Fig. 10: Trapped-ion \({{\mathbb{Z}}}_{2}\) tunneling in a rhomboidal plaquette.

Schematic representation of the gauge-invariant tunneling of a vibrational excitation, which is initially in the center of mass (c.o.m) mode along the x axis, and “tunnels” into the c.o.m mode along the y axis. The blue and green arrow represent the wavevectors of the lasers that induce the parametric excitation. In the upper insets, this tunneling is mediated by a spin flip in the Hadamard basis of the first ion qubit \({\left\vert -\right\rangle }_{1}\,\mapsto \,{\left\vert +\right\rangle }_{1}\), whereas in the lower inset it involves the second ion qubit \({\left\vert -\right\rangle }_{2}\,\mapsto \,{\left\vert +\right\rangle }_{2}\). These two paths can be interpreted as the two effective links of the synthetic rhomboidal plaquette displayed in Fig. 9.