Spin - From Basic Symmetries to Quantum Optimal Control

Preface
Contents
1 Mathematical Background
1.1 Sets and Maps
1.2 Topological Spaces
1.3 Fields
1.4 Linear Spaces
1.4.1 Inner Product Spaces
1.4.2 Linear Combinations and Basis Sets
1.4.3 Operators and Superoperators
1.4.4 Representations of Linear Spaces
1.4.5 Operator Norms and Inner Products
1.4.6 Representing Matrices with Vectors
1.5 Groups and Algebras
1.5.1 Finite, Discrete, and Continuous Groups
1.5.2 Conjugacy Classes and Centres
1.5.3 Group Actions, Orbits, and Stabilisers
1.5.4 Matrix Representations of Groups
1.5.5 Orthogonality Theorems and Characters
1.5.6 Algebras and Lie Algebras
1.5.7 Exponential and Tangent Maps
1.5.8 Ideals, Simple and Semisimple Algebras
1.5.9 Matrix Representations of Lie Algebras
1.5.10 Envelopes, Complexifications, and Covers
1.5.11 Cartan Subalgebras, Roots, and Weights
1.5.12 Killing Form and Casimir Elements
1.6 Building Blocks of Spin Physics
1.6.1 Euclidean and Minkowski Spaces
1.6.2 Special Orthogonal Group in Three Dimensions
1.6.2.1 Parametrisation of Rotations
1.6.2.2 Euler Angles Parametrisation
1.6.2.3 Angle-Axis Parametrisation
1.6.2.4 Irreducible Representations
1.6.3 Special Unitary Group in Two Dimensions
1.6.3.1 Parametrisation
1.6.3.2 Irreducible Representations
1.6.3.3 Normalisation-Commutation Dilemma
1.6.4 Relationship Between SU(2) and SO(3)
1.7 Linear Time-Invariant Systems
1.7.1 Pulse and Frequency Response
1.7.2 Properties of the Fourier Transform
1.7.3 Causality and Hilbert Transform
2 What Exactly Is Spin?
2.1 Time Translation Group
2.2 Full Translation Group
2.3 Rotation Group
2.4 Lorentz Group
2.4.1 Boost Generators
2.4.2 Irreps of Lorentz Group
2.4.3 Irreps of Lorentz Group with Parity
2.4.4 Poincare Group and the Emergence of Spin
2.5 Dirac’s Equation and Electron Spin
2.5.1 Dirac’s Equation
2.5.2 Total Angular Momentum and Spin
2.5.3 Total Angular Momentum Representation—Numerical
2.5.4 Total Angular Momentum Representation—Analytical
2.5.5 Benefits of the Individual Momentum Representation
2.6 Weakly Relativistic Limit of Dirac’s Equation
2.6.1 Zitterbewegung
2.6.2 Negative Energy Subspace Elimination
2.6.3 Zeeman Interactions and Langevin Susceptibility
2.6.4 Spin-Orbit Coupling
2.6.5 Spinning Charge Analogy
2.6.6 Spin as a Magnetic Moment
2.6.7 Spin Hamiltonian Approximation
2.6.8 Energy Derivative Formalism
2.6.9 Thermal Corrections
3 Bestiary of Spin Hamiltonians
3.1 Physical Side
3.1.1 Nuclear Spin and Magnetic Moment
3.1.2 Nuclear Electric Quadrupole Moment
3.1.3 Electronic Structure Derivative Table
3.1.4 Spin-Independent Susceptibility
3.1.5 Hyperfine Coupling
3.1.6 Electron and Nuclear Shielding
3.1.7 Nuclear Shielding by Susceptibility
3.1.8 Inter-nuclear Dipolar Interaction
3.1.9 Inter-nuclear J-coupling
3.1.10 Bilinear Inter-electron Couplings
3.1.10.1 Isotropic Exchange Coupling
3.1.10.2 Antisymmetric Exchange Coupling
3.1.10.3 Notes on Zero-Field Splitting
3.2 Algebraic Side
3.2.1 Interaction Classification
3.2.2 Irreducible Spherical Tensors
3.2.3 Stevens Operators
3.2.4 Hamiltonian Rotations
3.2.5 Rotational Invariants
3.3 Historical Conventions
3.3.1 Eigenvalue Order
3.3.2 Eigenvalue Reporting
3.3.3 Practical Considerations
3.3.4 Visualisation of Interactions
4 Coherent Spin Dynamics
4.1 Wavefunction Formalism
4.1.1 Example: Spin Precession
4.1.2 Example: Bloch Equations
4.2 Density Operator Formalism
4.2.1 Liouville–Von Neumann Equation
4.2.2 Calculation of Observables
4.2.3 Spin State Classification
4.2.4 Superoperators and Liouville Space
4.2.5 Treatment of Composite Systems
4.2.6 Frequency-Domain Solution
4.3 Effective Hamiltonians
4.3.1 Interaction Representation
4.3.2 Matrix Logarithm Method
4.3.3 Baker-Campbell-Hausdorff Formula
4.3.4 Zassenhaus Formula
4.3.5 Directional Taylor Expansion
4.3.6 Magnus and Fer Expansions
4.3.7 Combinations and Corollaries
4.3.8 Average Hamiltonian Theory
4.3.8.1 Zeeman Rotating Frames
4.3.8.2 Secular Coupling
4.3.8.3 Pseudosecular Coupling
4.3.8.4 Weak Coupling
4.3.8.5 Monochromatic Irradiation
4.3.8.6 Decoupling and Recoupling
4.3.9 Suzuki-Trotter Decomposition
4.4 Perturbation Theories
4.4.1 Rayleigh-Schrödinger Perturbation Theory
4.4.2 Van Vleck Perturbation Theory
4.4.3 Dyson Perturbation Theory
4.4.4 Fermi’s Golden Rule
4.5 Resonance Fields
4.5.1 Eigenfields Method
4.5.2 Adaptive Trisection Method
4.6 Symmetry Factorisation
4.6.1 Symmetry-Adapted Linear Combinations
4.6.2 Liouville Space Symmetry Treatment
4.6.3 Total Spin Representation
4.7 Product Operator Formalism
4.7.1 Evolution Under Zeeman Interactions
4.7.2 Evolution Under Spin–Spin Couplings
4.7.3 Example: Ideal Pulse
4.7.4 Example: Spin Echo
4.7.5 Example: Magnetisation Transfer
4.8 Floquet Theory
4.8.1 Single-Mode Floquet Theory
4.8.2 Effective Hamiltonian in Floquet Theory
4.8.3 Multi-mode Floquet Theory
4.8.4 Floquet-Magnus Expansion
4.9 Numerical Time Propagation
4.9.1 Product Integrals
4.9.2 Example: Time-Domain NMR
4.9.3 Lie-Group Methods, State-Independent Generator
4.9.4 Lie-Group Methods, State-Dependent Generator
4.9.5 Matrix Exponential and Logarithm
4.9.6 Matrix Exponential-Times-Vector
4.9.7 Bidirectional Propagation
4.9.8 Steady States of Dissipative Systems
4.9.9 Example: Steady-State DNP
5 Other Degrees of Freedom
5.1 Static Parameter Distributions
5.1.1 General Framework
5.1.2 Gaussian Quadratures
5.1.3 Gaussian Spherical Quadratures
5.1.4 Heuristic Spherical Quadratures
5.1.4.1 Geometric Pattern Girds
5.1.4.2 Optimisation Grids
5.1.4.3 Heuristic Weight Selection
5.1.5 Direct Product Quadratures
5.1.6 Adaptive Spherical Quadratures
5.1.6.1 Adaptive Spherical Grid Subdivision
5.1.6.2 Adaptive Transition Moment Integral
5.1.7 Example: DEER Kernel with Exchange
5.2 Dynamics in Classical Degrees of Freedom
5.2.1 Spatial Dynamics Generators
5.2.2 Algebraic Structure of the Problem
5.2.3 Matrix Representations
5.2.3.1 Distributed Initial Conditions and Detection States
5.2.3.2 Distributed Evolution Generators
5.2.3.3 Diffusion and Flow Generators
5.2.3.4 Phase Turning Generators
5.2.4 Periodic Motion, Diffusion, and Flow
5.2.5 Connection to Floquet Theory
5.3 Chemical Reactions
5.3.1 Networks of First-Order Spin-Independent Reactions
5.3.2 Networks of Arbitrary Spin-Independent Reactions
5.3.3 Chemical Transport of Multi-spin Orders
5.3.4 Flow, Diffusion, and Noise Limits
5.3.5 Spin-Selective Chemical Elimination
5.4 Spin-Rotation Coupling
5.4.1 Molecules as Rigid Rotors
5.4.2 Internal Rotations and Haupt Effect
5.5 Spin-Phonon Coupling
5.5.1 Harmonic Oscillator
5.5.2 Harmonic Crystal Lattice
5.5.3 Spin-Displacement Coupling
5.6 Coupling to Quantised Electromagnetic Fields
5.6.1 LC Circuit Quantisation
5.6.2 Spin-Cavity Coupling
6 Dissipative Spin Dynamics
6.1 Small Quantum Bath: Adiabatic Elimination
6.2 Large Quantum Bath: Hubbard Theory
6.3 Implicit Classical Bath: Redfield Theory
6.3.1 Redfield’s Relaxation Superoperator
6.3.2 Validity Range of Redfield Theory
6.3.3 Spectral Density Functions
6.3.4 A Simple Classical Example
6.3.5 Correlation Functions in General
6.3.6 Rotational Diffusion Correlation Functions
6.3.6.1 Sphere
6.3.6.2 Symmetric Top
6.3.6.3 Asymmetric Top
6.3.6.4 Lipari-Szabo Model
6.4 Explicit Classical Bath: Stochastic Liouville Equation
6.4.1 Derivation
6.4.2 General Solution
6.4.3 Rotational Diffusion
6.4.4 Solid Limit of SLE
6.5 Generalised Cumulant Expansion
6.5.1 Scalar Moments and Cumulants
6.5.2 Joint Moments and Cumulants
6.5.3 Connection to Redfield Theory
6.6 Secular and Diagonal Approximations
6.7 Group-Theoretical Aspects of Dissipation
6.8 Finite Temperature Effects
6.8.1 Equilibrium Density Matrix
6.8.2 Inhomogeneous Thermalisation
6.8.3 Homogeneous Thermalisation
6.9 Mechanisms of Spin Relaxation
6.9.1 Empirical: Extended T1/T2 Model
6.9.2 Coupling to Stochastic External Vectors
6.9.3 Scalar Relaxation: Noise in the Interaction
6.9.4 Scalar Relaxation: Noise in the Partner Spin
6.9.5 Isotropic Rotational Diffusion
6.9.5.1 Zeeman Interactions
6.9.5.2 Bilinear Interactions
6.9.5.3 Quadratic Interactions
6.9.5.4 Cross-Correlations
6.9.6 Nuclear Relaxation by Rapidly Relaxing Electrons
6.9.6.1 Contact Mechanism
6.9.6.2 Curie Mechanism
6.9.6.3 Dipolar Mechanism: Perturbative Treatment
6.9.6.4 Dipolar Mechanism: Adiabatic Elimination
6.9.7 Spin-Phonon Relaxation
6.9.7.1 High Temperature: Redfield-type Theories
6.9.7.2 Low Temperature: Hubbard-Type Theories
6.9.7.3 Populations Only: Dyson-Type Theories
6.9.7.4 Effects of Phonon Dynamics
6.9.8 Notes on Gas-Phase Relaxation
6.9.8.1 Diffusion Through Inhomogeneous Fields
6.9.8.2 Collisional Relaxation: Spin-Rotation Mechanism
7 Incomplete Basis Sets
7.1 Basis Set Indexing
7.2 Operator Representations
7.3 Basis Truncation Strategies
7.3.1 Correlation Order Hierarchy
7.3.2 Interaction Topology Analysis
7.3.3 Zero Track Elimination
7.3.4 Conservation Law Screening
7.3.5 Generator Path Tracing
7.3.6 Destination State Screening
7.4 Performance Illustrations
7.4.1 1H-1H NOESY Spectrum of Ubiquitin
7.4.2 19F and 1H NMR of Anti-3,5-Difluoroheptane
8 Optimal Control of Spin Systems
8.1 Gradient Ascent Pulse Engineering
8.2 Derivatives of Spin Dynamics Simulations
8.2.1 Elementary Matrix Calculus
8.2.2 Eigensystem Derivatives
8.2.2.1 Non-degenerate Eigenvalues
8.2.2.2 Degenerate Eigenvalues
8.2.3 Trajectory Derivatives
8.2.3.1 Derivative Superoperator
8.2.3.2 Derivative Co-propagation
8.2.3.3 Frequency-Domain Derivatives
8.2.4 Matrix Exponential Derivatives
8.2.4.1 Finite Differences
8.2.4.2 Complex Step Method
8.2.4.3 Eigensystem Differentiation
8.2.4.4 Differentiation of Power Series
8.2.4.5 Auxiliary Matrix Method
8.2.5 GRAPE Derivative Translation
8.3 Optional Components
8.3.1 Prefixes, Suffixes, and Dead Times
8.3.2 Keyholes and Freeze Masks
8.3.3 Multi-target and Ensemble Control
8.3.4 Cooperative Control and Phase Cycles
8.3.4.1 Single-Scan Cooperative Pulses
8.3.4.2 Multi-scan Cooperative Pulses
8.3.4.3 Conventional Phase Cycles
8.3.5 Fidelity and Penalty Functionals
8.3.6 Instrument Response
8.4 Optimisation Strategies
8.4.1 Gradient Descent
8.4.2 Quasi-Newton Methods
8.4.3 Newton–Raphson Methods
8.5 Trajectory Analysis
8.5.1 Trajectory Analysis Strategies
8.5.1.1 Correlation Order Populations
8.5.1.2 Coherence-Order Populations
8.5.1.3 Total Exclusive Population
8.5.1.4 Total Inclusive Population
8.5.2 Trajectory Similarity Scores
8.5.2.1 Selective State Grouping
8.5.2.2 Broad State Grouping
8.6 Pulse Shape Analysis
8.6.1 Frequency-Amplitude Plots
8.6.2 Spectrograms, Scalograms, etc.
9 Notes on Software Engineering
9.1 Sparse Arrays, Polyadics, and Opia
9.1.1 The Need to Compress
9.1.2 Sparsity of Spin Hamiltonians
9.1.3 Tensor Structure of Liouvillians
9.1.4 Kron-Times-Vector Operation
9.1.5 Polyadic Objects and Opia
9.2 Parallelisation and Coprocessor Cards
9.2.1 Obvious Parallelisation Modalities
9.2.2 Parallel Basis and Operator Construction
9.2.2.1 Basis Set Indexing
9.2.2.2 Hamiltonian and Relaxation Superoperator
9.2.3 Parallel Propagation
9.2.3.1 Liouville Space: Generator Diagonalisation
9.2.3.2 Liouville Space: Generator Block-Diagonalisation
9.2.3.3 Hilbert Space: Density Matrix Factorisation
9.2.3.4 Hilbert Space: Factorisation-Free Observables
9.2.3.5 Hilbert Space: Factorisation-Free Final State
9.3 Recycling and Cutting Corners
9.3.1 Efficient Norm Estimation
9.3.2 Hash-and-Cache Recycling
9.3.3 Analytical Coherence Selection
9.3.4 Implicit Linear Solvers
References
Index