Limit time optimal synthesis for a two-level quantum system

For ¿ ¿ (0, ¿/2), let (¿)_¿ be the control system x¿ = (F + uG)x, where x belongs to the two-dimensional unit sphere S², u ¿ [-1, 1] and F,G are 3 × 3 skew-symmetric matrices generating rotations with perpendicular axes and of respective norms cos(¿) and sin(¿). In this paper, we study the time optimal synthesis (TOS) from the north pole (0, 0, 1)^T associated to (¿)_¿, as the parameter ¿ tends to zero; this problem is motivated by specific issues in the control of two-level quantum systems subject to weak external fields. The TOS is characterized by a ¿two-snakes¿ configuration on the whole S², except for a neighborhood U_¿ of the south pole (0, 0,-1)^T of diameter at most O(¿). Inside U_¿, the TOS depends on the relationship between r(¿) := ¿/2¿-[¿/2¿] and ¿. More precisely, we characterize three main relationships, by considering sequences (¿_k)_k¿0 satisfying (a) r(¿_k) = r¿ (b) r(¿_k) = C¿_k and (c) r(¿_k) = 0, where r¿ ¿ (0, 1) and C ¿ 0. In each case we describe the TOS and, in the case (a), we provide, after a suitable rescaling, the limit behavior of the corresponding TOS inside U_¿, as ¿ tends to zero.