INVESTIGATION ON IDENTIFICATION AND CONTROL OF QUANTUM SYSTEMS XUE ZHENGUI (B.Eng and M.Eng.) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY NUS GRADUATE SCHOOL FOR INTEGRATIVE SCIENCES AND ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2012 Declaration I hereby declare that this thesis is my original work and it has been written by me in its entirety I have duly acknowledged all the sources of information which have been used in the thesis This thesis has also not been submitted for any degree in any university previously Xue Zhengui 25 February 2013 i Acknowledgments First of all, I would like to express my deepest gratitude to my main supervisor, Professor Tong Heng Lee, for his thoughtful guidance, unwavering support and kind help on all the troublesome matters despite his extremely full schedules My heartfelt appreciation goes to my co-supervisor, Professor Hai Lin, for his time and direct guidance on my research His rigorous scientific attitude and endless enthusiasm impressed me, inspired me and changed me Special thanks go to my co-supervisor, Professor Shuzhi Sam Ge for the learning opportunities, inspiring guidance as well as all the time and efforts on my training and education My appreciation also goes to the distinguished examiners for their time and constructive comments I am also grateful to all the fellow colleagues and friends for their kind companionship and friendship Special thanks to Dr Shouwei Zhao for the discussions and sharing in and beyond research Many thanks to Dr Yang Yang, Dr Mohammad Karimadini, Mr Xiaomeng Liu, Ms Xiaoyang Li, Ms Yajuan Sun, Mr Alireza Partovi, Mr Ali Karimoddini, Mr Jin Yao, Mr Lei Liu, Dr Chenguang Yang, Mr Hengsheng He, Mr Qun Zhang, and Mr Yanan Li Thanks are also extended to the NUS Graduate School for Integrative Sciences and Engineering for the financial support during the course of my PhD study Last but certainly not least, I am deeply indebted to my dear parents and brother for always being there with their selfless love, trust, support and encouragement, and to my beloved husband for his constant love, care, patience, support and encouragement during the past nine years ii Contents Contents Declaration i Acknowledgments ii Contents iii Summary viii List of Figures x List of Symbols xii Introduction 1.1 Background of Quantum Control 1.1.1 What makes a system quantum 1.1.2 Why quantum control iii Contents 1.1.3 What is the objective of quantum control theory and how to control in laboratories Open-loop control 1.2.2 1.3 Quantum Control Review 1.2.1 1.2 Closed-loop control Objectives and Structure of the Thesis 12 1.3.1 Analysis and control of closed quantum systems based on realvalued equations 12 1.3.2 Identification and control of a class of two-level quantum systems 14 1.3.3 Observer-based closed-loop control of two-level quantum systems with unknown initial conditions Quantum Mechanical Preliminaries 2.1 15 17 18 2.1.1 Quantum behavior 18 2.1.2 Mathematical representation 18 2.1.3 2.2 Quantum States Pure states and mixed states 22 Measurements 25 2.2.1 Hermitian operators 26 2.2.2 Hermitian operators and the spectral theorem 27 2.2.3 Measurement postulate 28 iv Contents 2.2.4 31 Quantum Dynamics 33 2.3.1 Schrădinger equation o 33 2.3.2 Liouville’s equation 38 2.3.3 2.3 Example of measurement: Measuring spin systems System evolution under continuous measurements 39 Analysis and Control of Closed Quantum Systems Based on RealValued Dynamics 43 3.1 Introduction 43 3.2 Pure State Identification Based on Measurement Outputs 45 3.2.1 Identification of two-level states 45 3.2.2 Identification of three-level states 49 3.2.3 Identification of n-level states 52 Analysis and Control of Two-Level Systems 54 3.3.1 System dynamics formulation 54 3.3.2 Analysis and control design 57 3.3.3 Simulation study 63 Control of Three-Level Systems 64 3.4.1 System dynamics formulation 64 3.4.2 Control design 71 3.4.3 Simulation study 74 3.3 3.4 v Contents 3.5 Control of N -Level Systems 77 3.6 Conclusion 78 Identification and Control of a Class of Two-Level Quantum Systems 80 4.1 Introduction 80 4.2 Problem Formulation 82 4.3 Parameter Estimation Based on Ensemble Average 84 4.4 Parameter Estimation Based on Single Implementation 88 4.4.1 Existence of a stationary process 88 4.4.2 Ergodicity of the solution with a stationary distribution 97 4.4.3 Convergence to the solution with a stationary distribution 101 4.5 Feedback Control 107 4.5.1 4.5.2 Control analysis in the presence of an estimation error 115 4.5.3 4.6 Markovian feedback control design 108 Simulation studies 116 Conclusion 118 Observer-Based Closed-Loop Control of Two-Level Quantum Systems with Unknown Initial Conditions 120 5.1 Introduction 120 5.2 Problem Formulation 122 vi Contents 5.3 Observer Design 123 5.4 Control Design 129 5.4.1 Control design for exactly known systems 129 5.4.2 Control of systems with unknown initial states 141 5.5 Simulation Studies 151 5.6 Conclusion 152 Conclusions and Future Work 157 6.1 Conclusions 157 6.2 Future Work 159 Bibliography 161 A Appendix for Chapter 176 A.1 Evolution of quantum systems in the unit ball 176 A.2 Existence of unique and regular solution of system (4.66) 179 B Author’s Publications 183 vii Summary Summary Promising applications of quantum phenomena have been proposed with the recognition of the quantum behavior and the advances of laboratory techniques, which consequently motivate the development of quantum control theory The unique properties of quantum systems, on the other hand, make quantum control a challenging research topic Despite the achievements in the literature, more effective and efficient approaches are in demand for the development of a systematic framework of quantum control theory The main purpose of this thesis is to develop new approaches to dealing with difficult identification and control problems in quantum technology applications Although the complex-valued Schrădinger equation provides an elegant description o of physical systems, it may bring unnecessary difficulties for the analysis and control of quantum systems, since the existing methods in the classical control theory are mainly for real-valued equations In this thesis, equivalent real-valued equations are first derived for closed quantum systems in order to facilitate the analysis and design of quantum control Based on the obtained real-valued equations, the Lyapunov control problem is then considered given the fact that it is not convenient to characterize invariant sets in the complex-valued picture and it is difficult to guarantee state transfer convergence The obtained results illustrate the capability of the real-valued viii Summary equations in the characterization of invariant sets and achievement of state transfer convergence Furthermore, the well-developed control strategies for exactly known systems may have limited capability in the manipulation of systems in the presence of uncertainties Thus, this thesis next considers the identification and control for a class of two-level systems with unknown decoherence rates Main concerns of the parameter estimation approaches in the literature are the heavy computation cost and the possibility of trapping into local optima during the iterative optimization processes In this thesis, a computationally efficient and easily implementable approach is proposed to estimate the decoherence rates by monitoring the ensemble average of identical systems Based on the result, a further step is taken to consider the parameter identification in terms of a single quantum system An efficient estimation approach is obtained by making use of the ergodic property of a single system With the estimation results, the control of high probability state transfers to the exited state is then studied via the Markovian feedback method Finally, an alternative real-time estimation and control approach is developed for two-level systems with unknown initial conditions An observer is constructed to estimate unknown system information, and feedback control signals are adjusted to achieve satisfactory control performance based on on-line estimated results Moreover, the positive effect of quantum measurements in state transfers is illustrated via the investigation of state transfers from mixed initial states to pure states The significance of the proposed real-time estimation and control approach lies in that it provides elementary support for the development of a general framework of on-line quantum identification and control ix Bibliography [75] J Wang and H Wiseman, “Feedback-stabilization of an arbitrary pure state of a two-level atom,” Physical Review A, vol 64, no 6, p 063810, 2001 [76] F Ticozzi and L Viola, “Analysis and synthesis of attractive quantum Markovian dynamics,” Automatica, vol 45, no 9, pp 2002–2009, 2009 [77] A Doherty and K Jacobs, “Feedback control of quantum systems using continuous state estimation,” Physical Review A, vol 60, no 4, p 2700, 1999 [78] A Doherty, S Habib, K Jacobs, H Mabuchi, and S Tan, “Quantum feedback control 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A.1.2 [119] Suppose that Xt is a random process and it satisfies the following stochastic differential equation dXt = (a(t, ω)Xt + c(t, ω)) dt + b(t, ω)Xt dWt , (A.2) where Wt is a Wiener process, and the non-anticipative functions a(t, ω) and b(t, ω) satisfy T |a(t, ω)|dt < ∞ =1 |b2 (t, ω)|dt < ∞ P = T P (A.3) Assume that the stochastic differential equation has a unique solution Then we have t e−ζs c(s, ω)ds , Xt = eζt X0 + (A.4) where t a(s, ω)ds − ζt = t t b2 (s, ω)ds + b(s, ω)dWs (A.5) For system (4.7), we have the following result showing that the quantum system evolves in the unit ball Theorem A.1.1 Consider the quantum system (4.7) with arbitrary admissible control law such that it has a unique solution We have that the system state almost 2 surely satisfies x2 + yt + zt ≤ 1, ∀ t ≥ t Proof Consider the following function 2 Vt = x2 + yt + zt − t 177 (A.6) A.1 Evolution of quantum systems in the unit ball Based on the system model (4.7) and the Ito’s formula (A.1), we have dVt = 2xt − γ+M γ+M xt + syt − uy (t)zt dt + 2yt − yt − sxt + ux (t)zt dt 2 2 2 2zt (γ(1 − zt ) − ux (t)yt + uy (t)xt ) dt + M x2 zt + M yt zt + M (1 − zt )2 dt t √ 2 (A.7) +2 M x2 zt + yt zt − (1 − zt )zt dWt t Note that (A.7) can be written as √ dVt = −γVt dt − M (1 − zt )Vt dt − γ(1 − zt )2 dt + M zt Vt dWt (A.8) According to Lemma A.1.2, we have t eζt −ζs γ(1 − zs )2 ds, Vt = eζt V0 − (A.9) where t ζt = −(γ + M )t − M √ zs ds + M t zs dWs (A.10) As mentioned in Section 2.1.3, the density matrix ρ of a quantum state should satisfy Tr(ρ2 ) ≤ (A.11) Based on the expression of ρ in (4.6), we have Tr(ρ2 ) = 2 xt + yt + zt + (A.12) Therefore, the following condition for the initial state of a quantum system holds 2 x2 + y0 + z0 ≤ 178 (A.13) A.2 Existence of unique and regular solution of system (4.66) Thus, from the definition of Vt in (A.6), we have V0 ≤ (A.14) From the derived expression of Vt in (A.9), we have the following inequality Vt ≤ 0, (A.15) i.e., 2 x2 + yt + zt ≤ 1, ∀ t > t Therefore, the quantum system (4.7) evolves in the unit ball A.2 Existence of unique and regular solution of system (4.66) Theorem A.2.1 There is a unique and regular solution of system (4.66) Proof Note that f (·) and Gc (·) are continuous functions For the function f (·), we have f (α) − f (β) ≤ f (α) ≤ Tr(AT A1 ) α − β , Tr(AT A1 ) α + γ, 179 ∀ α, β ∈ R3 , (A.16) (A.17) A.2 Existence of unique and regular solution of system (4.66) where α = [ α1 α2 α3 ]T , β = [ β1 β2 β3 ]T , and A1 is defined as    A1 =     − γ+M s −s − γ+M Uy −Ux −Uy   Ux    −γ For the function Gc (·), we have Gc (α) √ = max Kx (|α3 | + |α2 |), Ky (|α3 | + |α1 |), M (|g1 | + |g2 | + |1 − g3 |) = max (Kx (|α3 | + |α2 |), Ky (|α3 | + |α1 |), √ M (min(1, |α1 |)min(1, |α3 |) + min(1, |α2 |)min(1, |α3 |) + − min(α3 , 1)) √ ≤ max Kx α , Ky α , M (1 + α ) (A.18) Therefore, there exist a constant K1 such that Gc (α) ≤ K( α + 1) (A.19) Calculating Gc (α) − Gc (β), we have Gc (α) − Gc (β)  √ −Ky (α3 − β3 ) M (g1 (α) − g1 (β))   √ =  Kx (α3 − β3 ) M (g2 (α) − g2 (β))   √ −Kx (α2 − β2 ) Ky (α1 − β1 ) − M (−g3 (α) − g3 (β)) 180     (A.20)   A.2 Existence of unique and regular solution of system (4.66) The 1-norm of Gc (α) − Gc (β) is Gc (α)−Gc (β) = max (Kx (|α3 −β3 |+|α2 −β2 |), Ky (|α3 −β3 |+|α1 −β1 |), G(α)−G(β) ) , (A.21) where G(·) is defined in (4.21) In the proof of Theorem 4.4.1, it has been shown that there exist a constant K2 such that the following Lipschitz condition holds for G(·) G(α) − G(β) ≤ K2 α − β (A.22) With the equivalent property of the 1- and 2-norm, we have that a constant K3 exists such that G(α) − G(β) ≤ K3 α − β (A.23) By substituting (A.23) into (A.21), the following relation can be obtained Gc (α)−Gc (β) ≤ max (Kx (|α3 −β3 |+|α2 −β2 |), Ky (|α3 −β3 |+|α1 −β1 |), K3 α − β ) (A.24) Since |α3 −β3 |+|α2 −β2 | ≤ α − β |α3 −β3 |+|α1 −β1 | ≤ α − β , we can further obtain Gc (α)−Gc (β) ≤ K4 α − β 181 (A.25) A.2 Existence of unique and regular solution of system (4.66) for some constant K4 All the Lipschitz conditions of Lemma 4.4.1 are satisfied due to the equivalence between 1- and 2-norm Therefore, we can conclude that system (4.66) has a unique solution and it is regular 182 Appendix B Author’s Publications Journal Papers [1 ] Shouwei Zhao, Hai Lin, Jitao Sun and Zhengui Xue, “An implicit Lyapunov control for finite-dimensional closed quantum systems”, International Journal of Robust and Nonlinear Control, vol 22, no 11, pp 1212–1228, 2012 [2 ] Shouwei Zhao, Hai Lin and Zhengui Xue, “Switching control of closed quantum systems via the Lyapunov method”, Automatica, vol 48, no 8, pp 1833– 1838 [3 ] Zhengui Xue, Hai Lin and Tong Heng Lee, “Analysis and control of closed quantum systems based on real-valued dynamics”, IET Control Theory and Applications, vol 6, no 16, pp 2576–2584, 2012 [4 ] Zhengui Xue, Hai Lin and Tong Heng Lee, “Identification of unknown parameters for a class of two-level quantum systems”, IEEE Transactions on Automatic Control, accepted 183 Conference Papers [1 ] Shouwei Zhao, Hai Lin, Jitao Sun and Zhengui Xue, “Implicit Lyapunov control of closed quantum systems” in Proceedings of the IEEE Conference on Decision and Control, Shanghai, China, Dec 2009, pp 3811–3815 Presented the paper in the poster session and won the “General Chairs’ Recognition Award for Interactive Papers” [2 ] Zhengui Xue, Hai Lin and Tong Heng Lee, “Identification and control of quantum systems”, in Proceedings of the IEEE International conference on Robotics, Automation and Mechatronics, Singapore, June 2010, pp 30–35 [3 ] Zhengui Xue, Hai Lin and Tong Heng Lee, “Identification and control of a two-level open quantum system”, in Proceedings of the IEEE Conference on Decision and Control, Orlando, USA, Dec 2011, pp 6254–6259 184 ... stages of quantum control 1.2.1 Open-loop control Coherent control In many experimental setups of quantum control, control efforts are imposed on quantum systems in a classical manner One typical... real-time estimation and control of quantum systems The primary difficulty of the real-time estimation and control of quantum systems comes from the system information acquisition process Stochastic... the states of measured systems, closed-loop control of quantum systems is much more complex than that of classical systems 1.2 Quantum Control Review Learning control A learning control strategy