Contents

• Thesis Objectives
• List of Figures
• List of Tables
• List of Files
• Preface
  • A Quick Tour of Program TrapApp
  • A Map of the Thesis
• Theory and Simulation of a Single Paul-Trapped Ion
  • Motivation for Ion-Trapping
  • The Quadrupole Potential of Ion Traps
  • The Oscillating Paul Trap Potential
  • Motion in the Paul Trap: the Mathieu Equation
  • Classes of Solutions to Mathieu's Equation
  • Stability Regions for the 1-Dimensional Mathieu Equation
  • Stability Regions for a Single Ion in a Paul Trap
  • Approximate Solution of Mathieu Equations in Region A
  • Higher-Order Stability Regions
• Theory and Simulation of a Collection of Paul-Trapped Ions
  • Equations of Motion for a Collection of Paul-Trapped Particles
  • Behaviors Observed for Paul-Trapping in Region A
    • Mathieu Regime
    • Chaotic Regime
    • Quasiperiodic Regime
    • Crystalline Phase
    • The Coulomb Coupling Parameter
    • Orientation of Two-Ion Crystals
    • Equilibrium Spacing for the Two-Ion Crystal
  • Investigation of Crystal Stability in Region A
    • Viscous Damping and Crystal Stability in Region A
    • The Mathieu Parameters and Crystal Stability in Region A
      • Response of Crystal to Sudden Changes in Mathieu Parameters
      • Response of Crystal to Continuous Changes in Mathieu Parameters
  • Behaviors Simulated for Paul-Trapping in Region B
    • Mathieu Regime
    • Chaotic Regime
    • Ordered Phase
  • Investigation of Crystal Stability in Region B
• Theory and Simulation of Penning Traps
  • Motivation for Particle Confinement in Penning Traps
  • The Magnetic Field of Penning Traps
  • Stability Conditions for the Penning Trap
  • Behaviors Observed in Penning Trap Simulations
• Methods and Theory of the Computer Simulation
  • Structure of the Simulation
  • System of Units
  • Finite Difference Methods
  • Choosing a Time Step
    • Micromotion Scaling
    • Einstein Scaling
  • Initial Conditions
    • Initialization of Position
    • Initialization of Velocity
  • Damping/Reservoir Interaction
    • Viscous Drag
    • Rescaling for Fixed Temperature
    • Monte Carlo Collisions with Background Gas
• A Guide to Using Program TrapAppp
  • Menu Items
  • Buttons
  • Sliders
  • Mouse Clicks on Graph Objects
• Epilogue
• References
• About this document ...