 
Summary: ESAIM: COCV ESAIM: Control, Optimisation and Calculus of Variations
July 2006, Vol. 12, 409441 www.edpsciences.org/cocv
DOI: 10.1051/cocv:2006007
AN ESTIMATION OF THE CONTROLLABILITY TIME FOR SINGLEINPUT
SYSTEMS ON COMPACT LIE GROUPS
Andrei Agrachev1
and Thomas Chambrion2
Abstract. Geometric control theory and Riemannian techniques are used to describe the reachable
set at time t of left invariant singleinput control systems on semisimple compact Lie groups and to
estimate the minimal time needed to reach any point from identity. This method provides an effective
way to give an upper and a lower bound for the minimal time needed to transfer a controlled quantum
system with a drift from a given initial position to a given final position. The bounds include diameters
of the flag manifolds; the latter are also explicitly computed in the paper.
Mathematics Subject Classification. 22E46, 93B03.
Received September 4, 2004. Revised May 8, 2005.
1. Introduction
1.1. Physical context
Some recent papers [912, 14, 15, 22, 32] have studied the problem of transferring a population of quantum
particles from a quantum state to another, using an external field (laser). In many applications (e.g., MNR)
a crucial issue is to achieve this quantum transfer in the shortest possible time, in order to avoid relaxation
