Quantum simulation of Hubbard-type model with trapped ions

Quantum simulation provides important tools in studying strongly correlated many-body systems with controllable parameters. Among them two prototypical many-body models with spin-boson interactions are the Jaynes-Cummings-Hubbard(JCH) model and the Rabi-Hubbard(RH) model, both of which originate from cavity quantum electro-dynamics systems but are also well-suited for the trapped ions. Both demonstrate rich physics through the competition between local spin-boson interactions and long-range boson hopping. The RH model breaks the U(1) symmetry and thus shows non-trivial distinctions from other generalizations like Bose-Hubbard model and Jaynes-Cummings-Hubbard(JCH) model. The JCH model possessed U(1) symmetry thus demonstrates essentially different properties in the ground state phase diagram.

Here we report the quantum simulation of RH and JCH model with trapped Yb ions.We perform quantum simulation of the RH model for the first time with up to 16 ions and explore its equilibrium phase transition and quantum dynamical properties using spin observables. Then we focused on the change from the Markovian to non-Markovian dynamics by tuning the frequency of the spins into different locations of the phonon spectrum in JCH model. We further adjust the effective dimension of the system via the ion number and the excitation number, up to 32 ions 32 excitations, and observe that the non-Markovian dynamics persists for large systems. Compared with the spin models where phonons are only virtually excited, our inclusion of phonon modes greatly enlarges the effective dimension of the Hilbert space and thus demonstrates quantum simulation results that are intractable for the available classical computers.