Abstract We propose a laser-free method for creation of a phonon two-mode squeezed state and a beam-splitter transformation, using time-varying electric fields and non-linear couplings between the normal modes in a linear ion crystal. Such non-linear Coulomb-mediated interactions between the collective vibrational modes arise under specific trap-frequency conditions in an ion trap. We study the quantum metrological capability for parameter estimation of the two quantum states and show that a Heisenberg limit of precision can be achieved when the initial state with n phonons evolves under the action of the beam-splitter transformation. Furthermore, we show that the phonon non-linearity and the spin-dependent force can be used for creation of a three-qubit Fredkin gate.
We present a system for the simulation of Heisenberg models with spins
s=1/2 and
s=1 with a linear crystal of trapped ions. We show that the laser-ion interaction induces a Jaynes-Cummings-Hubbard interaction between the atomic
V-type level structure and the two phonon species. In the strong-coupling regime the collective atom and phonon excitations become localized at each lattice site and form an effective spin system with varying length. We show that the quantum-mechanical superexchange interaction caused by the second-order phonon hopping processes creates a Heisenberg-type coupling between the individual spins. Trapped ions allow control of the superexchange interactions by adjusting the trapping frequencies, the laser intensity, and the detuning.