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Chinese Science Bulletin

Trajectory tracking control of quantum systems

Shuang Cong, Jianxiu Liu

Published May 3, 2012• 6 citations
Publisher

Abstract

For quantum state trajectory tracking of density matrix in Liouville equation of quantum systems, with the help of concept in quantum system control, one can apply unitary transformation both to controlled system and free-evolutionary target system such as to change the time-variant and non-stationary target system into a stationary state. Therefore, the quantum state trajectory tracking problem becomes a steering one. State steering control law of the system transformed is designed by means of the Lyapunov stability theorem. Finally, numerical simulation experiments are given for a five-level energy quantum system. The comparison analysis of original system’s trajectory tracking with other method illustrates the advantage in control time of the method proposed.

DOI
10.1007/s11434-012-5194-z

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For quantum state trajectory tracking of density matrix in Liouville equation of quantum systems, with the help of concept in quantum system control, one can apply unitary transformation both to controlled system and free-evolutionary target system such as to change the time-variant and non-stationary target system into a stationary state. Therefore, the quantum state trajectory tracking problem becomes a steering one. State steering control law of the system transformed is designed by means of the Lyapunov stability theorem. Finally, numerical simulation experiments are given for a five-level energy quantum system. The comparison analysis of original system’s trajectory tracking with other method illustrates the advantage in control time of the method proposed.
A Lyapunov-based approach of trajectory tracking in Liouville equation for quantum systems is proposed. By selecting density matrix of controlled system as the system state, and defining the error between controlled state and target one as an error state, the paper regards this error as a new controlled state of system. In such a way the quantum state trajectory tracking problem is changed into the state regulation problem by driving the error state to zero. The control law is designed according to the Lyapunov stability theorem, which can be applied to track different kinds of quantum states including pure state, superposition state and mixed-state. Finally, numerical simulation experiments and results analysis are given.
A control method for trajectory tracking in non-Markovian open quantum systems is proposed in this paper. By inducing the error between the controlled state and the target state as the error state, the trajectory tracking problem of the quantum system is divided into two steps, which the regulation control of the error state and the keeping control of the error state. With the help of Lyapunov stability theorem, the tracking control law is designed to enable the non-Markovian open quantum system to track the evolutionary target quantum system over time. The control parameters are adjusted by simulation experiments, and the numerical simulation experiments are carried out. The tracking control of the quantum system when the initial states of the controlled system and the initial states of the target system are superposition states is implemented in simulation experiments. The experimental results demonstrate the feasibility of the proposed control method and the analysis is given.
The orbit tracking problem of a free-evolutionary target system in closed quantum systems is solved by changing it into the state transferring problem with the help of unitary transformation.The control law designed by the Lyapunov stability theorem employs a carefully constructed virtual mechanical quantity P to ensure the system convergence.The virtual mechanical quantity P is chosen by two approaches according to the forms of limit set,where P = —pf is suitable for regular limit set and a new different P is constructed for irregular one.The proposed tracking control theory is demonstrated on a four-level quantum system by means of numerical simulation experiments.
The orbit tracking of free-evolutionary target system in closed quantum systems is studied in this paper. Based on the concept of system control theory, the unitary transformation is applied to change the time-dependent target function into a stationary target state so that the orbit tracking problem is changed into the state transfer one. A Lyapunov function with virtual mechanical quantity P is employed to design a control law for such a state transferring. The target states in density matrix are grouped into two classes: diagonal and non-diagonal. The specific convergent conditions for target state of diagonal mixed-states are derived. In the case that the target state is a non-diagonal superposition state, we propose a non-diagonal P construction method; if the target state is a non-diagonal mixed-state we use a unitary transformation to change it into a diagonal state and design a diagonal P. In such a way, the orbit tracking problem with arbitrary initial state is properly solved. The explicit expressions of P are derived to obtain a convergent control law. At last, the system simulation experiments are performed on a two-level quantum system and the tracking process is illustrated on the Bloch sphere.
A Lyapunov-based unified control method for quantum state trajectory tracking control is proposed in this paper. By inducing the error between the controlled state and the target state as error state, the trajectory tracking control problem of the quantum system is transformed into the regular control problem of systematic error state. Through an interaction picture transformation, the spin effect is eliminated. The unified control method is designed in the trajectory tracking control for the quantum system which can also solve the problems caused by the non-ideal quantum systems, where the free Hamiltonian of the system is not strongly regular and (or) not commutes with the target state, and (or) the control Hamiltonian is not fully connected. The unified trajectory tracking control proposed can drive the system from an arbitrary initial state to an arbitrary target state and achieve the purpose of tracking. The numerical simulation experiments and results analyses are given.