Extended F_4-buildings and the Baby Monster

The Baby Monster group B acts naturally on a geometry E(B) with diagram c.F_4(t) for t=4 and the action of B on E(B) is flag-transitive. It possesses the following properties: (a) any two elements of type 1 are incident to at most one common element of type 2, and (b) three elements of type 1 are pairwise incident to common elements of type 2 iff they are incident to a common element of type 5. It is shown that E(B) is the only (non-necessary flag-transitive) c.F_4(t)-geometry, satisfying t=4, (a) and (b), thus obtaining the first characterization of B in terms of an incidence geometry, similar in vein to one known for classical groups acting on buildings. Further, it is shown that E(B) contains subgeometries E(^2E_6(2)) and E(Fi22) with diagrams c.F_4(2) and c.F_4(1). The stabilizers of these subgeometries induce on them flag-transitive actions of ^2E_6(2):2 and Fi22:2, respectively. Three further examples for t=2 with flag-transitive automorphism groups are constructed. A complete list of possibilities for the isomorphism type of the subgraph induced by the common neighbours of a pair of vertices at distance 2 in an arbitrary c.F_4(t) satisfying (a) and (b) is obtained.Comment: to appear in Inventiones Mathematica

Landau-Zener interferometry for qubits

One may probe intrinsic coherence of a qubit by periodically sweeping its control parameter. The qubit is then excited by Landau-Zener (LZ) mechanism. The interference between multiple LZ transition leads to the unusual oscillatory dependence of the energy absorption rate on sweeping amplitude and period. This interference pattern allows to determine the decoherence time of the qubit. We introduce a simple phenomenological model describing this "interferometer", and find the form of the interference pattern.Comment: 7 pages, 2 figures; added: microscopic derivation of the master equation, finite temperature of the reservoi