Independent individual addressing of multiple neutral atom qubits with a MEMS beam steering system

We demonstrate a scalable approach to addressing multiple atomic qubits for use in quantum information processing. Individually trapped 87Rb atoms in a linear array are selectively manipulated with a single laser guided by a MEMS beam steering system. Single qubit oscillations are shown on multiple sites at frequencies of ~3.5 MHz with negligible crosstalk to neighboring sites. Switching times between the central atom and its closest neighbor were measured to be 6-7 us while moving between the central atom and an atom two trap sites away took 10-14 us.Comment: 9 pages, 3 figure

Tilting exercises

This is an application of the theory of tilting objects to the geometric setting of perverse sheaves. We show that this theory is a natural framework for Beilinson's gluing of perverse sheaves construction. In the special case of Schubert stratification of a flag variety we get a short proof of Soergel's "Struktursatz", and describe (following a conjecture of Kapranov) Serre functor for category O. Some of our results were obtained independently by Rouquier.Comment: This final version to appear in Moscow Math Journal differs very slightly from the previous on

On the crystalline period map

The paper contains a proof of the Fontaine-Jannsen conjecture based on a crystalline version of the p-adic Poincar'e lemma (different proofs were found earlier by Faltings, Niziol and Tsuji).Comment: 38 pages. The previous version of the paper will appear in March 2013 issue of Cambridge Journal of Mathematics. In present version the exposition of Hyodo-Kato theory (sect. 1.15-1.16) is simplifie

Four dimensional topological quantum field theory, Hopf categories, and the canonical bases

We propose a new mwthod of constructing 4D-TQFTs. The method uses a new type of algebraic structure called a Hopf Category. We also outline the construction of a family of Hopf categories related to the quantum groups, using the canonical bases.Comment: 38 page

Mirror symmetry for Del Pezzo surfaces: Vanishing cycles and coherent sheaves

We study homological mirror symmetry for Del Pezzo surfaces and their mirror Landau-Ginzburg models. In particular, we show that the derived category of coherent sheaves on a Del Pezzo surface X_k obtained by blowing up CP^2 at k points is equivalent to the derived category of vanishing cycles of a certain elliptic fibration W_k:M_k\to\C with k+3 singular fibers, equipped with a suitable symplectic form. Moreover, we also show that this mirror correspondence between derived categories can be extended to noncommutative deformations of X_k, and give an explicit correspondence between the deformation parameters for X_k and the cohomology class [B+i\omega]\in H^2(M_k,C).Comment: 40 pages, 9 figure