Quantum Manipulation of an Ion Coupled to an LC Circuit

Although ion traps have several advantages as media for quantum simulation, interacting ions in different traps at a quantum mechanical level remains difficult. As a hybrid' solution to this challenge, we present a technique for coupling the spin of an individual ion to the mode of a nearby LC circuit. The circuit mode is engineered to be comparable to the spin splitting, which is on the order of GHz. In the spirit of the Molmer-Sorenson gate, the desired ``phonon-mediated'' coupling is achieved by driving interactions with the ion's radio frequency vibrational mode. We show how the spin-circuit coupling can be used to generate arbitrary, spin-dependent displacements of the circuit mode. In conjunction with quantum nondemolition measurements, these displacements can be used to coherently transfer information between the two degrees of freedom. Finally, we show how interactions with the ion can be used to generate squeezed states of the LC mode, and discuss the limits of this squeezing due to noise losses and nonlinear corrections.