Synthetic Gauge Fields for Vibrational Excitations of Trapped Ions

The vibrations of a collection of ions in a microtrap array can be described in terms of tunneling phonons. We show that the vibrational couplings may be tailored by using a gradient of the trap frequencies together with a periodic driving of the trapping potentials. These ingredients allow us to induce effective gauge fields on the vibrational excitations, such that phonons mimic the behavior of charged particles in a magnetic field. In particular, microtrap arrays are well suited to realize a quantum simulator of the famous Aharonov-Bohm effect and observe the paradigmatic edge states typical from quantum-Hall samples and topological insulators.

Figure 1

Arrangement of ion microtraps: Schematic representation of the microtrap layout for (a) two-ion link, (c) four-ion plaquette, (e) many-ion rhombic ladder [the relevant parameters of the effective Hamiltonian (13) are also shown]. Requirements for the photon-assisted tunneling of phonons: (b) The frequencies of two adjacent traps ${\omega }_1,{\omega }_2$ are shifted by $\Delta \omega$. By shining a pair of Raman lasers, one assists the phonon transfer. (d) For a plaquette, the gradient is along $x$ and the $\Delta \mathbf{k}$ has a component along the $x\mathrm{\text{−}}y\mathrm{\text{−}}z$ axes.

Figure 2

Photon-assisted tunneling and Aharonov-Bohm cages. (a) Effective hopping amplitude $|{\mathcal{F}}_{r=1}|$ as a function of $({\eta }_d,\Delta \varphi )$. The values of the dashed line at ${\eta }_d\approx 0.6$ are used in (b). (b) Photon-assisted hopping for a two-ion link of a single vibrational excitation $|{\psi }_0⟩=a_1^{†}|0⟩$ under the effective description (5) (solid red line) and the complete Hamiltonian (3) with the driving term (10) (yellow dots). We plot the vibrational population $n_2^{*}$ transferred to site 2 after time $t^{*}=\pi /|J_{[r]}(0.6,\varphi )|$. Maximum phonon transfer occurs at $\Delta \varphi =\pi$. (c),(d) Evolution of the phonon excitation in a rectangular plaquette. We plot the phonon populations under approximation (9) [$P_i(t)$], and under the exact Hamiltonian Eqs. (6, 7, 8) driven by (10) [$n_i(t)$]. In (c) ${\varphi }_{\circlearrowleft }=0$, ${\varphi }_x=\pi$, ${\varphi }_y=0$, $d_x=d_y|{\mathcal{F}}_1({\eta }_d,\pi ){|}^{2/3}$, $n_{\max }=2$, and other parameters same as (b). In (d), ${\varphi }_{\circlearrowleft }=\pi$, and there is an Aharonov-Bohm destructive interference that inhibits tunneling to site 3. ${\varphi }_x=\pi$, ${\varphi }_y=\pi$, ${\Omega }_L=0.25{\omega }_z$, and other values same as (c). (e) Energy spectrum $E$ for the $\pi$-flux regime of Hamiltonian (13) for $N=31$ microtraps. Flat bands appear at $E\approx \pm 2J_1$ (Aharonov-Bohm cages) (see also the schematic description of the eigenstates), and also at $E=0$ (which arise solely due to the geometry of the lattice). Also, in the middle of the gaps, single edge states localized to the boundaries of the ladder arise. (f) Energy level spectrum as a function of the effective flux, which displays a gap-vanishing point at $\varphi =0$.