Topological phenomena in trapped-ion systems

We propose and analyze a scheme to observe topological phenomena with ions in microtraps. We consider a set of trapped ions forming a regular structure in two spatial dimensions and interacting with lasers. We find phonon bands with nontrivial topological properties, which are caused by the breaking of time-reversal symmetry induced by the lasers. We investigate the appearance of edge modes, as well as their robustness against perturbations. Long-range hopping of phonons caused by the Coulomb interaction gives rise to flat bands which, together with induced phonon-phonon interactions, can be used to produce and explore strongly correlated states. Furthermore, some of these ideas can also be implemented with cold atoms in optical lattices.

Figure 1

(a) Ions trapped in microtraps, forming a planar hexagonal geometry. (b) Laser and magnetic field configuration for each ion in a microtrap. There is a homogenous magnetic field as well as a laser standing wave with circular polarization along the $x$ direction and two counterpropagating lasers in a lin$\perp$lin configuration along the $y$ direction. (c) Ion internal structure, where the quantization axis is chosen along the magnetic field ($x$ axis). For simplicity, we consider no nuclear spin (hyperfine structure). The magnetic field generates a Zeeman shift. The lasers propagating along the $x$ direction (red wavy arrows) produce different ac Stark shifts on the magnetic levels. Lasers along the $y$ direction induce Raman transitions. (d) Detail of the hexagonal lattice, with the unit vectors $e_{x,y}$ indicating the two oscillation directions of the ions (vibrational modes). Different colors are used for the $A$ and $B$ sublattices.

Figure 2

Different configurations considered in the text: (a) torus (PBC) and (b) cylinder.

Figure 3

Single-particle energy spectra in units of ${\tilde{\omega }}_x$ for the system with PBC and $N=400$. The left panels show the energy spectra as a function of the quasimomenta $k_x$ and $k_y$ (in the first Brillouin zone), whereas the right panels display the same spectra but as a function of $k_x$ only in order to highlight the different bands and the corresponding gaps. The Chern numbers of each band are shown to the right of these plots. (a) ${\beta }_x=0.02$ and $V_b=-0.1$, (b) ${\beta }_x=0.02$ and $V_b=-0.2$, (c) ${\beta }_x=0.04$ and $V_b=-0.1$, and (d) ${\beta }_x=0.04$ and $V_b=-0.2$.

Figure 4

As in Fig. 3 for the cylinder geometry and $N=400$: (a) ${\beta }_x=0.02$ and $V_b=-0.1$, (b) ${\beta }_x=0.02$ and $V_b=-0.2$, (c) ${\beta }_x=0.04$ and $V_b=-0.1$, and (d) ${\beta }_x=0.04$ and $V_b=-0.2$. The edge modes are marked in blue (dark gray).

Figure 5

As in Fig. 3 for the planar geometry and 384 sites: (a) ${\beta }_x=0.02$ and $V_b=-0.2$ and (b) ${\beta }_x=0.04$ and $V_b=-0.1$. Here, $l$ denotes the eigenmode. Note that for 384 sites, there are 768 eigenmodes.

Figure 6

Spatial distribution for typical edge and bulk eigenmodes corresponding to the system with planar geometry, with 384 sites, ${\beta }_x=0.04$ and $V_b=-0.1$. (a) and (c) For an energy $E=0.7{\tilde{\omega }}_x$, corresponding to an edge mode, and (c) and (d) for $E=0.9{\tilde{\omega }}_x$, corresponding to a bulk mode; (a) and (c) $x$ component and (b) and (d) $y$ component.

Figure 7

(a) and (b) Single-particle spectra in units of ${\tilde{\omega }}_x$ and (c) and (d) the spatial distribution for the $x$ and $y$ components of an eigenmode with energy $E=0.7{\tilde{\omega }}_x$ for a planar geometry with 384 lattice sites. We have added random numbers to the values of $V_b$ and ${\beta }_x$, as explained in the main text.

Figure 8

Density distribution of $x$ phonons after the evolution time $t_{\mathrm{f}}$ for ${\beta }_x=0.04$ and $V_b=-0.1$: (a) ${\omega }_{\mathrm{d}}=2.0{\tilde{\omega }}_x$, (b) ${\omega }_{\mathrm{d}}=0.6{\tilde{\omega }}_x$, (c) ${\omega }_{\mathrm{d}}=1.0{\tilde{\omega }}_x$, and (d) ${\omega }_{\mathrm{d}}=0.7{\tilde{\omega }}_x$.

Figure 9

Solid red curve: flatness $F$ of the first band as a function of $V_b$ and ${\beta }_x$. Dashed blue curve: Same for a model where phonon hopping is restricted to nearest neighbors.