Fig. 16: Wannier-Stark localization of a single \({{\mathbb{Z}}}_{2}\) charge in the chain.
From: Synthetic \({{\mathbb{Z}}}_{2}\) gauge theories based on parametric excitations of trapped ions
a Single bosonic particle (filled red circle), with an attached electric-field string that connects it to the background charge q1 = 1 at the left boundary. b Contour plots for the dynamics of the boson distribution \({\overline{n}}_{i}(t)=\langle {a}_{i}^{{{{\dagger}}} }{a}_{i}(t)\rangle\), the electric field on the links \({\overline{s}}_{i}^{x}(t)=\langle {\sigma }_{i,{{{{{{{{\bf{e}}}}}}}}}_{1}}^{x}(t)\rangle\), as well as the block entanglement entropy \(S({\rho }_{A})=-{{{{{{{\rm{Tr}}}}}}}}\{{\rho }_{A}\log {\rho }_{A}\}\). We consider a chain of N = 16 sites, and set the transverse electric field to \(h=0.6{t}_{1,{{{{{{{{\bf{e}}}}}}}}}_{1}}\) (upper row), and \(h=0.4{t}_{1,{{{{{{{{\bf{e}}}}}}}}}_{1}}\) (lower row). The initial state is \(\left\vert \Psi (0)\right\rangle =\left\vert \sim \!\! \sim {\bullet }_{7}\right\rangle\) in the notation of Eq. (62), which corresponds to a product state for the boson being localized at the center of the chain, and a domain wall of the gauge qubits with respect to the Hadamard x-basis magnetization.
