Lyapunov-based control of quantum systems

Transcription

1 Lyapunov-based control of quantum systems Symeon Grivopoulos Bassam Bamieh Department of Mechanical and Environmental Engineering University of California, Santa Barbara, CA Abstract We propose feedback control laws for quantum systems, based on Lyapunov design. These feedback laws locally asymptotically stabilize any desired eigenstate of the system Hamiltonian. The target eigenstate and the size of its region of attraction can be tailored by the choice of design parameters. These feedback control laws are used to design open loop controls for state transfer problems. We illustrate the method with a transfer problem between bound states of a molecular Morse oscillator. I. INTRODUCTION The model of a controlled quantum system (Schroedinger equation) is i ψ = ( H + α H α u α (t) ) ψ, () where ψ(t) is the state vector of the quantum system that belongs to an appropriate (finite or infinite-dimensional) complex Hilbert space, and H, H α are self-adjoint operators in this space, referred to as the system and control Hamiltonians respectively. H describes the natural dynamics of the system, and H α the coupling of the system to external classical control fields u α (t). The dynamics generated by () is unitary (norm preserving). By convention, we normalize ψ =. Perhaps the most significant question about controlled quantum systems concerns their controllability properties, such as controllability conditions on () (see [], [2]) and the design of controls for steering (see e.g. [3] for constructive control, [4], [5] for optimal control and [6] for time-optimal control). This work also concerns itself with a steering problem inspired by Quantum Chemistry: Design of controls that drive the system to an eigenstate of H. These states are very important in their own right: Each of them corresponds to a definite measurable energy of the system and being on one or the other energy eigenstate determines the chemical behavior of the system. The class of controls we propose are in state feedback form and are based on Lyapunov design. We prove that they asymptotically drive any initial state to an eigenstate of H. Although the controls are in feedback form, the collapse of the state associated with measurements in Quantum Mechanics would prevent us from using them as such in an experimental setting. Rather, we only use our method to generate open loop controls for specific transfer problems. For related approaches, see [7], [8]. In the following, we consider only finite-dimensional quantum systems, use units where = and consider () with only one control: ı ψ = ( H + H u(t) ) ψ. (2) II. CONTROL DESIGN We use V (ψ) = ψ P ψ, P = P, as a Lyapunov function. Our first task is to analyze the structure of its critical points on the sphere ψ =. Lemma : The set of critical points of V (ψ) = ψ P ψ on the sphere ψ ψ = is given by the normalized eigenvectors of P. The eigenvectors with the largest eigenvalue are the maxima, the eigenvectors with the smallest eigenvalue are the minima and all others are saddle points. Proof: To find the critical points of V (ψ) under the constraint ψ ψ =, consider the augmented function Ṽ (ψ, λ) = ψ P ψ λ (ψ ψ ), where λ is a Lagrange multiplier. Varying Ṽ with respect to ψ we have, ( ) Ṽ = = P ψ λψ, ψ so, ψ is an eigenvector of P and λ the corresponding eigenvalue. Varying Ṽ with respect to λ enforces the normalization of ψ. Denote the normalized eigenvectors of P by q i, i =,..., N, and the corresponding eigenvalues by p i. Then, P = N p iq i q i. To determine the structure of V around one of its critical points, q N for example, consider a finite variation ψ such that q N + ψ 2 =. Express ψ in the basis of the eigenvectors of P, ψ = N ψ i q i. The normalization condition q N + ψ 2 = implies that ( ψ N + ψn + ψ N ψn) + ψ i ψi =,

2 that is, not all components of the variation can be chosen arbitrarily. Consider the difference V (q N + ψ) V (q N ) = p N ( ψ N + ψn + ψ N ψn) + p i ψ i ψi = (p i p N ) ψ i ψi. Taking the ψ i, i =,..., N and arg( ψ N ), as the independent parameters of the variation, we see that the structure of V around q N depends on the ordering of the eigenvalues: q N is a local maximum iff p N is the largest eigenvalue, a local minimum iff p N is the smallest eigenvalue and a saddle point otherwise. We calculate the rate of change of V along a trajectory of (2) : dv dt = ıψ P (H + H u)ψ + ıψ (H + H u)p ψ = ıψ [H + H u, P ]ψ = ıψ [H, P ]ψ + ıψ [H, P ]ψ u, where [A, B] := A B B A, denotes the commutator of two matrices. Since tr([a, B]) = tr(ab) tr(ba) =, a commutator can never be sign definite. For this reason, we choose P to commute with H : [H, P ] =. This implies that the two matrices must have the same eigenvectors and hence, by Lemma, the critical points of V are the eigenvectors of H. Now, dv dt = ıψ [H, P ]ψ u. To make V, we choose the following (feedback) control law: u = f(ıψ [H, P ]ψ), (3) where f is a continuous function with f() = and xf(x) >, x. With this choice of u, (2) becomes ı ψ = ( H f(ıψ [H, P ]ψ) H ) ψ. (4) We show now that the set of eigenvectors of H is the largest invariant set of (4). For ψ to be in the invariant set of (4), V = u 2 =, which means that must be satisfied along with ı ψ (t) [H, P ] ψ(t) = (5) ı ψ(t) = ( H f( ı ψ (t)[h, P ] ψ(t) ) H ) ψ(t), with initial condition ψ. The solution is given by ψ(t) = e ıht ψ N = c i e ıeit q i, where ψ = N c i q i and the E i s are the eigenvalues of H. Substituting this in (5), we obtain N ( ı) (p i p j ) c i c j e ıωjit (q jh q i ) =, (6) i,j where ω ji := E j E i is the transition frequency between levels i and j. We make the following assumptions: () ω ji ω lk, (i, j) (k, l). (2) q j H q i, i j. (3) p i p j, i j. Assumption () guarantees that the exponential functions in (6) are linearly independent, while (2) and (3) ensure that the resulting equations, c i c j =, hold i j. The only non-trivial solutions of these have only one c i non-zero. This means that ψ is one of the q i s. Hence, the largest invariant set of (4) is the set of eigenvectors of H. LaSalle s invariance principle [9] guarantees its asymptotic stability. We summarize our result in Theorem : Consider system (2) with the feedback control law (3), where H and P commute. Given the assumptions () (3), this control law asymptotically drives any trajectory of (2) to an eigenspace of H. This analysis can be generalized: (i) One may also consider a P with multiple eigenvalues. Suppose that P has only M < N distinct eigenvalues. Group the eigenvectors of P in M corresponding eigenspaces. One can see that the conclusions of the theorem continue to hold if we substitute eigenspaces for eigenvectors. One still needs assumptions ()-(3), but the indices need only belong to levels in different eigenspaces of P. (ii) For systems with more controls, instead of (2) one would need the weaker condition (2 ) q j H αq i, i j, for some H α. This theorem only guarantees that all trajectories of (2) will approach asymptotically one of the eigenstates of H. For it to have any practical value, one should be able to specify the target energy eigenstate. One should also know the region of attraction of this eigenstate, that is the set of initial conditions that are asymptotically driven to it. Suppose that the desired eigenstate is q N. Make the corresponding eigenvalue, p N, strictly smaller than the others and then, by the Lemma, q N will be the unique minimum of V. Then, every state in the set A N = {ψ S 2 V (ψ) < min(p,..., p )} will asymptotically end up to q N, because energy decreases and the only critical point inside A N is q N. Incidentally, the region of attraction of q N may be bigger than A N but we can only prove convergence from A N. This is an open set whose volume depends on the sizes of the p i s. If one lumps p,..., p close together and makes p N much smaller, one can make it be

3 almost the whole sphere. As a limiting case, one can take p =... = p > p N. Then, V becomes a true Lyapunov function and A N (which will now contain the whole sphere except for the equator {ψ S 2 ψ q N = } = {ψ S 2 ψ span{q,..., q }}) is the whole region of attraction of q N, see also [8]. Nevertheless, the case of more than two distinct p i s is of interest in applications. This occurs, for example, when one wants to realize a transition between two levels and at the same time avoid populating a third level. Finally, 3 remarks: (i)numerical experimentation with these feedback laws lead us to suspect that the minimum of V is almost-globally asymptotically attractive, that is, with the exception of a set of initial states of measure zero, every trajectory of (2) will asymptotically approach the minimum of V. This is only an intuition which we cannot prove at the moment. (ii)assumptions () and (2) concern only the system and imply controllability of (2). As a matter of fact, they are somewhat stronger than controllability as given by the Lie Algebra Rank Condition [], [2]. Nevertheless, they are met by many realistic systems or, in some cases, can be satisfied by turning on a small constant control u and considering H = H + H u. Satisfying assumption (3) and, more generally, the ordering of the p i s, is part of the design. (iii) The controls generated by (3) are bounded, u < max f(x). x 2 H P III. APPLICATION TO A MOLECULAR SYSTEM We apply our method to transfer problems in a model that describes the vibrational dynamics of a two atom molecule, the Morse oscillator. The relevant Hamiltonian is with the Morse potential d 2 H = 2 2m dr 2 + V (r), V (r) = V {exp[ α(r r )] } 2 V, This is an infinite dimensional quantum system. r is the intramolecular distance, m is the reduced mass of the system, r the equilibrium distance of the molecule, α determines the width of the molecular well and V its depth. For these parameters, we take the values that correspond to the OH bond. The theory of the Morse oscillator predicts that the OH system supports 22 bound states as well as continuum states. The control is the time-varying electric field E(t) of a laser that enters through the semi-classical interaction Hamiltonian (in the electric dipole approximation) H int (t) = µ(r)e(t) and µ(r) is the molecular dipole function µ(r) = µ r exp( r/r ) (µ and r are parameters of the interaction). For the application of our technique, we truncate the (infinite dimensional) state space down to the subspace of bound states. Ignoring the continuous part of the spectrum, introduces a small error to our numerical results, but these are mostly meant to be an illustration. We consider a system initially in its ground state and construct a control that excites it to the th level and then to its last (22nd) level. Figures () and (2) display the control and the populations of some bound states during the transition. The population of state increases to 9% in about picoseconds. The final two figures display the control and the populations of some bound states for the transition 22. This transition is 9% complete also in picoseconds. IV. CONCLUSIONS We have presented a family of state feedback laws for the control of quantum systems. These feedback laws generate arbitrarily bounded controls that asymptotically drive the system to a desired eigenstate of H. The region of attraction of the target eigenspace can be specified at will. An application to a model molecular system was given. V. REFERENCES [] R. Brockett, Lie Theory and Control Systems defined on Spheres, SIAM Journal of Applied Mathematics, vol. 25, no. 2, p. 23, 973. [2] C. Altafini, Controllability of quantum mechanical systems by root space decomposition of su(n), Journal of Mathematical Physics, vol. 43, no. 5, p. 25, 983. [3] V. Ramakrishna, K. Flores, H. Rabitz, and R. Ober, Quantum control by decompositions of SU(2), Physical Review A, vol. 62, no. 5349, 2. [4] W. Zhu and H. Rabitz, A rapid monotonically convergent iteration algorithm for quantum optimal control over the expectation value of a positive definite operator, Journal of Chemical Physics, vol. 9, no. 2, p. 385, 998. [5] S. Grivopoulos and B. Bamieh, Iterative algorithms for optimal control of quantum systems, in Proceedings of the 4st IEEE Conference on Decision and Control, Dec 22. [6] N. Khaneja, R. Brockett, and S. Glaser, Time optimal control in spin systems, Physical Review A, vol. 63, no. 3238, 2. [7] A. Ferrante, M. Pavon, and G. Raccanelli, Driving the propagator of a spin system: a feedback approach, in Proceedings of the 4st IEEE Conference on Decision and Control, Dec 22. [8] P. Vettori, On the convergence of a feedback control strategy for multilevel quantum systems, in Proceedings of the MTNS Conference, 22. [9] H. Khalil, Nonlinear Systems. Prentice Hall, second ed., 996.

4 control (a.u) Fig.. transition : electric field.9.8 populations Fig. 2. transition : state populations

5 control (a.u) Fig. 3. transition 22: electric field populations Fig. 4. transition 22: state populations