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Chinese Control Conference (CCC)

Convergent control strategy of state in open quantum systems

Shuang Cong, Fei Yang

Published Jul 22, 2011• 0 citations
OpenAlex

Abstract

A control strategy of quantum states of open quantum system based on Lyapunov method is proposed. On the basis of choosing the average of observable operators as a Lyapunov function, the control laws are designed in interaction picture. We analyze the largest invariant set of the controlled system via Barbalat's Lemma. Based on some assumptions of Hamiltonians, we propose a sufficient condition of the observable operator to guarantee the convergence of the open quantum system, and proof that the proposed strategy can make any pure target state in decoherence-free subspace be globally asymptotic stable under the sufficient condition. Also, we provide a method to construct the observable operator via Schmidt orthogonalization. Numerical experiments on a three-level system validate the effectiveness of the proposed control strategy.

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A control strategy of quantum states of open quantum system based on Lyapunov method is proposed. On the basis of choosing the average of observable operators as a Lyapunov function, the control laws are designed in interaction picture. We analyze the largest invariant set of the controlled system via Barbalat's Lemma. Based on some assumptions of Hamiltonians, we propose a sufficient condition of the observable operator to guarantee the convergence of the open quantum system, and proof that the proposed strategy can make any pure target state in decoherence-free subspace be globally asymptotic stable under the sufficient condition. Also, we provide a method to construct the observable operator via Schmidt orthogonalization. Numerical experiments on a three-level system validate the effectiveness of the proposed control strategy.
A convergent state control strategy of open quantum system based on Lyapunov method is proposed for the target state being a pure state in decoherence-free subspaces. Based on the assumptions of all the transition frequencies being different and all the levels being directly coupled,a sufficient condition is proposed for the observation operator to make that the largest invariant set only contains target state.On the basis of choosing the average of observable operators as a Lyapunov function,the control laws are designed in interaction picture and the largest invariant set of the controlled system is analyzed via Barbalat's Lemma. It is shown that any pure target state in decoherence-free subspace is globally asymptotic stable under the sufficient condition given,and the system controlled could be transferred to target state from arbitrary initial state.Also,a method is provided to construct the observation operator via Schmidt orthogonalization.Numerical experiments on a three-level system verify the validity of the proposed control strategy.
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 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.
Under the ideal conditions that the controlled systems are non-degenerate,transitionally non-degenerate,and fully connected,the stabilization problem of mixed-state quantum systems is studied by using a Lyapunov function with free degrees. Based on the LaSalle's principle,the largest invariant set of the closed-loop systems and the convergent state set associated with any initial state are deduced,and the construction principles of free degrees that ensure the system can be asymptotically stabilized in any equilibrium state contained in the largest invariant set are given. A numerical simulation experiment on a two-level system shows the rationality of the obtained theoretical results.
We propose a Lyapunov-based control method for stochastic quantum systems. The method can make the quantum state converge to a separable mixed target state from an arbitrary initial state. Specifically, the target state can be written as the tensor product of two matrices. Based on a well designed virtual mechanical quantity, a Lyapunov function and two control laws are presented: one can make the system equilibria in the largest invariant set stable according to LaSalle's invariance principle, and the other one can make the target state globally asymptotically stable.