Simulation of a quantum phase transition of polaritons with trapped ions

We present a system for the simulation of quantum phase transitions of collective internal qubit and phononic states with a linear crystal of trapped ions. The laser-ion interaction creates an energy gap in the excitation spectrum, which induces an effective phonon-phonon repulsion and a Jaynes-Cummings-Hubbard interaction. This system shows features equivalent to phase transitions of polaritons in coupled cavity arrays. Trapped ions allow for easy tuning of the hopping frequency by adjusting the axial trapping frequency and the phonon-phonon repulsion via laser detuning and intensity. We propose an experimental protocol to access all observables of the system, which allows one to obtain signatures of the quantum phase transitions even with a small number of ions.

Figure 1

(Color online) Analogy between phase transitions in coupled cavities and trapped ions: (a) coupled cavities each containing photons and single two-state atoms. Intercavity hopping is provided by an optical fiber. The strong coupling between the atoms and the photons leads to a photon-photon repulsion. (b) All ions are simultaneously interacting with a traveling wave in the radial direction. The laser-ion interaction creates an effective on-site interaction between the local phonons. The phonon hopping appears due to the Coulomb interaction and can be adjusted by the mutual distance of the ions.

Figure 2

(Color online) Total $\left(\text{qubit}+\text{phonon}\right)$ variance $\mathcal{D}{N}_k$ (top) and the qubit variances $\mathcal{D}{N}_{a,k}\left(k=1,2,3\right)$ (bottom) for a chain of five ions with five excitations as a function of the laser detuning $\Delta$ for fixed hopping $\kappa =0.3g$. Negative values of $\Delta$ correspond to blue detuning with respect to the red-sideband transition.