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arXiv:1402.6768 · 系统科学与信息学报(英文)

A Convergent Control Strategy for Quantum Systems

Shuang Cong, Yuesheng Lou, Jianxiu Liu, Sen Kuang

Published Jun 25, 2014• 0 citations
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Abstract

Abstract In the interaction picture, a sufficient and necessary condition that guarantees the convergence of closed quantum control system is proposed in this paper. Theoretical derivation and the proof show that it is possible to achieve the convergence to the target state by constructing an observable operator in an energy function and selecting control Hamiltonians. Numerical simulation experiments on a four-level system verify the effectiveness of the proposed control strategy.

DOI
10.1515/jssi-2014-0255

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The state transfer of closed quantum systems in the interaction picture is studied. The convergent problem encountered in designing control laws based on the Lyapunov method is solved by the well constructed observable operator and a path programming control strategy. It is proved that the condition for the target state being a stable point in the Lyapunov's sense is the coherent vectors of the observable operator and the target state must be in opposite directions. For the local optimisation limitation of the Lyapunov-based method, the path programming control strategy is proposed, which is used to change the distribution of stationary points or choose a transition path by appropriately selecting intermediate target states. Comparative numerical system simulation experiments are implemented on a four-level quantum system and the experimental results are analysed.
Quantum ensemble systems arise in a variety of applications, including NMR spectroscopy and robust quantum control. While their theoretical properties have been extensively studied, relatively little attention has been given to the explicit construction of control inputs. In this paper, we address this gap by presenting a fully implementable control strategy for a one-parameter family of driftless two-level quantum systems. The proposed method is supported by rigorous analysis that guarantees accurate approximation of target distributions on SU(2). Convergence properties are established analytically, and numerical simulations are provided to demonstrate the effectiveness of the approach.
The problem of quantum state control in open quantum systems based on the Lyapunov method is studied. The control laws are designed to drive the system to a pure target state in decoherence-free subspace (DFS) in the interaction picture, where the average of an observable operator P is chosen as a Lyapunov function. By analyzing the positive limit set via Barbalat’s lemma, we propose a sufficient condition on the observable operator such that the positive limit set only contains the target state. Under this condition, it is proved that the control laws can drive the open system to any pure target state in DFS in the interaction picture. Also, we provide a method to construct the observable operator via Schmidt orthogonalization. Numerical simulations on a three-level system validate the effectiveness of the proposed control strategy.
The condition of a quantum Lyapunov-based control which can be well used in a closed quantum system is that the method can make the system convergent but not just stable. In the convergence study of the quantum Lyapunov control, two situations are classified: nondegenerate cases and degenerate cases. For these two situations, respectively, in this paper the target state is divided into four categories: the eigenstate, the mixed state which commutes with the internal Hamiltonian, the superposition state, and the mixed state which does not commute with the internal Hamiltonian. For these four categories, the quantum Lyapunov control methods for the closed quantum systems are summarized and analyzed. Particularly, the convergence of the control system to the different target states is reviewed, and how to make the convergence conditions be satisfied is summarized and analyzed.