Limit time optimal synthesis for a two-level quantum system
Résumé
For α ∈ (0, π/2), let (Σ)α be the control system ẋ = (F + uG)x, where × 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(α) and sin(α). In this paper, we study the time optimal synthesis (TOS) from the north pole (0, 0,1)T associated to (E) α, as the parameter α tends to zero; this problem is motivated by specific issues in the control of twolevel 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 α of the south pole (0,0,-1)T of diameter at most O(α). Inside Uα, the TOS depends on the relationship between r(α) = π/2α-[π/2α] and α. More precisely, we characterize three main relationships, by considering sequences (αk)k≥0 satisfying (a) r(αk) = r̄; (b) r(αk) = Cαk and (c) r(αk) = 0, where r̄ ∈ (0,1) and C andgt; 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 α, as α tends to zero. © 2008 IEEE.