Joint 48th IEEE Conference on Decision an Control an 28th Chinese Control Conference Shanghai, P.R. China, December 16-18, 29 ThAIn1.4 Implicit Lyapunov control of close quantum systems Shouwei Zhao, Hai Lin, Jitao Sun an Zhengui Xue Abstract In this paper, we investigate the state convergence problem for close quantum systems uner egenerate cases. An implicit Lyapunov-base control strategy is propose for the convergence analysis of finite imensional bilinear Schröinger equations. The egenerate cases that the systems o not satisfy the strong regular conition [17] an the conition φ i H 1 φ j =, i, j k for eigenstates φ i, φ j of H ifferent from target state φ k, are consiere. First the Lyapunov function is efine by the implicit function an the existence is guarantee by a fixe point theorem. Then the convergence analysis is investigate by the LaSalle invariance principle. Finally, an example is provie to show the effectiveness of propose results. I. INTRODUCTION Driven by scientific inquiry an the emans of avancing technology, the past ecaes have seen increasing theoretical an experimental research towars control of quantum systems. Control of quantum phenomena is essential in successful applications of quantum systems in a wie variety of areas such as quantum computation, quantum chemistry, nano-scale materials, NMR an Bose-Einstein conensates. Hence, recent years have seen a great eal of research efforts in the evelopment of quantum control theory an many results about controllability an control methos have been obtaine, see [1], [2] an references therein. Quantum control can be roughly ivie into two categories. The first category falls into the open-loop control scheme for which many control strategies utilize some forms of moel-base feeback, both geometry base an optimization base [3]-[4]. This metho is relatively simple to implement an many applications of open-loop Hamiltonian engineering in iverse areas from quantum chemistry to quantum information processing. But the question of when, i.e., for which systems an objectives, the metho is effective an when it is not, has not been answere satisfactorily. The secon category is of close-loop control, for example, state reuction an stabilization using feeback from measurement is applie through a combination of geometric control an classical probabilistic techniques[5]-[1]. This metho is Shouwei Zhao is with Dept. of Mathematics, Tongji University, 292, China an Dept. of Electrical an Computer Engineering, National University of Singapore, Singapore. zhaoshouwei@gmail.com Hai Lin is with Dept. of Electrical an Computer Engineering, National University of Singapore, Singapore. elelh@nus.eu.sg Jitao Sun is with Dept. of Mathematics, Tongji University, 292, China. sunjt@sh163.net Zhengui Xue is with NUS Grauate School for Integrative Sciences an Engineering, National University of Singapore, 117456, Singapore. zhenguixue@nus.eu.sg The financial supports from Singapore Ministry of Eucation s AcRF Tier 1 funing, TDSI, an TL an the National Natural Science Founation of China uner Grant 687427 are gratefully acknowlege. somehow more exact an intuitive to control the given quantum systems than the open-loop control. However, closeloop feeback control is a nontrivial problem as feeback requires measurements an any observation of a quantum system generally isturbs its state, an often results in a loss of coherence that can reuce the systems to mostly classical behavior. In orer to mitigate this backaction, measurement an feeback in quantum systems lea to much more complicate moels an ynamics than the Schröinger equations. In this paper, we consier Lyapunov open-loop control, where a Lyapunov function is efine an feeback from a moel is use to generate controls to minimize its value. Lyapunov control has been wiely use in feeback control to analyze the stability of close-loop systems. Several recent papers have propose the application of Lyapunov control esigns to quantum systems [11]-[19]. Since the quantum measurement an feeback woul lea to more complicate moel than Schröinger equations [15] an the super-short control require by some quantum ynamics restricts the application of observation an feeback [11], currently the open loop Lyapunov control remains ominant [11]. Such open loop control nees to be first simulate. From simulation one obtains a control signal which is then, in practice, applie in open loop control. Lyapunov open loop control has prove to be a simple an effective metho in achieving ieal control performance. Several papers on Lyapunov control for quantum systems only consiere the control of quantum systems with target states that are eigenstates of the free Hamiltonian H, an therefore fixe points of the ynamical system [11],[12]. While in the egenerate cases that target states are not eigenstates of H, i.e., evolve with, the issue of convergence analysis of such systems is investigate by (implicit) Lyapunov technique an the LaSalle invariance principle [2]. This means that the problem is reformulate to asymptotic convergence of the system s actual trajectory to that of the -epenent target state [13]-[16]. Moreover, some authors consiere the Lyapunov control for mixestate quantum systems in the notion of orbit convergence or the trajectory tracking problem [17]-[19]. It is well-known that Lyapunov functions base on the the average value of an imaginary mechanical quantity is of great significance in Lyapunov control of bilinear Schröinger equations [11]-[13]. In [11], the authors consiere the convergence analysis of Schröinger equations base on three kins of functions. But the invariant set is generally large, the invariant principle is not sufficient to conclue the asymptotic convergence. Particularly, the authors utilize Lyapunov function base on the average value of the imaginary mechanical quantity to give a generalize 978-1-4244-3872-3/9/$25. 29 IEEE 3811
theorem(theorem 6) on the largest invariant set of the closeloop system where the eigenvalue of H corresponing to the eigenvector λ i must satisfy the following strong regular conition ω ij ω lm, (i,j) (l, m), with ω ij = λ i λ j. It shoul be notice that this strong regular conition may fail to satisfy for many controllable systems [17]. On the other han, from Theorem 6 in [11], it can be seen that if λ i H 1 λ j = for i j k, the systems will be asymptotically stable using the propose control. But if this conition oes not hol, then it is ifficult to rive the systems to the goal state using the control law esigne in that paper. This motivates us to pursue another strategy for stability analysis an esign uner the above egenerate cases. Therefore, in the current paper, we investigate the asymptotic convergence of Schröinger equations by implicit Lyapunov techniques an the LaSalle invariant principle uner the above-mentione egenerate cases. The main ifficult lies in how to apply the LaSalle invariant principle combine with proper assumptions to rive systems asymptotically converge to the target state in the egenerate cases. The rest of this paper is organize as follows. Some preliminaries are presente in Section II. The feeback control law is esigne using implicit Lyapunov techniques, in which the Lyapunov function is base on the average value of the imaginary mechanical quantity an convergence analysis is erive by LaSalle invariance principle in Section III. A numerical example an simulation stuies are iscusse to emonstrate the effectiveness of the propose metho in Section IV. Some concluing remarks are rawn in Section V. II. PRELIMINARIES Consier the following bilinear Schröinger equation i Ψ = (H + u(t)h 1 )Ψ, Ψ t= = Ψ, Ψ = 1, where H is the free Hamiltonian, an H 1 is the interaction Hamiltonian; furthermore, both of them are Hermitian matrices. The state of the system verifies the conservation of probability: Ψ(t) = 1, t, which means that the state is on the unit sphere of C N : S = {x C N : x = 1}. In this paper, the main objective is to consier the asymptotic convergence of close quantum systems uner the above egenerate cases using implicit Lyapunov techniques. We choose the Lyapunov function base on the average value of the imaginary mechanical quantity. First, for r R, enote by (λ k,r ) 1 k N the eigenvalues of the operator H + rh 1, with λ 1,r λ N,r an by (φ k,r ) 1 k N the associate normalize eigenvectors: (H + rh 1 )φ k,r = λ k,r φ k,r. We assume that for any small r, φ i,r H 1 φ j,r = for i j k. Let φ k,r be the -varying target state instea (1) of φ k. For simplicity, the kth eigenvector φ k = φ for some k {1,2,,N} is enote to be the goal state, then we assume that the kth eigenspace of the free Hamiltonian H is of imension 1, so that the target state (i.e., kth eigenstate of the system) is efine without any ambiguity. Then the control strategy base on implicit Lyapunov metho is investigate uner the egenerate cases. By introucing the feeback controllers, the state of controlle systems is steere to a moving target state φ k,r(t) instea of φ k, where r(t) is efine implicitly by the state of the systems. The goal is to make φ k,r(t) converge slowly to φ k an at the same, using the feeback controller to stabilize the system state as fast as possible aroun the vector function φ k,r(t). The basic iea is shown by a irect way in Fig. 1 in [15]. III. THE LYAPUNOV METHOD BASED ON THE AVERAGE VALUE OF AN IMAGINARY MECHANICAL QUANTITY In this section, we first recall the Lyapunov function in the original reference [12] an evelop the iea of the controller esign. Suppose that the Hermitian operator P is a mechanical quantity of the quantum system. Accoring to quantum theory, if the system is in an eigenstate of P, then the average value of P is the eigenvalue corresponing to the eigenstate of P. From this point of view, it is reasonable to consier the average value of P as a Lyapunov function. V (Ψ) := Ψ P Ψ (2) In [12], S. Grivopoulos an B. Bamieh prove the following important lemma in control theory via variational calculus. Lemma 1: [12] With the constraint conition Ψ Ψ = 1, the set of critical points of the Lyapunov function V (Ψ) = Ψ P Ψ is given by the normalize eigenvectors of P. The eigenvectors with the largest eigenvalue are the maxima of V, the eigenvectors with the smallest eigenvalue are the minima an all others are sale points. Accoring to Lemma 1, if the goal state φ k correspons to the smallest eigenvalue l k of P, then V (Ψ) = Ψ P Ψ is equal to l k at Ψ = φ k. Thus, when the esigne control fiels make V ecrease continually to l k, the state of the system will be possibly riven to φ k, that is, the goal state. This iea will be use to esign the control fiels an construct the imaginary mechanical quantity P. A. Controller esign Corresponing to the target state φ k,r for any r (,r ], we construct the analytic matrix P r such that φ k,r is the eigenvector of P r an the corresponing eigenvalue l k,r is the smallest one of all the eigenvalues of P r. The process to fin such matrices can be foun in [11]. Now we efine the following Lyapunov function base on the average value of an imaginary mechanical quantity V 1 (Ψ) := Ψ P r Ψ, (3) An the function Ψ r(ψ) is implicitly efine as follows, r(ψ(t)) = r(ψ) := θ( Ψ P r Ψ l k ), (4) 3812
for a slowly varying real function θ. Noting that uner the assumption of non-egeneracy for the kth eigenstate of H + rh 1 for r [,r ], φ k,r an P r are analytic mappings of the parameter r [,r ] [16]. In particular, we can consier the erivative of the map r P r at least in the interval [,r ]. Denote by Pr r r the erivative of this map at the point r. Furthermore, as the epenence of φ k,r with respect to r is analytic, Pr r is boune on [,r ] an thus C := max{ P r r r ; r [,r ]} < A simple computation yiels that r θ(v 1(Ψ) l k ) = r θ( Ψ P r Ψ l k ) =θ ( Ψ P (5) r r Ψ ). Choosing the function θ such that θ is small enough an ue to the fact that C <, the function α [,r ] θ(v 1 (Ψ) l k ) := θ( Ψ P α Ψ l k ) will be contraction for fixe Ψ S. Thus, for any fixe point Ψ S, there exists a unique r(ψ) [,r ] such that (4) is satisfie. Let us explain the one-to-one corresponence, Ψ S r(ψ) [,r ] by the implicit function theorem. Consier the following function F(r,Ψ) := r θ(v 1 (Ψ) l k ). F is regular with respect to r an Ψ, an for a fixe Ψ S we have F(r(Ψ),Ψ) = ; furthermore, we have r F(r,Ψ) = 1 θ ( Ψ P r r Ψ ), which is non-zero for θ which ensures θ to be small enough. Thus, with the implicit function theorem an the uniqueness of the application Ψ r(ψ), we have the following existence result: Lemma 2: Let θ C (S;[,r ]) be such that θ() =, θ(s) >, s >, θ < 1 C where C := 1 + max{ Pr r r ; r [,r ]} <. Then there exists a unique map r C (S;[,r ]) such that for every Ψ S, r(ψ) = θ( Ψ P r Ψ l k ), with r(φ k ) =. Assumption 1: There exists a r such that for every r (,r ], we have λ 1,r < < λ N,r an the Hamiltonian H + rh 1 is not λ k,r -egenerate. Let φ k,r be an eigenstate of H +r(ψ)h 1 an be also the goal state. We assume that all the eigenstates of H + r(ψ)h 1 satisfying φ j,r H 1 φ i,r =, i, j {1,2,,N} i,j k. In the sequel, we assume that θ C (S;[,r ]) an θ < 1 2C. Accoring to the controller to be esigne with u(ψ(t)) = r(ψ(t)) + v(ψ(t)), for simplicity, u(t) = r(t) + v(t), the system (1) evolves as follows i Ψ = (H + (r(t) + v(t))h 1 )Ψ. Differentiating V 1 with respect to t yiels that An V 1(Ψ(t)) = Ψ P r Ψ + Ψ P r Ψ + ṙ(t) Ψ P r r Ψ =i Ψ [H + r(t)h 1,P r ] Ψ + i Ψ [H 1,P r ] Ψ v(t) + ṙ(t) Ψ P r r Ψ. (6) ṙ(t) =θ (V 1 l k ){i Ψ [H + r(t)h 1,P r ] Ψ + i Ψ [H 1,P r ] Ψ v(t) + ṙ(t) Ψ P r r Ψ }. (7) Let us enote by K(t) := θ (V 1 l k )( Ψ P r r Ψ ). From the assumption that θ < 1 2C, K(t) 1 2 for any t [,+ ). Accoring to (6) an (7) an the conition that [H + r(t)h 1,P r ] =, we have (1 K(t))ṙ(t) = θ (V 1 l k )i Ψ [H 1,P r ] Ψ v(t), which means that ṙ(t) = θ (V 1 l k ) 1 K(t) i Ψ [H 1,P r ] Ψ v(t). Now, we rewrite (6) as follows: V 1(Ψ(t)) = i Ψ [H 1,P r ] Ψ v(t) + θ (V 1 l k ) 1 K(t) i Ψ [H 1,P r ] Ψ v(t) Ψ P r r Ψ = (1 + K(t) 1 K(t) )i Ψ [H 1,P r ] Ψ v(t), where 1 + K(t) 1 K(t) > for every t. Thus, esign a feeback law as follows (8) v(t) = v(ψ(t)) := cf(i Ψ [H 1,P r ] Ψ ) (9) with a positive constant c, where the image of function y = f(x) passes the origin of plane x y monotonically an lies in quarant I or III. It is clear that with the above controller we have V1. The main purpose of the next section is to provie the convergence analysis of this feeback esign uner some suitable assumptions. Characterization of the ω-limit set for the close-loop system will be propose by the LaSalle invariance principle. B. Convergence analysis In this section, we use the LaSalle invariance principle to analyze the convergence of the system (1) with the implicit feeback function. First, let us recall the LaSalle invariance principle [2]: Lemma 3: [2] For an autonomous ynamical system, ẋ = f(x), let V (x) be a Lyapunov function on the phase space Ω = {x}, satisfying V (x) > for all x x an V (x), an let O(x(t)) be the orbit of x(t) in the phase 3813
space. Then the invariant set E = {O V (x(t)) = } contains the positive limiting sets of all boune solutions, i.e., any boune solution converges to E as t +. Base on the controller esign above, the convergence analysis of the controlle quantum system is presente. Theorem 1: Consier the system (1) with the feeback esign u(ψ(t)) := r(ψ(t)) + v(ψ(t)) where r(ψ) is given by Lemma 2 an v(ψ(t)) := cf(i Ψ [H 1,P r ] Ψ ) with a positive constant c. Moreover, let θ C (S;[,r ]) be such that the conitions in Lemma 2 are satisfie. Let us also suppose that (i) [H + r(t)h 1,P r ] =, (ii) ωij r ωr lm, (i,j) (l, m), (iii) l r,i l r,j, i j, where ωij r = λr i λr j, λr i (i = 1,2,,N) is the eigenvalue of H + r(t)h 1 corresponing to the eigenvector φ i,r. Then the solution of system (1) converges towar S := {φ k e iθ ;θ R} uner Assumption 1 in the sense of lim t ist(ψ(t), S) =. Proof: From the Lyapunov function V 1 (Ψ) efine in the previous section an the feeback controller u(ψ(t)) = r(ψ(t))+v(ψ(t)) esigne in the Theorem, we have V1. From Lemma 3, the trajectories of the close-loop system converge to the largest invariant set containe in V1 =. Let us characterize this invariant set. Suppose that Ψ is a solution of the system (1) such that V1 =. Then there exists a constant V such that V 1 (Ψ) = V. This implies that r(ψ) is a constant enote by r(ψ) = r where r := θ( V ). The equation V1 = satisfies if an only if v(ψ(t)) := cf(i Ψ [H 1,P r ] Ψ ) =. (1) Thus the controlle system can be represente by i Ψ = (H + rh 1 )Ψ, Ψ t= = Ψ, Ψ = 1. (11) There are two cases to be consiere. (i) r =, then θ( V ) =, which means that V =, so we have Ψ S from Lemma 1 an the efinition of V 1 (Ψ). Then we complete the proof in this case. (ii) r, which means that < r < r. Without loss of generality, we assume that when t = t, (1) is satisfie. Now to verify that Ψ(t ) is the point in the invariant set of the close-loop system, we only nee to verify that for any t (,+ ), v(ψ(t + t)) = v(t + t) =. Denote the state of the system at t by Ψ(t ) = c i (t )φ i, r, (12) i=1 where φ i, r is the ith eigenvector of H + r(ψ)h 1. By the property of invariance, Ψ(t + t) shoul also satisfy (1) i.e., Ψ(t + t) [H 1,P r ] Ψ(t + t) =. (13) Since Ψ solves the equation (11), we can erive Ψ(t + t) as follows: Ψ(t + t) = e i(h+ rh1) t Ψ(t ) (14) = c i (t )e i(h+ rh1) t φ i, r i=1 Using (13), it follows that (l r,i l r,j )c i (t )c j(t )e iω r ji t φ j, r H 1 φ i, r =, i,j=1 (15) where ω r ij = λ r i λ r j, λ r i (i = 1,2,,N) is the eigenvalue of H + r(t)h 1 corresponing to the eigenvector φ i, r. Furthermore, it can be written in the following simple form (l r,i l r,j ) φ j, r ρ(t ) φ i, r φ j, r H 1 φ i, r e iω r ji t = i,j=1 (16) where ρ(t ) := Ψ(t ) Ψ(t ). From conition (ii) of the Theorem, by the arbitrary of t, e iω r ji t are linear inepenent with each other. Then for the eigenvectors φ i, r, φ j, r such that φ j, r H 1 φ i, r =, we only nee to construct the matrix P r such that l r,i l r,j. Thus equation (16) can be expresse by φ j, r ρ(t ) φ i, r φ j, r H 1 φ i, r =, i,j {1,2,,N} (17) If the system satisfies Assumption 1, then Ψ S an we finish the proof of the theorem. IV. SIMULATIONS Consier the following system with H an H 1 H := 1,H 1 := 1 1 1 1 1. 2 1 1 1 Let the secon eigenstate of H, φ 2 = ( 1 ) T be the target state. An we can see that this system oes not satisfy the strong regular conition. If we apply the implicit Lyapunov technique base on the state istance using the feeback control law propose in [15], simulaitons in Fig. 1 show that the systems can not be stabilize when c = 1 an the initial state Ψ = 1 3 (1 1 1) T. Now we aopt the Lyapunov metho base on the average value of the imaginary mechanical quantity. Simple computation yiels that Assumption 1 hols an all the conitions in Theorem 1 are satisfie. The first part of the control fiel r(ψ(t)) is efine implicitly by (4). In orer to fin this function at each step, we use a fixe point algorithm by computing iteratively the value of θ(v 1 ) an the function θ(s) is chosen to be θ(s) = s/2. The secon part of feeback law is given by (9) with v(ψ(t)) = i( Ψ [H 1,P r ] Ψ ). The simulations in Fig.2 illustrate the performance of this approach. An for real quantum systems, we use the above control signals from the close-loop simulations to achieve the state steering. 3814
(a) populations (b) control fiel 1.8.6 population of the 1st eigenstate 2n eigenstate 3r eigenstate 5 1 15 2 25 3 35 4 5 1 15 2 25 3 35 4 Fig. 1: (a): The population of the system trajectory Ψ(t) solution of the system (1) with feeback esign (9) an (15) in [15]. It can be seen that the system can not converge to φ 2 the secon eigenstate of the internal Hamiltonian as t + ; (b): the control fiel γ(ψ) + v(ψ). (a) populations (b) control fiel 1.8.6 population of the 1st eigenstate 2n eigenstate 3r eigenstate 5 1 15 2 25 3 35 4 5 1 15 2 25 3 35 4 Fig. 2: (a): The population of the system trajectory Ψ(t) solution of the system (1) with feeback esign u(ψ) = r(ψ) + v(ψ). It can be seen that the system reaches φ 2 the secon eigenstate of the internal Hamiltonian; (b): the control fiel r(ψ(t)) + v(ψ(t)). REFERENCES [1] M.A. Nielsen an I.L. Chuang, Quantum computation an quantum information, Cambrige University Press, Cambrige; 2. [2] D. D Alessanro, Introuction to quantum control an ynamics, Chapman & Hall/CRC; 27. [3] P. Vettori, On the convergence of a feeback control strategy for multilevel quantum systems, in Proceeings of the MTNS Conference, Notre Dame, USA, 22. [4] S. Shi, A. Wooy an H. Rabitz, Optimal control of selective vibrational excitation in harmonic linear chain molecules, J. Chem. Phys., vol. 88, 1988, pp 687-6883. [5] H.M. Wiseman an A.C. Doherty, Optimal unravellings for feeback control in linear quantum systems, Phys. Rev. Lett., vol. 94, 25, 745. [6] Z.R. Xi an G.S. Jin, Classical an quantum control of a simple quantum system, International Journal of Quantum Information, vol. 5, 27, pp 857-884. [7] M. Mirrahimi an R. Van Hanel, Stabilizing feeback controls for quantum systems, SIAM J. Cont. Opt., vol. 46, 27, pp 445-467. [8] R. van Hanel, J.K. Stockton an H. Mabuchi, Feeback control of quantum state reuction, IEEE Trans. Auto. contr., vol. 5, 25, pp 768-78. [9] H.M. Wiseman, Quantum theory of continuous feeback, Phys. Rev. A, vol. 49, 1994, pp 2133-215. [1] K. Jacobs an A.P. Lun, Feeback control of nonlinear quantum systems: A rule of thumb, Phys. Rev. Lett., vol. 99, 27, 251. [11] S. Kuang an S. Cong, Lyapunov control methos of close quantum systems, Automatica, vol. 44, 28, pp 98-18. [12] S. Grivopoulos an B. Bamieh, Lyapunov-base control of quantum systems, in Proc. 42th IEEE Conf. Decision an Control, 1, 434-438, 23. [13] M. Mirrahimi an P. Rouchon, Trajectory generation for quantum systems base on Lyapunov techniques, in Proceeings of IFAC symposium NOLCOS, Stuttgart, Germany, 24. [14] M. Mirrahimi an P. Rouchon, Trajectory tracking for quantum systems: A Lyapunov approach, in Proceeings of the international symposium MTNS, Leuven, Belgium, 24. [15] K. Beauchara, J.M. Coron, M. Mirrahimi an P. Rouchon, Implicit Lyapunov control of finite imensional Schröinger equations, System Control Lett., vol. 56, 27, pp 388-395. [16] M. Mirrahimi, P. Rouchon an G. Turinici, Lyapunov control of bilinear Schröinger equations, Automatica, vol. 41, 25, pp 1987-1994. [17] X.T. Wang an S.G. Schirmer, Analysis of Lyapunov metho for control of quantum states, arxiv: 81.72v1. [18] C. Altafini, Feeback stabilization of quantum ensembles: a global convergence analysis on complex flag manifols, in Proc. 45th IEEE Conf. Decision an Control, San Diego, USA, 2471-2476, 26. [19] C. Altafini, Feeback control of spin systems, Quantum Information Processing, vol. 6, 27, pp 9-36. [2] J. LaSalle an S. Lefschetz, Stability by Liapunov s irect metho with applications, Acaemic Press, New York; 1961. V. CONCLUSIONS A stabilization metho for finite imensional quantum systems an its convergence analysis have been propose uner the egenerate cases that the strong regular conition as well as the conition φ i H 1 φ j, i,j k o not hol. By aopting the Lyapunov function base on the average value of the imaginary mechanical quantity, feeback control laws base on implicit Lyapunov functions have been esigne. Moreover, convergence analysis has been investigate via the LaSalle invariance principle. Aitionally, simulation stuies have been provie to show the effectiveness of the propose results. 3815