Superoperators for NMR Quantum Information Processing. Osama Usman June 15, 2012

Transcription

1 Superoperators for NMR Quantum Information Processing Osama Usman June 15, 2012

2 Outline 1 Prerequisites 2 Relaxation and spin Echo 3 Spherical Tensor Operators 4 Superoperators 5 My research work 6 References.

3 NMR NMR is a phenomenon in which the resonance frequencies of nuclear magnetic systems are investigated. NMR always employs some form of magnetic field (usually a strong externally applied field B 0 and a RF field) Nucleis have a magnetic moment and spin angular momentum

4 Random direction of spin polarization in the absence of magnetic field.

5 Net magnetic moment from small z excess of Nuclei in +1/2 state. -1/2 B 0 Applied magnetic field Spin States Split in the presence of B 0 M 0 y x +1/2

6 Longitudinal and Transverse Magnetizations

7 Most commonly studied nuclei

8 Applications of NMR Physics Condensed matter physics Chemistry Identification of material Biophysics Analysis of Protein structure Medical MRI (Magnetic Resonance Image)

9 Interactions in NMR Internal Interactions NRM Interactions External Interactions 1. Chemical shift 1. Applied Magnetic field 2. J-Coupling 2. RF field 3. DD-Coupling

10 Relaxation T1 spin-lattice (relaxing back to precessing about the z axis) Recovery of Z component of magnetization. T2 spin-spin (fanning out) Decay of x, y component of magnetization.

11 Spin Echo

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13 Density Matrix Formalism A tool used to describe the state of a spin ensemble, as well as its evolution in time. Average of any observable For any state Diagonal elements = probabilities [1] Off-diagonal elements = "coherences" (provide info. about relative phase)

14 Spherical tensor operators. Tensors are very useful simplifying tools that encountered in spherical symmetric problems. Any spherical tensor operator can be found by Commutation relations with angular momentum operator. [2]

15 Transformation of Spherical tensor operators Wigner rotation matrices Reduced rotation matrix elements [2]

16 Total Hamiltonian in terms of Spherical tensor operators. [3]

17 Superoperators Liouville-von Neumann equation We define a superoperator [4]

18 Matrix representation of Superoperators Superoperators belong to the superoperator space. The difference between the Superoperator space and Hilbert space is dimensionality. Different physical conditions. Matrix representation

19 Table Name Continuous representation Continuous scalar product Discrete representation Discrete scalar product Superoperator Space superoperators n² n² matrices Liouville space operators, Density matrices n n matrices Hilbert space wavefunctions n vectors

20 Long lived Singlet states in solution NMR Singlet and Triplet states Two coupled spins

21 Density operator for Spin pair

22 Equation of Motion Liouvillian Space operator Coherent Effects Incoherent Effects Dipole-dipole relaxation External random field relaxation

23 For DD-relaxation Relaxation Superoperator For ERF-relaxation [5] Autocorrelation function Correlation time

24 Relaxation of singlet-state population

25 Final matrix of the coherent Liouvillian Superoperator. Shows the conversion of singlet population to singlettriplet zero-quantum coherence then to triplet population.

26 Final matrix of the evolution of spin density operator.

27 Signal Obtained

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30 Advantages and applications of longlived states Store information up to 40 times longer than T1. Can be created in both high or low magnetic fields, the latter case very interesting for applications on humans. For the study of molecular transportation and storage of polarized nuclear spin coherence. Can be used for the investigation of slow-cross relaxation between different molecules in solutions.

31 NMR for Quantum Information Advantages Disadvantages 1. Physics of NMR 1. Noisy signal is well described. 2. Long decoherence times 2. No entanglement. 3. Small quantum computer 3. Limited measurement. is easy to construct.

32 Why NMR? A major requirement of a quantum information/computer is that the coherence should last long. Nuclear spins in liquids retain coherence ~ 100 s millisec and their longitudinal state for several seconds. A system of N coupled spins (each spin 1/2) form an N qubit Quantum Computer. Unitary Transform can be applied using R.F. Pulses and various logical operations and quantum algorithms can be implemented. [7]

33 References 1. Levitt M.H, Spin Dynamics Basics of Nuclear Magnetic Resonance, John Wiley & Sons Ltd, C.P. Slichter. Principles of Magnetic Resonance. Harper, Row, Melinda J. D, Introduction to Solid-State NMR Spectroscopy, John Wiley & Sons Ltd, Mayne C.L, Liouville Equation of Motion, Encycl. Nucl. Mag. Res, M. Carravetta and M. H. Levitt. Theory of long-lived nuclear spin states in solution nuclear magnetic resonance. i. singlet states in low magnetic field. J. Chem. Phys., 122:214505, Jeener J, Superoperators in Magnetic Resonance, Adv. Mag. Res, 10 (1982), Jones J.A, Quantum Computing with NMR, Prog. NMR Spectrosc. 38, (2001).

34 THANKS FOR YOUR PATIENCE

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