Abstract:
We explore the elementary observation that a Markov chain with values in a finite space $M$ with $|M| = m$, $m\geq 2$, has many different extensions to a compatible $n$-point Markov chain in $M^n$, for all $1< n\leq m$. Embedding this phenomenon into the context of stochastic L\'evy flows of diffeomorphisms in Euclidean spaces, we introduce the notion of an $n$-point bifurcation of a stochastic flow. Roughly speaking a $n$-point bifurcation takes place, when a small perturbation of the stochastic flow does not change the characteristics at lower level $k$-point motions, $k

Abstract:
This article shows a strong averaging principle for diffusions driven by discontinuous heavy-tailed L\'evy noise, which are invariant on the compact horizontal leaves of a foliated manifold subject to small transversal random perturbations. We extend a result for such diffusions with exponential moments and bounded, deterministic perturbations to diffusions with polynomial moments of order $p \geq 2$, perturbed by deterministic and stochastic integrals with unbounded coefficients and polynomial moments. The main argument relies on a result of the dynamical system for each individual jump increments of the corresponding canonical Marcus equation. The example of L\'evy rotations on the unit circle subject to perturbations by a planar Ornstein-Uhlenbeck process is carried out in detail.

Abstract:
We used resonant laser spectroscopy of multiple confocal InGaAs quantum dots to spatially locate charge fluctuators in the surrounding semiconductor matrix. By mapping out the resonance condition between a narrow-band laser and the neutral exciton transitions of individual dots in a field effect device, we identified spectral discontinuities as arising from charging and discharging events that take place within the volume adjacent to the quantum dots. Our analysis suggests that residual carbon dopants are a major source of charge-fluctuating traps in quantum dot heterostructures.

Abstract:
Resonant optical excitation of lowest-energy excitonic transitions in self-assembled quantum dots lead to nuclear spin polarization that is qualitatively different from the well known optical orientation phenomena. By carrying out a comprehensive set of experiments, we demonstrate that nuclear spin polarization manifests itself in quantum dots subjected to finite external magnetic field as locking of the higher energy Zeeman transition to the driving laser field, as well as the avoidance of the resonance condition for the lower energy Zeeman branch. We interpret our findings on the basis of dynamic nuclear spin polarization originating from non-collinear hyperfine interaction and find an excellent agreement between the experimental results and the theoretical model.

Abstract:
We show how resonant laser spectroscopy of the trion optical transitions in a self-assembled quantum dot can be used to determine the temperature of a nearby electron reservoir. At finite magnetic field the spin-state occupation of the Zeeman-split quantum dot electron ground states is governed by thermalization with the electron reservoir via co-tunneling. With resonant spectroscopy of the corresponding excited trion states we map out the spin occupation as a function of magnetic field to establish optical thermometry for the electron reservoir. We demonstrate the implementation of the technique in the sub-Kelvin temperature range where it is most sensitive, and where the electron temperature is not necessarily given by the cryostat base temperature.

Abstract:
A Quantum Point Contact (QPC) causes a one-dimensional constriction on the spatial potential landscape of a two-dimensional electron system. By tuning the voltage applied on a QPC at low temperatures the resulting regular step-like electron conductance quantization can show an additional kink near pinch-off around 0.7(2$e^2$/h), called 0.7-anomaly. In a recent publication, we presented a combination of theoretical calculations and transport measurements that lead to a detailed understanding of the microscopic origin of the 0.7-anomaly. Functional Renormalization Group-based calculations were performed exhibiting the 0.7-anomaly even when no symmetry-breaking external magnetic fields are involved. According to the calculations the electron spin susceptibility is enhanced within a QPC that is tuned in the region of the 0.7-anomaly. Moderate externally applied magnetic fields impose a corresponding enhancement in the spin magnetization. In principle, it should be possible to map out this spin distribution optically by means of the Faraday rotation technique. Here we report the initial steps of an experimental project aimed at realizing such measurements. Simulations were performed on a particularly pre-designed semiconductor heterostructure. Based on the simulation results a sample was built and its basic transport and optical properties were investigated. Finally, we introduce a sample gate design, suitable for combined transport and optical studies.

Abstract:
We show how the optical properties of a single semiconductor quantum dot can be controlled with a small dc voltage applied to a gate electrode. We find that the transmission spectrum of the neutral exciton exhibits two narrow lines with $\sim 2$ $\mu$eV linewidth. The splitting into two linearly polarized components arises through an exchange interaction within the exciton. The exchange interaction can be turned off by choosing a gate voltage where the dot is occupied with an additional electron. Saturation spectroscopy demonstrates that the neutral exciton behaves as a two-level system. Our experiments show that the remaining problem for manipulating excitonic quantum states in this system is spectral fluctuation on a $\mu$eV energy scale.

Abstract:
We introduce a subexponential algorithm for geometric solving of multivariate polynomial equation systems whose bit complexity depends mainly on intrinsic geometric invariants of the solution set. From this algorithm, we derive a new procedure for the decision of consistency of polynomial equation systems whose bit complexity is subexponential, too. As a byproduct, we analyze the division of a polynomial modulo a reduced complete intersection ideal and from this, we obtain an intrinsic lower bound for the logarithmic height of diophantine approximations to a given solution of a zero--dimensional polynomial equation system. This result represents a multivariate version of Liouville's classical theorem on approximation of algebraic numbers by rationals. A special feature of our procedures is their {\em polynomial} character with respect to the mentioned geometric invariants when instead of bit operations only arithmetic operations are counted at unit cost. Technically our paper relies on the use of straight--line programs as a data structure for the encoding of polynomials, on a new symbolic application of Newton's algorithm to the Implicit Function Theorem and on a special, basis independent trace formula for affine Gorenstein algebras.

Abstract:
We consider a finite dimensional deterministic dynamical system with a global attractor A with a unique ergodic measure P concentrated on it, which is uniformly parametrized by the mean of the trajectories in a bounded set D containing A. We perturbe this dynamical system by a multiplicative heavy tailed L\'evy noise of small intensity \epsilon>0 and solve the asymptotic first exit time and location problem from a bounded domain D around the attractor A in the limit of {\epsilon} to 0. In contrast to the case of Gaussian perturbations, the exit time has the asymptotically algebraic exit rate as a function of \epsilon, just as in the case when A is a stable fixed point. In the small noise limit, we determine the joint law of the first time and the exit location on the complement of D. As an example, we study the first exit problem from a neighbourhood of a stable limit cycle for the Van der Pol oscillator perturbed by multiplicative \alpha-stable L\'evy noise.

Abstract:
The zero-noise limit of differential equations with singular coefficients is investigated for the first time in the case when the noise is an $\alpha $-stable process. It is proved that extremal solutions are selected and the respective probability of selection is computed. For this purpose an exit time problem from the half-line, which is of interest in its own right, is formulated and studied by means of a suitable decomposition in small and large jumps adapted to the singular drift.