consider one-term, two-term, three-term, d-term, and infinite-term superposition states. The number of terms under consideration corresponds to the level of electronic near-degeneracy in the system of interest. We derive a generalized Bloch transformation that is valid for any finite-term superposition and numerically simulate three-state superpositions of the radially vibrating spherical quantum billiard with null angular-momentum eigenstates. We discuss the physical interpretation of our Galërkin approach and thereby justify its use for vibrating quantum billiards. For example, d-term superposition states of one degree-of-vibration quantum billiards may be used to study nonadiabatic behavior in polyatomic molecules with one excited nuclear mode and a d-fold electronic near-degeneracy. Finally, we apply geometric methods to analyze the symmetries of vibrating quantum billiards.

Introduction to Quantum Mechanics in Chemistry (Ratner, Mark A.; Schatz, George C.)

1. Introduction and Background to Quantum Mechanics. Aim of Theoretical Chemistry. Key Concepts from Classical Physics. Classical Mechanics. Classical Wave Theory. Early History of Quantum Mechanics. Particle Nature of Light. Wave Nature of Particles. Uncertainty Principle. Discovery of Quantum Mechanics. Concepts in Quantum Mechanics. 2. Quantum Theory. Postulates of Quantum Mechanics. Definitions of ...Y and ...|Y|2. Operators. Time-Dependent and Time-Independent Schrodinger Equations. Eigenvalues. Expectation Values. Properties of the Time-Independent Schrodinger Eigenfunctions. 3. Particle-in-Box Models. Particle in a One-Dimensional Box. Particle in a Two-Dimensional Box. Particle in a Three-Dimensional Box. Free-Electron Molecular Orbital Model. 4. Rigid-Rotor Models and Angular Momentum Eigenstates. Motions of a Diatomic Molecule: Separation of the Center of Mass. Rigid-Rotor Model in Two Dimensions. Three-Dimensional Rigid Rotor. Spherical Harmonics. Rotational Spectra. Angular Momentum. Dirac Notation. Raising and Lowering Angular-Momentum Operators. 5. Molecular Vibrations and Time-Independent Perturbation Theory. Diatomic Molecule Vibrations. Raising and Lowering Operators for the Harmonic Oscillator. Polyatomic Molecule Vibrations. Time-Independent Perturbation Theory. Examples. 6. The Hydrogen Atom. The Schrodinger Equation. Radial Solutions and Eigenvalues. Energy Eigenvalues Spectroscopy of the H Atom. Properties of Hydrogen and Hydrogenlike Wavefunctions. Atomic Units. 7. The Helium Atom. Schrodinger Equation. Independent-Particle Model. The Variational Method. Better Wavefunctions. 8. Electron Spin and the Pauli Principle. Electron Spin. The Pauli Principle. He-Atom Wavefunctions, Including Spin. Excited State of He. Energies of He(1s2s) States. Interaction of Electron Spin with Magnetic Fields. EPR and NMR. 9. Many-Electron Atoms. Many-Electron Hamiltonian and Schrodinger Equation. Slater Determinants. Hartree Method. Hartree-Fock Method. Koopmans' Theorem. Electron Correlation. Constants of the Motion. Angular-Momentum Operators for Many-Electron Atoms. Relativistic Effects. 10. Homonuclear Diatomic Molecules. Hydrogen Molecular Ion: Born-Oppenheimer Approximation. LCAO-MO Treatment of H2 +. Other H2 + States. Electronic Structure of Homonuclear Diatomics. Electronic Structure of H2: Molecular Orbital and Valence Bond Wavefunctions. Improvements to MO and VB Results for H2. 11. Ab Initio and Density Functional Methods. LCAO-MO-SCF Theory for Molecules. Atomic Orbitals. Hartree-Fock Calculations. Beyond Hartree-Fock. Density Functional Theory Methods. 12. Semiempirical Methods. Huckel Model. Extended Huckel Method. PPP Method. NDO Methods. 13. Applications of Group Theory. Group Theory for Point Groups. Applications of Group Theory to Molecular Quantum Mechanics. Symmetry Properties of Many-Electron Wavefunctions. Symmetry Properties of Molecular Vibrations. 14. Applications of Electronic Structure Theory. Potential-Energy Functions. Optimized Geometries and Frequencies. IR Spectra. Barriers to Reaction. Excited States. Molecular Clusters. Remarks on Other Methods. 15. Time Dependence and Spectroscopy. Transition Probabilities and the Golden Rule. Electronic Spectroscopy of Molecules. Vibration (Infrared) Spectroscopy. Appendices. Mathematical Background. Two-Electron Repulsion Integral. Character Tables. Atomic Units, Energy Conversion Factors, and Physical Constants. Solutions to Odd-Numbered Problems. Index.

Basis for Breakup States of Three Identical Particles

A new basis for expanding three-body momentum-space states for three identical particles is studied. The basis states are simultaneously eigenstates of the total angular momentum and the total antisymmetrization operator. The total kinetic energy and two Dalitz-Fabri variables are chosen as the remaining three continuous variables. Zernike polynomials are used as a basis set for a generalized Fourier expansion in the Dalitz-Fabri variables. Born approximations to the nucleon-deuteron breakup amplitude (zero total orbital angular momentum) are calculated for Malfliet-Tjon I–III potentials and displayed in a Dalitz plot that shows the global structures of the reaction probabilities. Numerical results are presented, which indicate favorable convergence properties of the generalized Fourier expansion. These results suggest that the new basis set may be attractive in more realistic calculations.

Independent eigenstates of angular momentum in a quantumN-body system

The global rotational degrees of freedom in the Schr\"{o}dinger equation for an $N$-body system are completely separated from the internal ones. After removing the motion of center of mass, we find a complete set of $(2\ell+1)$ independent base functions with the angular momentum $\ell$. These are homogeneous polynomials in the components of the coordinate vectors and the solutions of the Laplace equation, where the Euler angles do not appear explicitly. Any function with given angular momentum and given parity in the system can be expanded with respect to the base functions, where the coefficients are the functions of the internal variables. With the right choice of the base functions and the internal variables, we explicitly establish the equations for those functions. Only (3N-6) internal variables are involved both in the functions and in the equations. The permutation symmetry of the wave functions for identical particles is discussed.

ANGULAR MOMENTUM-PHASE WIGNER OPERATOR FOR DESCRIBING ELECTRONS IN UNIFORM MAGNETIC FIELDS

By introducing the angular momentum-phase Wigner operator both in &lt;λ| representation and &lt;l,r| representation, which are respectively the eigenstates of position and angular momentum of an electron in a uniform magnetic field, we establish the angular momentum-phase Wigner function whose marginal distributions are physically meaningful.

Introduction to Quantum Mechanics in Chemistry

1. Introduction and Background to Quantum Mechanics. Aim of Theoretical Chemistry. Key Concepts from Classical Physics. Classical Mechanics. Classical Wave Theory. Early History of Quantum Mechanics. Particle Nature of Light. Wave Nature of Particles. Uncertainty Principle. Discovery of Quantum Mechanics. Concepts in Quantum Mechanics. 2. Quantum Theory. Postulates of Quantum Mechanics. Definitions of ...Y and ...|Y|2. Operators. Time-Dependent and Time-Independent Schroedinger Equations. Eigenvalues. Expectation Values. Properties of the Time-Independent Schroedinger Eigenfunctions. 3. Particle-in-Box Models. Particle in a One-Dimensional Box. Particle in a Two-Dimensional Box. Particle in a Three-Dimensional Box. Free-Electron Molecular Orbital Model. 4. Rigid-Rotor Models and Angular Momentum Eigenstates. Motions of a Diatomic Molecule: Separation of the Center of Mass. Rigid-Rotor Model in Two Dimensions. Three-Dimensional Rigid Rotor. Spherical Harmonics. Rotational Spectra. Angular Momentum. Dirac Notation. Raising and Lowering Angular-Momentum Operators. 5. Molecular Vibrations and Time-Independent Perturbation Theory. Diatomic Molecule Vibrations. Raising and Lowering Operators for the Harmonic Oscillator. Polyatomic Molecule Vibrations. Time-Independent Perturbation Theory. Examples. 6. The Hydrogen Atom. The Schroedinger Equation. Radial Solutions and Eigenvalues. Energy Eigenvalues Spectroscopy of the H Atom. Properties of Hydrogen and Hydrogenlike Wavefunctions. Atomic Units. 7. The Helium Atom. Schroedinger Equation. Independent-Particle Model. The Variational Method. Better Wavefunctions. 8. Electron Spin and the Pauli Principle. Electron Spin. The Pauli Principle. He-Atom Wavefunctions, Including Spin. Excited State of He. Energies of He(1 s2s) States. Interaction of Electron Spin with Magnetic Fields. EPR and NMR. 9. Many-Electron Atoms. Many-Electron Hamiltonian and Schroedinger Equation. Slater Determinants. Hartree Method. Hartree-Fock Method. Koopmans' Theorem. Electron Correlation. Constants of the Motion. Angular-Momentum Operators for Many-Electron Atoms. Relativistic Effects. 10. Homonuclear Diatomic Molecules. Hydrogen Molecular Ion: Born-Oppenheimer Approximation. LCAO-MO Treatment of H 2 +. Other H 2 + States. Electronic Structure of Homonuclear Diatomics. Electronic Structure of H 2: Molecular Orbital and Valence Bond Wavefunctions. Improvements to MO and VB Results for H 2. 11. Ab Initio and Density Functional Methods. LCAO-MO-SCF Theory for Molecules. Atomic Orbitals. Hartree-Fock Calculations. Beyond Hartree-Fock. Density Functional Theory Methods. 12. Semiempirical Methods. Huckel Model. Extended Huckel Method. PPP Method. NDO Methods. 13. Applications of Group Theory. Group Theory for Point Groups. Applications of Group Theory to Molecular Quantum Mechanics. Symmetry Properties of Many-Electron Wavefunctions. Symmetry Properties of Molecular Vibrations. 14. Applications of Electronic Structure Theory. Potential-Energy Functions. Optimized Geometries and Frequencies. IR Spectra. Barriers to Reaction. Excited States. Molecular Clusters. Remarks on Other Methods. 15. Time Dependence and Spectroscopy. Transition Probabilities and the Golden Rule. Electronic Spectroscopy of Molecules. Vibration (Infrared) Spectroscopy. Appendices. Mathematical Background. Two-Electron Repulsion Integral. Character Tables. Atomic Units, Energy Conversion Factors, and Physical Constants. Solutions to Odd-Numbered Problems. Index.