time optimal control of spin systems - syscon · ωω ω∈− +[, ] 00. bb. ε δδ∈− +[1 ,1 ]...
TRANSCRIPT
![Page 1: Time Optimal Control of Spin Systems - SysCon · ωω ω∈− +[, ] 00. BB. ε δδ∈− +[1 ,1 ] 0 0 () 0. x ut x d y vt y dt z ut vt z. ωε ωε εε −∆ − ∆−= f ()ε](https://reader034.vdocuments.us/reader034/viewer/2022042217/5ec1ae29004f764de2504864/html5/thumbnails/1.jpg)
Control of Spin Systems
![Page 2: Time Optimal Control of Spin Systems - SysCon · ωω ω∈− +[, ] 00. BB. ε δδ∈− +[1 ,1 ] 0 0 () 0. x ut x d y vt y dt z ut vt z. ωε ωε εε −∆ − ∆−= f ()ε](https://reader034.vdocuments.us/reader034/viewer/2022042217/5ec1ae29004f764de2504864/html5/thumbnails/2.jpg)
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.
![Page 3: Time Optimal Control of Spin Systems - SysCon · ωω ω∈− +[, ] 00. BB. ε δδ∈− +[1 ,1 ] 0 0 () 0. x ut x d y vt y dt z ut vt z. ωε ωε εε −∆ − ∆−= f ()ε](https://reader034.vdocuments.us/reader034/viewer/2022042217/5ec1ae29004f764de2504864/html5/thumbnails/3.jpg)
• 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.
![Page 4: Time Optimal Control of Spin Systems - SysCon · ωω ω∈− +[, ] 00. BB. ε δδ∈− +[1 ,1 ] 0 0 () 0. x ut x d y vt y dt z ut vt z. ωε ωε εε −∆ − ∆−= f ()ε](https://reader034.vdocuments.us/reader034/viewer/2022042217/5ec1ae29004f764de2504864/html5/thumbnails/4.jpg)
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= −
![Page 5: Time Optimal Control of Spin Systems - SysCon · ωω ω∈− +[, ] 00. BB. ε δδ∈− +[1 ,1 ] 0 0 () 0. x ut x d y vt y dt z ut vt z. ωε ωε εε −∆ − ∆−= f ()ε](https://reader034.vdocuments.us/reader034/viewer/2022042217/5ec1ae29004f764de2504864/html5/thumbnails/5.jpg)
Chemistry Pharmaceuticals
Life Science Imaging
Food Material Research Process Control
![Page 6: Time Optimal Control of Spin Systems - SysCon · ωω ω∈− +[, ] 00. BB. ε δδ∈− +[1 ,1 ] 0 0 () 0. x ut x d y vt y dt z ut vt z. ωε ωε εε −∆ − ∆−= f ()ε](https://reader034.vdocuments.us/reader034/viewer/2022042217/5ec1ae29004f764de2504864/html5/thumbnails/6.jpg)
![Page 7: Time Optimal Control of Spin Systems - SysCon · ωω ω∈− +[, ] 00. BB. ε δδ∈− +[1 ,1 ] 0 0 () 0. x ut x d y vt y dt z ut vt z. ωε ωε εε −∆ − ∆−= f ()ε](https://reader034.vdocuments.us/reader034/viewer/2022042217/5ec1ae29004f764de2504864/html5/thumbnails/7.jpg)
One dimensional spectrum
![Page 8: Time Optimal Control of Spin Systems - SysCon · ωω ω∈− +[, ] 00. BB. ε δδ∈− +[1 ,1 ] 0 0 () 0. x ut x d y vt y dt z ut vt z. ωε ωε εε −∆ − ∆−= f ()ε](https://reader034.vdocuments.us/reader034/viewer/2022042217/5ec1ae29004f764de2504864/html5/thumbnails/8.jpg)
2 1 1 2( ) exp( )exp( )cos(2 )cos(2 )I s S Is t R t R t t tη πν πν= − −
![Page 9: Time Optimal Control of Spin Systems - SysCon · ωω ω∈− +[, ] 00. BB. ε δδ∈− +[1 ,1 ] 0 0 () 0. x ut x d y vt y dt z ut vt z. ωε ωε εε −∆ − ∆−= f ()ε](https://reader034.vdocuments.us/reader034/viewer/2022042217/5ec1ae29004f764de2504864/html5/thumbnails/9.jpg)
![Page 10: Time Optimal Control of Spin Systems - SysCon · ωω ω∈− +[, ] 00. BB. ε δδ∈− +[1 ,1 ] 0 0 () 0. x ut x d y vt y dt z ut vt z. ωε ωε εε −∆ − ∆−= f ()ε](https://reader034.vdocuments.us/reader034/viewer/2022042217/5ec1ae29004f764de2504864/html5/thumbnails/10.jpg)
Relaxation Optimized Coherent Spectroscopy Singular Optimal Control Problems
![Page 11: Time Optimal Control of Spin Systems - SysCon · ωω ω∈− +[, ] 00. BB. ε δδ∈− +[1 ,1 ] 0 0 () 0. x ut x d y vt y dt z ut vt z. ωε ωε εε −∆ − ∆−= f ()ε](https://reader034.vdocuments.us/reader034/viewer/2022042217/5ec1ae29004f764de2504864/html5/thumbnails/11.jpg)
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
![Page 12: Time Optimal Control of Spin Systems - SysCon · ωω ω∈− +[, ] 00. BB. ε δδ∈− +[1 ,1 ] 0 0 () 0. x ut x d y vt y dt z ut vt z. ωε ωε εε −∆ − ∆−= f ()ε](https://reader034.vdocuments.us/reader034/viewer/2022042217/5ec1ae29004f764de2504864/html5/thumbnails/12.jpg)
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
= ≈
![Page 13: Time Optimal Control of Spin Systems - SysCon · ωω ω∈− +[, ] 00. BB. ε δδ∈− +[1 ,1 ] 0 0 () 0. x ut x d y vt y dt z ut vt z. ωε ωε εε −∆ − ∆−= f ()ε](https://reader034.vdocuments.us/reader034/viewer/2022042217/5ec1ae29004f764de2504864/html5/thumbnails/13.jpg)
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
− − − = − −
![Page 14: Time Optimal Control of Spin Systems - SysCon · ωω ω∈− +[, ] 00. BB. ε δδ∈− +[1 ,1 ] 0 0 () 0. x ut x d y vt y dt z ut vt z. ωε ωε εε −∆ − ∆−= f ()ε](https://reader034.vdocuments.us/reader034/viewer/2022042217/5ec1ae29004f764de2504864/html5/thumbnails/14.jpg)
The control problem
![Page 15: Time Optimal Control of Spin Systems - SysCon · ωω ω∈− +[, ] 00. BB. ε δδ∈− +[1 ,1 ] 0 0 () 0. x ut x d y vt y dt z ut vt z. ωε ωε εε −∆ − ∆−= f ()ε](https://reader034.vdocuments.us/reader034/viewer/2022042217/5ec1ae29004f764de2504864/html5/thumbnails/15.jpg)
Relaxation Optimized Pulse Elements (ROPE)
2 2
1 1
u ru r
η=
![Page 16: Time Optimal Control of Spin Systems - SysCon · ωω ω∈− +[, ] 00. BB. ε δδ∈− +[1 ,1 ] 0 0 () 0. x ut x d y vt y dt z ut vt z. ωε ωε εε −∆ − ∆−= f ()ε](https://reader034.vdocuments.us/reader034/viewer/2022042217/5ec1ae29004f764de2504864/html5/thumbnails/16.jpg)
Comparison
S x
![Page 17: Time Optimal Control of Spin Systems - SysCon · ωω ω∈− +[, ] 00. BB. ε δδ∈− +[1 ,1 ] 0 0 () 0. x ut x d y vt y dt z ut vt z. ωε ωε εε −∆ − ∆−= f ()ε](https://reader034.vdocuments.us/reader034/viewer/2022042217/5ec1ae29004f764de2504864/html5/thumbnails/17.jpg)
![Page 18: Time Optimal Control of Spin Systems - SysCon · ωω ω∈− +[, ] 00. BB. ε δδ∈− +[1 ,1 ] 0 0 () 0. x ut x d y vt y dt z ut vt z. ωε ωε εε −∆ − ∆−= f ()ε](https://reader034.vdocuments.us/reader034/viewer/2022042217/5ec1ae29004f764de2504864/html5/thumbnails/18.jpg)
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
− − − = − −
![Page 19: Time Optimal Control of Spin Systems - SysCon · ωω ω∈− +[, ] 00. BB. ε δδ∈− +[1 ,1 ] 0 0 () 0. x ut x d y vt y dt z ut vt z. ωε ωε εε −∆ − ∆−= f ()ε](https://reader034.vdocuments.us/reader034/viewer/2022042217/5ec1ae29004f764de2504864/html5/thumbnails/19.jpg)
Finite Horizon Problem
![Page 20: Time Optimal Control of Spin Systems - SysCon · ωω ω∈− +[, ] 00. BB. ε δδ∈− +[1 ,1 ] 0 0 () 0. x ut x d y vt y dt z ut vt z. ωε ωε εε −∆ − ∆−= f ()ε](https://reader034.vdocuments.us/reader034/viewer/2022042217/5ec1ae29004f764de2504864/html5/thumbnails/20.jpg)
Experimental Results
Khaneja, Reiss, Luy, Glaser JMR(2003)
z z zH H C→
![Page 21: Time Optimal Control of Spin Systems - SysCon · ωω ω∈− +[, ] 00. BB. ε δδ∈− +[1 ,1 ] 0 0 () 0. x ut x d y vt y dt z ut vt z. ωε ωε εε −∆ − ∆−= f ()ε](https://reader034.vdocuments.us/reader034/viewer/2022042217/5ec1ae29004f764de2504864/html5/thumbnails/21.jpg)
Cross-Correlated Relaxation
θ0B I
( )rfB t
S
I S
a ck k− a ck k+
2IJν +
αα
βα
2IJν −
αβ
ββ
![Page 22: Time Optimal Control of Spin Systems - SysCon · ωω ω∈− +[, ] 00. BB. ε δδ∈− +[1 ,1 ] 0 0 () 0. x ut x d y vt y dt z ut vt z. ωε ωε εε −∆ − ∆−= f ()ε](https://reader034.vdocuments.us/reader034/viewer/2022042217/5ec1ae29004f764de2504864/html5/thumbnails/22.jpg)
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
![Page 23: Time Optimal Control of Spin Systems - SysCon · ωω ω∈− +[, ] 00. BB. ε δδ∈− +[1 ,1 ] 0 0 () 0. x ut x d y vt y dt z ut vt z. ωε ωε εε −∆ − ∆−= f ()ε](https://reader034.vdocuments.us/reader034/viewer/2022042217/5ec1ae29004f764de2504864/html5/thumbnails/23.jpg)
ξξη −+== 2
1
2 1ll
22
22
Jkkk
c
ca
+−
=ξ
Khaneja, Luy, Glaser PNAS(2003)
![Page 24: Time Optimal Control of Spin Systems - SysCon · ωω ω∈− +[, ] 00. BB. ε δδ∈− +[1 ,1 ] 0 0 () 0. x ut x d y vt y dt z ut vt z. ωε ωε εε −∆ − ∆−= f ()ε](https://reader034.vdocuments.us/reader034/viewer/2022042217/5ec1ae29004f764de2504864/html5/thumbnails/24.jpg)
![Page 25: Time Optimal Control of Spin Systems - SysCon · ωω ω∈− +[, ] 00. BB. ε δδ∈− +[1 ,1 ] 0 0 () 0. x ut x d y vt y dt z ut vt z. ωε ωε εε −∆ − ∆−= f ()ε](https://reader034.vdocuments.us/reader034/viewer/2022042217/5ec1ae29004f764de2504864/html5/thumbnails/25.jpg)
Khaneja, Luy, Glaser PNAS(2003)
![Page 26: Time Optimal Control of Spin Systems - SysCon · ωω ω∈− +[, ] 00. BB. ε δδ∈− +[1 ,1 ] 0 0 () 0. x ut x d y vt y dt z ut vt z. ωε ωε εε −∆ − ∆−= f ()ε](https://reader034.vdocuments.us/reader034/viewer/2022042217/5ec1ae29004f764de2504864/html5/thumbnails/26.jpg)
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)
![Page 27: Time Optimal Control of Spin Systems - SysCon · ωω ω∈− +[, ] 00. BB. ε δδ∈− +[1 ,1 ] 0 0 () 0. x ut x d y vt y dt z ut vt z. ωε ωε εε −∆ − ∆−= f ()ε](https://reader034.vdocuments.us/reader034/viewer/2022042217/5ec1ae29004f764de2504864/html5/thumbnails/27.jpg)
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
≪
![Page 28: Time Optimal Control of Spin Systems - SysCon · ωω ω∈− +[, ] 00. BB. ε δδ∈− +[1 ,1 ] 0 0 () 0. x ut x d y vt y dt z ut vt z. ωε ωε εε −∆ − ∆−= f ()ε](https://reader034.vdocuments.us/reader034/viewer/2022042217/5ec1ae29004f764de2504864/html5/thumbnails/28.jpg)
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
![Page 29: Time Optimal Control of Spin Systems - SysCon · ωω ω∈− +[, ] 00. BB. ε δδ∈− +[1 ,1 ] 0 0 () 0. x ut x d y vt y dt z ut vt z. ωε ωε εε −∆ − ∆−= f ()ε](https://reader034.vdocuments.us/reader034/viewer/2022042217/5ec1ae29004f764de2504864/html5/thumbnails/29.jpg)
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
![Page 30: Time Optimal Control of Spin Systems - SysCon · ωω ω∈− +[, ] 00. BB. ε δδ∈− +[1 ,1 ] 0 0 () 0. x ut x d y vt y dt z ut vt z. ωε ωε εε −∆ − ∆−= f ()ε](https://reader034.vdocuments.us/reader034/viewer/2022042217/5ec1ae29004f764de2504864/html5/thumbnails/30.jpg)
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− −
![Page 31: Time Optimal Control of Spin Systems - SysCon · ωω ω∈− +[, ] 00. BB. ε δδ∈− +[1 ,1 ] 0 0 () 0. x ut x d y vt y dt z ut vt z. ωε ωε εε −∆ − ∆−= f ()ε](https://reader034.vdocuments.us/reader034/viewer/2022042217/5ec1ae29004f764de2504864/html5/thumbnails/31.jpg)
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
![Page 32: Time Optimal Control of Spin Systems - SysCon · ωω ω∈− +[, ] 00. BB. ε δδ∈− +[1 ,1 ] 0 0 () 0. x ut x d y vt y dt z ut vt z. ωε ωε εε −∆ − ∆−= f ()ε](https://reader034.vdocuments.us/reader034/viewer/2022042217/5ec1ae29004f764de2504864/html5/thumbnails/32.jpg)
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
![Page 33: Time Optimal Control of Spin Systems - SysCon · ωω ω∈− +[, ] 00. BB. ε δδ∈− +[1 ,1 ] 0 0 () 0. x ut x d y vt y dt z ut vt z. ωε ωε εε −∆ − ∆−= f ()ε](https://reader034.vdocuments.us/reader034/viewer/2022042217/5ec1ae29004f764de2504864/html5/thumbnails/33.jpg)
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).
![Page 34: Time Optimal Control of Spin Systems - SysCon · ωω ω∈− +[, ] 00. BB. ε δδ∈− +[1 ,1 ] 0 0 () 0. x ut x d y vt y dt z ut vt z. ωε ωε εε −∆ − ∆−= f ()ε](https://reader034.vdocuments.us/reader034/viewer/2022042217/5ec1ae29004f764de2504864/html5/thumbnails/34.jpg)
Fourier Synthesis Methods for Compensation
![Page 35: Time Optimal Control of Spin Systems - SysCon · ωω ω∈− +[, ] 00. BB. ε δδ∈− +[1 ,1 ] 0 0 () 0. x ut x d y vt y dt z ut vt z. ωε ωε εε −∆ − ∆−= f ()ε](https://reader034.vdocuments.us/reader034/viewer/2022042217/5ec1ae29004f764de2504864/html5/thumbnails/35.jpg)
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
![Page 36: Time Optimal Control of Spin Systems - SysCon · ωω ω∈− +[, ] 00. BB. ε δδ∈− +[1 ,1 ] 0 0 () 0. x ut x d y vt y dt z ut vt z. ωε ωε εε −∆ − ∆−= f ()ε](https://reader034.vdocuments.us/reader034/viewer/2022042217/5ec1ae29004f764de2504864/html5/thumbnails/36.jpg)
Example
= -i λ1
⋱λ𝑛
+∑ 𝑢𝑖𝐵𝑖𝑖 U
𝐵𝑖 ∈ 𝑠𝑠 𝑛 ,𝑈 ∈ SU(n)
![Page 37: Time Optimal Control of Spin Systems - SysCon · ωω ω∈− +[, ] 00. BB. ε δδ∈− +[1 ,1 ] 0 0 () 0. x ut x d y vt y dt z ut vt z. ωε ωε εε −∆ − ∆−= f ()ε](https://reader034.vdocuments.us/reader034/viewer/2022042217/5ec1ae29004f764de2504864/html5/thumbnails/37.jpg)
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!
![Page 38: Time Optimal Control of Spin Systems - SysCon · ωω ω∈− +[, ] 00. BB. ε δδ∈− +[1 ,1 ] 0 0 () 0. x ut x d y vt y dt z ut vt z. ωε ωε εε −∆ − ∆−= f ()ε](https://reader034.vdocuments.us/reader034/viewer/2022042217/5ec1ae29004f764de2504864/html5/thumbnails/38.jpg)
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
![Page 39: Time Optimal Control of Spin Systems - SysCon · ωω ω∈− +[, ] 00. BB. ε δδ∈− +[1 ,1 ] 0 0 () 0. x ut x d y vt y dt z ut vt z. ωε ωε εε −∆ − ∆−= f ()ε](https://reader034.vdocuments.us/reader034/viewer/2022042217/5ec1ae29004f764de2504864/html5/thumbnails/39.jpg)
Geometry, Control and NMR
![Page 40: Time Optimal Control of Spin Systems - SysCon · ωω ω∈− +[, ] 00. BB. ε δδ∈− +[1 ,1 ] 0 0 () 0. x ut x d y vt y dt z ut vt z. ωε ωε εε −∆ − ∆−= f ()ε](https://reader034.vdocuments.us/reader034/viewer/2022042217/5ec1ae29004f764de2504864/html5/thumbnails/40.jpg)
Reiss, Khaneja, Glaser J. Mag. Reson. 165 (2003)
Khaneja, et. al PRA(2007)
![Page 41: Time Optimal Control of Spin Systems - SysCon · ωω ω∈− +[, ] 00. BB. ε δδ∈− +[1 ,1 ] 0 0 () 0. x ut x d y vt y dt z ut vt z. ωε ωε εε −∆ − ∆−= f ()ε](https://reader034.vdocuments.us/reader034/viewer/2022042217/5ec1ae29004f764de2504864/html5/thumbnails/41.jpg)
![Page 42: Time Optimal Control of Spin Systems - SysCon · ωω ω∈− +[, ] 00. BB. ε δδ∈− +[1 ,1 ] 0 0 () 0. x ut x d y vt y dt z ut vt z. ωε ωε εε −∆ − ∆−= f ()ε](https://reader034.vdocuments.us/reader034/viewer/2022042217/5ec1ae29004f764de2504864/html5/thumbnails/42.jpg)
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