Recent advances in quantum technology enabled us to simulate a certain quantum system with another well-controlled system (quantum simulation or analog quantum computation). Systems of trapped ions are among the best suited for that purpose. In this talk, we report an experiment on quantum simulation of the Jaynes-Cummings-Hubbard (JCH) model [1,2] using ions in a linear Paul trap .
The Hubbard model was originally introduced to model the strongly correlated behavior of electrons in solid-state materials. The JCH model  is a variant of the Hubbard model, and describes a system of coupled cavity arrays, each containing a two-level atom, which can be considered as an "artificial solid-state material". It has a similarity to the Bose-Hubbard model, which has been extensively studied with cold atoms in optical lattices.
We have realized the JCH model using the internal and radial vibrational states of ions in a linear Paul trap based on the proposal by Iwanov et al.  We observed quantum dynamics and adiabatic transfer between many-body ground states. Observation of a polaritonic Mott insulator ('frozen phonons') and the possibility of scaling up the system to several ions will be also discussed.
In addition, we would like to mention our recent progress on the realization of a bosonic-particle system using the radial vibrational motion, including an experiment on the Hong-Ou-Mandel interference of two phonons, and holonomic single-qubit operations , which aims at the realization of robust quantum computation.
 A. D. Greentree et al., Nature Phys. 2, 856-861 (2006).; M. J. Hartmann et al., Nature Phys. 2, 849-855 (2006); D. G. Angelakis et al., Phys. Rev. A 76, 031805 (2007).
 P. A. Ivanov et al., Phys. Rev. A 80, 060301 (2009).
 K. Toyoda, Y. Matsuno, A. Noguchi, S. Haze, and S. Urabe, Phys. Rev. Lett. 111,160501 (2013)
 K. Toyoda, K. Uchida, A. Noguchi, S. Haze, and S. Urabe, Phys. Rev. A 87, 052307 (2013)
Kenji Toyoda is an assistant professor at Osaka University in the Graduate School of Engineering Science. He received his Ph.D. from the Physics Department at Kyoto University in 2002. His research interests include quantum information processing using trapped ions. His recent research subjects include quantum simulation of coupled optical cavities (the Jaynes-Cummings-Hubbard model), analog quantum computation using phonons in trapped ions, and implementation of adiabatic quantum gates.