Limit time optimal synthesis for a two-level quantum system

• Published in: 2008 47th IEEE Conference on Decision and Control
• Main Authors: Mason, Paolo, Salmoni, Rebecca, Boscain, Ugo, Chitour, Yacine
• Other Authors: Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Laboratoire des signaux et systèmes (L2S), Université Paris-Sud - Paris 11 (UP11)-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS), SISSA MathLab Trieste
• Format: Conference Object
• Language: English
• Published: HAL CCSD 2008
• Subjects: Asymptotics; Control of quantum systems; Control-affine systems; Minimum time; Optimal synthesis; [MATH]Mathematics [math]; North Pole; South Pole; South pole
• Online Access: https://hal.science/hal-02320807; https://doi.org/10.1109/CDC.2008.4738782

International audience 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.