Feedback stabilization of N-dimensional stochastic quantum systems based on bang-bang control

Control Theory and Technology - Tập 15 - Trang 206-218 - 2017
Xiaqing Sun1, Sen Kuang1, Yanan Liu1, Juan Zhou1, Shuang Cong1
1Department of Automation, University of Science and Technology of China, Hefei Anhui, China

Tóm tắt

For an N-dimensional quantum system under the influence of continuous measurement, this paper presents a switching control scheme where the control law is of bang-bang type and achieves asymptotic preparation of an arbitrarily given eigenstate of a non-degenerate and degenerate measurement operator, respectively. In the switching control strategy, we divide the state space into two parts: a set containing a target state, and its complementary set. By analyzing the stability of the stochastic system model under consideration, we design a constant control law and give some conditions that the control Hamiltonian satisfies so that the system trajectories in the complementary set converge to the set which contains the target state. Further, for the case of a non-degenerate measurement operator, we show that the system trajectories in the set containing the target state will automatically converge to the target state via quantum continuous measurement theory; while for the case of a degenerate measurement operator, the corresponding system trajectories will also converge to the target state via the construction of the control Hamiltonians. The convergence of the whole closed-loop systems under the cases of a non-degenerate and a degenerate measurement operator is strictly proved. The effectiveness of the proposed switching control scheme is verified by the simulation experiments on a finite-dimensional angular momentum system and a two-qubit system.

Tài liệu tham khảo

D. Dong, I. R. Petersen. Quantum control theory and applications: A survey. IET Control Theory and Applications, 2010, 4(12): 2651–2671.

H. M. Wiseman, G. J. Milburn. Quantum Measurement and Control. Cambridge: Cambridge University Press, 2009.

D. D’Alessandro. Introduction to Quantum Control and Dynamics. Boca Raton: Taylor and Francis Group, 2007.

D. Stefanatos. Optimal shortcuts to adiabaticity for a quantum piston. Automatica, 2013, 49(10): 3079–3083.

Q. M. Chen, R. B. Wu, T. M. Zhang, et al. Near-time-optimal control for quantum systems. Physical Review A, 2015, 92(6): DOI 10.1103/PhysRevA.92.063415.

S. Zhao, H. Lin, Z. Xue. Switching control of closed quantum systems via the Lyapunov method. Automatica, 2012, 48(8): 1833–1838.

S. Kuang, S. Cong. Lyapunov control methods of closed quantum systems. Automatica, 2008, 44(1): 98–108.

X. Wang, S. G. Schirmer. Analysis of Lyapunov method for control of quantum states. IEEE Transactions on Automatic Control, 2010, 55(10): 2259–2270.

D. Dong, I. R. Petersen. Sliding mode control of two-level quantum systems. Automatica, 2012, 48(5): 725–735.

D. Dong, I. R. Petersen. Notes on sliding mode control of twolevel quantum systems. Automatica, 2012, 48(12): 3089–3097.

M. R. James, H. I. Nurdin, I. R. Petersen. H∞ control of linear quantum stochastic systems. IEEE Transactions on Automatic Control, 2008, 53(8): 1787–1803.

J. Zhang, Y. X. Liu, R. B. Wu, et al. Quantum feedback: theory, experiments, and applications. Physics Reports, 2017, 679: 1–60.

B. Qi, H. Pan, L. Guo. Further results on stabilizing control of quantum systems. IEEE Transactions on Automatic Control, 2013, 58(5): 1349–1354.

W. Cui, F. Nori. Feedback control of Rabi oscillations in circuit QED. Physical Review A, 2013, 88(6): DOI 10.1103/PhysRevA.88.063823.

S. Wang, D. Dong. Fault-tolerant control of linear quantum stochastic systems. IEEE Transactions on Automatic Control, 2017, 62(6): 2929–2935.

J. Gong, S. A. Rice. Measurement-assisted coherent control. The Journal of Chemical Physics, 2004, 120(21): 9984–9988.

D. Dong, J. Lam, T. J. Tarn. Rapid incoherent control of quantum systems based on continuous measurements and reference model. IET Control Theory and Applications, 2009, 3(2): 161–169.

D. Dong, I. R. Petersen, H. Rabitz. Sampled-data design for robust control of a single qubit. IEEE Transactions on Automatic Control, 2013, 58(10): 2654–2659.

J. Zhang, R. B. Wu, C. W. Li, et al. Protecting coherence and entanglement by quantum feedback controls. IEEE Transactions on Automatic Control, 2010, 55(3): 619–633.

S. S. Ge, T. L. Vu, C. C. Hang. Non-smooth Lyapunov functionbased global stabilization for quantum filter. Automatica, 2012, 48(6): 1031–1044.

R. V. Handel, J. K. Stockton, H. Mabuchi. Feedback control of quantum state reduction. IEEE Transactions on Automatic Control, 2005, 50(6): 768–780.

M. Mirrahimi, R. V. Handel. Stabilizing feedback controls for quantum systems. SIAM Journal on Control and Optimization, 2007, 46(2): 445–467.

J. Zhou, S. Kuang. Feedback preparation of maximally entangled states of two-qubit systems. IET Control Theory and Applications, 2016, 10(3): 339–345.

Y. Liu, S. Kuang, S. Cong. Lyapunov-based feedback preparation of GHZ entanglement of N-qubit systems. IEEE Transactions on Cybernetics: DOI 10.1109/TCYB.2016.2584698 (in press).

W. Zhou, S. G. Schirmer, M. Zhang, et al. Bang-bang control design for quantum state transfer based on hyperspherical coordinates and optimal time-energy control. Journal of Physics A: Mathematical and Theoretical, 2011, 44(10): DOI 10.1088/1751-8113/44/10/105303.

S. Kuang, D. Dong, I. R. Petersen. Approximate bang-bang Lyapunov control for closed quantum systems. Proceedings of the Australian Control Conference, Canberra, Australia: IEEE, 2014: 130–135.

S. Kuang, D. Dong, I. R. Petersen. Rapid Lyapunov control of finite-dimensional quantum systems. Automatica, 2017, 81: 164–175.

J. Alonso, F. M. Leupold, Z. U. Solèr, et al. Generation of large coherent states by bang-bang control of a trapped-ion oscillator. Nature Communications, 2016, 7: DOI 10.1038/ncomms11243.

T. L. Vu, S. S. Ge, C. C. Hang. Real-time deterministic generation of maximally entangled two-qubit and three-qubit states via bang-bang control. Physical Review A, 2012, 85(1): DOI 10.1103/PhysRevA.85.012332.

J. H. Wei, B. Qi, H. Y. Dai, et al. Deterministic generation of symmetric multi-qubit Dicke states: An application of quantum feedback control. IET Control Theory and Applications, 2015, 9(17): 2500–2505.

J. Zhou, S. Kuang, S. Cong. Bell state preparation based on switching between quantum system models. Journal of Systems Science and Complexity, 2017, 30(2): 347–356.

K. Jacobs, D. A. Steck. A straightforward introduction to continuous quantum measurement. Contemporary Physics, 2006, 47(5): 279–303.

H. J. Kushner. Stability of Stochastic Dynamical Systems. Berlin: Springer, 1972.

K. Tsumura. Global stabilization of N-dimensional quantum spin systems via continuous feedback. Proceedings of the American Control Conference, New York: IEEE, 2007: 2129–2134.

Thông tin
Thông tin xuất bản

Control Theory and Technology

Tập 15

206-218

Thông tin tác giả