Geometric analysis of minimum time trajectories for a two-level quantum system

We consider the problem of controlling in minimum time a two-level quantum system which can be subject to a drift. The control is assumed to be bounded in magnitude, and to affect two or three independent generators of the dynamics. We describe the time optimal trajectories in $SU(2)$, the Lie group of possible evolutions for the system, by means of a particularly simple parametrization of the group. A key ingredient of our analysis is the introduction of the optimal front line. This tool allows us to fully characterize the time-evolution of the reachable sets, and to derive the worst-case operators and the corresponding times. The analysis is performed in any regime: controlled dynamics stronger, of the same magnitude or weaker than the drift term, and gives a method to synthesize quantum logic operations on a two-level system in minimum time.Comment: 27 pages. Formulas (75), (76) and (78) have been correcte

Growing spherulitic calcite grains in saline, hyperalkaline lakes: experimental evaluation of the effects of Mg-clays and organic acids

The origin of spherical-radial calcite bodies – spherulites – in sublacustrine, hyperalkaline and saline systems is unclear, and therefore their palaeoenvironmental significance as allochems is disputed. Here, we experimentally investigate two hypotheses concerning the origin of spherulites. The first is that spherulites precipitate from solutions super-saturated with respect to magnesium-silicate clays, such as stevensite. The second is that spherulite precipitation happens in the presence of dissolved, organic acid molecules. In both cases, experiments were performed under sterile conditions using large batches of a synthetic and cell-free solution replicating waters found in hyperalkaline, saline lakes (such as Mono Lake, California). Our experimental results show that a highly alkaline and highly saline solution supersaturated with respect to calcite (control solution) will precipitate euhedral to subhedral rhombic and trigonal bladed calcite crystals. The same solution supersaturated with respect to stevensite precipitates sheet-like stevensite crystals rather than a gel, and calcite precipitation is reduced by ~ 50% compared to the control solution, producing a mixture of patchy prismatic subhedral to euhedral, and minor needle-like, calcite crystals. Enhanced magnesium concentration in solution is the likely the cause of decreased volumes of calcite precipitation, as this raised equilibrium ion activity ratio in the solution. On the other hand, when alginic acid was present then the result was widespread development of micron-size calcium carbonate spherulite bodies. With further growth time, but falling supersaturation, these spherules fused into botryoidal-topped crusts made of micron-size fibro-radial calcite crystals. We conclude that the simplest tested mechanism to deposit significant spherical-radial calcite bodies is to begin with a strongly supersaturated solution that contains specific but environmentally-common organic acids. Furthermore, we found that this morphology is not a universal consequence of having organic acids dissolved in the solution, but rather spherulite development requires specific binding behaviour. Finally, we found that the location of calcite precipitation was altered from the air:water interface to the surface of the glassware when organic acids were present, implying that attached calcite precipitates reflect precipitation via metal–organic intermediaries, rather than direct forcing via gas exchange

Archaean Soils, Lakes and Springs: Looking for Signs of Life

Microbial life in Archaean non-marine settings like soils, lakes and springs would have faced several challenges. These would have included exposure to UV light; aridity, salinity and temperature changes; and nutrient availability. Current understanding is that none of these challenges would have been insurmountable. Microbial organisms of Archaean marine environments are likely to have been similar in their lifestyles and habits to those of the Archaean terrestrial world. Non-marine stromatolites, microbial filaments, microbial borings and microbially-induced sedimentary structures might therefore have been preserved. But Archaean subaerial surfaces would have been very prone to erosion by wind and rain, so the oldest fossil ‘soils’ of subaerially weathered surfaces (up to 3.47 Ga) are mostly identified using geochemistry. However, some ancient duricrusts like calcretes have been reported. Archaean lacustrine microbial life may have included stromatolites of the Tumbiana Formation of Western Australia. The case that these were lacustrine rather than marine is critically assessed, with the conclusion that the stratigraphy provides the strongest supporting evidence here. Archaean terrestrial hot springs, though often mentioned in origin of life studies, are not yet known from the rock record. In the Palaeoproterozoic to present these silica and carbonate-precipitating environments are commonly found in proximity to volcanic sediments and faults, where the deposits form terraced mounds, fissure ridges and hydrothermal lakes. It remains plausible that life could have existed and even evolved in these hypothesised Archaean hot-spring settings, and there is cause for optimism that the evidence for this might one day be found

Travertine precipitation in the Paleoproterozoic Kuetsjärvi Sedimentary Formation, Pechenga Greenstone Belt, NE Fennoscandian Shield

Precambrian travertines, tufas and speleothems either formed rarely or they have not been identified in previous studies. In the absence of high pC