Trudy Mat. Inst. Steklova, 2021, Volume 313, Pages 161–177
On Reachable and Controllability Sets for Minimum-Time Control of an Open Two-Level Quantum System
Oleg V. Morzhin, Alexander N. Pechen
Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia
We consider a two-level open quantum system whose dynamics is governed by the Gorini–Kossakowski–Sudarshan–Lindblad equation with Hamiltonian and dissipation superoperator depending, respectively, on coherent and incoherent controls. Results about reachability, controllability, and minimum-time control are obtained in terms of the Bloch parametrization. First, we consider the case when the zero coherent and incoherent controls satisfy the Pontryagin maximum principle in the class of piecewise continuous controls. Second, for zero coherent control and for incoherent control lying in the class of constant functions, the reachability and controllability sets of the system are exactly described and some analytical results on the minimum-time control are found. Third, we consider a series of increasing values of the final time and the corresponding classes of controls with zero incoherent control and with coherent control equal to zero until a switching time instant and to a cosine function after it. The corresponding reachable points in the Bloch ball are numerically obtained and visualized. Fourth, a known method for estimating reachable sets is adapted and used to analyze the situation where the zero coherent and incoherent controls satisfy the Pontryagin maximum principle in the class of piecewise continuous controls while, as shown numerically, are not optimal.
|Russian Science Foundation
|This work is supported by the Russian Science Foundation under grant 17-11-01388.
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Proceedings of the Steklov Institute of Mathematics, 2021, 313, 149–164
Received: August 25, 2020
Revised: December 13, 2020
Accepted: December 26, 2020
Oleg V. Morzhin, Alexander N. Pechen, “On Reachable and Controllability Sets for Minimum-Time Control of an Open Two-Level Quantum System”, Mathematics of Quantum Technologies, Collected papers, Trudy Mat. Inst. Steklova, 313, Steklov Math. Inst., Moscow, 2021, 161–177; Proc. Steklov Inst. Math., 313 (2021), 149–164
Citation in format AMSBIB
\by Oleg~V.~Morzhin, Alexander~N.~Pechen
\paper On Reachable and Controllability Sets for Minimum-Time Control of an Open Two-Level Quantum System
\inbook Mathematics of Quantum Technologies
\bookinfo Collected papers
\serial Trudy Mat. Inst. Steklova
\publ Steklov Math. Inst.
\jour Proc. Steklov Inst. Math.
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