1. Control, Constraints, and Quanta Bedlewo, Poland, October 10-16, 2007 Optimal Control for Design and Optimization of Solid-State NMR Experiments Control Quantum System

2. Our tool is Nuclear Magnetic Resonance M agnetic R esonance N uclear

3. How does it work? Macroscopic magnetic dipole moment – Ensemble of spins One spin

4. The nuclear spin Hamiltonian: Key to information and control Internal Hamiltonian H = H  + H J + H D + H Q H Z =  0 I z (Zeeman Interaction) H  =   I z (Chemical Shielding) H J =   J 2I z S z (J coupling) H D =  D 2I z S z (Heteronuclear Dipole-Dipole coupling) H D =  D (3I z S z -I·S) (Homenuclear Dipole-Dipole coupling) H Q =  Q (3I z 2 -I 2 ) (Quadrupole coupling) External Hamiltonian H = H Z + H rf,I + H rf,J H rf,I =  rf,I I x H rf,S =  rf,S S x

5. Density Operators & The Equation of Motion Liouville-von-Neumanequation Our handle to information and manipulation

6. NMR is more that direct read out of the internal Hamiltonian – control of information! Rf pulses Many channels J couplings

7. Cartesian rotations in spin space {I x, I y, I z } {I x, 2I y S z, 2I z S z } {2I z S x, S y, 2I z S z } Cyclic commutation Known as ”product operator formalism” (Sørensen et al.) Density matrix:  =|  > <  |

8. Solid-state NMR Additional control by sample rotation

9. Tayloring the Internal Hamiltonian Magic Angle-Spinning & Dipolar Recoupling H=H  + H  + H D + H D iso aniso IS II, SS H=H  + H  + H D + H D iso aniso IS II, SS H=  H D IS + H=H  iso

10. Biological solid-state NMR Combination of pulses and spinning Assignment Structural contraints

11. Quantum control? Optimal control => Design of Ū Hamiltonian H(t): External parameters B 0, B 1, Rf pulses, gradients, sample spinning Internal couplings H J, H D H , H Q  i Initial Spin State  f Final Spin State U  f = U  i U + U=exp{-i  H(t)dt}

12. State-to-state or Effective Hamiltonian optimization  i Initial Spin State  f Destination Spin State U  rf /2  T 0 Control fields U i = exp {-i(H rf + H int )  t i } U=  U i i a. State-to-state: A→C Final cost b. Effective Hamiltonian H D (or U D ) Final cost

13. What is the prize to get there! Cost Function  i Initial Spin State  f Destination Spin State U T 0 Control fields Final cost Running cost GRAPE Krotov’s algorithms

14. What is needed for GRAPE? I A -I B spin pair example A Hamiltonian & a guess An equation of motion A gradient... A correction... Khaneja, Glaser et al.

15. Optimum control in biological solid-state NMR Optimization of building blocks - Optimal sensitivity - Lowest cost - Reduction of instrumental errors LGCP OC LGCP DCP OC DCP OC HORROR HORROR

16. Primary gain from inhomogeneity compensation! OPT CONTROL Gain: ~ 100% DCP DCP, 10% L DCP, 5% L RAMP DCP Excitation time (ms) Efficiency + + + x x x + adiabatic x adiabatic w. 5%L Standard approach (DCP): 13 C 15 N 35 kHz 35 – r kHz Kehlet, Sivertsen, Reiss, Bjerring, Khaneja, Glaser, Nielsen, J. Am. Chem. Soc. 126, 10202 (2004)

17. OC Experiment more Economical Rf Inhomogeneity / Power OC DCP DCP 25 kHz 13 C 15 N 35 kHz DCP DCP We have to get used to pulse sequences looking like NOISE: ”Signals out of noise by noise A factor of 2 in signal & reduction of a factor of 4-10 in Rf Power!! Kehlet and Nielsen, Bruker Report 157, 31 (2006)

18. OC Experiments for Real Applications 15 N 13 C Kehlet, Bjerring, Sivertsen, Glaser, Khaneja, Nielsen, J. Magn. Reson 188, 216 (2007)

19. 3D NCOCX: U- 13 C, 15 N-ubiquitin Kehlet, Bjerring, Sivertsen, Glaser, Khaneja, Nielsen, J. Magn. Reson 188, 216 (2007) Optimal Control 50-100% Higher sensitivity (less sample/less time) Less sample heating (in the order of (in the order of 1/10 – 1/4 power 1/10 – 1/4 power

20. Band-selective Double-Quantum Dipolar Recoupling Bodholt, Bjerring, Nielsen, Poster

21. Optimization on the level of Effective Hamiltonians – or Propagators T 0   (0)   (T ) ) EFFECTIVE HAMILTONIAN Gradient of target function: Target function Target function (to be maximied): Desired Tosner, Glaser, Khaneja, Nielsen, J. Chem. Phys. 125, 184502 (2006)

22. An example: Mixing in 2D solid-state NMR experiments Most dipolar recoupling experiments based on PLANAR mixing : 2Q (i.e., 2I x S x -2I y S y )........ CN, HORROR, DREAM.... 2Q (i.e., 2I x S x -2I y S y )........ CN, HORROR, DREAM.... ZQ (i.e., 2I x S x +2I y S y )........ DCP, R 2, RIL.... ZQ (i.e., 2I x S x +2I y S y )........ DCP, R 2, RIL.... This is associated with a loss of a factor √2 in sensitivity!! z z Only transfer of x- or y- component Solution: Isotropic mixing (I x S x +I y S y +I z S z ) takes I x → S x I y → S y Tosner, Glaser, Khaneja, Nielsen, J. Chem. Phys. 125, 184502 (2006)

23. Planar and Isotropic Mixing Tosner, Glaser, Khaneja, Nielsen, J. Chem. Phys. 125, 184502 (2006)

24. Theoretical/Experimental Transfer Efficiencies RFDR planar mixing Isotropic mixing Tosner, Glaser, Khaneja, Nielsen, J. Chem. Phys. 125, 184502 (2006)

25. What about Krotov’s algorithm: Isotropic Mixing in Two-Spin System Lagrange Multiplier method:  semipositive definite n: iteration step; ,  optimization parameters Maximov, Tosner, Nielsen, in prep (2007)

26. Isotropic Mixing by Krotov Maximov, Tosner, Nielsen, in prep (2007)

27. GRAPE vs Krotov Krotov: Blue GRAPE: Red The two methods treats running costs differently Computational speed Krotov: Red GRAPE: Blue Depending on application, Grape may need a little fewer iterations – implying that major gains are expected in cases with several spins Major asset of Krotov: It handles running severe running costs better, thereby being good for optimizing lower-power experiments Maximov, Tosner, Nielsen, in prep (2007)

28. Krotov and DNP S(Electron)-I(Nuclei) spin system spin system S z → I x A: secular part of hyperfine interaction hyperfine interaction B: Pseudo secular part of hyperfine interaction hyperfine interaction ss ss

29. Learning Message from OC design! - Better sequences can be found!! Improved characteristics due to better handling of: -rf inhomogeneity -Powder dependency New inspiration to experiment design? Great help if you know what is possible!

30. Composite Rotations A great tool for compensation  RL  PR +  r t  PR Offset/Rf Rf/Dipole Pulse / Recoupling

31. Composite Dipolar Recoupling DCP COMB 3 DCP COMB 6 DCP IIIIIIIVV

32. Experimental Performance Simulations Experiments Khaneja, Kehlet, Glaser, Nielsen,J. Chem. Phys. 124, 114503-1 – 114503-7 (2006)

33. Composite Dipolar Recoupling OC → Analytics → OC Hansen, Kehlet, Vosegaard, Glaser, Khaneja, Nielsen, Chem. Phys. Lett. 447, 154 (2007) Rf pulses: Dipolar recoupling: WANTED An order of magnitude faster – and Fewer parameters (2x2 matrices Instead of 4x4 matrices) OPTIMIZATION

34. Dipolar Recoupling by Single-Spin OC: OC COMB Rf Pulse Recoupling Hansen, Kehlet, Vosegaard, Glaser, Khaneja, Nielsen, Chem. Phys. Lett. 447, 154 (2007)

35. Dipolar Recoupling by Single-Spin OC: OC COMB DCP COMB 3 (2  )COMB 6 (6  ) Analytical from known composite pulses OC COMB 30 (3  ) OC COMB 18 (3  ) OC COMB 9 (2.4  ) Numerical from OC optimized Rf pulses Hansen, Kehlet, Vosegaard, Glaser, Khaneja, Nielsen, Chem. Phys. Lett. 447, 154 (2007)

36. Optimal Control with Reduced Dimensionality Excitation time (ms) Transfer Efficiency DCP COMB 3 COMB 6 OC COMB 9 Hansen, Kehlet, Vosegaard, Glaser,Khaneja, Nielsen, Chem. Phys. Lett. 447, 154 (2007)

37. 2D NCO COMB 9 13 C, 15 N-ubiquitin Compensation of Rf Inhomogeneity/ HH mismatch > 50% gain in sensitivity Complete scalable to any spinning speed, recoupling sequence COMB 9 DCP Hansen, Kehlet, Vosegaard, Glaser, Khaneja, Nielsen, Chem. Phys. Lett. 447, 154 (2007)

38. Conclusions Optimal control Optimal control - gain sensitivity (much if many steps optimized) - gain sensitivity (much if many steps optimized) - robustness towards rf inhomogeneity - robustness towards rf inhomogeneity - sequences with much lower rf power (less heating) - sequences with much lower rf power (less heating) - robustness towards offsets, parameter variation etc - robustness towards offsets, parameter variation etc Optimal control provides inspiration to better design procedures – IF OC CAN DO IT, A GOOD BRAIN MAY DO IT? Optimal control provides inspiration to better design procedures – IF OC CAN DO IT, A GOOD BRAIN MAY DO IT? Circular design priniciple Circular design priniciple OC →Analytics → OC →.....

39. Thanks to …. Navin Khaneja, Harvard Steffen Glaser, München Zdenek Tosner, Praque Aarhus OC group: Cindie Kehlet Ivan Maximov Morten Bjerring Astrid Colding Sivertsen Thomas Vosegaard Jonas Ørbæk Hansen Anders Bodholt Nielsen Lasse Arnt... and YOU for your attention