Non-Abelian Aharonov-Bohm Caging in Synthetic Dimensions with a Trapped Ion

Abstract

Aharonov-Bohm (AB) caging is a complete localization phenomenon in two-dimensional lattices due to destructive interference induced by the background gauge fields. However, current investigations of AB caging are mostly restricted to the Abelian gauge field case, and the observation of AB caging under non-Abelian gauge fields in a quantum system still remains elusive. Here, we report experimental realization of tunable synthetic non-Abelian SU(2) gauge fields in a rhombic lattice, engineered within the synthetic dimensions of a vibrating trapped ion with multiple levels. We realize AB caging under both Abelian and non-Abelian gauge fields and systematically investigate the distinctive transport properties of the non-Abelian case. In particular, we observe typical emergent quantum dynamics unique to non-Abelian AB caging, including initial-state-dependent dynamics, second-order effects, and asymmetric caging behavior. These observations demonstrate the trapped ion system as a powerful platform for simulating emergent phenomena in high-dimensional quantum systems with exotic synthetic gauge fields.

Figures (7)

• Figure 1: Synthesis of $SU(2)$ gauge fields in the spin-phonon synthetic dimensions. (a) The two-layer rhombic lattice. Each lattice site is encoded in a spin level with a specific phonon number $n$. The inter-site couplings are realized via couplings between the internal spin states (blue bonds) and the spin-motion couplings (red bonds). (b) The effective rhombic lattice with internal spins. The gauge field $U_i\in SU(2)$ is synthesized on the link by controlling the magnitude and the phase of the couplings. The Wilson loop is defined by $|{\rm Tr}[U(\gamma)]|:=|{\rm Tr}(U_3U_4U_2U_1)|$ of loop $\gamma$ in a plaquette. (c) Encoding the lattice into a ${}^{40}{\rm Ca}^+$ ion. Four levels in the $D_{5/2}$ manifold are used to encode sites B (blue balls in (a)) and sites C (orange balls). Two levels in the $S_{1/2}$ manifold are used to encode sites A (light green balls). Four lasers resonant with the quadrupole transition drive the carrier transition (blue arrows). Another four lasers detuned by $-\omega_m$ drive the red sideband transition (red arrows).
• Figure 2: Realization of AB caging. (a-d) Dynamical evolution diagrams under the Abelian (left panel) and the non-Abelian gauge field (right panel). We initialize the system with an out-of-phase superposition state (a)(b) and an in-phase superposition state (c)(d) at site $A_0$, respectively. (e)(f) The initial-state dependent dynamics. We continuously tune the relative phase $\phi$ of the initial state. $P_0$ denotes the sum of the population of all the $n=0$ sites. (e) Numerical simulated plot of $P_0$ versus time $t$ and the initial relative phase $\phi$ for the Abelian (left panel) and the non-Abelian gauge field (right panel). (f) The experiment is carried out at the time of $t=0.15$ ms (dashed lines in (e)). The orange dots (blue diamonds) are the experimental results for the Abelian (non-Abelian) case. The theory lines are as in (e). Error bars stand for 1 standard deviation.
• Figure 3: Unique dynamics of non-Abelian AB caging. (a) Illustration of the second-order caging. The solid ball indicates the initial state. The red (blue) lines are the couplings with phase $\pi$ (0). (b) The experimental results (left) and the numerical simulations (right) show that the population localizes inside the $n\leq1$ sites. (c) Illustration of the asymmetry caging. The colors of the lines are set in the same way as in (a). (d) The experimental results (left) and the numerical simulations (right) show that the destructive interference occurs at $A_0$ and $A_3$.
• Figure 4: $SU(2)$ interference dynamics. We simultaneously tune the phase $\varphi$ of two couplings. (a) Numerical calculated dynamics of $P_0$. (b) Experimental results at the time of $t=0.2\ {\rm ms}$ (dashed lines in (a)). The red circles (green squares) are the results of the Abelian (non-Abelian) gauge fields. The theory lines are as in (a). Error bars stand for 1 standard deviation, and are smaller than markers.
• Figure S1: Examples of the blue sideband transition curves and the reconstruction of the phonon states. $p$ is the population of the ion bright state. $p(n)$ is the population of the phonon state $|n\rangle$ obtained from the fitting. (a)(b) Each data point is the mean result from 100 repetitions of the experiment. (c)(d) Each data point is the mean result from 400 repetitions of the experiment. We slightly shorten the total time of the sideband transition in (c)(d) to increase the number of data points in one period. Error bars stand for 1 standard deviation.