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Time Optimal Control of Nonlinear Bloch Equations

Irigo Edouard Zibo, Rachida El Assoudi-Baikari, Nicolas Forcadel

Published Sep 23, 2020• 0 citations
OpenAlex

Abstract

We deal with time optimal control of nonlinear Bloch equations describing the influence of Coulomb parameters on particle dynamics. The dynamic is analyzed using tools of geometric optimal control theory. We present different time optimal syntheses by varying the parameters of a two energy levels quantum dots system. We show the nontrivial role of Coulomb parameters on the time minimal trajectories.

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The time optimal problem for a two level quantum system is studied. We compare two different control strategies of bang-bang control and the geometric control, respectively, especially in the case of minimizing the time of steering the state from North Pole to South Pole on the Bloch sphere with bounded control. The time performances are compared for different parameters by the individual numerical simulation experiments, and the experimental results are analyzed. The results show that the geometric control spends less time than the bang-bang control does.
We propose an analysis of the time-optimal control of a dissipative two-level quantum system whose dynamics is governed by the Lindblad equation. This simple system allows one to use tools of geometric control theory and to construct its optimal synthesis, i.e., to determine the set of all the optimal trajectories starting from a given initial point. We study different processes such as conversion of a pure state into a mixed state and purification of a mixed state. In particular cases, we show that dissipation is not undesirable and can help accelerating the control.
We study time-minimum optimal control for a class of quantum two-dimensional dissipative systems whose dynamics are governed by the Lindblad equation and where control inputs acts only in the Hamiltonian. The dynamics of the control system are analyzed as a bi-linear control system on the Bloch ball after a decoupling of such dynamics into intra- and inter-unitary orbits. The (singular) control problem consists of finding a trajectory of the state variables solving a radial equation in the minimum amount of time, starting at the completely mixed state and ending at the state with the maximum achievable purity.The boundary value problem determined by the time-minimum singular optimal control problem is studied numerically. If controls are unbounded, simulations show that multiple local minimal solutions might exist. To find the unique globally minimal solution, we must repeat the algorithm for various initial conditions and find the best solution out of all of the candidates. If controls are bounded, optimal controls are given by bang-bang controls using the Pontryagin minimum principle. Using a switching map we construct optimal solutions consisting of singular arcs. If controls are bounded, the analysis of our model also implies classical analysis done previously for this problem.