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Chinese Control Conference (CCC)

State transfer for closed quantum systems with dipole and polarizing coupling via implicit Lyapunov control

Zhao Shouwei, Hai Lin, Sun Jitao, Zhou Lei

Published Jul 1, 2015• 1 citation
Publisher

Abstract

In this paper, the state transfer problem of finite dimensional closed quantum systems with dipole and polarizability coupling in non-ideal cases is investigated. The non-ideal cases are considered where the internal Hamiltonian of the system is not strongly regular [1, 2] and not all the eigenvectors of the internal Hamiltonian are directly coupled with the target state. Both cases are closely related to practical systems such as the one-dimensional oscillator. A class of Lyapunov functions are first defined by implicit functions and their existences are guaranteed by a fixed point theorem. Then, the local convergence analysis of the controlled system is investigated by the LaSalle invariance principle. Finally, the characterization of the largest invariant set is presented. Moreover, simulation studies are discussed to show the effectiveness and advantage of proposed methods.

DOI
10.1109/chicc.2015.7260954

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In this paper, the state transfer problem of finite dimensional closed quantum systems with dipole and polarizability coupling in non-ideal cases is investigated. The non-ideal cases are considered where the internal Hamiltonian of the system is not strongly regular [1, 2] and not all the eigenvectors of the internal Hamiltonian are directly coupled with the target state. Both cases are closely related to practical systems such as the one-dimensional oscillator. A class of Lyapunov functions are first defined by implicit functions and their existences are guaranteed by a fixed point theorem. Then, the local convergence analysis of the controlled system is investigated by the LaSalle invariance principle. Finally, the characterization of the largest invariant set is presented. Moreover, simulation studies are discussed to show the effectiveness and advantage of proposed methods.
Summary In this paper, the state transfer of finite dimensional closed quantum systems with dipole and polarizability coupling in non‐ideal cases is investigated. Two kinds of non‐ideal systems are considered, where the internal Hamiltonian of the system is not strongly regular and not all the eigenvectors of the internal Hamiltonian are directly coupled with the target state. Such systems often exist in practical quantum systems such as the one‐dimensional oscillator and coupled two‐spin system. An implicit Lyapunov‐based control strategy is proposed here with convergence analysis for quantum systems modeled by finite dimensional bilinear Schrödinger equations. Specifically, two kinds of Lyapunov functions are defined via implicit functions, and their existences are guaranteed with the help of a fixed point theorem. Then, the local convergence analysis is investigated with the explicit characterization of the largest invariant set by LaSalle invariance principle. Finally, the performance of the feedback design is illustrated by numerical simulations. Copyright © 2017 John Wiley & Sons, Ltd.
SUMMARY In this paper, the state convergence problem for closed quantum systems is investigated. We consider two degenerate cases, where the internal Hamiltonian of the system is not strongly regular or the linearized system around the target state is not controllable. Both the cases are closely related to practical systems such as one‐dimensional oscillators and coupled two spin systems. An implicit Lyapunov‐based control strategy is adopted for the convergence analysis. In particular, two kinds of Lyapunov functions are defined by implicit functions and their existences are guaranteed by a fixed point theorem. The convergence analysis is investigated by the LaSalle invariance principle for both cases. Moreover, the two Lyapunov functions are unified in a general form, and the characterization of the largest invariant set is presented. Finally, simulation studies are included to show the effectiveness and advantage of the proposed methods. Copyright © 2011 John Wiley & Sons, Ltd.
For closed quantum systems, if controlled systems are strongly regular and all other eigenstates are directly coupled to a target state, then such control systems can be asymptotically stabilized by means of the Lyapunov-based control. However, when the controlled systems are not strongly regular, or when there exists at least one eigenstate that is directly uncoupled to the target state, the situations will become complicated. This paper proposed a method based on an implicit Lyapunov function to overcome these two degenerate cases. Also, the proposed method is suitable for multi-control Hamiltonian systems. The convergence of closed-loop systems is analyzed by the LaSalle invariance principle. Finally, numerical simulation experiments on a 4-level system are done. The experiment results show the effectiveness of the implicit Lyapunov control method for degenerate cases and multi-control Hamiltonians.
In closed quantum systems, if controlled systems are not strongly regular or there exists at least one eigenstate that is directly uncoupled to the target state, then such systems control will become degenerate cases. This paper proposes a quantum control method designed by an implicit Lyapunov function to transfer states of the quantum systems in these two degenerate cases, especially the method proposed is suitable for multi-control Hamiltonian systems. The state distance is selected as the Lyapunov function in this paper. Numerical simulation experiments on a 4-level system are done, and the experiment results indicate the effectiveness of the implicit Lyapunov control method proposed for degenerate cases and multi-control Hamiltonians.
A quantum system whose internal Hamiltonian is not strongly regular or/and control Hamiltonians are not full connected, are thought to be in the degenerate cases. In this paper, convergence problems of the multi-control Hamiltonians closed quantum systems in the degenerate cases are solved by introducing implicit function perturbations and choosing an implicit Lyapunov function based on the average value of an imaginary mechanical quantity. For the diagonal and non-diagonal tar-get states, respectively, control laws are designed. The convergence of the control system is proved, and an explicit design principle of the imaginary mechanical quantity is proposed. By using the proposed method, the multi-control Hamiltonians closed quantum systems in the degenerate cases can converge from any initial state to an arbitrary target state unitarily equivalent to the initial state. Finally, numerical simulations are studied to verify the effectiveness of the proposed control method.
In the closed quantum system, if the control system is strongly regular and all other eigenstates are directly coupled to the target state, the control system can be asymptotically stabilized at the target eigenstate by the Lyapunov control based on the state error. However, if the control system is not strongly regular or as long as there is one eigenstate not directly coupled to the target state, the situations will become complicated. In this paper, we propose an implicit Lyapunov control method based on the state error to solve the convergence problems for these two degenerate cases. And at the same time, we expand the target state from the eigenstate to the arbitrary pure state. Especially, the proposed method is also applicable in the control system with multi-control Hamiltonians. On this basis, the convergence of the control systems is analyzed using the LaSalle invariance principle. Furthermore, the relation between the implicit Lyapunov functions of the state distance and the state error is investigated. Finally, numerical simulations are carried out to verify the effectiveness of the proposed implicit Lyapunov control method. The comparisons of the control effect using the implicit Lyapunov control method based on the state distance with that of the state error are given.