Quantization of the Coulomb Chain in an External Focusing Field

With the appropriate choice of parameters and sufficient cooling, charged particles in a circular accelerator are believed to undergo a transition to a highly-ordered crystalline state. The simplest possible crystalline configuration is a one-dimensional chain of particles. In this paper, we write down the quantized version of its dynamics. We show that in a low-density limit, the dynamics is that of a theory of interacting phonons. There is an infinite sequence of $n$-phonon interaction terms, of which we write down the first orders, which involve phonon scattering and decay processes. The quantum formulation developed here can serve as a first step towards a quantum-mechanical treatment of the system at finite temperatures.