Control of Inhomogeneous Spin Ensembles

2 2u v A

[1 ,1 ]

Robust Control of Inhomogeneous Spin Ensembles

0 ( )

0 ( )

( ) ( ) 0

x u t xd

y v t ydt

z u t v t z

0B

M

x

y

d

M

dt

M

B

( )rfB t M

0B0 B0

0 0 ( )

0 0 ( )

( ) ( ) 0

x v t xd

y u t ydt

z v t u t z

[1 ,1 ]

Compensation and Composite Pulses

z

x

y

2 2u v A

[1 ,1 ]

Robust Control of Inhomogeneous Spin Ensembles

0 ( )

0 ( )

( ) ( ) 0

x u t xd

y v t ydt

z u t v t z

0B

M

x

y

d

M

dt

M

B

( )rfB t M

0B0 B0

The problem of manipulating quantum systems with uncertainities or inhomogeneities in parameters govering the system dynamics is ubiquitous in spectroscopy and information processing .

a) Understanding what aspect of system dynamics makes compensation possible.b) What kind of inhomogeneities or errors can or cannot be corrected.

Typical settings includea) Resonance offsetsb) Inhomogeneities in the strength of excitation field (systematic errors)c) Time dependent noise (nonsystematic errors)d) Addressing errors or cross talk

Widespread use of composite pulse sequences and pulse shaping first to correct for errors or compensate for inhomogeneties

0 0 ( )

0 0 ( )

( ) ( ) 0

x v t xd

y u t ydt

z v t u t z

[1 ,1 ]

Broadband Control

[ ( ) ( ) ]x y

dXu t v t X

dt

Lie Algebras and Polynomial Approximations

2

2,

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

( ) [ ]

z

y x y x

x y

U t t t t t

I t

[ ( ) ( ) ]x y

dXu t v t X

dt

5,[ [ [ [ ]]x x x x y y

3

2,

( ) exp( ) ( ) exp( )

( ) [ [ ]]

y

x x

x x y

U t t U t t

I t

Lie Algebras and Polynomial Approximations

exp( ( ) )yf

2 1( ) kk

k

f c ( )f Choose such that it is approx. constant for

[1 ,1 ]

Using3 2 1

,, , ky y y as generators

Lie Algebras and Polynomial Approximations and Ensemble Controllability

2 1( ) kk

k

f c

1 2 3( ) exp( ( ) ) exp( ( ) ) exp( ( ) )x y xf f f

Create Unitary Evolution as a function of inhomogeneity to desired level of accuracy

Basic Mathematical Structure:Non commutativity of generators and an

underlying semi-simple Lie-algebra

Repeated Lie brackets (commutators) will raise the dispersion parameter to higher powers. The various powers of can be combined to form polynomials that approximate any desired evolution with continuous dependency on

[ ( ) ( ) ]x y

dXu t v t X

dt

exp( ( ) )yh

0 ( )

0 ( )

( ) ( ) 0

x u t xd

y v t ydt

z u t v t z

[ , ]B B

Ensemble Controllability of Bloch Equations

2,

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

( ) [ ]

y

x z x z

z x

U t t t t t

I t

2

,[ [ ]]z z y y ( ) kkf c

exp( ( ) )yf

Larmor Dispersion and Strong Fields

exp( ) exp( )exp( )exp( )z x z xt t

2,

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

( ) [ ]

y

x z x z

z x

U t t t t t

I t

( ) kkf c

exp( ( ) )yf

Larmor Dispersion and Bounded Controls

exp( ) exp( )exp( )exp( )z x z xt t

Adiabatic Passage

2

exp( ( ) ) exp( )exp( ( ) )z x zU

U I

exp( ) exp( )z zt U t U

Adiabatic Passage is Robust to rf-inhomogeneity

0 ( )

0 ( )

( ) ( ) 0

x u t xd

y v t ydt

z u t v t z

Ensemble Controllability of Bloch Equations

exp( )exp( )exp( )exp( )x z x zt t t t

k jkjc

1 2 3( , ) exp( ( , ) ) exp( ( , ) ) exp( ( , ) )y x yf f f

Some Negative Results

Nil-Potent Systems Cannot be Compensated

1 0

( ) 0 ( ) 1

f g

xd

y u t v tdt

z y x

0

[ , ] 0

1

f g

x

y

Some Negative Results

Linear systems cannot be compensatedfor field inhomogeneities

dXAX Bu

dt

( )( ) (0) ( ) ( )At A tX t e X e B u d

Some Negative Results

( )[cos( ( ) ) sin( ( ) ) ]x y

dXA t t t X

dt

Phase Dispersions Cannot be Compensated

( )[cos( ( )) sin( ( )) ]x y

dXA t t t X

dt

cos sinx x y

cos siny y x

,[ ]z x y

Ensemble Controllability of Coupled Spins with Inhomogeneous Couplings

Interactions

SI

S

J

I

B

(D)

Spin Hamiltonian: H + H (t)

B (t)rf

0

0 rf

1 2c z zH J

31 2 1 2 1 2 1 2[ [ , ]z x z x z z z zJ J J J

21 2 1 2 2[ , ]z z z x yJ J J

1 1 2 1 2 1 2 2( ) ( ) exp( ( ) ( ) ( ) ) ( )x x y y z zU J V J a J b J c J V J

1 2cH J

I

S

ISr

6

1

ISrNOE

0B

M

x

y

d

M

dt

M

B

( )rfB t M

0B0 B0

H

0 (1 )B

2 1 1 2( ) exp( )exp( )cos(2 )cos(2 )I s S Is t R t R t t t

One dimensional spectrum

Relaxation Optimized Coherent SpectroscopySingular Optimal Control Problems

Anisotropy Compensated Experiments in Solid state NMRTheory of Broadband Control

Inhomogeneous Broadening due to Dipolar Coupling Dispersion

B0

S

Broadband control in biological solid-state NMR

DCP OCDCPOCHORRORHORROR

J. Am. Chem. Soc., 126 (2005)

Chem. Phys. Letter (2005)

Time Optimal control of inhomogeneous quantum ensembles

0 ( )

0 ( )

( ) ( ) 0

x u t xd

y v t ydt

z u t v t z

Find the shortest pulse sequence (shape) that produces a coherent excitation over [ , ]B B

2 2u v A

Optimal control of inhomogeneous quantum ensembles

0 ( )

0 ( )

( ) ( ) 0

x u t xd

y v t ydt

z u t v t z

2 2

0

minT

u v dt(0,0,1) (0, ( ), ( ))x y

Create desired excitation profile as a function of

Minimum energy pulses for desired excitations (SLR algorithm)

0

22

u ivd i

u ivdt

11

1 0;

0j j i t

jj j

C SU z e

S C z

( ) ( )

( ) ( )n n

n n

A z B zU

B z A z

1

0

1

0

( ) ;

( )

nj

n jj

nj

n jj

A z a z

B z b z

2 2

00

1 ( )n

j jj

a c u v

Constructive Controllability

exp( ( ) )yf

2 1( ) kk

k

f c

( ) cos( )kk

f c k

Applications in NMR and MRI

• Time optimal selective excitation, inversion and saturation pulses.

• Imaging and Spectroscopy in inhomogeneous fields.

Phase correcting pulses for NMR in Inhomogeneous Static Fields

( , , )r x y z

0( ) ( )B r B B r

0( ) (1 )[ ( )]r B B r T

0

( )

( ) (1 ) ( ) ( )r

r B T B r T B r T

exp ( ) zr

( )r

X

Y

NMR in Inhomogeneous Static Fields

exp ( ) zr

0

1 2 3

( ) ( )r c c r r

c r c x c y c z

( ) cos( )

( , , )

ik rk k k

k k

x y z

r c e A k r

k k k k

NMR in Inhomogeneous Static Fields

exp exp exp

exp cos( ) sin( )

z x z

x y

k r k r

k r k r

exp exp exp 2 exp exp

exp 2 cos( )

z x z x z

x

k r k r k r

k r

exp( cos( ) ) exp 2 cos( ) kn

k k x xU A k r k r

xk

yk

zk

exp( cos( ) )k k xkk

U A k r exp ( )k zk r

0 ( )

0 ( )

( ) ( ) 0

x k v t xd

y k u t ydt

z v t u t z

Relaxation Specific Excitation

(0,0,1) ( ( ), ( ), ( ))x k y k z k

CollaboratorsSteffen GlaserBurkhard LuyFrank KramerTimo ReissKyryl KobzarAndreas SpoerlBjoern Heitmann

Gerhard WagnerDominique FruehTakuhiro Ito

Niels NielsenAstrid SivertsenCindie KehletMorten Bjerring

Technische Universitaet Muenchen

Harvard Medical School

University of Aarhus

NSF Career, NSF Qubic, Sloan, DARPA, AFRL, ONR, AFOSR, Humboldt

Haidong YuanDionisis StefanatosBrent PryorDan IancuAndrew JohnsonNavin KhanejaJr-Shin Li