Observation of Topological Magnon Insulator States in a Superconducting Circuit

Searching topological states in artificial systems has recently become a rapidly growing field of research. Meanwhile, significant experimental progress on observing topological phenomena has been made in superconducting circuits. However, topological insulator states have not yet been reported in this system. Here, for the first time, we experimentally realize a tunable dimerized spin chain model and observe the topological magnon insulator states in a superconducting qubit chain. Via parametric modulations of the qubit frequencies, we show that the qubit chain can be flexibly tuned into topologically trivial or nontrivial magnon insulator states. Based on monitoring the quantum dynamics of a single-qubit excitation in the chain, we not only measure the topological winding numbers, but also observe the topological magnon edge and defect states. Our experiment exhibits the great potential of tunable superconducting qubit chain as a versatile platform for exploring noninteracting and interacting symmetry-protected topological states.

Figure 1

A dimerized qubit chain. (a) The schematic setup of a qubit chain, where $x$ denotes the unit cell index. Each unit cell has two qubits $a$ and $b$. The intra- and interunit cell couplings are $J_1$ and $J_2$, respectively. (b) Five cross-shaped transmon qubits ($X$ mons, $a_1$, $b_1$, $a_2$, $b_2$, and $a_3$) arranged in a linear array. Each qubit is coupled to a separate $\lambda /4$ resonator for simultaneous and individual readout and has independent $XY$ and $Z$ controls (labeled as $“x”$ and $“f”$, respectively).

Figure 2

Topological winding number measurements. (a),(b) Schematic of the experiments in which only $a_1$, $b_1$, $a_2$, and $b_2$ are used without $a_3$. The couplings between neighboring qubits are configured into $J_1\text{−}{J}_2\text{−}{J}_1=5\text{−}1\text{−}5(\mathrm{MHz})$ [(a) topologically trivial] and $J_1\text{−}{J}_2\text{−}{J}_1=1\text{−}5\text{−}1(\mathrm{MHz})$ [(b) topologically nontrivial], respectively. (c),(d) Time evolution of the qubit excitation $P_{id}^e$ for $id=a_1,b_1,a_2,b_2$ for the two different coupling configurations. Dots are experimental data, while solid lines are calculated from the ideal Hamiltonian [Eq. (1)] with the measured system decoherence for an initial state $|gegg⟩$. (e),(f) Time evolution of ${\overline{P}}_d=(P_{a_1}^e-P_{b_1}^e)+2(P_{a_2}^e-P_{b_2}^e)$ for the two cases. Dots are experimental data (averaged 5000 times), red dashed lines are from numerical simulations, and the black horizontal lines represent the oscillation centers.

Figure 3

Observation of topological magnon edge states. (a) Schematic of the experiment in which all five qubits have been used. The couplings between neighboring qubits are configured into $J_1\text{−}{J}_2\text{−}{J}_1\text{−}{J}_2=5\text{−}1\text{−}5\text{−}1(\mathrm{MHz})$ (topologically trivial) and $J_1\text{−}{J}_2\text{−}{J}_1\text{−}{J}_2=1\text{−}5\text{−}1\text{−}5(\mathrm{MHz})$ (topologically nontrivial), respectively. (b),(c) Two-dimensional representation of time evolutions of all qubits’ excited state populations [$P_{id}^e(t)$]. (d),(e) Time evolution of $P_{id}^e$. Dots are experimental data, while solid lines are calculated from the ideal Hamiltonian [Eq. (1)] with the measured system decoherence for an initial state $|egggg⟩$.

Figure 4

Observation of topological magnon defect states. (a) Schematic of the experiment. The couplings between neighboring qubits are tuned into $J_1\text{−}{J}_2\text{−}{J}_1\text{−}{J}_2=4\text{−}1\text{−}1\text{−}4(\mathrm{MHz})$. (b) Two-dimensional representation of time evolutions of all qubits’ excited state populations. (c) Time evolutions of $P_{id}^e$. Dots are experimental data, while solid lines are calculated from the ideal Hamiltonian [Eq. (1)] with the measured system decoherence for an initial state $|ggegg⟩$.