The reachable set of single-mode quadratic Hamiltonians

Open-loop controllability in quantum mechanics refers to finding conditions on time-varying Hamiltonians such that a full group of unitary transformations can be enacted with them. For compact groups controllability is well understood and is dealt with using the Lie algebra rank criterion. Gaussian systems, however, evolve under Hamiltonians generating the non-compact symplectic group, rendering the rank criterion necessary but no longer sufficient. In this setting it is possible to satisfy the rank criterion without the ability to enact all symplectic transformations. We refer to such systems as 'unstable' and explore the set of symplectic transformations that remain reachable. We provide a partial analytical characterisation for the reachable set of a single-mode unstable system. From this it is proven that no orthogonal-symplectic operations ('energy-preserving' or 'passive' in the literature) may be reached with such controls. We then apply numerical optimal control algorithms to demonstrate a complete characterisation of the set in specific cases. These results suggest approaches to the long-standing open problem of controllability in n modes

From z>6 to z~2: Unearthing Galaxies at the Edge of the Dark Ages

Galaxies undergoing formation and evolution can now be observed over a time baseline of some 12 Gyr. An inherent difficulty with high-redshift observations is that the objects are very faint and the best resolution (HST) is only ~0.5 kpc. Such studies thereby combine in a highly synergistic way with the great detail that can be obtained for nearby galaxies. 3 new developments are highlighted. First is the derivation of stellar masses for galaxies from SEDs using HST and now Spitzer data, and dynamical masses from both sub-mm observations of CO lines and near-IR observations of optical lines like Halpha. A major step has been taken with evidence that points to the z~2-3 LBGs having masses that are a few x 10^10 Msolar. Second is the discovery of a population of evolved red galaxies at z~2-3 which appear to be the progenitors of the more massive early-type galaxies of today, with dynamical masses around a few x 10^11 Msolar. Third are the remarkable advances that have occurred in characterizing dropout galaxies to z~6 and beyond, < 1 Gyr from recombination. The HST ACS has played a key role here, with the dropout technique being applied to i & z images in several deep ACS fields, yielding large samples of these objects. This has allowed a detailed determination of their properties and meaningful comparisons against lower-z samples. The use of cloning techniques has overcome many of the strong selection biases affecting the study of these objects. A clear trend of size with redshift has been identified, and its impact on the luminosity density and SFR estimated. There is a significant though modest decrease in the SFR from z~2.5 to z~6. The latest data also allow for the first robust determination of the LF at z~6. Finally, the latest UDF ACS and NICMOS data has resulted in the detection of some galaxies at z~7-8.Comment: 18 pages, 8 figures. To appear in Penetrating Bars through Masks of Cosmic Dust: The Hubble Tuning Fork Strikes a New Note, eds. D. Block, K. Freeman, R. Groess, I. Puerari, & E.K. Block (Dordrecht: Kluwer), in pres