Geometric Optimal Control Theory in spin systems

Transcription

1 Geometric Optimal Control Theory in spin systems Laboratoire Interdisciplinaire Carnot de Bourgogne, Dijon, France Workshop quantum control IHP, Paris 2 9/2/2

2 Optimal control theory A joint work between mathematicians, physicists and chemists. B. Bonnard, J.-B. Caillau, O. Cots, N. Scherbakova (IMB, Dijon) Group of S. J. Glaser, Y. Zhang, M. Braun (Chemistry department, Munich) M. Lapert (PhD, ICB, Dijon), E. Assémat (PhD, ICB, Dijon) Rem.: Other results of our research project will be given in the conferences of B. Bonnard and S. J. Glaser

3 Quantum control Application of tools of control theory to quantum dynamics. Photochemistry, femtosecond laser fields Condensed matter: superconducting Josephson junction cold atoms Nuclear Magnetic Resonance liquid phase solid phase

4 NMR in liquid phase A promising field of applications. fundamental problems quantum computing NMR spectroscopy and imaging (increase of the sensitivity and resolution of standard NMR techniques) Advantages of the NMR domain Accuracy of the models, even in presence of dissipation low and finite-dimensional problems = geometry very good agreement between theory and experiment Rem.: Such points are not true in Photochemistry

5 Optimal control in NMR Numerical approaches, e.g. the GRAPE and monotonic algorithms Geometric approach Ex.: Lie group methods for unbounded controls N. Khaneja, S. J. Glaser et al. Our work: use of geometric optimal control theory to analyze NMR systems (Pontryagin Maximum Principle, indirect methods, analytic and numerical computations) From mathematics (group of B. Bonnard) to experiments (group of S. J. Glaser)

6 The Bloch equations Ṁ x Ṁ y Ṁ z = ωm y + ω y M z ωm x ω x M z ω x M y ω y M x + M x /T 2 M y /T 2 (M M z )/T + RDE with T, T 2 dissipation terms, ω: offset. RDE = M T r M x M z M y M z M 2 x + M 2 y RDE: Radiation damping effect T r : Radiation damping constant.

7 Time-optimal control of a spin /2 particle Saturation control problem in NMR: North pole (equilibrium point) center of the Bloch ball. Constraint: ω ω max, ω = Rem.: standard problem in NMR (remove the corresponding spin contribution). Geometric analysis: symmetry of revolution = use of one control: ẏ = uz y/t 2 yz/t r ż = uy + ( z)/t + y 2 /T r (y, z): reduced coordinates and u u.

8 Optimal synthesis PMP and geometric analysis of the extremals: u = ±u or singular. Rem.: structure depends on the dissipative parameters and on the bound of the control. Figure: Schematic representation of the optimal synthesis

9 A concrete example Ex.: The proton spins of H 2 O in an organic solvent (T = 74 ms, T 2 = 6 ms and ω max = 32.3 Hz) z y u τ Figure: Time-optimal saturation of a spin /2 particle Rem.: Comparison with the inversion recovery sequence: gain of 6% in the control duration.

10 Analysis in the unbounded case Rem.: In the limit ω max +, the structure is B-S-S..8 T opt /T IR ω max /2π.5 z - optimal.5 IR.5.5 y

11 Analytic computation in the unbounded case Neglecting the duration of the first bang pulse, we get: T opt T 2 2 ln[ 2 ] + T ln[ 2T T 2 αt 2 2(T T 2 ) ] T IR T ln 2 Rem.: The physical limit of the control process is due to the dissipative parameters: Use of the dissipation to reach the target z y Figure: Contour plot of dṙ/dθ.

12 The GRAPE algorithm Iterative algorithm to solve the optimal equations: Cost: Φ t = y 2 + z 2. Fixed control duration T and bound ω max on the control field. 2 4 Log (Φ t ) t Figure: Φ t when T T geom.

13 Comparison of the different optimal solutions z y u u t (a) (b) Figure: Smooth regularization of the geometric solution. Results: Geom.: T geom. = 6.58, Φ t = Grape: T grape = 6.6, Φ t = Grape: T grape = 6.58, Φ t =

14 Generalization to other systems Energy minimization problem (no singular extremal, a smooth solution) Radiation damping effect (smooth effect if T r > ). Optimal control of uncoupled spin systems Two spins with different offsets The contrast problem

15 The radiation damping effect Experimental constraint: Inhomogeneous ensemble with different offsets..5 Z Y Figure: Optimal trajectory computed with the GRAPE algorithm. Rem. Example of coupling between geometric methods and the GRAPE algorithm.

16 Two spins with different offsets: the symmetric case Ṁ ax Ṁ ay Ṁ az Ṁ bx Ṁ by Ṁ bz = = ωm ay + ω y M az ωm ax ω x M az ω x M ay ω y M ax ωm by + ω y M bz ωm bx ω x M bz ω x M by ω y M bx One spin, one control: P. Mason, U. Boscain, JMP (26) Optimal solution: bang-bang controls

17 Inversion of two spins two spins with two controls two spins with one control Two proton spins of methyl acetate CH 3 OOCCH 3. ωx /2 π [Hz] M x (a),m y (a).5.5 (a) 3 6 t (ms) (b) t (ms) (c) M z (a) t (ms)

18 The contrast problem: the blood Two uncoupled spins describing the oxygeneated/desoxygeneated blood Structure of the optimal control: Bang-Singular M z /M ω x (Hz) M /M x t(s)

19 Conclusion Mathematical developments Global understanding of the contrast problem Classification of the different structure in the case of RDE Numerical applications limit of indirect methods, continuation approach coupling between GRAPE and geometric methods Experimental implementation of geometric solutions Applications in NMR spectroscopy and NMR imaging.

20 References B. Bonnard et al., IEEE Trans. AC 54, 2598 (29) B. Bonnard et al., J. Math. Phys. 5, 9275 (2) M. Lapert et al., Phys. Rev. Lett. 4, 83 (2) E. Assémat et al., Phys. Rev. A 82, 345 (2) M. Lapert et al., to be published in Phys. Rev. A (2) Y. Zhang et al., to be published in J. Chem. Phys. (2) page web: