The quantum theory of the Penning trap

We present the quantum theory of the Penning trap in terms of individual x and y radial modes of the motion of a single charged particle in the trap, and demonstrate how the conventional rotating frame used to examine these individual dynamics fails in the quantum regime. In solving the radial Hamiltonian in the {x, y} basis, we show how canonical transformation of the variables must take place after quantization, in order that these separate motions can be consistently tracked. This is in contrast to previous work. The results of the discussion lend themselves to a fully quantum treatment of mode coupling in the trap, leading to an avoided crossing between the coupled energy levels of the system. Exploiting the algebraic structure of the problem allows employment of a dressed-atom formalism within quantum Penning trap theory, and future applications resulting from this are proposed.