Nuclear Spin Systems: New applications of control

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Nuclear Spin Systems New applications of control
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Nuclear Magnetic Resonance

• Many Atomic Nuclei have intrinsic angular
  momentum called spin.
• The spin gives the nucleus a magnetic moment
  (like a small bar magnet).
• The spins can be manipulated with radio-frequency
  electromagnetic fields.
• Probing spins with rf 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 chemistry.

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• The magnetic moment of a single nuclear spin is
  too weak to detect.
• The spins are therefore usually detected in the
  Bulk.

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Control of Bloch Equations
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Need for an Oscillatory Control
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Control of Ensemble of Bloch Equations(Ensemble
Controllability)
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Compensation and Composite Pulses
z
y
x
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Lie Algebras and Polynomial Approximations
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Lie Algebras and Polynomial Approximations
as generators
Using
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Fourier Synthesis Methods for Compensation
B. Pryor, N. Khaneja Journal of Chemical Physics
(2006).
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Fourier Synthesis Methods for Compensation
Ongoing work with M. Nimbalkar and S.J. Glaser
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Some Negative Results
Nil-Potent Systems Cannot be Compensated
y
x
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Some Negative Results
Linear systems cannot be compensated for field
inhomogeneities
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Ensemble Controllability of Bloch Equations
J.S. Li and N. Khaneja, PRA (2006)
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Ensemble Controllability of Bloch Equations
J.S. Li and N. Khaneja, PRA (2006)
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Larmor Dispersion and Bounded Controls
Adiabatic Passage
Adiabatic Passage is Robust to rf-inhomogeneity
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Control of Inhomogeneous Spin Ensembles
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• Ensemble Control of Switched Bilinear Systems

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Switched Control of Electron Nuclear Spin Systems
N. Khaneja PRA(2007)
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Control of Nuclear Spin Dynamics by Flipping
Electrons and Switching Quantization Axis
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Switching Sequence
N. Khaneja, Switched Control of Electron Nuclear
Spin Dynamics PRA 2007
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How many Switchings
Algorithms known to find minimum number of
switches
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How many switchings if a controlled operations is
desired ? Shortest Switched Loops with a given
holonomy
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Electron coupled to many Nuclear Spins
See Jamin Sheriffs Talk
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Dispersion in the Hyperfine Couplings in a Powder
Is the System Ensemble Controllable ?
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Manipulation of Coupled Spin Dynamics
Relaxation Optimized Coherent Spectroscopy Singula
r Optimal Control Problems
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Optimal Control in Presence of Relaxation
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Reducing the singular control problem
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Experimental Results Maximum Principle at Atomic
Scale
Khaneja, Reiss, Luy, Glaser JMR(2003)
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Control of Spin Networks
Spin Networks are excellent examples of Bilinear
Control systems with Dissipation. Computing
reachable sets and optimal controls are of both
practical and theoretical interest.
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Time Optimal Control of Quantum Systems
U
F
K
Minimum time to go from U(0) to U is the
same as minimum time to go from KU(0) to KU
F
G
U(0)
F
Khaneja, Brockett, Glaser, Physical Review A ,
63, 032308, 2001
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Control Systems on Coset Spaces
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Reachable Set
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Cartan Decompositions , Two-Spin Systems and
Canonical Decomposition of SU(4)
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Time Optimal Quantum Information Processing
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Reiss, Khaneja, Glaser J. Mag. Reson. 165 (2003)
Khaneja, et. al PRA(2007)
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Control of Indirectly Coupled Qubits
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Control of Indirectly Coupled Qubits
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Dynamics of n-coupled Spins
The dynamics of coupled spin ½ particles is
described by an element of
A basis of Lie algebra of can
be expressed as tensor product of pauli spin
matrices
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• Control of Bilinear Control Systems with Periodic
  Drift

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Inhomogeneous Broadening due to Dipolar Coupling
Dispersion
S

I
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Inhomogeneous broadening by chemical shift
anisotropy
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Magic-angle spinning
Static sample
54.7 (magic angle)
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Broadband Control
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Broadband control in biological solid-state NMR
J. Am. Chem. Soc., 126 (2005)
Chem. Phys. Letter (2005)
Journal of Chemical Physics (2006)
OCDCP
DCP
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Conclusion

• I hope, I may have convinced you that control and
  manipulation of spins is a fertile ground for new
  problems and developments in bilinear/geometric
  control theory.
• We introduced and motivated the notion of
  ensemble controllability.
• We showed how problems of control of dissipative
  bilinear control systems and time optimal control
  on Lie Groups arise naturally in control of spin
  systems.
• If you find all this exciting, walk over to the
  chemistry or physics department and convince a
  colleague that you have a better pulse sequence
  than anything else in the field as it comes from
  maximum principle !
• We showed examples of Switched Control Systems
  and Bilinear Control Systems
• with periodic drift.
• Thanks for your attention.