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arXiv:1810.07134 · Physical Review A

Time-optimal selective pulses of two uncoupled spin-1/2 particles

Léo Van Damme, Quentin Ansel, Steffen J. Glaser, Dominique Sugny

Published Oct 16, 2018• 28 citations
arXivHTMLPDF

Abstract

We investigate the time-optimal solution of the selective control of two uncoupled spin 1/2 particles. Using the Pontryagin maximum principle, we derive the global time-optimal pulses for two spins with different offsets. We show that the Pontryagin Hamiltonian can be written as a one-dimensional effective Hamiltonian. The optimal fields can be expressed analytically in terms of elliptic integrals. The time-optimal control problem is solved for the selective inversion and excitation processes. A bifurcation in the structure of the control fields occurs for a specific offset threshold. In particular, we show that for small offsets, the optimal solution is the concatenation of regular and singular extremals.

DOI
10.1103/physreva.98.043421

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We derive the explicit solution of the problem of time-optimal control by a common magnetic field for two independent spin-12\frac{1}{2}21​ particles. Our approach is based on the Pontryagin maximum principle and a symmetry reduction technique. We experimentally implement the optimal control using zero-field nuclear magnetic resonance. This reveals an average gate error of 11%1 and a 70--8080%80 decrease in the experiment duration as compared to existing methods. This analytical solution and experimental demonstration of time-optimal control in such a system provides a route to achieve time-optimal control in more general quantum systems.
We investigate the time- and energy-minimum optimal solutions for the robust control of two-level quantum systems against offset or control-field uncertainties. Using the Pontryagin maximum principle, we derive the global optimal pulses for the first robustness orders. We show that the dimension of the control landscape is lower than or equal to 2N2N2N for a field robust to the NthN\mathrm{th}Nth order, which leads to an estimate of its complexity.
We study the time-optimal robust control of a two-level quantum system subjected to field inhomogeneities. We apply the Pontryagin maximum principle and we introduce a reduced space onto which the optimal dynamics is projected down. This reduction leads to a complete analytical derivation of the optimal solution in terms of elliptic functions and elliptic integrals. Necessary optimality conditions are then obtained for the original system. These conditions are verified numerically and lead to the optimal control protocol. Various examples, ranging from state-to-state transfer to the generation of a NOT gate, illustrate this study. The connection with other geometric optimization approaches that have been used to solve this problem is also discussed. Finally, we show how this method can be combined with numerical optimization algorithms to design robust control for higher-dimensional quantum systems, as illustrated by a Bose-Einstein condensate in an optical lattice.
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