Controlled random matrix, optimal control

This chapter is organized as follows. In Section II, we show how quantum chaos systems can be controlled under the optimal fields obtained by OCT. The examples are a random matrix system and a quantum kicked rotor. (The former is considered as a strong-chaos-limit case, and the latter is considered as mixed regular-chaotic cases.) In Section III, a coarse-grained Rabi state is introduced to analyze the controlled dynamics in quantum chaos systems. We numerically obtain a smooth transition between time-dependent states, which justifies the use of such a picture. In Section IV, we derive an analytic expression for the optimal field under the assumption of the CG Rabi state, and we numerically show that the field can really steer an initial state to a target state in random matrix systems. Finally, we summarize the chapter and discuss further aspects of controlling quantum chaos. 

Figure 1. Optimal control between Gaussian random vectors in a 64 x 64 random matrix system by the Zhu-Botina-Rabitz scheme with T — 20 and a = 1. a) The optimal field after 100 iterations b) its power spectrum (c) the optimal evolution of the squared overlap with the target (([)(r) (py) as well as its magnified values near the target time in the inset (d) the convergence behavior of the overlap Jq (solid curve) and the functional J (dashed curve) versus the number of iteration steps.

In Section II.A, we have already obtained the optimal field e t) by the numerical calculation for the random matrix systems, Eq. (6). However, only the overlap between the time-evolving controlled state < )(t)) and the target state (pj) was shown there. In this section, we show the overlaps between the time-dependent states defined by Eq. (15) and < )(t)), and we find a smooth transition picture. 

Figure 1 illustrates the excellent agreement between the experimental data on electron tunneling and Eq. (7 a) for the decay of e,7 by the reaction with Cu(en) + (en represents here ethylenediamine) in a water-alkaline (10 M NaOH) vitreous matrix at 77 K. The random character of Cu(en)2 + spatial distribution was controlled in this experiment by special measurements. In Fig. 1 the solid lines represent theoretical curves calculated by means of Eq. (7 a) and the optimal values of the parameters ve = 1015 2 s 1 and ae = 1.83 A selected so as to fit best all the four experimental curves simultaneously. Equation (7a) is seen to describe quite well the reaction kinetics over 13 orders of magnitude variation of time and 1 order of magnitude variation of acceptor concentration. 

Figure 10. The target-time dependence of the final overlap 7o by the analytic optimal field with k — 1 is shown. The residual probability 1 — 7o from perfect control = 1 is depicted for various matrix sizes N of GOE random matrices. The initial and the final states are Gaussian random vectors.

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