Abstract
The global stabilization control of arbitrary eigenstates for finite dimensional stochastic quantum systems with non-diagonal free Hamiltonian and non-regular measurement operator is studied in this paper. We propose a switching feedback control law, in which a constant control is used to steer the system state to a convergence domain, and another control law designed based on Lyapunov stability theorem, is used to attract the states in the convergence domain to the desired target state. The convergence to an arbitrary target eigenstate from any initial state is strictly proved. Moreover, numerical simulation experiments on a three-dimensional stochastic quantum system are implemented to demonstrate the effectiveness of the proposed control.
概要
中文概要
研究了对具有非对角自由哈密顿量以及非规则测量算符的有限维随机量子系统任意本征态控制的全局稳定性. 提出了一种开关反馈控制律, 其中一个常量控制用来驱动系统状态到收敛域中; 另一个基于李雅谱诺夫稳定性定理设计出的控制律被用来吸引状态到收敛域中的期望目标态. 严格证明了从任意初始态到任意目标本征态的收敛性, 并且通过3维随机量子系统的数值仿真实验展现了所提控制方法的有效性.
创新点
-
1)
首次对具有非对角自由哈密顿量以及非规则测量算符的有限维随机量子系统采用李普诺夫控制理论进行了研究;
-
2)
成功地推导出非规则的有限维随机量子系统任意本征态的全局稳定控制律, 并给出了严格证明;
-
3)
所提出的控制方法简单、易于实现; 由所采用的控制理论所获得的控制律函数能够确保量子控制系统的状态以概率1转移并稳定到期望的目标状态, 对实际量子系统实验中调控性能的进一步提高具有重要意义.
Similar content being viewed by others
References
Abe T, Tsumura K. Generation of quantum entangled state via continuous feedback control. In: Proceedings of SICE Annual Conference, Chofu, 2008. 3305–3308
D’Alessandro D, Dahleh M. Optimal control of two-level quantum systems. IEEE Trans Automa Control, 2001, 46: 866–876
Tsumura K. Global stabilization at arbitrary eigenstates of N-dimensional quantum spin systems via continuous feedback. In: Proceedings of the American Control Conference, Washington, 2008. 4148–4153
Ticozzi F, Schirmer S G, Wang X. Stabilizing quantum states by constructive design of open quantum dynamics. IEEE Trans Autom Control, 2010, 55: 2901–2905
Lou Y, Cong S. State transfer control of quantum systems on the Bloch Sphere. J Syst Sci Complex, 2011, 24: 506–518
Rangelov A A, Vitanov N V. Complete population transfer in a three-state quantum system by a train of pairs of coincident pulses. Phys Rev A, 2012, 85: 043407
Zhang J, Liu Y X, Wu R B, et al. Quantum feedback: theory, experiments, and applications. arXiv.org/abs/1407.8536. 2014
Krstic M, Deng H. Stabilization of Nonlinear Uncertain Systems. Berlin: Springer-Verlag, 1999
Aharonov Y, Vaidman L. Properties of a quantum system during the time interval between two measurements. Phys Rev A, 1990, 41: 11–20
Belavkin V. Quantum stochastic calculus and quantum nonlinear filtering. J Multivar Anal, 1992, 44: 171–201
van Handel R, Stockton J K, Mabuchi H. Feedback control of quantum state reduction. IEEE Trans Autom Control, 2005, 50: 768–780
Jacobs K, Steck D A. A straightforward introduction to continuous quantum measurement. Contemp Phys, 2006, 47: 279–303
Ge S S, Vu T L, Hang C C. Non-smooth Lyapunov function-based global stabilization for quantum filters. Automatica, 2012, 48: 1031–1044
Belavkin V. Nondemolition Measurement and Control in Quantum Dynamical Systems. Vienna: Springer, 1985
Wiseman H. Quantum theory of continuous feedback. Phys Rev A, 1994, 49: 2133–2150
Belavkin V. On the theory of controlling observable quantum systems. Autom Remote Control, 1983, 44: 178–188
Belavkin V, Hirota O, Hudson R. Quantum Communications and Measurement. Vienna: Springer, 1995
Bouten L, Edwards S, Belavkin V. Bellman equations for optimal feedback control of qubit states. J Phys B: Atomic Mol Opt Phys, 2005, 38: 151–160
Bouten L, van Hande R. On the separation principle in QUANTUM CONTROL. In: Quantum Stochastics and Information. Singapore: World Scientific, 2008. 206–238
Mirrahimi M, van Handel R. Stabilizing feedback controls for quantum systems. SIAM J Control Optim, 2007, 46: 445–467
Wang J, Wiseman H M. Feedback-stabilization of an arbitrary pure state of a two-level atom. Phys Rev A, 2001, 64: 063810
Geremia J, Stockton J K, Mabuchi H. Real-time quantum feedback control of atomic spin-squeezing. Science, 2004, 304: 270–273
Bouten L, Edwards S, Belavkin V, et al. An introduction to quantum filtering. SIAM J Control Optim, 2007, 46: 2199–2241
Tsumura K. Global stabilization of n-dimensional quantum spin systems via continuous feedback. In: Proceedings of the American Control Conference, New York City, 2007. 2119–2134
Altafini C, Ticozzi F. Almost global stochastic feedback stabilization of conditional quantum dynamics. arXiv.org/abs/quant-ph/0510222v1. 2008
Ge S S, Vu T L, Lee T H. Quantum measurement-based feedback control: a nonsmooth time delay control approach. SIAM J Control Optim, 2012, 50: 845–863
Kushner H J. Stochastic Stability and Control. New York: Academic Press, 1967
Øksendal B. Stochastic Differential Equations: An Introduction With Applications. Berlin: Springer-Verlag, 1993
Mao X. Stochastic Differential Equations and Applications. Chichester: Horwood, 1998
Klenke A. Probability Theory. Berlin: Springer-Verlag, 2006
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Cong, S., Wen, J., Kuang, S. et al. Global stabilization control of stochastic quantum systems. Sci. China Inf. Sci. 59, 112502 (2016). https://doi.org/10.1007/s11432-015-0911-7
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11432-015-0911-7