Control of Spin Systems

Control of Spin Systems

The Nuclear Spin Sensor Many Atomic Nuclei have intrinsic angular momentum called spin. The spin gives the nucleus a magnetic moment (like a small bar magnet). Magnetic moments precess in a magnetic field at a precession frequency that depends on the magnetic field strength. The spins are therefore beautiful, very localized (angstrom resolution) probes of local magnetic fields. The chemical environment of a nucleus in a molecule effects the local magnetic field on the nucleus. Probing spins with radio frequency magnetic fields and observing them gives information about chemical environment of the nuclei in a non-invasive way. Therefore field of nuclear magnetic resonance is the single most important analytical tool in science.

The magnetic moment of a single nuclear spin is too weak to detect. The spins are generally detected in the Bulk by making them precess coherently. The precession of nuclear spins is detected or observed by a Magnetic resonance technique.

NOE 1 6 r IS ω = γb (1 0 S σ) I H r IS B 0 M dm = γ M B dt B0 ω0 = γ B 0 y Brf () t M x

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One dimensional spectrum

st ( ) = η exp( Rt )exp( Rt)cos(2 πν t)cos(2 πν t ) I 2 s 1 S 1 I 2

Relaxation Optimized Coherent Spectroscopy Singular Optimal Control Problems

Transfer of Polarization Interactions ν I B 0 I z IS z z S ν S (D) J I B (t) rf Spin Hamiltonian: H + H (t) 0 rf

Random collisions with solvent molecules causes stochastic tumbling of the protein molecules S B0 I θ Brf () t C( τ) exp( τ / τ c ) τ J ( ω) 1 +ωτ ωτ c >> 2 1 c 2 2 c 1 πt = k J(0) Spin Diffusion Limit

Optimal Control in Presence of Relaxation d dt d dt x 0 ut () x 1Iz 0 ut () Iz1 I x ut () x I 2 ut () k J x 2 = = 2IS y z () 2IS y z x 3 J k vt () x 3 2IS 0 x z z vt () 0 2 IS x z z 4 4 1 0 0 0 0 0 0 η x x 3 2 2IS y z = =η I x

The control problem

Relaxation Optimized Pulse Elements (ROPE) ur ur 2 2 11 =η

Comparison S x

Iz 0 ut () Iz I x ut () k J d Ix = dt 2IS y z J k vt () 2IS y z 2IS vt () 0 z z 2IS z z

Finite Horizon Problem

Experimental Results H HC z z z Khaneja, Reiss, Luy, Glaser JMR(2003)

Cross-Correlated Relaxation S I S k a k + k kc a c B0 I θ Brf () t ββ βα ν I J 2 ν + I J 2 αβ αα

Optimal control of spin dynamics in the presence of Cross-correlated Relaxation I z IS z z r 1 r 2 β 1 I y β 2 IS y z l 1 l 2 I x IS x z Iz Iz 0 u v 0 0 0 Ix I u k 0 0 x a J k c d Iy v 0 ka kc J 0 Iy dt = IS 0 J kc ka 0 v y z IS y z 0 kc J 0 IS ka u x z IS x z 0 0 0 v u 0 IS z z IS z z

l l 2 1 = η = 1 + ξ 2 ξ ξ = k k 2 a 2 c + k J 2 c 2 Khaneja, Luy, Glaser PNAS(2003)

Khaneja, Luy, Glaser PNAS(2003)

TROPIC: Transverse relaxation optimized polarization transfer induced by cross-correlated relaxation. Groel Protein: 800KDa Room Temperature J = 93 Hz Proton Freq = 750Mhz Ka = 446 Hz Kc = 326 Hz Frueh et. al Journal of Biomolecular NMR (2005)

Control of Bloch Equations d dt x 0 ω0 ut () x y = ω0 0 vt () y z ut () vt () 0 z A 2 2 u v A + ω 0 M dm = γ M B dt B0 ω0 = γ B 0 B 0 x y Brf () t M

d dt Broadband Excitation x 0 ω εut () x y = ω 0 εvt () y z εut () εvt () 0 z 2 2 u + v A ε [1 δ,1 + δ] M dm = γ M B dt B0 ω0 = γ B 0 B 0 x y Brf () t M

Inhomogeneous Ensemble of Bloch Equations d dt x 0 ω εut () x y = ω 0 εvt () y z εut () εvt () 0 z g( ω) ω [ ω B, ω + B] 0 0 f ( ε ) ε [1 δ,1 + δ]

Robust Control Design d dt x 0 0 εvt () x y = 0 0 εut () y z εvt () εut () 0 z ( π ) y π π ( ) ( π ) ( ) 2 2 y x y dx dt = ε [ ut () Ω + vt () Ω ] X x y z ε [1 δ,1 + δ] y x

Robust Control Design by Area Generation dx dt = ε [ ut () Ω + vt () Ω ] X x y U ( t) = exp( Ω ε t)exp( Ω ε t)exp( εω t)exp( εω t) ε I + Ω Ω 2 ( t)[ ε x, ε y] 2 ε Ω z y x y x U ( t)exp( Ω ε tu ) ( t)exp( εω t) ε I + Ω Ω Ω 2 ( t) [ ε x[ ε x, ε y]] 3 x ε Ω y ε x

Lie Algebras, Areas and Robust Control Design Using εω, ε Ω, ε + Ω 3 2k 1 y y, y as generators Choose f ( ε ) f ( ε) = ckε + k 2k 1 such that it is approx. constant for ε [1 δ,1 + δ] exp( f ( ε ) Ω y ) Θ ( ε) = exp( f ( ε) Ω )exp( f ( ε) Ω )exp( f ( ε) Ω ) 1 x 2 y 3 x

B. Pryor, N. Khaneja Journal of Chemical Physics (2006). Fourier Synthesis Methods for Robust Control Design βk U1 = exp( kπεωx)exp( ε Ωy)exp( kπεωx) 2 βk = exp( ε (cos( kπε ) Ω y + sin( kπε ) Ωz)) 2 βk U2 = exp( kπεωx)exp( ε Ωy)exp( kπεωx) 2 βk = exp( ε (cos( kπε ) Ωy sin( kπε ) Ωz)) 2 UU 1 2 = exp( εβk cos( kπε ) Ω y) 1 ε k β cos( kπε ) k ε θ = ε

Fourier Synthesis Methods for Compensation

Time Optimal Control of Quantum Systems K U F k = { ih j } LA K = exp(k) du(t) dt U(0) m G = i[h d + u j H j ]U(t); U(0) = I j=1 Minimum time to go from U(0) to U is the F same as minimum time to go from KU(0) to KU F Khaneja, Brockett, Glaser, Physical Review A, 63, 032308, 2001

Example U = -i λ 1 + i u i B i U λ n B i ss n, U SU(n)

This image cannot currently be displayed. Control Systems on Coset Spaces G/K is a Riemannian Symmetric Space The velocities of the shortest paths in G/K always commute!

Cartan Decompositions, Two-Spin Systems and Canonical Decomposition of SU(4) Interactions S ν S (D) J I α = σ α I ; S α = I σ α ; I α S β = σ α σ β ; I ν I Spin Hamiltonian: H + H (t) 0 rf B 0 B (t) rf G = SU(4); K = SU(2) SU(2) σ x = 1 2 0 1 1 0 ; σ = 1 y 2 0 i i 0 ; σ z = 1 1 0 2 0 1

Geometry, Control and NMR

Reiss, Khaneja, Glaser J. Mag. Reson. 165 (2003) Khaneja, et. al PRA(2007)

Collaborators Steffen Glaser Niels Nielsen Gerhard Wagner Robert Griffin James Lin Jamin Sheriff Philip Owrutsky Mai Van Do Paul Coote Haidong Yuan Jr Shin Li Brent pryor Arthanari Haribabu