Lyapunov control methods of closed quantum systems

Abstract

According to special geometric or physical meanings, the paper summarizes three Lyapunov functions in controlling closed quantum systems and their controller designing processes. Specially, for the average value-based method, the paper gives the generalized condition of the largest invariant set in the original reference and develops the construction method of the imaginary mechanical quantity; for the error-based method, this paper gives its strict mathematical proof train of thought on the asymptotic stability and the corresponding physical meaning. Also, we study the relations among the three Lyapunov functions and give a unified form of these Lyapunov functions. Finally, we compare the control effects of three Lyapunov methods by doing some simulation experiments.

Introduction

Since the mid-1990s, with the development of femtosecond laser technology and the appearance of a series of research results (Armen et al., 2002, Geremia et al., 2003, Zhang et al., 2000) on quantum information technology, quantum measurement, and quantum clone technology, people have shown great interests in the feedback control of quantum systems, for example, Markovian feedback in quantum optics (Wiseman & Milburn, 1993), feedback based on estimation and filtering (Doherty and Jacobs, 1999, Edwards and Belavkin, 2005), etc. But, the super-short control time required by some quantum dynamics (e.g., molecular dynamics) restricts the application of observation and feedback, so the prevailing control style has been open loop control. In fact, for the same reason and some limitations of quantum measurement itself, currently open loop control remains dominant. Generally speaking, quantum systems may be divided into two classes: closed quantum systems and open quantum systems. A closed quantum system evolves unitarily, whose dynamics is governed by the Schrödinger equation or the non-dissipative Liouville–von Neumann equation. The dynamics of an open quantum system is no longer unitary due to interaction with its environment. In open loop quantum control, the state steering of closed systems is a class of important problems, i.e., given an initial state and a goal one, how to find some realizable controls to drive the initial state to the goal one (Cong, 2006).

Undoubtedly, optimal control techniques are important approaches. In the techniques, the control aims are often to solve steering problems in a complex numerical iterative fashion to search for the control field while minimizing an energy-type cost functional of the control that usually requires a maximal transition probability from an initial state to a particular goal state (D’Alessandro and Dahleh, 2001, Kuang and Cong, 2006, Schirmer et al., 1999, Shi and Rabitz, 1990, Zhu and Rabitz, 1998) (see also, the references herein). Sugawara developed a general local control theory for manipulating quantum dynamics, which is a generalization of the local optimization approaches and can be used in transition path control, population distribution control, and wave packet shaping (Sugawara, 2003). The decoupling techniques based on an adaptive tracking algorithm can be used to address quantum control problems (Zhu & Rabitz, 2003). With tracking, the control is obtained without iteration by following an explicit pathway, but the price paid for eliminating iteration is the need to manage possible field singularities arising along the control pathway. The factorization techniques of the unitary group also can be used to implement state steering (Altafini, 2002, Constantinescu and Ramakrishna, 2003, Ramakrishna et al., 2002). They are based on explicit generation of unitary operators by using Lie group decompositions. In addition, Lyapunov-based techniques are good approaches (Cong & Kuang, 2007; Ferrante, Pavon, & Raccanelli, 2002a; Ferrante, Pavon, & Raccanelli, 2002b; Grivopoulos & Bamieh, 2003; Mirrahimi, Rouchon, & Turinici, 2005; Paolo, 2002). In such methods, the control laws can be obtained when the first-order time derivative of a selected Lyapunov function is kept non-positive.

In principle, as long as a quantum system is controllable, one can design the desired control laws. An advantage of the Lyapunov method is that the designed control laws would not make the closed-loop system divergent. A key problem in such a method is to select appropriate Lyapunov functions. Generally speaking, different Lyapunov functions will lead to different designed control laws and different control effects. Designing Lyapunov functions from some special geometric or physical meanings is a good approach. In Cong and Kuang (2007), we selected a distance-based Lyapunov function to design the corresponding control laws and obtained the conditions for the asymptotic stability of the closed-loop system by linearizing the unitary operator of the state. On the basis of such a study, the main works in this paper include summarizing the Lyapunov-based design methods for the control of closed quantum systems, analyzing the corresponding asymptotic stability of the closed-loop systems, and comparing several involved Lyapunov methods by doing some simulation experiments.

This paper only treats finite dimensional quantum systems and assumes that the systems of interest are controllable. Two state vectors $|{\psi }_1〉$ and $|{\psi }_2〉$ which satisfy $|{\psi }_1〉={\mathrm{e}}^{\mathrm{i}\theta }|{\psi }_2〉$ are called equivalent state vectors. In order to avoid the trouble caused by the global phase in the equivalence class, we will disregard the global phase between equivalent states and think of all the states in an equivalence class as the same state. For the state steering problems in closed quantum systems, this paper will study the following three Lyapunov functions: Lyapunov functions based on the state distance, on the average value of an imaginary mechanical quantity, and on the state error. In Section 2, the Lyapunov method based on the state distance is given, which contains the design of control laws and the main results on stability. In Section 3, the Lyapunov method based on the average value is studied, where we relax some conditions for the largest invariant set in Grivopoulos and Bamieh (2003), and analyze and give the construction method of an imaginary mechanical quantity P. Section 4 contains some results of the Lyapunov method based on the state error. Especially, we study relations among three Lyapunov functions and give a unified form of these Lyapunov functions. The control laws in the three methods are applied to a spin-$\frac{1}{2}$ particle system and a five-level system in Section 5, in which the control effects of the three Lyapunov methods are compared and analyzed. In Section 6, we conclude the paper with brief remarks.