control of inhomogeneous spin ensembles. robust control of inhomogeneous spin ensembles m x y m
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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
Relaxation Optimized Coherent SpectroscopySingular Optimal Control Problems
Anisotropy Compensated Experiments in Solid state NMRTheory of Broadband Control
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
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