Symmetry reduction and recovery of trajectories of optimal control problems via measure relaxations

We address the problem of symmetry reduction of optimal control problems under the action of a finite group from a measure relaxation viewpoint. We propose a method based on the moment-SOS aka Lasserre hierarchy which allows one to significantly reduce the computation time and memory requirements compared to the case without symmetry reduction. We show that the recovery of optimal trajectories boils down to solving a symmetric parametric polynomial system. Then we illustrate our method on the symmetric integrator and the time-optimal inversion of qubits.Comment: 38 pages, 23 figure

Dirichlet Form Theory and its Applications

Theory of Dirichlet forms is one of the main achievements in modern probability theory. It provides a powerful connection between probabilistic and analytic potential theory. It is also an effective machinery for studying various stochastic models, especially those with non-smooth data, on fractal-like spaces or spaces of infinite dimensions. The Dirichlet form theory has numerous interactions with other areas of mathematics and sciences. This workshop brought together top experts in Dirichlet form theory and related fields as well as promising young researchers, with the common theme of developing new foundational methods and their applications to specific areas of probability. It provided a unique opportunity for the interaction between the established scholars and young researchers

Models of wave-function collapse, underlying theories, and experimental tests

We describe the state of the art in preparing, manipulating and detecting coherent molecular matter. We focus on experimental methods for handling the quantum motion of compound systems from diatomic molecules to clusters or biomolecules.Molecular quantum optics offers many challenges and innovative prospects: already the combination of two atoms into one molecule takes several well-established methods from atomic physics, such as for instance laser cooling, to their limits. The enormous internal complexity that arises when hundreds or thousands of atoms are bound in a single organic molecule, cluster or nanocrystal provides a richness that can only be tackled by combining methods from atomic physics, chemistry, cluster physics, nanotechnology and the life sciences.We review various molecular beam sources and their suitability for matter-wave experiments. We discuss numerous molecular detection schemes and give an overview over diffraction and interference experiments that have already been performed with molecules or clusters.Applications of de Broglie studies with composite systems range from fundamental tests of physics up to quantum-enhanced metrology in physical chemistry, biophysics and the surface sciences.Nanoparticle quantum optics is a growing field, which will intrigue researchers still for many years to come. This review can, therefore, only be a snapshot of a very dynamical process

Exact analytical solutions to the master equation of quantum Brownian motion for a general environment

We revisit the model of a quantum Brownian oscillator linearly coupled to an environment of quantum oscillators at finite temperature. By introducing a compact and particularly well-suited formulation, we give a rather quick and direct derivation of the master equation and its solutions for general spectral functions and arbitrary temperatures. The flexibility of our approach allows for an immediate generalization to cases with an external force and with an arbitrary number of Brownian oscillators. More importantly, we point out an important mathematical subtlety concerning boundary-value problems for integro-differential equations which led to incorrect master equation coefficients and impacts on the description of nonlocal dissipation effects in all earlier derivations. Furthermore, we provide explicit, exact analytical results for the master equation coefficients and its solutions in a wide variety of cases, including ohmic, sub-ohmic and supra-ohmic environments with a finite cut-off.Comment: 37 pages (26 + appendices), 14 figures; this paper is an evolution of arXiv:0705.2766v1, but contains far more general and significant results; v2 minor changes, double column, improved Appendix

Generalized quantum mean-field systems and their application to ultracold atoms

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