Abstract. Certain solutions to Harper's equation are discrete analogues of (and approximations to) the Hermite-Gaussian functions. They are the energy eigenfunctions of a discrete algebraic analogue of the harmonic oscillator, and they lead to a definition of a discrete fractional Fourier transform (FT). The discrete fractional FT is essentially the time-evolution operator of the discrete harmonic oscillator.

Schwinger's finite (D) dimensional periodic Hilbert Space representations are studied on the toroidal lattice with specific emphasis on the deformed oscillator subalgebras and the generalized representations of the Wigner function. These subalgebras are shown to be admissible endowed with the non-negative norm of Hilbert space vectors. Hence, they provide the desired canonical basis for the algebraic formulation of the quantum phase problem. Certain equivalence classes in the space of labels are identified within each subalgebra, and connections with area preserving canonical transformations are studied. The generalised representations of the Wigner function are examined in the finite dimensional cyclic Schwinger basis. These representations are shown to conform to all fundamental conditions of the generalized phase space Wigner distribution. As a specific application of the Schwinger basis, the number-phase unitary operator pair is studied and, based on the admissibility of the underlying q-oscillator subalgebra, an algebraic approach to the unitary quantum phase operator is established. Connections with the Susskind-Glogower-Carruthers-Nieto phase operator formalism as well as the standard action-angle Wigner function formalisms are examined in the infinite period limit. The concept of continuously shifted Fock basis is introduced to facilitate the Fock space representations of the Wigner function.

The influence of the Rashba spin-orbit coupling (RSOC) on the two-dimensional (2D) electrons and holes in a strong perpendicular magnetic field leads to different results for the Landau quantization in different spin projections. In the Landau gauge the unidimensional wave vector describing the free motion in one in-plane direction is the same for both spin projections, whereas the numbers of Landau quantization levels are different. For an electron in an s-type conduction band they differ by one, as was established earlier by Rashba (1960 Fiz. Tverd. Tela 2 1224), whereas for heavy holes in a p-type valence band influenced by the 2D symmetry of the layer they differ by three. The shifts and the rearrangements of the 2D hole Landau quantization levels on the energy scale are much larger in comparison with the case of conduction electron Landau levels. This is due to the strong influence of the magnetic field on the RSOC parameter. At sufficiently large values of this parameter the shifts and rearrangements are comparable with the hole cyclotron energy. There are two lowest spin-split Landau levels for electrons as well as four lowest ones for holes in the case of small RSOC parameters. They give rise to eight lowest energy bands of the 2D magnetoexcitons, as well as of the band-to-band quantum transitions. It is shown that three of them are dipole-active, three are quadrupole-active and two are forbidden. The optical orientation under the influence of circularly polarized light leads to optical alignment of the magnetoexcitons with different orbital momentum projections in the direction of the external magnetic field.

The condensation of electron-hole pairs is studied at zero temperature and in the presence of a weak spin-orbit coupling (SOC) in coupled quantum wells. Under realistic conditions, a perturbative SOC can have observable effects in the order parameter of the condensate. First, the fermion exchange symmetry is absent. As a result, the condensate spin has no definite parity. Additionally, the excitonic SOC breaks the rotational symmetry yielding a complex order parameter in an unconventional way; i.e., the phase pattern of the order parameter is a function of the condensate density. This is manifested through finite off-diagonal components of the static spin susceptibility, suggesting a new experimental method to confirm an excitonic condensate.

The Landau quantization of the two-dimensional (2D) heavy holes, its influence on the energy spectrum of 2D magnetoexcitons, as well as their optical orientation are studied. The Hamiltonian of the heavy holes is written in two-band model taking into account the Rashba spin-orbit coupling (RSOC) with two spin projections, but with nonparabolic dispersion law and third-order chirality terms. The most Landau levels, except three with m=0,1,2, are characterized by two quantum numbers m-3 and m for m≥3 for two spin projections correspondingly. The difference between them is determined by the third-order chirality. Four lowest Landau levels (LLLs) for heavy holes were combined with two LLLs for conduction electron, which were taken the same as they were deduced by Rashba in his theory of spin-orbit coupling (SOC) based on the initial parabolic dispersion law and first-order chirality terms. As a result of these combinations eight 2D magnetoexciton states were formed. Their energy spectrum and the selection rules for the quantum transitions from the ground state of the crystal to exciton states were determined. On this base such optical orientation effects as spin polarization and magnetoexciton alignment are discussed. The continuous transformation of the shake-up (SU) into the shake-down (SD) recombination lines is explained on the base of nonmonotonous dependence of the heavy hole Landau quantization levels as a function of applied magnetic field. © 2013 Elsevier B.V. All rights reserved