Strong Stationarity Conditions for Optimal Control of Hybrid Systems

We present necessary and sufficient optimality conditions for finite time optimal control problems for a class of hybrid systems described by linear complementarity models. Although these optimal control problems are difficult in general due to the presence of complementarity constraints, we provide a set of structural assumptions ensuring that the tangent cone of the constraints possesses geometric regularity properties. These imply that the classical Karush-Kuhn-Tucker conditions of nonlinear programming theory are both necessary and sufficient for local optimality, which is not the case for general mathematical programs with complementarity constraints. We also present sufficient conditions for global optimality. We proceed to show that the dynamics of every continuous piecewise affine system can be written as the optimizer of a mathematical program which results in a linear complementarity model satisfying our structural assumptions. Hence, our stationarity results apply to a large class of hybrid systems with piecewise affine dynamics. We present simulation results showing the substantial benefits possible from using a nonlinear programming approach to the optimal control problem with complementarity constraints instead of a more traditional mixed-integer formulation.Comment: 30 pages, 4 figure

The outer crust of non-accreting cold neutron stars

The properties of the outer crust of non-accreting cold neutron stars are studied by using modern nuclear data and theoretical mass tables updating in particular the classic work of Baym, Pethick and Sutherland. Experimental data from the atomic mass table from Audi, Wapstra, and Thibault of 2003 is used and a thorough comparison of many modern theoretical nuclear models, relativistic and non-relativistic ones, is performed for the first time. In addition, the influences of pairing and deformation are investigated. State-of-the-art theoretical nuclear mass tables are compared in order to check their differences concerning the neutron dripline, magic neutron numbers, the equation of state, and the sequence of neutron-rich nuclei up to the dripline in the outer crust of non-accreting cold neutron stars.Comment: 20 pages, 10 figures, accepted for publication in Phys. Rev.

EIT ground-state cooling of long ion strings

Electromagnetically-induced-transparency (EIT) cooling is a ground-state cooling technique for trapped particles. EIT offers a broader cooling range in frequency space compared to more established methods. In this work, we experimentally investigate EIT cooling in strings of trapped atomic ions. In strings of up to 18 ions, we demonstrate simultaneous ground state cooling of all radial modes in under 1 ms. This is a particularly important capability in view of emerging quantum simulation experiments with large numbers of trapped ions. Our analysis of the EIT cooling dynamics is based on a novel technique enabling single-shot measurements of phonon numbers, by rapid adiabatic passage on a vibrational sideband of a narrow transition

Quantum simulation of the Klein paradox with trapped ions

We report on quantum simulations of relativistic scattering dynamics using trapped ions. The simulated state of a scattering particle is encoded in both the electronic and vibrational state of an ion, representing the discrete and continuous components of relativistic wave functions. Multiple laser fields and an auxiliary ion simulate the dynamics generated by the Dirac equation in the presence of a scattering potential. Measurement and reconstruction of the particle wave packet enables a frame-by-frame visualization of the scattering processes. By precisely engineering a range of external potentials we are able to simulate text book relativistic scattering experiments and study Klein tunneling in an analogue quantum simulator. We describe extensions to solve problems that are beyond current classical computing capabilities.Comment: 3 figures, accepted for publication in PR

Observation of Entangled States of a Fully Controlled 20-Qubit System

We generate and characterise entangled states of a register of 20 individually controlled qubits, where each qubit is encoded into the electronic state of a trapped atomic ion. Entanglement is generated amongst the qubits during the out-of-equilibrium dynamics of an Ising-type Hamiltonian, engineered via laser fields. Since the qubit-qubit interactions decay with distance, entanglement is generated at early times predominantly between neighbouring groups of qubits. We characterise entanglement between these groups by designing and applying witnesses for genuine multipartite entanglement. Our results show that, during the dynamical evolution, all neighbouring qubit pairs, triplets, most quadruplets, and some quintuplets simultaneously develop genuine multipartite entanglement. Witnessing genuine multipartite entanglement in larger groups of qubits in our system remains an open challenge.Comment: 20 pages, 4 figure