Energy-level structure of ion cloud and crystal in a linear Paul trap

Ion cloud and crystal are recognized as two distinct states in ion traps, but there is no theory that can describe the energy-level structures of these two states from the perspective of quantum theory. In this paper, we construct a model describing ions in a linear Paul trap to investigate the energy-level structure with the linear response function, which is often used in the field of condensed matter to compute the energy-level of electrons or plasmas. We support a method to calculate the energy-level of both the crystal and cloudy states of trapped ions in a linear Paul trap. Furthermore, we present the energy-level diagram of two trapped ions and give a quantum interpretation for the fluorescence hysteresis loops that were observed in our laboratory. This is a fundamental theory that will help us to understand the crystallization and cooling processes of ions in a linear Paul trap.