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Formulas to calculate multivector exponentials in a base-free representation and in a orthonormal basis are presented for an arbitrary Clifford geometric algebra Cl(p,q). The formulas are based on the analysis of roots of characteristic polynomial of a multivector. Elaborate examples how to use the formulas in practice are presented. The results may be useful in the quantum circuits or in the problems of analysis of evolution of the entangled quantum states., Comment: 9 pages

Closed form expressions in real Clifford geometric algebras Cl(0,3), Cl(3,0), Cl(1,2), and Cl(2,1) are presented in a coordinate-free form for exponential function when the exponent is a general multivector. The main difficulty in solving the problem is connected with an entanglement (or mixing) of vector and bivector components a and a in a form (a-a), i≠ j≠ k . After disentanglement, the obtained formulas simplify to the well-known Moivre-type trigonometric/hyperbolic function for vector or bivector exponentials. The presented formulas may find wide application in solving GA differential equations, in signal processing, automatic control and robotics.

The problem of square root of multivector (MV) in real 3D (n = 3) Clifford algebras Cl3;0, Cl2;1, Cl1;2 and Cl0;3 is considered. It is shown that the square root of general 3D MV can be extracted in radicals. Also, the article presents basis-free roots of MV grades such as scalars, vectors, bivectors, pseudoscalars and their combinations, which may be useful in applied Clifford algebras. It is shown that in mentioned Clifford algebras, there appear isolated square roots and continuum of roots on hypersurfaces (infinitely many roots). Possible numerical methods to extract square root from the MV are discussed too. As an illustration, the Riccati equation formulated in terms of Clifford algebra is solved.

Majorana spinors are constructed in terms of the multivectors of relativistic Cl1,3 algebra. Such spinors are found to be multiplied by primitive idempotents which drastically change spinor properties. Running electronic waves are used to solve the real Dirac–Majorana equation transformed to Cl1,3 algebra. From the analysis of the solution it is concluded that free Majorana particles do not exist, because relativistic Cl1,3 algebra requires the massive Majorana particle to move with light velocity.

The knowledge of exact wave functions is required in calculating physical parameters such as optical dipole moments, scattering matrix elements, or in wave function engineering. In this report we describe how a system of algebraic equations that follows from the Schrodinger equation can be reduced to a Grobner basis from which the exact wave function can be easily constructed. As an example, closed form solutions for a cylindrical electronic waveguide and double quantum well nanostructures are presented.

We apply Clifford algebra to investigate 2D electron spin reflection off and transmission through a stepped discontinuity of physical parameters in semiconducting quantum well. The discontinuity may be due to change of spin-orbit interaction constants, effective masses, or electrostatic potential. In the paper the posed problem has been solved exactly. It is shown that the reflected electronic beam has identical spin polarization as the incident one. However, the transmitted beam suffers spin flipping and in general case consists of a mixture of up and down spin states. Optimal conditions for total reversion of 2D electron spin polarization are found. Special attention is paid to correct boundary conditions in the presence of spin-orbit interaction. A simple formula that connects spin polarization of the transmitted beam and SO interaction constants is presented. Keywords: Clifford algebra, geometric algebra, semiconductors, spintronics, spin polarization, spin flipping, quantum well

To have a closed system, the Maxwell electromagnetic equations should be supplemented by constitutive relations which describe medium properties and connect primary fields (E, B) with secondary ones (D, H). J.W. Gibbs and O. Heaviside introduced the basis vectors {i, j, k} to represent the fields and constitutive relations in the three-dimensional vectorial space. In this paper the constitutive relations are presented in a form of Cl3,0 algebra which describes the vector space by three basis vectors {σ1, σ2, σ3} that satisfy Pauli commutation relations. It is shown that the classification of electromagnetic wave propagation phenomena with the help of constitutive relations in this case comes from the structure of Cl3,0 itself. Concrete expressions for classical constitutive relations are presented including electromagnetic wave propagation in a moving dielectric.

A practical computation method to find the eigenvalues and eigenspinors of quantum mechanical Hamiltonian is presented. The method is based on reduction of the eigenvalue equation to well known geometric algebra rotor equation and, therefore, allows to replace the usual det(H-E)=0 quantization condition by much simple vector norm preserving requirement. In order to show how it works in practice a number of examples are worked out in Cl_{3,0} (monolayer graphene and spin in the quantum well) and in Cl_{3,1} (two coupled two-level atoms and bilayer graphene) algebras., Comment: 12 pages, 3 figures in Advances in Applied Clifford Algebras, 2015

The scattering of valence band hole spin by acoustical and optical phonons due to deformation potential interaction is considered. Six-band deformation potential matrix, which takes into account doubly degenerate heavy-mass, light-mass, and split-off energy bands, is used to evaluate the matrix elements for spin conserving and flipping transitions. The concept of the spin surface was addressed to define the initial and final states of the hole spin. It was found that, in agreement with experiment, the spin conserving and flipping transitions are of comparable magnitude in both intravalence and intervalence band scattering mediated by acoustical as well as optical phonons.

Spin properties of holes in the wurtzite semiconductors, when the hole with a given wave vector ballistically propagates in either heavy-, light-mass or crystal-field split-off band, are considered. Analytical solutions for an averaged hole spin are found for two important cases, when the hole wave vector is parallel and perpendicular to the hexagonal axis. Shapes of the spin surfaces are presented for these cases. It is shown that the surfaces, in general, are spheroids. However, depending on the wave vector magnitude and valence band parameters, the surfaces may also be spheres or line-shaped. The properties of the spin surfaces are illustrated for parameters of the wurtzite GaN, where the crystal field dominates in the splitting of the valence band.

We present the results of time-resolved photoluminescence (TRPL) and optically detected microwave resonance (ODMR) spectroscopy investigations of semiconductor quantum dots and quantum wells. The ODMR spectra of InAs/GaAs QDs were detected via modulation of the total intensity of the QDs emission induced by 95 GHz microwave excitation and exciton fine structure was studied. Very long life times (up to 10 ns) of photoexcited carriers were observed in this system using TRPL at low temperatures and excitation intensities promising higher responsitivity of such QDs for quantum dot infrared photodetector development. The effects of proton and alpha particles irradiation on carrier dynamics were investigated on different InGaAs/GaAs, InAlAs/AlGaAs and GaAs/AlGaAs QD and QW systems. The obtained results demonstrated that carrier lifetimes in the QDs are much less affected by proton irradiation than that in QWs. A strong influence of irradiation on the PL intensity was observed in multiple QWs after high-energy alpha particles irradiation.

Multiple quantum well (MQW) samples grown by the molecular beam epitaxy method were irradiated by alpha particles from isotope 239.Pu. The photoluminescence (PL) spectra and PL integrated intensity dependencies are presented at various alpha particle fluences, up to 10(exp 11) cm(exp -2). The experimental results are in agreement with a model which assumes that point centers (residual impurities and point defects introduced during irradiation) are responsible for PL intensity decrease with the alpha particle fluence. It was found that annealing of irradiated MQW samples at a temperature above 650 K nearly restores the PL intensity. An enhancement of PL by more than an order of magnitude was observed at annealing temperatures higher than 850 K, just before GaAs and Al(0.35)Ga(0.65)As interdiffusion begins.

Undoped multiple quantum well (MQW) samples grown by the molecular beam epitaxy method were irradiated by 5.15 MeV energy alpha particles. A strong influence of irradiation on the photoluminescence (PL) intensity was observed in these nearly intrinsic samples: the PL intensity at liquid nitrogen and room temperatures decreased by more than two orders of magnitude at alpha particle fluence of 10(exp 11) cm(exp -2). The dependence of the PL intensity on irradiation dose is described by a rate equation model that takes into account the generation of nonradiative point centers in barriers and wells of the MQW structure during irradiation.

We present a detailed experimental study on photoluminescence quenching due to exciton and donor impact ionization by accelerated electrons under an in-plane nanosecond duration electric field created in $\mathrm{GaAs}/{\mathrm{Al}}_{0.35}{\mathrm{Ga}}_{0.65}\mathrm{As}$ quantum wells. From the photoluminescence transients measured by the time-correlated single-photon counting technique, we have determined the experimental conditions under which donor impact ionization can have an influence on quenching of the excitonic photoluminescence. The coefficient of two-dimensional exciton impact ionization has been estimated; its dependences on the applied electric field, lattice temperature, and width of the quantum wells are given.

Spin precession of a ballistic-hole when external magnetic field is absent and when the hole freely propagates in either heavy- or light-mass band is considered. The spin precession is due to linear in wave vector terms in the Luttinger-Kohn Hamiltonian. It was found that in contrast to conduction-band the valence band spin precession orbits are not circular. They are either ellipses or lines for light- and heavy-holes, respectively. The results of the paper can be applied to describe hole spin-dynamics in A3B5 compounds, for example, in simulation of spin-FETs by Monte Carlo method.© (2006) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Impact neutralization coefficient for the reaction D−+e→D0+2e is reported, where D− and D0 are the negatively charged and neutral donor, respectively, and e is the conduction band electron. The coefficient was calculated using the Boltzmann transport equation and the scaled neutralization cross section of the negatively charged hydrogen atom. At liquid helium and lower temperatures the above reaction for n-GaAs and n-InP doped with shallow donors was found to be characterized by a relatively large impact neutralization coefficient. At weak electric fields, from about 0.2 to 1 V/cm, the neutralization coefficient increases exponentially and at higher fields saturates at the value of about 1.5×10−3 cm3/s for n-GaAs and 9×10−4 cm3/s for n-InP.

Investigations of transient photoluminescence induced by external electric fields parallel to the layers of GaAs/Al0.35Ga0.65As quantum wells are reviewed. The photoluminescence was detected by time-correlated single-photon counting technique at liquid nitrogen and liquid helium temperatures applying electric fields of nanosecond duration to the wells of different width. It is shown how experimentally one can resolve between excitonic and donor impact ionization processes in combining spectral and time domains. From the study of the spectral-temporal dynamics at initial moments we have determined the coefficient of exciton impact ionization as a function of electric field for various widths of the quantum wells.© (2003) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

The control of optical transitions induced by an ultrashort infrared radiation pulse between energy bands is considered. The optical control problem is treated as a direct constrained minimization problem based on the nonlinear programming. As an example, in the report the transitions of the hole between heavy and light mass valence bands is analyzed in detail. The described optimization procedure allows to find the shape of femtosecond infrared light pulse that transfers the hole from the heavy to light mass valence band with a probability equal or nearly equal one.

The luminescence induced by alpha particles in GaAs/Al0.35Ga0.65As multiple quantum well (MQW) semiconductor was detected and investigated. The luminescence was observed at liquid nitrogen and at room temperatures as light flashes induced by individual (alpha) -particles. The MQW samples studied, in addition, were investigated using standard photoluminescence method to show that (alpha) - particle created flashes are associated with a radiative decay of 2D excitons in quantum wells. Since the number of detected light quanta in a flash is critical, a discussion of the collection efficiency of photons created by a single (alpha) -particle is presented.© (2001) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.