Decisive Solution to a Long-standing Controversy in Paul-Trap Physics

The study of charged particles' dynamics in a Paul trap is the foundation of its wide-ranging applications, including analyzing proteins, determining isotope ratios, and constructing a quantum computer. However, in the simplest case of two-particle dynamics, there remains a controversy on whether a two-ion planar crystal undergoes an order-to-chaos transition at a critical trap parameter value. Via linearization and separation of the Mathieu-Coulomb equations, this thesis shows analytically and numerically that the transition does not exist. Furthermore, the method developed here can be extended to determine the stability and existence of two-ion crystals in the whole parameter space of both the hyperbolic and linear Paul trap.