Improved description of trapped ions as an electro-mechanical system

Trapped ions are among the leading candidates for quantum computing technologies. Interfacing ion qubits in separate traps and interfacing ion qubits with superconducting qubits are two of the many challenges to scale up quantum computers. One approach to overcome both problems is to use a conducting wire to mediate the Coulomb interaction between ions in different traps, or between ions and superconducting qubits. To this end, a trapped charged particle inducing charge on a conductor has long been modeled as a system of equivalent lumped element electronic components. Careful consideration reveals two assumptions in the derivation of this model which are unjustified in many situations of interest. We identify these assumptions and explain their implications. In addition, we introduce an improved way to use linear relationships to describe the interaction of trapped ions with nearby conductors. The new method is based on realistic assumptions and reproduces results from other works that are not based on the circuit element model. It is targeted for trouble-shooting experimental designs and allows experiments to test and compare the accuracy of different theoretical models.