Control of bilinear systems: multiple systems and perturbations

Transcription

1 Control of bilinear systems: multiple systems and perturbations Gabriel Turinici CEREMADE, Université Paris Dauphine ESF OPTPDE Workshop InterDyn2013, Modeling and Control of Large Interacting Dynamical Systems September 10-12, Paris, 2013 Gabriel Turinici (U Paris Dauphine) Multiple and perturbed bi-linear control InterDyn2013, Paris 1 / 42

2 Motivation Outline 1 Motivation Mathematical questions Optical and magnetic manipulation of quantum dynamics 2 Modelization Single quantum system: wavefunction formulation Several quantum systems: density matrix formulation Simultaneous control of quantum systems Evolution semigroup Observables 3 Controllability Background on controllability criteria Simultaneous controllability of quantum systems Controllability of a set of identical molecules Controllability in presence of known perturbations Gabriel Turinici (U Paris Dauphine) Multiple and perturbed bi-linear control InterDyn2013, Paris 2 / 42

3 Motivation Mathematical questions Simultaneous controllability d Isolated system: dt x(t) = f (t, x(t), u(t)). Collection of similar systems d dt x k(t) = f k (t, x k (t), u(t)), k = 1,..., K. (1) Similar may mean f k (t, x, u) = f (t, x, α k u), α k R. The systems differ in their interaction with the control e.g. for instance are placed spatially differently with respect to the controlling action. Can one simultaneously control all systems with same u(t)? Gabriel Turinici (U Paris Dauphine) Multiple and perturbed bi-linear control InterDyn2013, Paris 3 / 42

4 Motivation Mathematical questions Controllability of large perturbations d Un-perturbed system: dt x(t) = f (t, x(t), u(t)). Perturbed system: d dt x k(t) = f (t, x k (t), u(t) + δu k (t)), k = 1,..., K. (2) Large perturbations: δu k (t), k = 1,..., K in a fixed list (alphabet) but there is no knowledge about which k will actually occur. Can one still control the system? Gabriel Turinici (U Paris Dauphine) Multiple and perturbed bi-linear control InterDyn2013, Paris 4 / 42

5 Motivation Optical and magnetic manipulation of quantum dynamics Laser SELECTIVE control over quantum dynamics Figure: Optimized laser pulses can be used to control molecular dynamics. a, An optimized laser pulse excites benzene into the superposition state S0 + S1 with bidirectional electron motion that results in switching between two discrete Kekulé structures on a subfemtosecond timescale; b: A different optimized laser pulse excites benzene into the superposition state S0 + S2, which is triply ionic; c: The next frontier in simulating control is to excite both electronic and nuclear dynamics simultaneously. Credits: NATURE CHEMISTRY, VOL 4, FEBRUARY 2012, p 72. Gabriel Turinici (U Paris Dauphine) Multiple and perturbed bi-linear control InterDyn2013, Paris 5 / 42

6 Motivation Optical and magnetic manipulation of quantum dynamics Figure: Studying the excited states of proteins. F. Courvoisier et al., App.Phys.Lett. Gabriel Turinici (U Paris Dauphine) Multiple and perturbed bi-linear control InterDyn2013, Paris 6 / 42

7 Motivation Optical and magnetic manipulation of quantum dynamics Quantum non-demolition measurements Figure: Physics Nobel prize 2012 to Haroche and Wineland for ground-breaking experimental methods that enable measuring and manipulation of individual quantum systems. Serge Haroche and David Wineland have independently invented and developed methods for measuring and manipulating individual particles while preserving their quantum-mechanical nature, in ways that were previously thought unattainable ; [these are ] the first tiny steps towards building a quantum computer. Picture credits: Nature 492, 55 (06 December 2012) Gabriel Turinici (U Paris Dauphine) Multiple and perturbed bi-linear control InterDyn2013, Paris 7 / 42

8 Motivation Optical and magnetic manipulation of quantum dynamics Other applications EMERGENT technology NMR: spin interacting with magnetic fields; control by magnetic fields creation of particular molecular states fast switch in semiconductors... Gabriel Turinici (U Paris Dauphine) Multiple and perturbed bi-linear control InterDyn2013, Paris 8 / 42

9 Modelization Outline 1 Motivation Mathematical questions Optical and magnetic manipulation of quantum dynamics 2 Modelization Single quantum system: wavefunction formulation Several quantum systems: density matrix formulation Simultaneous control of quantum systems Evolution semigroup Observables 3 Controllability Background on controllability criteria Simultaneous controllability of quantum systems Controllability of a set of identical molecules Controllability in presence of known perturbations Gabriel Turinici (U Paris Dauphine) Multiple and perturbed bi-linear control InterDyn2013, Paris 9 / 42

10 Modelization Single quantum system: wavefunction formulation Single quantum system Time dependent Schrödinger equation { i t Ψ(x, t) = H(t)Ψ(x, t) Ψ(x, t = 0) = Ψ 0 (x). (3) H(t) = H 0 + interaction terms E.g. H 0 = + V (x) H(t) = H(t) thus Ψ(t) L 2 = 1, t 0. dipole approximation: H(t) = H 0 ɛ(t)µ(x) E.g. O H bond, H 0 = 2m + V, m = reduced mass V (x) = D 0 [e β(x x0) 1] 2 D 0, µ(x) = µ 0 xe x/x higher order approximation: H(t) = H 0 + k ɛk (t)µ k (x) misc. approximations (rigid rotor interacting with two-color linearly polarized pulse): H(t) = H 0 + (E 1 (t) 2 + E 2 (t) 2 )µ 1 + E 1 (t) 2 E 2 (t)µ 2 Gabriel Turinici (U Paris Dauphine) Multiple and perturbed bi-linear control InterDyn2013, Paris 10 / 42

11 Modelization Several quantum systems: density matrix formulation Several quantum systems: density matrix formulation Evolution equation for a projector Let P ψ(t) = Ψ(t)Ψ (t) i.e. the projector on Ψ(t), also noted in bra-ket notation Ψ(t) Ψ(t) ; then i t P ψ(t) = (i ) ( t Ψ(t) Ψ (t) + Ψ(t) i ) t Ψ(t) (4) ( ) ( H(t)Ψ(t) Ψ (t) + Ψ(t) H(t)Ψ(t)) = (5) [H(t), Ψ(t)Ψ (t)] = [H(t), P ψ(t) ]. (6) Gabriel Turinici (U Paris Dauphine) Multiple and perturbed bi-linear control InterDyn2013, Paris 11 / 42

12 Modelization Several quantum systems: density matrix formulation Several quantum systems: density matrix formulation For a sum of projectors (density matrix): ρ(t) = k η kp ψ k (t). By linearity the equation for the density matrix evolution { i t ρ(x, t) = [H(t), ρ(x, t)] ρ(x, t = 0) = ρ 0 (x). (7) Usually η k R + define a discrete probability law with η k the probability to be at t = 0 in state Ψ k (0). This gives interpretation in terms of observables. Gabriel Turinici (U Paris Dauphine) Multiple and perturbed bi-linear control InterDyn2013, Paris 12 / 42

13 Modelization Simultaneous control of quantum systems Simultaneous control of quantum systems Under simultaneous control of a unique laser field L 2 molecular species. Initial state Ψ(0) = L l=1 Ψ l(0). Any molecule evolves by its own Schrödinger equation i t Ψ l(t) = [H l 0 µl ɛ(t)] Ψ l (t) Gabriel Turinici (U Paris Dauphine) Multiple and perturbed bi-linear control InterDyn2013, Paris 13 / 42

14 Modelization Simultaneous control of quantum systems A set of identical molecules with different spatial positions N 2 identical molecules, DIFFERENT orientations, simultaneous control by one laser field. Interaction with a field µɛ(t)α k where α k depends on R = (r, θ, ζ) which characterizes the localization of the molecule in the ensemble. Gabriel Turinici (U Paris Dauphine) Multiple and perturbed bi-linear control InterDyn2013, Paris 14 / 42

15 Modelization Evolution semigroup Evolution semigroup { i t U(t) = H(t)U(t) U(0) = Id. Relationship with - wavefunction forumulation Ψ(t) = U(t)Ψ(0) - density matrix version: ρ(t) = U(t)ρ(0)U (t) (8) Gabriel Turinici (U Paris Dauphine) Multiple and perturbed bi-linear control InterDyn2013, Paris 15 / 42

16 Modelization Observables Observables: localization Measurable quantities: Ψ(t), OΨ(t) for self-adjoint operators O (rq: phase invariance). For density matrix: ρ(t) = k η kp ψ k (t) η k Ψ k (t), OΨ k (t) = Tr(ρO). (9) k coherent with the probabilistic interpretation. Important example: O = (sum of) projection(s) to some (eigen)state. Gabriel Turinici (U Paris Dauphine) Multiple and perturbed bi-linear control InterDyn2013, Paris 16 / 42

17 Modelization Observables Observables: localization e.g. O-H bond : O(x) = γ 0 π e γ2 0 (x x ) 2 Gabriel Turinici (U Paris Dauphine) Multiple and perturbed bi-linear control InterDyn2013, Paris 17 / 42

18 Outline 1 Motivation Mathematical questions Optical and magnetic manipulation of quantum dynamics 2 Modelization Single quantum system: wavefunction formulation Several quantum systems: density matrix formulation Simultaneous control of quantum systems Evolution semigroup Observables 3 Controllability Background on controllability criteria Simultaneous controllability of quantum systems Controllability of a set of identical molecules Controllability in presence of known perturbations Gabriel Turinici (U Paris Dauphine) Multiple and perturbed bi-linear control InterDyn2013, Paris 18 / 42

19 Background on controllability criteria Single quantum system, bilinear control Time dependent Schrödinger equation { i t Ψ(x, t) = H 0Ψ(x, t) Ψ(x, t = 0) = Ψ 0 (x). Add external BILINEAR interaction (e.g. laser) { i t Ψ(x, t) = (H 0 ɛ(t)µ(x))ψ(x, t) Ψ(x, t = 0) = Ψ 0 (x) (10) (11) Ex.: H 0 = + V (x), unbounded domain Evolution on the unit sphere: Ψ(t) L 2 = 1, t 0. Gabriel Turinici (U Paris Dauphine) Multiple and perturbed bi-linear control InterDyn2013, Paris 19 / 42

20 Background on controllability criteria Controllability A system is controllable if for two arbitrary points Ψ 1 and Ψ 2 on the unit sphere (or other ensemble of admissible states) it can be steered from Ψ 1 to Ψ 2 with an admissible control. Norm conservation : controllability is equivalent, up to a phase, to say that the projection to a target is = 1. Gabriel Turinici (U Paris Dauphine) Multiple and perturbed bi-linear control InterDyn2013, Paris 20 / 42

21 Background on controllability criteria Galerkin discretization of the Time Dependent Schrödinger equation i t Ψ(x, t) = (H 0 ɛ(t)µ)ψ(x, t) basis functions {ψ i ; i = 1,..., N}, e.g. the eigenfunctions of the H 0 : H 0 ψ k = e k ψ k wavefunction written as Ψ = N k=1 c kψ k We will still denote by H 0 and µ the matrices (N N) associated to the operators H 0 and µ : H 0kl = ψ k H 0 ψ l, µ kl = ψ k µ ψ l, Gabriel Turinici (U Paris Dauphine) Multiple and perturbed bi-linear control InterDyn2013, Paris 21 / 42

22 Background on controllability criteria Lie algebra approaches To assess controllability of i t Ψ(x, t) = (H 0 ɛ(t)µ)ψ(x, t) construct the dynamic Lie algebra L = Lie( ih 0, iµ): { M1, M 2 L, α, β IR : αm 1 + βm 2 L M 1, M 2 L, [M 1, M 2 ] = M 1 M 2 M 2 M 1 L Theorem If the group e L is compact any e M ψ 0, M L can be attained. Proof M = iat : trivial by free evolution Trotter formula: e i(ab BA) = lim n [e ib/ n e ia/ n e ib/ n e ia/ n ] n Gabriel Turinici (U Paris Dauphine) Multiple and perturbed bi-linear control InterDyn2013, Paris 22 / 42

23 Background on controllability criteria Operator synthesis ( lateral parking ) Trotter formula: e i[a,b] = lim [e ] ib/ n e ia/ n e ib/ n e ia/ n n n e ±ia = advance/reverse ; e ±ib = turn left/right Gabriel Turinici (U Paris Dauphine) Multiple and perturbed bi-linear control InterDyn2013, Paris 23 / 42

24 Background on controllability criteria Corollary. If L = u(n) or L = su(n) (the (null-traced) skew-hermitian matrices) then the system is controllable. Proof For any Ψ 0, Ψ T there exists a rotation U in U(N) = e u(n) (or in SU(N) = e su(n) ) such that Ψ T = UΨ 0. (Albertini & D Alessandro 2001) Controllability also true for L isomorphic to sp(n/2) (unicity). sp(n/2) = {M( : M + M = ) 0, M t J + JM = 0} where J is a matrix unitary 0 IN/2 equivalent to and I I N/2 0 N/2 is the identity matrix of dimension N/2 Gabriel Turinici (U Paris Dauphine) Multiple and perturbed bi-linear control InterDyn2013, Paris 24 / 42

25 Background on controllability criteria Results by the connectivity graph (G.T. & H. Rabitz) Let us define the connectivity graph A = G = (V, E), V = {ψ 1,..., ψ N }; E = {(ψ i, ψ j ), i j, B ij 0} , B = Gabriel Turinici (U Paris Dauphine) Multiple and perturbed bi-linear control InterDyn2013, Paris 25 / 42

26 Background on controllability criteria Look for degenerate transitions : list all e kl = e k e l and look for repetitions. A = , B = e 41 = = 1.0, e 51 = = 1.15 e 42 = = 0.8, e 52 = = 0.95 e 43 = = 0.7, e 53 = = 0.85 Gabriel Turinici (U Paris Dauphine) Multiple and perturbed bi-linear control InterDyn2013, Paris 26 / 42

27 Background on controllability criteria Theorem (G.T. & H.Rabitz 2000, C.Altafini 2001) If the connectivity graph is connected and if there are no degenerate transitions then the system is controllable. Note: non connected = independent quantum systems. A = , B = Gabriel Turinici (U Paris Dauphine) Multiple and perturbed bi-linear control InterDyn2013, Paris 27 / 42

28 Simultaneous controllability of quantum systems Simultaneous controllability of quantum systems Under simultaneous control of a unique laser field L 2 molecular species. Initial state Ψ(0) = L l=1 Ψ l(0). Total controllability = can simultaneously and arbitrarily steer any state Ψ l (0) Ψ l (t), t 0 under the influence of only one laser field ɛ(t). Any molecule evolves by its own Schrödinger equation i t Ψ l(t) = [H l 0 µl ɛ(t)] Ψ l (t) Gabriel Turinici (U Paris Dauphine) Multiple and perturbed bi-linear control InterDyn2013, Paris 28 / 42

29 Simultaneous controllability of quantum systems Discretization spaces D l = {ψ l i (x); i = 1,.., N l}, N l, N l 3 eigenstates of H l 0 A l and B l = matrices of the operators H0 l and µl respectively, with respect to D l ; N = L l=1 N l, A B A A =......, B = 0 B A L B L Note: the system evolves on the product of spheres S = L l=1 SN l 1 C, S k 1 C = complex unit sphere of C k. Gabriel Turinici (U Paris Dauphine) Multiple and perturbed bi-linear control InterDyn2013, Paris 29 / 42

30 Simultaneous controllability of quantum systems Theorem (B. Li, G. T., V. Ramakhrishna, H. Rabitz. 2002, 2003) : If dim R Lie( ia, ib) = 1 + L (Nl 2 1), then the system is controllable (the dimension of Lie( ia, ib) is computed over R). Moreover, if the system is controllable, there exists a time T > 0 such that any target can be reached at or before time T (and thereafter for all t > T ), i.e. for any c 0 S et t T the set of reachable states from c 0 is S. l=1 Gabriel Turinici (U Paris Dauphine) Multiple and perturbed bi-linear control InterDyn2013, Paris 30 / 42

31 Controllability of a set of identical molecules Controllability of a set of identical molecules N 2 identical molecules, placed under simultaneous control of only one laser field. The molecules have DIFFERENT orientations. Gabriel Turinici (U Paris Dauphine) Multiple and perturbed bi-linear control InterDyn2013, Paris 31 / 42

32 Controllability of a set of identical molecules Controllability of a set of identical molecules Linear case: d dt x 1 = Ax 1 + Bu(t), x 1 (0) = 0 (12) d dt x 2 = Ax 2 + 2Bu(t), x 2 (0) = 0. (13) (14) Then for any control u(t) we have x 2 (t) = 2x 1 (t), thus there are restrictions on the attainable set; no simultaneous control in the linear case. Gabriel Turinici (U Paris Dauphine) Multiple and perturbed bi-linear control InterDyn2013, Paris 32 / 42

33 Controllability of a set of identical molecules Controllability of a set of identical molecules Interaction with the control field is µɛ(t)α k where α k depends on the localization R = (r, θ, ζ) of the molecule in the ensemble. Theorem (GT & H. Rabitz, PRA 2004): Suppose α k α j, H 0 is with non-degenerate transitions and graph of µ is connected. Then if one (arbitrary) molecule is controllable then the whole (discrete) ensemble is. Gabriel Turinici (U Paris Dauphine) Multiple and perturbed bi-linear control InterDyn2013, Paris 33 / 42

34 Controllability of a set of identical molecules Controllability of a set of identical molecules Interaction with the control field is µɛ(t)α k where α k depends on the localization R = (r, θ, ζ) of the molecule in the ensemble. Theorem (GT & H. Rabitz, PRA 2004): Suppose H 0 is with non-degenerate transitions and that the connectivity graph of the system is connected but not bi-partite. Then if a molecule is controllable then the whole (discrete) ensemble is. Gabriel Turinici (U Paris Dauphine) Multiple and perturbed bi-linear control InterDyn2013, Paris 34 / 42

35 Controllability of a set of identical molecules P(x) x=1 single molecule full ensemble x Figure: The target yield P(x) for different orientations of x = cos(θ). The yield P(x) from a field optimized at x = 1 produced the quality control index Q(ɛ) = 49%. The yield P(x) for full ensemble control of was required to optimize a sample of M = 31 orientations uniformly distributed over the interval [ 1, 1]. The resultant quality index is Q(ɛ) = 85% (G.T. and H. Rabitz. PRA 2004 Gabriel Turinici (U Paris Dauphine) Multiple and perturbed bi-linear control InterDyn2013, Paris 35 / 42

36 Controllability of a set of identical molecules Other results from the literature: control of PDE: T. Chambrion, K. Beauchard, P. Rouchon: mostly for A + α k u(t)b; recent works by M. Morancey, V. Nersesyan finite-dimensional, mostly for spin systems: C. Altafini, N. Khaneja: α k A + u(t)b Gabriel Turinici (U Paris Dauphine) Multiple and perturbed bi-linear control InterDyn2013, Paris 36 / 42

37 Controllability in presence of known perturbations Controllability in presence of known perturbations: joint work with M. Belhadj, J. Salomon, C. Lefter, B. Gavrilei d x = (A + u(t)b)x. (15) dt What if u(t) is submitted to a random perturbation in a predefined (discrete) list {δu k, k = 1,..., K}? Linear systems: d dt x 1 = Ax 1 + Bu(t), x 1 (0) = 0. (16) d dt x 2 = Ax 2 + B[u(t) + α], x 2 (0) = 0. (17) The dynamics of x 2 (t) x 1 (t) is not influenced by the control: d dt (x 2 x 1 ) = A(x 2 x 1 ) + Bα, x 2 (0) x 1 (0) = 0. (18) Thus: no control of perturbations in the linear case. Gabriel Turinici (U Paris Dauphine) Multiple and perturbed bi-linear control InterDyn2013, Paris 37 / 42

38 Controllability in presence of known perturbations Controllability in presence of known perturbations: joint work with M. Belhadj, J. Salomon, C. Lefter, B. Gavrilei Theorem (M. Belhadj, J. Salomon, GT 2013) Suppose the bi-linear system on SU(N) d x = (A + u(t)b)x (19) dt is such that the Lie algebra generated by [A, B] and B is the whole Lie algebra su(n). Then for any distinct α k R, k = 1,.., K, the collection of systems d dt x k = (A + [u(t) + α k ]B)x, k = 1,..., K, (20) is controllable. Gabriel Turinici (U Paris Dauphine) Multiple and perturbed bi-linear control InterDyn2013, Paris 38 / 42

39 Controllability in presence of known perturbations Controllability in presence of known perturbations: joint work with M. Belhadj, J. Salomon, C. Lefter, B. Gavrilei Theorem (M. Belhadj, J. Salomon, GT 2013) Consider the collection of control systems on SU(N): { } A + (u(t) + δ k u(t))b Y k (t), dy k (t) dt = Y k (0) = Y k,0 SU(N). (21) Suppose that there exists 0 < t 1 < t 2 < such that δ k u(t) = α k (constant) t [t 1, t 2 ]. Then there exists T A,B,α1,,α K such that if t 2 t 1 T A,B,α1,,α K the collection of systems (21) is simultaneously controllable at any time T t 2. Gabriel Turinici (U Paris Dauphine) Multiple and perturbed bi-linear control InterDyn2013, Paris 39 / 42

40 Controllability in presence of known perturbations Controllability in presence of known perturbations: joint work with M. Belhadj, J. Salomon, C. Lefter, B. Gavrilei Different model: dz k (t) dt = AZ k (t) + [u(t) + α k ]ξ(t)bz k (t), Z k (0) = Z k,0 SU(N), (22) Controls : u(t), ξ(t) with ξ(t) {0, 1}, t 0 (t ξ(t) measurable). Theorem The system (22) is simultaneously controllable if and only if L A,B = su(n). Gabriel Turinici (U Paris Dauphine) Multiple and perturbed bi-linear control InterDyn2013, Paris 40 / 42

41 Controllability in presence of known perturbations Controllability in presence of known perturbations: extensions and perspectives slowly varying perturbations, (beyond piecewise constant ) control of more non-linear situations: Rigid rotor interacting with linearly polarized pulse: i t Ψ(x, t) = [ H 0 + u(t)µ 1 + u(t) 2 µ 2 + u(t) 3 µ 3 ] Ψ(x, t). (23) Rigid rotor interacting with two-color linearly polarized pulse: i t Ψ(x, t) = [ H 0 + (E 1 (t) 2 + E 2 (t) 2 )µ 1 + E 1 (t) 2 E 2 (t)µ 2 ] Ψ(x, t). (24) Gabriel Turinici (U Paris Dauphine) Multiple and perturbed bi-linear control InterDyn2013, Paris 41 / 42

42 Controllability in presence of known perturbations Controllability in presence of known perturbations: extensions and perspectives control of rotational motion: time dependent Schrödinger equation (θ, φ = polar coordinates): i t ψ(θ, φ, t) = (BĴ2 u(t) d ) ψ(θ, φ, t) (25) ψ(0) = ψ 0, (26) numerics (joint works with M. Belhadj, J. Salomon, C. Lefter, B. Gavrilei) : several approaches : monotonic algorithms, Lyapunov approaches, multi-criterion optimization. In general it is difficult to find the field robust to control. Gabriel Turinici (U Paris Dauphine) Multiple and perturbed bi-linear control InterDyn2013, Paris 42 / 42