Realizing Synthetic Dimensions and Artificial Magnetic Flux in a Trapped-Ion Quantum Simulator

Synthetic dimension is a potent tool in quantum simulation of topological phases of matter. Here we propose and demonstrate a scheme to simulate an anisotropic Harper-Hofstadter model with controllable magnetic flux on a two-leg ladder using the spin and motional states of a single trapped ion. We verify the successful simulation of this model by comparing the measured dynamics with theoretical predictions under various coupling strength and magnetic flux, and we observe the chiral motion of wave packets on the ladder as evidence of the topological chiral edge modes. We develop a quench path to adiabatically prepare the ground states for varying magnetic flux and coupling strength, and we measure the chiral current on the ladder for the prepared ground states, which allows us to probe the quantum phase transition between the Meissner phase and the vortex phase. Our work demonstrates the trapped ion as a powerful quantum simulation platform for topological quantum matter.

Figure 1

Experimental scheme. (a) We use the spin and a phonon mode of a trapped ion to simulate the two-leg ladder model. Two counterpropagating laser beams are shined on the ion to drive three pairs of Raman transitions near the carrier and the blue and the red motional sidebands. (b) The simulated two-leg ladder model. Its $x$ direction is represented by the phonon number $|n⟩$ and the $y$ direction by the spin states $|\uparrow ⟩$ and $|\downarrow ⟩$. The carrier driving provides the coupling along the spin direction, while the driving near the two motional sidebands generates a spin-dependent force, which translates into the site-dependent coupling along the phonon direction with an adjustable magnetic flux in a suitable frame of reference.

Figure 2

Comparison of experimental and theoretical dynamics under various parameters for the HH Hamiltonian in Eq. (1) from an initial state $|\downarrow ⟩|\alpha =2⟩$. We choose $\mathrm{\Delta }=2\pi \times 2\mathrm{kHz}$ and $\theta =\pi /2$. (a) $J_x=2\pi \times 4\mathrm{kHz}$, $J_y=2\pi \times 3.6\mathrm{kHz}$, and $\mathrm{\Phi }=2\phi =0$. (b) $J_x=2\pi \times 6\mathrm{kHz}$, $\mathrm{\Phi }=2\phi =\pi /2$ and various $J_y$.

Figure 3

Chiral motion of an initial wave packet $|\downarrow ⟩|\alpha =2⟩$ under the HH model with $\mathrm{\Delta }=2\pi \times 2\mathrm{kHz}$, $J_x=J_y=2\pi \times 6\mathrm{kHz}$, $\mathrm{\Phi }=2\phi =\pi /2$, and $\theta =\pi /2$. (a)–(f) Time evolution of phonon population in the lower leg ($|\downarrow ⟩$, first column), the upper leg ($|\uparrow ⟩$, second column) and their combination (third column). The experimental results (first row) agree well with the theoretical prediction (second row). (g) Time evolution of average phonon number in the lower and upper legs. (h) Schematic dynamics of a wave packet initially located on the lower leg centered around $⟨n⟩=|\alpha {|}^2=4$. First it stays on the lower leg and moves to the left, then starts to transfer to the upper leg at the boundary $n=0$, and finally moves to the right on the upper leg.

Figure 4

Quantum phase transition in the HH model. (a) Energy band structure for an isotropic HH model at $\mathrm{\Delta }=0$ and $J_x=J_y=J$ under small ($\phi =\pi /8$, dashed curves, Meissner phase) or large ($\phi =3\pi /8$, solid curves, vortex phase) magnetic flux, respectively. The color represents the spin component $⟨{\sigma }_z⟩$ of the eigenstates. (b) Energy band structure for an isotropic HH model at $\mathrm{\Delta }=0$ and $\phi =\pi /4$ under large ($J_x=0.1J$, $J_y=J$, dashed curves, Meissner phase) or small ($J_x=0.5J$, $J_y=0.1J$, solid curves, vortex phase) coupling ratio, respectively. (c) Chiral current for the adiabatically prepared ground state at $\mathrm{\Delta }=2\pi \times 3$, $J_x=2\pi \times 6$, and $J_y=2\pi \times 9\mathrm{kHz}$ (black circles for experiment and blue dash-dotted curve for theory), the ideal ground state for the anisotropic HH model (red dashed curve) and the isotropic HH model (green solid curve) vs magnetic flux $\phi$. (d) Similar plot as (c) under various detuning $\mathrm{\Delta }$. Dots represent experimental data and curves represent ideal ground states for the anisotropic HH model.