Emulating Solid-State Physics with a Hybrid System of Ultracold Ions and Atoms

We propose and theoretically investigate a hybrid system composed of a crystal of trapped ions coupled to a cloud of ultracold fermions. The ions form a periodic lattice and induce a band structure in the atoms. This system combines the advantages of high fidelity operations and detection offered by trapped ion systems with ultracold atomic systems. It also features close analogies to natural solid-state systems, as the atomic degrees of freedom couple to phonons of the ion lattice, thereby emulating a solid-state system. Starting from the microscopic many-body Hamiltonian, we derive the low energy Hamiltonian, including the atomic band structure, and give an expression for the atom-phonon coupling. We discuss possible experimental implementations such as a Peierls-like transition into a period-doubled dimerized state.

Figure 1

(a) A cloud of ultracold atoms is brought close to a linear crystal of ions, which are trapped in a Paul trap. Because of the strong Coulomb repulsion, the ions are strongly pinned to a periodic lattice and the excitations are bosonic phonons, with a typical but highly tunable spectrum shown in (b) in units of $E^{*}$ for ${}^6\mathrm{Li}\mathrm{\text{−}}{}^{174}\mathrm{Yb}^{+}$. To lowest order and for transversal atom-ion separation ${\Delta }_r=0$, mobile atoms experience a periodic spatial potential via the attractive atom-ion potential, forming Bloch states [shown in (d)] and energy bands (c) for $d=15R^{*}$ and 1D scattering phases ${\phi }_o=-{\phi }_e=\pi /4$.

Figure 2

The interaction between atoms and ions is expressed as an atom-phonon coupling $|{\lambda }_{k,k^{\prime }}^{(q=k-k^{\prime },x/z)}|$ in the low energy limit. For atoms displaced by ${\Delta }_r=0.75R^{*}$ and tuning to a small ${\omega }_{k=\pi /d,x}=0.0093{\Omega }_x=30E^{*}$, (a) the longitudinal coupling is several orders of magnitude smaller than (b) the transverse coupling, and the latter may exceed the bandwidth while still being smaller than the band gap.

Figure 3

Experimentally tuning the ion trapping allows for the observation of a Peierls-like phase transition when atoms are confined in an effective 1D dipole trap at a separation ${\Delta }_r>0$ from the ion string. (a) The discrete translational symmetry of the linear chain (c) is spontaneously broken towards a zigzag pattern of the ions, and a band gap opens at $k=\pi /2d$ for the atoms. At different atomic fillings $n$ (per ion), and below a critical temperature $T_c$, (b) a phase transition occurs. Unlike the Peierls effect in solid-state materials, the optimal filling that generates the largest $T_c$ can (d) be larger than $n=1/2$.