We have investigated Bose-Einstein condensates and ultra cold atoms in the vicinity of a surface of a magnetic microtrap. The atoms are prepared along copper conductors at distances to the surface between 300 µm and 20 µm. In this range, the lifetime decreases from 20 s to 0.7 s showing a linear dependence on the distance to the surface. The atoms manifest a weak thermal coupling to the surface, with measured heating rates remaining below 500 nK/s. In addition, we observe a periodic fragmentation of the condensate and thermal clouds when the surface is approached.PACS numbers: 03.75. Fi, 03.75.Be, 34.50.Dy, Micropotentials have proven to be a powerful tool to manipulate and structure the shape of a Bose-Einstein condensate [1] on a length scale shorter than the coherence length of the condensate. Besides the manipulation with light [2,3,4,5,6], current carrying microstructures [7] are particularly interesting since they can be tailored in an arbitrary way, providing a variety of potential geometries. In previous experiments with magnetic microtraps the work was mainly focused on the demonstration of different trapping geometries, loading schemes and guiding principles [8,9,10,11,12,13,14]. The recent realization of Bose-Einstein condensates in magnetic microtraps [15,16] however provides new possibilities to control coherent matter on the micrometer scale. Coherent beam splitters, on-chip interferometers or quantum dots may become feasible. In current experimental setups, the dimensions of the conductors vary from 1 µm to 100 µm and the distance between the trap minimum and the surface is typically of the same size. At such small distances the atoms are affected by the nearby surface. For experiments with coherent matter waves or even single atoms in microfabricated traps an understanding of the mutual influences of the atoms and the surface is highly desirable.In this letter, we describe three effects on ultra cold atoms which appear in the vicinity of the surface of a magnetic microtrap. We observe a decrease of the lifetime of the atomic cloud which scales roughly linearly with the distance to the surface. At 20 µm, the lifetime is reduced to less than 1 s, which has to be compared to the "far distance" value of 100 s. Simultaneously, an increased heating rate is observed which, however, does not exceed 500 nK/s. Furthermore, a periodic fragmentation of both, the thermal cloud and the condensate occurs when the surface is approached at distances of about 250 µm. This gives strong evidence for additional potentials arising from the nearby surface.In our experiment, the microtrap is generated by a microstructure which consists of seven parallel copper conductors with widths of 3 µm, 11 µm and 30 µm, a height of 2.5 µm and a length of 25 mm [17]. The conductors are electroplated on an Al 2 O 3 ceramic substrate.An additional copper wire with a circular diameter of 90 µm is mounted parallel to the microstructure, allowing for reference measurements. The free surface of the wire is in the plane of the fabricated c...

We have used ultracold atoms to characterize the magnetic field near the surface of copper conductors at room temperature carrying currents between 0.045 A and 2 A. In addition to the usual circular field we find an additional, 1000 − 10000 times smaller longitudinal field. The field changes its strength periodically with a period of 200 − 300 µm.

Scanning probe microscopes are widely used to study surfaces with atomic resolution in many areas of nanoscience. Ultracold atomic gases trapped in electromagnetic potentials can be used to study electromagnetic interactions between the atoms and nearby surfaces in chip-based systems. Here we demonstrate a new type of scanning probe microscope that combines these two areas of research by using an ultracold gas as the tip in a scanning probe microscope. This cold-atom scanning probe microscope offers a large scanning volume, an ultrasoft tip of well-defined shape and high purity, and sensitivity to electromagnetic forces (including dispersion forces near nanostructured surfaces). We use the cold-atom scanning probe microscope to non-destructively measure the position and height of carbon nanotube structures and individual free-standing nanotubes. Cooling the atoms in the gas to form a Bose-Einstein condensate increases the resolution of the device.

We consider the variational discretization of elliptic Dirichlet optimal control problems with constraints on the control. The underlying state equation, which is considered on smooth twoand three-dimensional domains, is discretized by linear finite elements taking into account domain approximation. The control variable is not discretized. We obtain optimal error bounds for the optimal control in two and three space dimensions and prove a superconvergence result in two dimensions provided that the underlying mesh satisfies some additional condition. We confirm our analytical findings by numerical experiments.

We consider an elliptic optimal control problem with control constraints and pointwise bounds on the gradient of the state. We present a tailored finite element approximation to this optimal control problem, where the cost functional is approximated by a sequence of functionals which are obtained by discretizing the state equation with the help of the lowest order Raviart-Thomas mixed finite element. Pointwise bounds on the gradient variable are enforced in the elements of the triangulation. Controls are not discretized. Error bounds for control and state are obtained in two and three space dimensions. A numerical example confirms our analytical findings.