The state transfer problem of a class of nonideal quantum systems is investigated. It is known that traditional Lyapunov methods may fail to guarantee convergence for the nonideal case. Hence, a hybrid impulsive control is proposed to accomplish a more accurate convergence. In particular, the largest invariant sets are explicitly characterized, and the convergence of quantum impulsive control systems is analyzed accordingly. Numerical simulation is also presented to demonstrate the improvement of the control performance.
One of major concerns in quantum control is how to steer quantum states to a desired target state precisely and efficiently. A solution to this quantum state transfer problem will help us to advance some promising applications such as quantum computation and quantum chemistry. The main difficulty in quantum control is due to the limitations on the application of observation and feedback in quantum systems. Open-loop control has therefore been a commonly adopted approach in quantum control, where recorded control signals obtained from numerical simulations are implemented to real quantum systems. Among existing open-loop control design methods, the Lyapunov method could be the most popular one and has been tested in real applications [
In particular, this paper will study the state transfer for closed quantum systems modeled as the following Schrödinger equation:
The basic idea of the hybrid impulsive control is to divide the control into a piecewise continuous open-loop coherent control
In practical implementations, the
With this understanding, this paper will focus on the development of hybrid impulsive control design itself to achieve more accurate convergence under the nonideal cases. Moreover, the invariant set of the controlled system is characterized explicitly, which is shown to be strictly smaller than that obtained using classical Lyapunov methods. The convergence analysis is then obtained via an extending LaSalle invariance principle for impulsive systems in [
The rest of this paper is organized as follows. In Section
In practice, we do not have much freedom to choose the control Hamiltonian due to the structure limitations of the control fields [
In quantum control, the goal state
In order that the designed control can work in the case of the initial state being orthogonal to the goal state, we rewrite (
For control law ( if there exists if
(i) For a sufficiently small
(ii) Initially, the system evolves freely because
For the characterization of invariant sets, properties of the states such that
If there exists
Lemma
Consider the following differential impulsive system on an open set
The following theorem presents the characterization of the invariant set for the nonideal systems under the hybrid impulsive control, by which the invariant set is smaller compared with that obtained by the conventional Lyapunov method. In the following, the unitary matrix
Consider system (
When
In the following, we will discuss the conditions on the initial states from which the trajectories stay in the set
In conclusion, all the states which stay in the intersection
It should be noticed that the basis of the set
Consider system (
If
It is known that different Lyapunov functions may have different control effects. The relations among them were studied in our previous work [
Consider system (
Let
Next, we characterize the initial states from which the system trajectories stay in
In conclusion, all the states which remain in the intersection
Similar to the discussion in Corollary
Consider the five-level system with the internal Hamiltonian and impulsive control Hamiltonian given by
Now we compare performance of the hybrid impulsive control with that of classical Lyapunov control. If the impulsive control is not applied to the system, then the hybrid impulsive control is reduced to the classical Lyapunov control, by which the performance of the controlled system is shown in Figure
The population of the system state by the hybrid impulsive control based on the state distance.
The population of the system state by the Lyapunov control without impulsive control.
Consider the five-level system with the same internal Hamiltonian as the previous example. Let the target state and the initial state be
When the hybrid impulsive control is reduced to the classical Lyapunov control, the trajectory of the controlled system is plotted in Figure
The population of the system state by the hybrid impulsive control based on the state error.
The population of the system state by the Lyapunov control without impulsive control.
The population of the system state by the implicit Lyapunov control in [
In this paper, the coherent hybrid impulsive control for closed quantum systems has been investigated for the nonideal case that
This work is supported by the NNSF of China under Grant nos. 61174039, 61203128, the Connotative Construction Project of Shanghai University of Engineering Science (nhky-2012-13), the Fundamental Research Funds for the Central Universities of China, the “Chen Guang” Project supported by Shanghai Municipal Education Commission and Shanghai Education Development Foundation (no. 12CG65), and the Research Funds of Shanghai University of Engineering Science no. 2012td19.