time optimal control of spin systems - syscon · ωω ω∈− +[, ] 00. bb. ε δδ∈− +[1 ,1 ]...

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.

I

S

ISr

6

1

ISrNOE

0BM

x

y

d

M dt

= γ

M ×B

( )rfB t M

0Bω0 = γ B0

H

0 (1 )Bω γ σ= −

Chemistry Pharmaceuticals

Life Science Imaging

Food Material Research Process Control

One dimensional spectrum

2 1 1 2( ) exp( )exp( )cos(2 )cos(2 )I s S Is t R t R t t tη πν πν= − −

Relaxation Optimized Coherent Spectroscopy Singular Optimal Control Problems

Transfer of Polarization Interactions

SI

νSJ

ν I

B

(D)

Spin Hamiltonian: H + H (t)

B (t)rf

0

0 rf

zI z zI S

Random collisions with solvent molecules causes stochastic tumbling of the protein molecules

θ0B I

( )rfB t

S

2 2( )1

c

c

J τωω τ

≈+

1cωτ >>Spin Diffusion Limit

( ) exp( / )cC τ τ τ− 2

1 (0)k JTπ

= ≈

0 ( )( )

2 2( )( ) 02 2

z z

x x

y z y z

z z z z

I Iu tI Iu t k JdI S I SJ k v tdt

v tI S I S

− − − = − −

Optimal Control in Presence of Relaxation

1 00 00 00 η

→

3

2

2 y z

x

I Sxx I

η= =

1 1

2 2

3 3

4 4

0 ( )( )

( )( ) 0

x xu tx xu t k Jdx xJ k v tdtx xv t

− − − = − −

The control problem

Relaxation Optimized Pulse Elements (ROPE)

2 2

1 1

u ru r

η=

Comparison

S x

0 ( )( )

2 2( )( ) 02 2

z z

x x

y z y z

z z z z

I Iu tI Iu t k JdI S I SJ k v tdt

v tI S I S

− − − = − −

Finite Horizon Problem

Experimental Results

Khaneja, Reiss, Luy, Glaser JMR(2003)

z z zH H C→

Cross-Correlated Relaxation

θ0B I

( )rfB t

S

I S

a ck k− a ck k+

2IJν +

αα

βα

2IJν −

αβ

ββ

Optimal control of spin dynamics in the presence of Cross-correlated Relaxation

1r

1β

1l

xI

2r

2β

2l

x zI S

0 0 0 00 0

0 00 00 00 0 0 0

z z

x xa c

y ya c

c ay z y z

c ax z x z

z z z z

I Iu v

I Iu k J kI Iv k k Jd

J k k vdt I S I Sk J k uI S I S

v uI S I S

− − − − − − −

= − − − − − − −

zI

yI

z zI S

y zI S

ξξη −+== 2

1

2 1ll

22

22

Jkkk

c

ca

+−

=ξ

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)

2 2

0

u v AA ω

+ ≤

Control of Bloch Equations

0

0

0 ( )0 ( )

( ) ( ) 0

x u t xd y v t ydt

z u t v t z

ωω

− − = −

0B

M

x

y

d

M dt

= γ

M ×B

( )rfB t M

0Bω0 = γ B0

≪

2 2u v A+ ≤

[1 ,1 ]ε δ δ∈ − +

Broadband Excitation

0 ( )0 ( )

( ) ( ) 0

x u t xd y v t ydt

z u t v t z

ω εω ε

ε ε

−∆ − = ∆ −

0B

M

x

y

d

M dt

= γ

M ×B

( )rfB t M

0Bω0 = γ B0

Inhomogeneous Ensemble of Bloch Equations

0 0[ , ]B Bω ω ω∈ − +

[1 ,1 ]ε δ δ∈ − +

0 ( )0 ( )

( ) ( ) 0

x u t xd y v t ydt

z u t v t z

ω εω ε

ε ε

−∆ − = ∆ −

( )f ε

( )g ω

0 0 ( )0 0 ( )

( ) ( ) 0

x v t xd y u t ydt

z v t u t z

εε

ε ε

= − −

Robust Control Design

[1 ,1 ]ε δ δ∈ − +

z

x

y

[ ( ) ( ) ]x ydX u t v t Xdt

ε= Ω + Ω( )

( ) ( ) ( )2 2

y

y x y

π

π ππ− −

Robust Control Design by Area Generation

2

2,

( ) exp( )exp( )exp( )exp( )

( ) [ ]

z

y x y x

x y

U t t t t t

I tε

ε

ε ε ε ε

ε εΩ

∆ = − Ω ∆ − Ω ∆ Ω ∆ Ω ∆

≈ + ∆ Ω Ω

[ ( ) ( ) ]x ydX u t v t Xdt

ε= Ω + Ω

3

2,

( ) exp( ) ( ) exp( )

( ) [ [ ]]

y

x x

x x y

U t t U t tI tε ε

ε

ε ε

ε ε ε− Ω

− ∆ − Ω ∆ ∆ Ω ∆

≈ + ∆ Ω Ω Ω

Lie Algebras, Areas and Robust Control Design

exp( ( ) )yf ε Ω

2 1( ) kk

kf cε ε += ∑

( )f εChoose such that it is approx. constant for

[1 ,1 ]ε δ δ∈ − +

Using 3 2 1

,, , ky y yε ε ε +Ω Ω Ω as generators

1 2 3( ) exp( ( ) ) exp( ( ) ) exp( ( ) )x y xf f fε ε ε εΘ = Ω Ω Ω

Fourier Synthesis Methods for Robust Control Design

cos( )kk

k θβ πεε

=∑

1 exp( )exp( )exp( )2

exp( (cos( ) sin( ) ))2

kx y x

ky z

U k k

k k

βπε ε πε

βε πε πε

= Ω Ω − Ω

= Ω + Ω

2 exp( )exp( )exp( )2

exp( (cos( ) sin( ) ))2

kx y x

ky z

U k k

k k

βπε ε πε

βε πε πε

= − Ω Ω Ω

= Ω − Ω

1 2 exp( cos( ) )k yU U kεβ πε= Ω

1ε

ε

B. Pryor, N. Khaneja Journal of Chemical Physics (2006).

Fourier Synthesis Methods for Compensation

Time Optimal Control of Quantum Systems

U(0)

U F

G

K

dU (t)dt

= −i[Hd + uj Hjj=1

m

∑ ]U(t); U(0) = I

k = −iHjLA

K = exp(k)

Minimum time to go from U(0) to U is the same as minimum time

to go from KU(0) to KU F

F

Khaneja, Brockett, Glaser, Physical Review A , 63, 032308, 2001

Example

= -i λ1

⋱λ𝑛

+∑ 𝑢𝑖𝐵𝑖𝑖 U

𝐵𝑖 ∈ 𝑠𝑠 𝑛 ,𝑈 ∈ SU(n)

Control Systems on Coset Spaces This image cannot currently be displayed.

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)

σx =12

0 11 0

; σ y =

12

0 −ii 0

; σ z =

12

1 00 −1

G = SU(4); K = SU(2) ⊗ SU (2)

Iα = σα ⊗ I ;Sα = I ⊗ σα ;IαSβ = σα ⊗ σβ ;

Interactions

SI

νSJ

ν I

B

(D)

Spin Hamiltonian: H + H (t)

B (t)rf

0

0 rf

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