Signatures of hermitian forms and the Knebusch Trace Formula

Signatures of quadratic forms have been generalized to hermitian forms over algebras with involution. In the literature this is done via Morita theory, which causes sign ambiguities in certain cases. In this paper, a hermitian version of the Knebusch Trace Formula is established and used as a main tool to resolve these ambiguities. The last page is an erratum for the published version. We inadvertently (I) gave an incorrect definition of adjoint involutions; (II) omitted dealing with the case $(H\times H, \widehat{\phantom{m}}\,)$. As $W(H\times H, \widehat{\phantom{m}}\,)= W(R\times R, \widehat{\phantom{m}}\,)=0$, the omission does not affect our reasoning or our results. For the sake of completeness we point out where some small changes should be made in the published version.Comment: This is the final version before publication. The last page is an updated erratum for the published versio

Division, adjoints, and dualities of bilinear maps

The distributive property can be studied through bilinear maps and various morphisms between these maps. The adjoint-morphisms between bilinear maps establish a complete abelian category with projectives and admits a duality. Thus the adjoint category is not a module category but nevertheless it is suitably familiar. The universal properties have geometric perspectives. For example, products are orthogonal sums. The bilinear division maps are the simple bimaps with respect to nondegenerate adjoint-morphisms. That formalizes the understanding that the atoms of linear geometries are algebraic objects with no zero-divisors. Adjoint-isomorphism coincides with principal isotopism; hence, nonassociative division rings can be studied within this framework. This also corrects an error in an earlier pre-print; see Remark 2.11

Spin coherence of holes in GaAs/AlGaAs quantum wells

The carrier spin coherence in a p-doped GaAs/(Al,Ga)As quantum well with a diluted hole gas has been studied by picosecond pump-probe Kerr rotation with an in-plane magnetic field. For resonant optical excitation of the positively charged exciton the spin precession shows two types of oscillations. Fast oscillating electron spin beats decay with the radiative lifetime of the charged exciton of 50 ps. Long lived spin coherence of the holes with dephasing times up to 650 ps. The spin dephasing time as well as the in-plane hole g factor show strong temperature dependence, underlining the importance of hole localization at cryogenic temperatures.Comment: 5 pages, 4 figures in PostScript forma

Longitudinal and transversal spin dynamics of donor-bound electrons in fluorine-doped ZnSe: spin inertia versus Hanle effect

The spin dynamics of the strongly localized, donor-bound electrons in fluorine-doped ZnSe epilayers is studied by pump-probe Kerr rotation techniques. A method exploiting the spin inertia is developed and used to measure the longitudinal spin relaxation time, $T_1$, in a wide range of magnetic fields, temperatures, and pump densities. The $T_1$ time of the donor-bound electron spin of about 1.6 $\mu$s remains nearly constant for external magnetic fields varied from zero up to 2.5 T (Faraday geometry) and in a temperature range $1.8-45$ K. The inhomogeneous spin dephasing time, $T_2^*=8-33$ ns, is measured using the resonant spin amplification and Hanle effects under pulsed and steady-state pumping, respectively. These findings impose severe restrictions on possible spin relaxation mechanisms.Comment: 10 pages, 7 figure

Spin dynamics and magnetic-field-induced polarization of excitons in ultrathin GaAs/AlAs quantum wells with indirect band gap and type-II band alignment

The exciton spin dynamics are investigated both experimentally and theoretically in two-monolayer-thick GaAs/AlAs quantum wells with an indirect band gap and a type-II band alignment. The magnetic-field-induced circular polarization of photoluminescence, $P_c$, is studied as function of the magnetic field strength and direction as well as sample temperature. The observed nonmonotonic behaviour of these functions is provided by the interplay of bright and dark exciton states contributing to the emission. To interpret the experiment, we have developed a kinetic master equation model which accounts for the dynamics of the spin states in this exciton quartet, radiative and nonradiative recombination processes, and redistribution of excitons between these states as result of spin relaxation. The model offers quantitative agreement with experiment and allows us to evaluate, for the studied structure, the heavy-hole $g$ factor, $g_{hh}=+3.5$, and the spin relaxation times of electron, $\tau_{se} = 33~\mu$s, and hole, $\tau_{sh} = 3~\mu$s, bound in the exciton.Comment: 17 pages, 16 figure

Level Eulerian Posets

The notion of level posets is introduced. This class of infinite posets has the property that between every two adjacent ranks the same bipartite graph occurs. When the adjacency matrix is indecomposable, we determine the length of the longest interval one needs to check to verify Eulerianness. Furthermore, we show that every level Eulerian poset associated to an indecomposable matrix has even order. A condition for verifying shellability is introduced and is automated using the algebra of walks. Applying the Skolem--Mahler--Lech theorem, the ${\bf ab}$-series of a level poset is shown to be a rational generating function in the non-commutative variables ${\bf a}$ and ${\bf b}$. In the case the poset is also Eulerian, the analogous result holds for the ${\bf cd}$-series. Using coalgebraic techniques a method is developed to recognize the ${\bf cd}$-series matrix of a level Eulerian poset

Optical control of electron spin coherence in CdTe/(Cd,Mg)Te quantum wells

Optical control of the spin coherence of quantum well electrons by short laser pulses with circular or linear polarization is studied experimentally and theoretically. For that purpose the coherent electron spin dynamics in a n-doped CdTe/(Cd,Mg)Te quantum well structure was measured by time-resolved pump-probe Kerr rotation, using resonant excitation of the negatively charged exciton (trion) state. The amplitude and phase shifts of the electron spin beat signal in an external magnetic field, that are induced by laser control pulses, depend on the pump-control delay and polarization of the control relative to the pump pulse. Additive and non-additive contributions to pump-induced signal due to the control are isolated experimentally. These contributions can be well described in the framework of a two-level model for the optical excitation of the resident electron to the trion.Comment: 15 pages, 18 figure