Hierarchical adaptive polynomial chaos expansions

Polynomial chaos expansions (PCE) are widely used in the framework of uncertainty quantification. However, when dealing with high dimensional complex problems, challenging issues need to be faced. For instance, high-order polynomials may be required, which leads to a large polynomial basis whereas usually only a few of the basis functions are in fact significant. Taking into account the sparse structure of the model, advanced techniques such as sparse PCE (SPCE), have been recently proposed to alleviate the computational issue. In this paper, we propose a novel approach to SPCE, which allows one to exploit the model's hierarchical structure. The proposed approach is based on the adaptive enrichment of the polynomial basis using the so-called principle of heredity. As a result, one can reduce the computational burden related to a large pre-defined candidate set while obtaining higher accuracy with the same computational budget

Some Aspects of the Exact Foldy-Wouthuysen Transformation for a Dirac Fermion

The Foldy-Wouthuysen transformation (FWT) is used to separate distinct components of relativistic spinor field, e.g. electron and positron. Usually, the FWT is perturbative, but in some cases there is an involution operator and the transformation can be done exactly. We consider some aspects of an exact FWT and show that, even if the theory does not admit an involution operator, one can use the technique of exact FWT to obtain the conventional perturbative result. Several particular cases can be elaborated as examples

A complete characterisation of the heralded noiseless amplification of photons

Heralded noiseless amplifcation of photons has recently been shown to provide a means to overcome losses in complex quantum communication tasks. In particular, to overcome transmission losses that could allow for the violation of a Bell inequality free from the detection loophole, for Device Independent Quantum Key Distribution (DI-QKD). Several implementations of a heralded photon amplifier have been proposed and the first proof of principle experiments realised. Here we present the first full characterisation of such a device to test its functional limits and potential for DI-QKD. This device is tested at telecom wavelengths and is shown to be capable of overcoming losses corresponding to a transmission through $20\, \rm km$ of single mode telecom fibre. We demonstrate heralded photon amplifier with a gain $>100$ and a heralding probability $>83$, required by DI-QKD protocols that use the Clauser-Horne-Shimony-Holt (CHSH) inequality. The heralded photon amplifier clearly represents a key technology for the realisation of DI-QKD in the real world and over typical network distances.Comment: 9 pages, 4 figure

Non-quantized Dirac monopoles and strings in the Berry phase of anisotropic spin systems

The Berry phase of an anisotropic spin system that is adiabatically rotated along a closed circuit C is investigated. It is shown that the Berry phase consists of two contributions: (i) a geometric contribution which can be interpreted as the flux through C of a non-quantized Dirac monopole, and (ii) a topological contribution which can be interpreted as the flux through C of a Dirac string carrying a non-quantized flux, i.e., a spin analogue of the Aharonov-Bohm effect. Various experimental consequences of this novel effect are discussed.Comment: 4 pages, 3 figures (RevTeX + eps); v2 (revised paper): 4 pages, 4 figure

Dynamic Active Constraints for Surgical Robots using Vector Field Inequalities

Robotic assistance allows surgeons to perform dexterous and tremor-free procedures, but robotic aid is still underrepresented in procedures with constrained workspaces, such as deep brain neurosurgery and endonasal surgery. In these procedures, surgeons have restricted vision to areas near the surgical tooltips, which increases the risk of unexpected collisions between the shafts of the instruments and their surroundings. In this work, our vector-field-inequalities method is extended to provide dynamic active-constraints to any number of robots and moving objects sharing the same workspace. The method is evaluated with experiments and simulations in which robot tools have to avoid collisions autonomously and in real-time, in a constrained endonasal surgical environment. Simulations show that with our method the combined trajectory error of two robotic systems is optimal. Experiments using a real robotic system show that the method can autonomously prevent collisions between the moving robots themselves and between the robots and the environment. Moreover, the framework is also successfully verified under teleoperation with tool-tissue interactions.Comment: Accepted on T-RO 2019, 19 Page

Equilibrium spin currents and magnetoelectric effect in magnetic nanostructures

We discuss the problem of equilibrium spin currents in ferromagnets with inhomogeneous magnetization. Using simple microscopic models we explain the physical origin of equilibrium spin currents. Next we derive the equilibrium spin current from the Hamiltonian with a gauge field associated with local rotations in the spin space. Several examples of magnetic systems are studied in details, and the persistent spin current is found to exist in the ground state of these systems. We also demonstrate the possibility to measure the equilibrium spin current using the magnetoelectrically induced electric field near the ring.Comment: RevTex 4 (4 pages