Coupling the motional quantum states of spatially distant ions using a conducting wire

Interfacing ion qubits in separate traps is among the challenges towards scaling up ion quantum computing. This theoretical study focuses on using a conducting wire to couple the motional quantum states of ions in separate planar traps. This approach of interfacing ion traps provides an alternative to coupling distant qubits with lasers. We include the effects of $1/f^{\tilde{\alpha}}$ (Anomalous) surface heating noise, using aggregate and recent experimental findings as the basis for an analytical model of the motional state decoherence time $t_{\mathrm{deco.}}$. Our optimized design for the coupling system can be used to exchange quantum information with a time $t_{\mathrm{ex.}}$ less than one tenth of the information decay time $t_{\mathrm{deco.}}$. We derive a coefficient $\zeta$ which relates the capacitances of each part of the coupling system and corrects an oversight common to several previous works. Where possible, we calculate the classical signal strength and classical noise strength, and use the criterion (classical) signal-to-noise-ratio $\ge 10$ to further constrain design parameters. Ranges for all parameters are discussed, and the ratio $t_{\mathrm{deco.}} /t_{\mathrm{ex.}}$ and the signal-to-noise ratio for thermal noise are plotted to assess specific parameter ranges for which transfer of quantum information is possible. Although $1/f^{\tilde{\alpha}}$ surface noise significantly constrains parameter ranges, we find no barriers to exchanging quantum information between ion qubits in separate surface traps using a conducting wire. Moreover, this should be possible using existing technologies and materials, and singly-charged ions.