Observation of topological magnon insulator states in a superconducting circuit

TL;DR

This paper reports the first experimental observation of topological magnon insulator states in a superconducting qubit chain, demonstrating tunable topological phases and edge states, advancing the exploration of topological phenomena in artificial quantum systems.

Contribution

It introduces a tunable superconducting qubit chain model that realizes topological magnon insulator states and measures their topological properties for the first time.

Findings

Observation of topological magnon insulator states

Measurement of topological winding numbers

Detection of edge and defect states

Abstract

Searching topological states in artificial systems has recently become a rapidly growing field of research. Meanwhile, significant experimental progresses on observing topological phenomena have 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 non-interacting and interacting symmetry-protected topological states.

Figures

Equations

$$\hat{H}=\sum _{x=1}^N(J_1{\hat{\sigma }}_{a_x}^{+}{\hat{\sigma }}_{b_x}^{-}+J_2{\hat{\sigma }}_{b_x}^{+}{\hat{\sigma }}_{a_{x+1}}^{-}+\text{H.c.),}$$

$${\omega }_{id}={\omega }_{\mathrm{o}\_id}+{\epsilon }_{id}\sin ({\mu }_{id}t+{\phi }_{id}),$$

$$\begin{array}{rl}{J}_1& =g_{a_x,b_x}{\mathcal{J}}_1({\alpha }_{b_x}){\mathcal{J}}_0({\alpha }_{a_x})e^{i({\phi }_{b_x}+\pi /2)},\\ J_2& =g_{b_{x-1},a_x}{\mathcal{J}}_1({\alpha }_{a_x}){\mathcal{J}}_0({\alpha }_{b_{x-1}})e^{-i({\phi }_{a_x}-\pi /2)},\end{array}$$

Full text

Topological insulators are new states of matter beyond Landau symmetry breaking theory, signified by topological invariants and topological edge states, and now lie at the forefront of condensed matter physics Hasan and Kane (2010); Qi and Zhang (2011). The concept of topological insulators recently has been intensively studied and also been expanded to artificial systems, including ultracold atomic Goldman et al. (2016); Zhang et al. (2019); Cooper et al. (2019), photonic Lu et al. (2014); Ozawa et al. (2019) and mechanical Huber (2016); Ma et al. (2019) systems. Nevertheless, as a result of the difficulties in extracting the Berry curvature in photonic and mechanical systems and in engineering edges in optical lattices, it is still challenging to experimentally observe both the topological invariants and the topological edge states in a separate artificial topological system.

Superconducting circuits now have become one of the leading quantum platforms for implementing scalable quantum computation You and Nori (2011); Devoret and Schoelkopf (2013); Neill et al. (2018) and large-scale quantum simulation Buluta and Nori (2009); Houck et al. (2012); Georgescu et al. (2014). In particular, topological effects recently have also been experimentally studied in superconducting circuits. Specifically, topological concepts have been investigated in the parameter space of superconducting qubits Schroer et al. (2014); Roushan et al. (2014); Wang et al. (2018); Tan et al. (2018) and the phase space of superconducting resonators Ramasesh et al. (2017); Flurin et al. (2017); Synthetic gauge fields Roushan et al. (2017a); Wang et al. (2015, 2016); Owens et al. (2018) and Hofstadter butterfly Roushan et al. (2017b) have been realized in a superconducting qubit chain; Topological phenomena have also been observed in a network of superconducting flux qubits King et al. (2018). However, due to the challenges in engineering a topological chain with tunable qubit couplings and the lack of methods in detecting the topology of the qubit chain system, topological insulator states still have not been experimentally observed before in the superconducting system.

In this paper, we experimentally demonstrate the first observation of topological insulator states in a tunable superconducting transmon qubit chain, which exhibits both the nontrivial topological invariants and topological edge states. Our experiment is based on realizing a dimerized spin chain model, which supports topologically trivial or nontrivial magnon insulator states dependent on the qubit coupling configurations. We demonstrate that such configurations can be flexibly tuned via parametrical modulations of the qubit frequencies in situ Zhou et al. (2009); Strand et al. (2013); Wu et al. (2018); Reagor et al. (2018); Li et al. (2018). Through exciting one of the qubits in the chain and then monitoring its quantum dynamics, we further show that the topological winding numbers can be directly measured. By tuning the qubit chain with odd and even number of qubits, the localization and hybridization of topological edge states are observed respectively. Via locally tuning the qubit couplings, we also exhibit that a topological defect can be easily created and probed.

Distinct from previous systems studying the topological states of non-interacting bosons in a lattice Goldman et al. (2016); Zhang et al. (2019); Cooper et al. (2019); Lu et al. (2014); Ozawa et al. (2019); Huber (2016); Ma et al. (2019), the superconducting system allows the study of the topological states of magnons (qubit excitations) in an interacting spin (qubit) chain, where magnons are bosonic quasiparticle excitations around the ground state of the spin chain  Fukuhara et al. (2013a, b) and are interacting hard-core bosons. Although our experiment investigates the single-excitation case and observes the resulted non-interacting symmetry-protected topological magnon insulator states, it paves the way for further studying bosonic interacting symmetry-protected topological states Chen et al. (2013, 2012) when introducing multiple excitations into a longer qubit chain. Realizing interacting symmetry-protected topological states currently is still a great challenge de Léséleuc et al. (2018) and can not be achieved in previously reported non-interacting topological systems Goldman et al. (2016); Zhang et al. (2019); Cooper et al. (2019); Lu et al. (2014); Ozawa et al. (2019); Huber (2016); Ma et al. (2019). Our experiment represents the first step towards realizing such states with a qubit chain system.

The experimental model is based on a dimerized spin chain which describes a one-dimensional spin lattice with two spins per unit cell and different intracell and intercell hopping amplitudes, as shown in Fig. 1a. We implement such a model in a superconducting qubit chain Mei et al. (2018a), where each unit cell contains two qubits labelled by $a$ and $b$. The resulted qubit chain can be described by the dimerized spin chain Hamiltonian after rotating wave approximation

[TABLE]

where $x$ is the unit cell index, $N$ is the number of the unit cells, $J_{1}$ and $J_{2}$ are the intracell and intercell qubit couplings, respectively, and $\sigma^{+}_{a_{x}}$ ($\sigma^{-}_{a_{x}}$) is the raising (lowering) operator associated with qubit $a_{x}$. The single-qubit excitation in this spin chain is called a magnon in condensed matter physics Fukuhara et al. (2013a, b). In the single-qubit excitation case, its topology is same as of the Su-Schrieffer-Heeger model Su et al. (1979); Atala et al. (2013); Meier et al. (2016); St-Jean et al. (2017), which has two distinct topological insulator states characterized by topological winding numbers. However, the difference from the Su-Schrieffer-Heeger model is that the system studied in our experiment is an interacting spin chain and can be further used to realize interacting symmetry-protected topological states. Moreover, magnons are bosonic quasiparticle excitations around the ground state of the spin chain, therefore the topological states observed here are associated with excited states instead of ground states. When the qubit couplings are tuned into $J_{1}<J_{2}$ ($J_{1}>J_{2}$), the topological winding number $\nu=1$ ($\nu=0$) and the system supports a topologically nontrivial (trivial) magnon insulator state (see Supplementary Materials).

We implement the experiment in a superconducting circuit You and Nori (2011); Devoret and Schoelkopf (2013); Gu et al. (2017) consisting of five cross-shaped transmon qubits (Xmons, $a_{1},b_{1},a_{2},b_{2}$, and $a_{3}$) Barends et al. (2013, 2014) arranged in a linear array with fixed capacitive nearest-neighbor couplings, as shown in Fig. 1b. Each qubit has independent $XY$ and $Z$ controls. Separate $\lambda/4$ resonators with different frequencies couple to individual qubits for independent readouts. The average qubit $T_{1}\approx 18~{}\mu$s and $T^{*}_{2}\approx 17~{}\mu$s at the frequency sweet spots. We use a Josephson parametric amplifier Hatridge et al. (2011); Roy et al. (2015), a gain over 20 dB and a bandwidth about 260 MHz, for high-fidelity single-shot measurements of the qubits. To overcome the readout imperfections, we in addition use a calibration matrix to reconstruct the readout results based on Bayes’ rule.

The topologically trivial and nontrivial phases require different qubit-qubit coupling configurations, necessitating full control of the effective couplings between neighboring qubits. Tunable couplings through parametrical modulations of the qubit frequencies can be realized in situ without increasing circuit complexity Zhou et al. (2009); Strand et al. (2013); Wu et al. (2018); Reagor et al. (2018); Li et al. (2018), therefore are ideal for topological simulations. We adopt this technique throughout our experiment to realize the required neighboring qubit coupling strengths as described in Eq. 1.

Explicitly, we apply

[TABLE]

where $\omega_{\mathrm{o}\_{id}}$ is the mean operating frequency, $\varepsilon_{id}$, $\mu_{id}$, and $\varphi_{id}$ are the modulation amplitude, frequency, and phase, respectively for the qubit $id=a_{x},b_{x}$ in the chain. By neglecting the higher order oscillating terms and under the resonant conditions $\omega_{\mathrm{o}\_{b_{x}}}-\omega_{\mathrm{o}\_{a_{x}}}=\mu_{b_{x}}$ or $\omega_{\mathrm{o}\_{b_{x-1}}}-\omega_{\mathrm{o}\_{a_{x}}}=\mu_{a_{x}}$, the effective coupling strengths are

[TABLE]

where $\mathcal{J}_{m}(\alpha)$ is the $m$th Bessel function of the first kind and $g_{a_{x},b_{x}}$ is the static capacitive coupling strength between neighboring qubits. Both $J_{1}$ and $J_{2}$ can be conveniently tuned via changing $\alpha_{id}=\varepsilon_{id}/\mu_{id}$ of the external modulation. Note that the qubit at the edge (for example, $a_{1}$) could be stationary without parametric modulation, while the middle qubit can be parametrically modulated with two independent sinusoidal drives in order to tune the coupling strengths with its two neighboring qubits respectively. The experimental setup, device parameters, and parametric modulation parameters are all presented in detail in Supplementary Materials.

We firstly demonstrate that the topological winding number can be measured by single-magnon quantum dynamics in a chain of four transmon qubits, provided the qubit chain is initially prepared in a single-magnon bulk state. This dynamic method for measuring topological winding number was originally proposed in a linear-optics system for studying discrete-time quantum walk Cardano et al. (2017). We choose to excite one of the middle qubits to the excited state $|e\rangle$ and leave the other qubits in the ground state $|g\rangle$, leading to an initial state of the system $|\psi(t=0)\rangle=|gegg\rangle$. After an evolution time $t$, the state of the system becomes $|\psi(t)\rangle=e^{-i\hat{H}t}|\psi(t=0)\rangle$. To reveal the relationship between this dynamics and the topological winding number, we introduce the chiral displacement (CD) operator $\hat{P}_{d}=\sum_{x=1}^{2}x(\hat{P}_{a_{x}}^{e}-\hat{P}_{b_{x}}^{e})$ with $\hat{P}_{id}^{e}=|e\rangle_{id}\langle e|$ ($id=a_{x},b_{x}$). In the long-time limit, the topological winding number $\nu$ can be extracted from the time-averaged CD, $\nu={\lim_{T\rightarrow\infty}}\frac{2}{T}\int_{0}^{T}dt\,\bar{P}_{d}(t)$, where $T$ is the evolution time and $\bar{P}_{d}(t)=\langle\psi(t)|\hat{P}_{d}|\psi(t)\rangle$ is the CD associated with the dynamics of the single-magnon state (see Supplementary Materials). As we can see, the topological winding number is two times the time-averaged CD, i.e., the oscillation center of the CD versus time. Experimentally, to measure the time-averaged CD we only need to track the time evolution of the excitation for each qubit.

In the experiment, as shown in Figs. 2a and 2b, we tune the qubit chain into two configurations with the qubit coupling dimerization $J_{1}>J_{2}$ and $J_{1}<J_{2}$, corresponding to the topologically trivial and nontrivial magnon insulator states, respectively. After preparing the initial state $|\psi(0)\rangle$, we measure the time evolution (with an interval of 1 ns) of the qubit excitation of the four qubits and show the experimental data in Figs. 2c and 2d. The measured excitation evolutions agree well with the theoretical predictions. Based on these time-resolved excitation data for each qubit, we directly derive the time evolution of CDs and plot them in Figs. 2e and 2f. Clearly the two curves oscillate around two different center values, qualitatively giving the signature of different topological winding numbers. The evolution time in our experiment is chosen as 1 $\mu$s, during which the experimentally measured time-averaged CDs are $0.015$ and $0.359$ for the topologically trivial and nontrivial cases, respectively. Both experiments agree very well with the theoretically expected values [math] and $0.378$, giving the experimentally measured topological winding numbers $v=0.030$ and $\nu=0.718$ for the two cases. The measured winding number for the topologically trivial case is quite close to the ideal value.

The reasons for the difference in the topologically nontrivial case between the measured winding number $\nu=0.718$ and the ideal value $\nu=1$ are that both the evolution time and the qubit chain we choose are not long enough and there is also inevitable system decoherence. Nevertheless, our experimental data within 1 $\mu$s agrees excellently with the theoretical expectation and demonstrates the validness of the method using single-magnon dynamics to measure topological winding number. The clear distinction between the measured nontrivial and trivial topological winding numbers thus can unambiguously distinguish the topologically nontrivial and trivial magnon insulator states.

The second hallmark for topological magnon insulator states is the existence of topological magnon edge states at the boundary. When the qubit chain is tuned into the topological magnon insulator state, according to bulk-edge correspondence Hasan and Kane (2010); Qi and Zhang (2011), topological magnon edge state will emerge at the edges of the qubit chain. The wavefunctions of the left and right magnon edge states can be analytically derived as $|\psi_{L}\rangle=\sum_{x}(-1)^{x}(J_{1}/J_{2})^{x}\sigma_{a_{x}}^{+}|gg\cdot\cdot gg\rangle$ and $|\psi_{R}\rangle=\sum_{x}(-1)^{N-x}(J_{1}/J_{2})^{N-x}\sigma_{b_{x}}^{+}|gg\cdot\cdot gg\rangle$, respectively (see Supplementary Materials). It turns out that the magnon in the left (right) edge state only occupies the $a$-type ($b$-type) qubit and is maximally distributed in the leftmost (rightmost) qubit. Such two features provide a mean to observe the topological magnon edge states. However, the coupling between the left and right magnon edge states is very large due to finite lattice size effect, we cannot unambiguously observe the left or right magnon edge state localization in a short qubit chain. This problem can be solved by tuning the qubit chain with an odd number of qubits, where the right topological magnon edge state has been artificially removed (see Supplementary Materials).

Now we show that the left topological magnon edge state can be clearly observed in a chain of five qubits where there is no right topological magnon edge state (see Supplementary Materials). As shown in Fig. 3a, we can tune the qubit couplings in a chain of five qubits to make the system topologically trivial and nontrivial, respectively. Initially, the leftmost $a$-type qubit is excited and a single magnon has been put on the leftmost qubit with the initial system state $|\psi(t=0)\rangle=|egggg\rangle$. Then, we let this magnon state evolve for certain time and measure the time evolution of the magnon density in the qubit chain. The results for the qubit chain being tuned into the topologically trivial state are shown in Fig. 3b. As expected, there is no magnon edge state localization and the wavepacket has a ballistic spread versus time, which is a typical feature of bulk Bloch state. The reason is that the initial magnon state in this case is a superposition of different bulk states, therefore, it evolves in the qubit chain via the bulk state wavepackets and does not support edge state localization.

In contrast, if the qubit chain is tuned into the topologically nontrival state that can support left magnon edge states, as shown in Fig. 3c, the measured magnon density is always maximal in the leftmost qubit. This is because the initial magnon state $|\psi(t=0)\rangle$ has a large overlap with the left magnon edge state $|\psi_{L}\rangle$. The magnon state thus mainly evolves in the qubit chain based on the edge state wavepacket and always maximally localizes in the leftmost qubit. Moreover, the magnon only populates the $a$-type qubits, also satisfying the feature of left topological magnon edge state as mentioned before. These two features prove the existence of left topological magnon edge state and clearly indicate that the system is topologically nontrivial. In Figs. 3d and 3e, we also find that the measured qubit excitation evolutions agree excellently with the theoretical predictions.

The third important topological aspect is the emergence of a topological defect state at the interface between topologically trivial and nontrivial regions Su et al. (1979). When the qubit chain is tuned with two different topological configurations, a topological interface separating the topologically trivial ($J_{1}>J_{2}$) and nontrivial ($J_{1}<J_{2}$) regions can be created, where a topological magnon defect state is trapped. As shown in Fig. 4a, we can create such a defect state at qubit $a_{2}$ in a chain of five qubits. The magnon in the topological defect state should only occupy $a$-type qubits and its density should be maximally distributed in qubit $a_{2}$ (see Supplementary Materials). Experimentally, we initially excite qubit $a_{2}$ and prepare the system in $|\psi(t=0)\rangle=|ggegg\rangle$. Such an initial state has a large overlap with the wavefunction of the topological magnon defect state. If the system has the topological defect state, the magnon will propagate in the qubit chain via the defect state wavepacket. In the experiment, after evolving $\left|\psi(t=0)\right\rangle$ for certain time, we measure the final magnon density distribution in the qubit chain. The experimental results are shown in Fig. 4b and indeed indicate that the magnon is maximally localized in the center qubit $a_{2}$ and only has populations in the $a$-type qubits, unambiguously demonstrating the existence of a topological magnon defect state. The time evolutions of qubit excitation for the five qubits are also shown in Fig. 4c, agreeing well with the theoretical expectations.

In conclusion, our experiment has demonstrated the potential of a tunable superconducting qubit chain as a versatile platform for exploring topology, including measuring topological invariants and observing topological edge and defect states. Since multiple-qubit excitations can be precisely prepared in this system, our study paves the way for further using a longer qubit chain to realize interacting symmetry-protected topological states Chen et al. (2013, 2012) and probe symmetry-protected ground state degeneracy de Léséleuc et al. (2018). Through periodically driving the qubit frequencies, non-equilibrium interacting symmetry-protected topological states also can be studied Potter et al. (2016). Besides, it is also quite interesting to study how the topological states in the qubit chain helps to accomplish topologically protected quantum information processing tasks Mei et al. (2018b); Nayak et al. (2008). In addition to superconducting qubits, our experiment can be generalized to other qubit systems and could attract broad interests in exploring symmetry-protected topological states with different quantum computing platforms.