Fig. 1: Experimental scheme for observing topological Euler insulators with a trapped ion.

a Schematic of our experimental setup. A single ¹⁷¹Yb⁺ ion is trapped in a surface-electrode chip trap. b The energy level structure of the ¹⁷¹Yb⁺ ion. The three states \(\left|1\right\rangle\), \(\left|2\right\rangle\), and \(\left|3\right\rangle\) for the Hamiltonian are encoded in the hyperfine states \(\left|F=0,{m}_{F}=0\right\rangle\), \(\left|F=1,{m}_{F}=-1\right\rangle\), and \(\left|F=1,{m}_{F}=0\right\rangle\), respectively. Quantum operations and adiabatic evolutions are implemented by microwaves. Two far-detuned microwave pulses (denoted by yellow and green arrows) are used for Raman transitions between the levels \(\left|2\right\rangle\) and \(\left|3\right\rangle\). The quantum state projected to \(\left|2\right\rangle\) or \(\left|3\right\rangle\) generates fluorescence detected by a 370 nm detection beam. c Schematic of our microwave setup. The microwaves are generated by an arbitrary waveform generator (AWG) controlled by a computer according to the sequence mixed with a high-frequency microwave signal. They are shone on the ion through a microwave horn. d Experimental sequences. The ion is firstly cooled through the Doppler cooling for 2 ms and then initialized to the dark state by optical pumping, which typically takes 2−3 μs. We then follow the adiabatic passage to slowly tune the Hamiltonian by microwave operations so as to drive the state to an eigenstate of H(k) at any momentum point in the 2D Brillouin zone. The adiabatic passage typically takes 200−300 μs, and at some momentum points it may take up to 500 μs (see Subsections "Microwave operations in the trapped-ion system" and "Adiabatic passage" in Methods for more details). At the end, we perform the quantum state tomography to obtain the full density matrix of the final state (see Subsection "Quantum state tomography for a qutrit system" in Methods for more details).