Open Access
Add to BibSonomy
Abstract
Higher-order topological insulator (HOTI) occupies an important position in topological band theory due to its exotic bulk-edge correspondence. Recently, it has been predicted that external magnetic field can induce novel topological phases in 2D HOTIs. However, up to now the theoretical description is still incomplete and the experimental realization is still lacking. Here we proposed a superconducting quantum circuit simulator of 2D Su-Schriffer-Heeger lattice, which is one of the most celebrated HOTI models, and investigate consequently the influence of the continuously varying magnetic field. By using the parametric conversion coupling method, we can establish in principle the time- and site-resolved tunable hopping constants in the proposed architecture, thus providing an ideal platform for investigating the higher-order topological phase transitions induced by continuously varying magnetic field. Our numerical calculation further shows that the higher-order topology of the lattice, which manifests itself through the existence of the zero energy corner modes, exhibit exotic and rich dependence on the imposed magnetic field and the inhomogeneous hopping strength. To probe the proposed magnetic-field-induced topological phase transition, we study the response of the lattice to the corner site pumping in the steady state limit, with results implying that the predicted topological phase boundaries can be unambiguously identified by the measurement of the corner sites and their few neighbors. Requiring only current level of technology, our scheme can be readily tested in experiment and may pave an alternative way towards the future investigation of HOTIs under various mechanisms including magnetic field, disorder, and strong correlation.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Higher-order topological insulator (HOTI) has attracted extensive research attention in the past few years due to its exotic boundary states [1,2]. Generally speaking, a $d$-D $n$th-order HOTI has gapless $(d-n)$-D boundary states, which is not only determined by but also a representation of its higher-order band topology [3–5]. A typical example is the 2D Su-Schrieffer-Heeger (SSH) lattice shown in Fig. 1(a), which has four-site unit-cells coupled through homogeneous intra-cell hopping strength $\gamma$ and inhomogeneous inter-cell hopping strengths $\lambda _i$ [1,6–12]. This lattice can host 0D zero-energy corner modes (ZECMs) instead of 1D gapless edge modes under certain circumstances, thus belonging to 2D second-order HOTI. Recently, the influence of external magnetic field on 2D HOTI has been investigated, and a variety of novel topological phases have been predicted [1,13]. The motivation and the physics behind is that the imposed magnetic field can change the symmetry of the Hamiltonian [14]. It has been pointed out that the higher-order topology of the 2D SSH model depends on both the imposed magnetic field $\phi$ and the ratio $\lambda _i/\gamma$ (Fig. 1(a)). For $\lambda _i \gt \gamma$ and $\phi =\pi$ [1,15], this model enters topological non-trivial region characterized by the appearance of ZECMs [16]. However, theoretical research up to now has focused mainly on several discrete values of the imposed magnetic field, e.g., $0$-flux and $\pi$-flux in Ref. [1] and $2\pi /3$-flux and $2\pi /10$-flux in Ref. [13]. The change of higher-order topology induced by continuously varying magnetic field has been considered only in very recent papers [17–19], and the description of the role of magnetic field in HOTI is still far from detailed and complete. Therefore, building a corresponding analog quantum simulator [20] and investigating the behavior of the boundary states [21,22] can offer us insight and inspiration for further research. Meanwhile, despite the experimental progress of realizing HOTIs in various metamaterials [17,23–30], the proposed magnetic-field-induced topological phase transition has not been experimentally demonstrated yet, partially due to several experimental challenges. For instance, artificial gauge field for photons has been demonstrated in coupled ring resonators [31] and coupled waveguides [32]. However, the realized magnetic field is fabrication-determined and its tunability is to some extent limited. From this point of view, a flexible quantum simulator which can provide simultaneously the tunable inhomogeneous hopping strength and tunable synthetic gauge field with time- and site-resolved tunability is highly desirable.
Fig. 1. (a) Sketch of the 2D SSH lattice composed of unit-cells with four sites labeled A—D (shadowed area). The lattice sites are coupled through uniform intra-cell coupling strength $\gamma$ and inhomogeneous inter-cell coupling strengths $\lambda _i$. In each transverse rectangle loops (i.e. the irreducible loops in the big transverse red rectangle) a uniform synthetic magnetic flux $\phi$ is penetrated, while in other irreducible rectangle loops the penetrated synthetic magnetic flux vanish. (b) SQC simulator of the 2D SSH model. The lattice is built by TLRs grounded at their common ends (big dots) by coupling SQUIDs (crossed squares). The colors of the TLRs label their different eigenfrequencies. In the loop of each SQUID, an external time-dependent magnetic flux bias is added to induce the tunable effective coupling between neighboring TLRs.
On the other hand, superconducting quantum circuit (SQC) has been regarded as a promising platform of quantum simulation [33–36]. The lattice sites in this setup are constructed by superconducting transmissionline resonators (TLRs) and superconducting qubits [37], and electrons in conventional materials are replaced by microwave photons. Compared with other physical systems, SQC takes the advantage of flexible engineering of a large scaled lattice, which stems from the tunability and scalability in the circuit design and control [38,39]. With advances in theory and technology, controllable coupling between SQC elements has been experimentally demonstrated [40,41], leading to the on-demand synthesization of continuously tunable artificial gauge field [42] and many-body localization [43]. Moreover, the strong coupling between SQC elements [37,44] allows the introduction of Kerr [45] and Jaynes-Cummings-Hubbard type [46] nonlinearity which can be regarded as effective photon-photon interaction, and consequently the quantum simulation of strongly correlated photonic liquids [47–49].
In this manuscript, we investigate the SQC quantum simulation of higher-order topological phase transition in 2D SSH lattice induced by continuously varying magnetic field. In the first step, we propose a flexible SQC quantum simulator of 2D SSH lattice. Here the lattice sites are constructed by superconducting TLRs and coupled to their neighbors by grounding superconducting quantum interference devices (SQUIDs) (Fig. 1(b)). The photon hopping on this lattice is induced through the parametric modulation of the grounding SQUIDs [42,50–53]. The distinct merit of this scheme is that it can provide time- and site-resolved tunability of both the hopping strengths and the hopping phases between the adjacent sites. The tunable hopping strengths enable the establishment of tunable inhomogeneous hopping which is critical for the lattice to have non-trivial higher-order topology, and the tunable non-trivial hopping phases lead directly to the synthetic magnetic field for microwave photons with time- and site-resolved tunability. From this point of view, our architecture can therefore be regarded as an efficient platform of investigating the magnetic field effect on 2D HOTIs. Due to the flexibility and scalability offered by the SQC system, our proposal can be readily generalized to other HOTI lattice configurations [54,55], and can allow the further incorporation of non-Hermicity [56,57], disorder [58], and strong correlation [34,47–49,59], thus providing an alternative route of exploring HOTI in the future.
The tunability of the proposed simulator in turn stimulates us to calculate the energy spectrum of the 2D SSH lattice versus continuous external magnetic field and the ratio between the intra- and inter-cell hopping strengths. Our results show that the spectrum and the ZECMs (and thus the higher-order topology of the lattice) exhibit rich and complicated behavior. In particular, the HOTI phase diagram of the lattice can be generally divided into five parts (see detailed discussion in Section 3) based on the existence of the ZECMs. In certain phases regions, the non-trivial HOTI phase is robust against the imposed magnetic field, while in other regions the imposed magnetic field can lead to band gap closing and thus induce HOTI phase transition. Our estimation based on recently reported experimental data [42,51,60–62] further pinpoints that the parameters of all the predicted HOTI phase regions can in principle be achieved in the proposed SQC simulator.
As the higher-order band topology of the proposed model can be characterized by the ZECMs, we further investigate the discrimination of the predicted HOTI phases through the corner-site pumping and the consequent steady-state photon number (SSPN) distribution measurement. The essential physics is that the spatial and spectral localization of the ZECMs can lead to exotic spatially localized SSPN distribution, and the latter can be used as strong evidence of the existence of the ZECMs. Our numerical results imply that with current level of technology, one can clearly identify the existence of the ZECMs and consequently the boundaries of the predicted HOTI phase regions by only measuring few sites near the corner.
2. SQC lattice simulator of 2D SSH model
As shown in Fig. 1(b), the proposed SQC lattice simulator is built up by four kinds of TLRs differed by their eigenfrequencies and placed in an interlaced square pattern [52]. These four kinds of TLRs correspond to the A-D sublattice sites in Fig. 1(a), respectively. In addition, we ground these TLRs at their common ends by grounding SQUIDs with effective Josephson inductances much smaller than those of the TLRs [51]. The roles of the grounding SQUIDs are two-fold. Firstly, their small inductances impose shortcut boundary conditions for the TLRs. Due to the current dividing mechanism, a current flowing out from a particular TLR will prefer flowing to ground rather than to its neighbors [63]. Therefore, we can exploit the lowest $\lambda /2$ eigenmodes of the TLRs as the uncoupled localized Wannier modes of the lattice, and write the on-site part of the lattice Hamiltonian as
The second function of the grounding SQUIDs is to implement the following effective tunnel Hamiltonian
- 1. Let us consider a particular neighboring TLR pair $\left \langle (\mathbf {r}, \alpha ),\left (\mathbf {r}^{\prime }, \beta \right )\right \rangle$. Due to the very small inductances of their common grounding SQUID, the voltage across the SQUID is very small, and the grounding SQUID works effectively as a linear inductance modulated by its flux bias. As the currents of the two TLRs flow through the same grounding SQUID, an inductive current-current coupling$$\mathcal{H}_\mathrm{S}^{\mathbf{r}\alpha,\mathbf{r}^{\prime}\beta}= \mathcal{T}^{\mathrm{ac}}_{\mathbf{r}\alpha, \mathbf{r^{\prime}}\beta}(t)(\alpha_{\mathbf{r}}+\alpha_{\mathbf{r}}^{{\dagger}}) (\beta_{\mathbf{r^{\prime}}}+\beta_{\mathbf{r^{\prime}}}^{{\dagger}}),$$can be induced, with the time-dependent coupling constant $\mathcal {T}^{\mathrm {ac}}_{\mathbf {r}\alpha, \mathbf {r^{\prime }}\beta }(t)$ proportional to the Josephson coupling energy (and consequently controlled by the flux bias) of the grounding SQUID. We then a.c. modulate the grounding SQUID with frequency bridging the frequency gap between the two TLRs. With this modulation, a parametric frequency conversion process between the TLRs can be induced, described in the rotating frame of $\mathcal {H}_\mathrm {S}$ by$$\mathcal{H}_{\mathrm{T}}^{\mathbf{r} \alpha,\mathbf{r}^{\prime}\beta}=\mathcal{T}_{\mathbf{r},\alpha}^{\mathbf{r}^{\prime},\beta} \beta_{\mathbf{r}^{\prime}}^{{\dagger}} \alpha_{\mathbf{r}} e^{i \theta_{\mathbf{r}, \alpha}^{\mathbf{r}^{\prime}, \beta}}+\mathrm{H.C.},$$where the hopping amplitude $\mathcal {T}_{\mathbf {r},\alpha }^{\mathbf {r}^{\prime },\beta }$ is proportional to the amplitude of the modulating tone, and the hopping phase $\theta _{\mathbf {r}, \alpha }^{\mathbf {r}^{\prime }, \beta }$ is exactly the initial phase of the modulating tone. That is, for the $(\mathbf {r}, \alpha ) \Leftrightarrow \left (\mathbf {r}^{\prime }, \beta \right )$ hoping bond, both the amplitude and the phase of the hopping constant can be tuned by the corresponding external flux bias in a time-dependent way. With reported experimental data [42,51,60–62], the hopping strength $\mathcal {T}_{\mathbf {r},\alpha }^{\mathbf {r}^{\prime },\beta }$ can further be estimated as $\mathcal {T}_{\mathbf {r},\alpha }^{\mathbf {r}^{\prime },\beta } \in \left [ 10, 20 \right ] \mathrm { MHz}$. Moreover, previous discussions [52,53] have suggested that such parametric frequency conversion formalism is robust against fabrication errors and various types of $1/f$ noise in SQC [65], implying that the $\mathcal {H}_{\mathrm {T}}^{\mathbf {r} \alpha,\mathbf {r}^{\prime }\beta }$ term can be stably and precisely synthesized.
- 2. We then generalize the described parametric frequency conversion method to each of the hopping link on the lattice: We modulate each of the grounding SQUIDs on the lattice with pulses containing three tones $\Delta$, $2\Delta$, and $4\Delta$. A close inspection indicates that we can independently control every vertical hopping bond and every pair of horizontal hopping bonds by a modulating tone threaded in one of the grounding SQUIDs [52]. The summation of all these hopping terms thus leads to the establishment of the effective Hamiltonian in Eq. (2) with time- and site-resolved tunability. That is, for each hoping bonds shown in Fig. 1(a), both the amplitude and the phase of the hopping constant can be tuned by the corresponding external bias in a site- and time-dependent way. In particular, the site-dependent hopping strength meets the requirement of inhomogeneous hopping amplitudes, which is important for the 2D SSH lattice to enter the topological non-trivial region [3,66], while the controllable hopping phases pinpoint the creation of arbitrary synthetic magnetic field with Landau gauge for microwave photons.
We should emphasize that such time- and site-resolved tunability is the distinct advantage of the exploited parametric frequency conversion approach in the sense that the synthetic gauge field here can be tuned on demand, while in the experiments based on coupled ring resonators and waveguide arrays [17,32] the tunability of the synthetic gauge field there is to some extent limited (theoretically one can heat or inject carriers to fine tuning the synthetic magnetic field). Moreover, due to the flexibility of SQC system, the proposed architecture here can be generalized in a variety of manners. Firstly, by noticing the flexible wiring advantage of the SQC system [67], we can expect that other HOTI lattice configuration (e.g. kagome and honeycomb [54,55,68]) can also be established by using our method. Secondly, the parametric frequency conversion method allows the further combination with many other mechanisms, including on-site Hubbard interaction [34,48], disorder, and non-Hermicity [56]. In particular, the tunability of the proposed scheme can be exploited to controllably introduce the mechanism of disorder. The diagonal, off-diagonal, and random magnetic field type disorder [69] can be induced by adjusting the frequencies, amplitudes, and initial phases of the modulating pulses in the SQUIDs, respectively (and choosing accordingly a different rotating frame). Another expectation comes from the fact that the effective photon-photon interaction, i.e. the nonlinearity of microwave photons, can be incorporated by coupling the TLRs with superconducting qubits and exploiting mechanisms including electromagnetically induced transparency [45], Jaynes-Cummings-Hubbard nonlinearity [46], and nonlinear Josephson coupling [70,71]. Our third perspective comes from the fact that the decoherence in superconducting quantum circuit can now be suppressed and efficiently controlled [72]. This technique can be used to manipulate the gain and loss of the system, leading to the quantum simulation of non-Hermitian HOTI [73,74]. Therefore, our proposed architecture can be regarded as a versatile and flexible platform of simulating HOTIs under various mechanisms.
3. Higher-order topological phases versus varying magnetic field in 2D SSH lattice
In the rest of this manuscript, we consider the corner excitation physics on a SQC lattice consisting of $8 \times 8$ unit-cells (i.e. $16 \times 16$ sites). In the first situation, we consider the uniform inter-cell hopping and set $\lambda _i$ as unity, i. e. $\lambda _i=1$ for $i=1,2,3,4$. The intra-cell hopping strength $\gamma$ and the magnetic flux $\phi$ is set in the range $\gamma \in \left [0,1.2 \right ]$ and $\phi \in \left [0,2\pi \right ]$, respectively.
Currently the detailed-and-ultimate characterization of higher-order band topology in the presence of magnetic field is still under discussion [1,13–15,54,55]. We should notice that our model is different from that calculated in the supplementary information of Ref. [17] in the sense that the net flux in a physical unit-cell there is zero and the lattice translation symmetry is preserved. Meanwhile, in our model the nonzero net flux $\phi$ in a physical unit-cell manifestly breaks the lattice translation symmetry, and the consequent definition of polarization of quadrupole moment needs further consideration (especially for irrational-valued $\phi$). Therefore, motivated by the principle of bulk-edge correspondence [75], we follow an alternative route of investigation in this manuscript, that is, to study the boundary behavior of the lattice. The Hofstadter butterfly energy spectrum under open boundary condition is calculated and shown in Fig. 2 (the truncation is placed between the unit-cells). Compared with that under periodic boundary condition, the Hofstadter butterfly spectrum under open boundary condition has edge modes (blue lines) and ZECMs (green lines) emerged from the bulk modes (red lines). In what follows, we characterize the topology of the considered lattice by the behavior of the ZECMs. As shown in Fig. 2, the Hofstadter butterfly spectrum and the topological phase of the lattice versus $\gamma$ and $\phi$ can be generally divided into five situations (here we only describe the range $\phi \in \left [0,\pi \right ]$ because the spectrum exhibits mirror symmetry with respect to $\phi =\pi$):
Fig. 2. (a–h) Numerically simulated Hofstadter butterfly spectrum under open boundary condition for varying $\gamma$. The bulk, edge, and corner modes are labelled by the red, blue, and green lines, respectively. i HOTI phase diagram of the lattice. Here we characterize the topological trivial nodal point semimetal (NPSM)/non-trivial HOTI region by the absence/existence of the ZECMs, and label them by the blue/yellow colors, respectively. The phase diagram is mirror-symmetric with respect to $\phi =\pi$ (the red dashed line). The red solid lines denote the gap closing points $\gamma _{c1}=0.41$, $\gamma _{c2}=0.45$, $\gamma _{c3}=0.76$, and $\gamma _{c4}=0.90$ from bottom to top (see main text for details). The black line with red dots at $\phi =2\pi /3$ and $\phi =2\pi /10$ corresponds to $\gamma =0.5$ which is used for the numerical simulation in Section 4. (j–l) Zoom-in spectrum of the lattice near $E=0$. These three panels corresponds to rectangles in the panels c-e, respectively.
- 1. As shown in Fig. 2(a), The Hofstadter butterfly spectrum at $\gamma =0$ can be classified into bulk, edge, and corner state. The eigenvalues of the edge states are 28-fold degenerate in our $8\times 8$ unit-cells, with the degeneracy depending on the lattice size, while the zero eigenvalue of the corner states are four-fold degenerate, with the degeneracy irrelevant of the lattice size. The existence of the ZECMs pinpoints that the 2D SSH model now is a non-trivial HOTI. As $\gamma$ increases in the range $\gamma \in (0,0.41)$ (Fig. 2(b)), the degeneracy of the edge states is destroyed and the energy band of the edge states becomes broad. Meanwhile, the corner states still remain zero-energy and four-fold degenerate. The band gap at $E=0$ between the bulk modes and the ZECMs exists in all range of $\phi$ and decreases with increasing $\gamma$, implying that the HOTI phase now is robust against the imposed magnetic fields. Here the upper bound $\gamma _{c1}=0.41$ (and the lower and upper bounds in the other situations) is not an analytic result but obtained numerically.
- 2. As $\gamma$ passes the point $\gamma _{c1}=0.41$ and varies in the range $\gamma \in \left [ 0.41,0.45 \right ]$, the band gap and consequently the ZECMs always exist for $\phi \in [\pi /2,\pi ]$. However, the band gap is closed at points located in $\phi \in [0,\pi /2]$ (Fig. 2(c)). Here the gap-closing is not complete in the sense that the gap is not closed for all $\phi \in [0,\pi /2]$. The ZECMs in this range behaves much more complicated: the ZECMs still exist for those $\phi$ where the band gap is still open. Meanwhile, for those $\phi$ where the band gap is closed, the ZECMs may either disappear or even co-exist with the bulk modes.
- 3. After $\gamma$ goes across the point $\gamma _{c2}=0.45$ and moves into the range $\gamma \in [0.45,0.76]$, the band gap closes completely in the range $\phi \in \left [0, \pi /2 \right ]$ but still exist in the whole range of $\phi \in \left [\pi /2,\pi \right ]$, as shown in Fig. 2(d). The band width of the edge state now becomes larger, and approaches the zero-energy value in the range $\phi \in \left [0, \pi /2 \right ]$. Meanwhile, the ZECMs exist in the whole range of $\phi \in \left [\pi /2,\pi \right ]$ and disappear completely in the whole range of $\phi \in \left [0,\pi /2 \right ]$. It implies that the system with $0\lt \phi \lt \pi /2$ can not be continuously deformed into that with $\pi /2\lt \phi \lt \pi$ due to the gap-closing at the $\phi =\pi /2$ [13,14], and pinpoint a HOTI phase transition occurring at $\phi =\pi /2$.
As a summary of the above results, we further plot in Fig. 2(i) the HOTI phase diagram of the lattice, where we discriminate the trivial/non-trivial HOTI phases by the existence of the ZECMs. Notice that our result is consistent with the previous work which considered only several discrete values of $\phi$ [13].
Before proceeding, we offer several remarks on the ZECMs. The first is that, although there exist in-gap corner states, these might not be topologically protected. To verify the topological origin of the ZECMs, we investigate the robustness of the corner states against disorder. We calculate the spectrum and the eigenmodes of the lattice in the presence of on-site disorder, which is Gaussian distributed with mean-square root set as large as $0.15$. We further choose $\gamma =0.41$ and $\phi =\sqrt {2}\pi /2$ with which the in-gap corner modes exist. We perform numerical calculations with more than $1000$ realizations. We find out that in all these realizations the four corner modes still exist. The energy spectrum and the density distribution of the corner modes in a particular disorder realization is displayed in Fig. 3. Our calculation thus verifies the robustness of the corner modes against disorder and indicate its topological origin.
Fig. 3. (a) Energy spectrum of the lattice in the presence of an on-site disorder realization at $\gamma =0.41$ and $\phi =\sqrt {2}\pi /2$. The on-site disorder is Gaussian distributed with mean-square root $0.15$. The existence of the four in-gap corner modes can be discriminated (see the inset). (b) The corresponding density distribution of the corner modes. Here we plot the distribution of the four modes together because they are localized at the four individual corners and do not overlap with each other.
The potential symmetry that protects the ZECMs should also be addressed. The symmetry issue of 2D SSH lattice has already been addressed in Refs. [1,3,66], which state that the corner modes of the 2D SSH lattice is protected by the rotation symmetry $C_{4}$ of the hopping strengths on the lattice when the penetrated magnetic field is $\pi$. When the $C_{4}$ symmetry is broken by anisotropic hopping strength, the higher-order topology of the lattice is changed, manifested by the shrinking of the number of corner modes from four to two. Meanwhile, our calculation of the continuous magnetic field case indicates that, by merely tuning the magnitude of the penetrated flux, we can have corner modes existing in certain parameter regions and vanishing in other regions. Although the introduction of the continuous magnetic field breaks various symmetries, e.g. time reversal and discrete lattice translation, the continuous tuning does not alter the preserving or breaking of all the above-mentioned symmetries. From this point of view, we think that the symmetry issue of the continuously-varying magnetic field case still needs further investigation and should be our further research direction.
Our final remark is offered to the observed coexistence of the corner modes and the bulk modes in certain parameter regions. The coexistence of the corner modes and the bulk modes in 2D SSH lattice has already been discussed in Ref. [76,77] with the penetrated magnetic flux set as $\phi =\pi$ (and also in earlier Ref. [78] in the context of bound states in continuum). It is shown that the corner modes in this situation do not hybridize with the surrounding bulk states of the lattice. Moreover, the existence/absence of the corner modes is coincidence with the nontrivial/trivial polarization vector of the lattice [66,79,80] (notice that the specific $\pi$ flux preserves the discrete translation symmetry and consequently the concepts of Bloch bands and their polarizations). These points thus indicate that the in-bulk corner states are of topological origin. Turning to our model, we find out the situation is much more complicated due to the introduced continuous magnetic field. For instance, the discrete translation symmetry is broken and consequently the concepts of Bloch bands and their polarizations need further discussion. However, we have found that the corner states still exist in certain parameter regions, and its robustness against disorder has been demonstrated in our previous numerical simulation, pinpointing the topological origin of the corner modes. Therefore, our understanding of the in-bulk corner states up to now is based to some extent on the principle of adiabatic continuation: the corner-bulk coexistence has already existed in specific magnetic flux value, and when the flux is tuned, this coexistence is still preserved in certain range of the parameters.
4. Measurement scheme of the proposed HOTI phase
In the next step, we consider the measurement scheme of the proposed topological phases based on the corner site excitation of the proposed lattice. The basic picture is that the proposed higher-order topology is characterized by the existence of the ZECMs and the ZECMs are spatially localized near the corners and spectrally located at $E = 0$. Then the intuitive way of detecting whether the ZECMs exist is that we pump the corner sites with pulse matching the eigenfrequency of the ZECM. We can capacitively couple external pump coil to the corner sites (Fig. 1(b)) and inject pulses matching the zero frequency in the rotating frame of $\mathcal {H}_\mathrm {S}$. If the ZECMs exist, the external pump will excite the ZECMs, and the response of the lattice will accordingly exhibit spatially localized patterns, otherwise the external pump will excite only the extended bulk modes or even only excite the lattice off-resonantly, and the response will extend over a wide range on the lattice. In this sense, we can determine the existence of the ZECMs and consequently the higher-order topology of the lattice by measuring only few sites near the corners of the lattice.
Based on the above picture, two candidate physical observables can be considered. The first method is to measure the steady state of the lattice: We pump the corner sites for sufficiently long time such that the lattice approaches its steady state, and then we measure the energy leaked out from the corner sites (Fig. 1(b)) and their neighbors in a given time interval, which are proportional to the steady state photon distribution in these sites. The information of the ZECMs can then be extracted from the energy leaked out, which is equivalent to the steady state photon number distribution. Actually this measurement scheme has already been exploited in an experiment in which the parametric conversion between two superconducting cavity modes has been achieved [81]. The second method is that we inject a pulse with frequency detuning $\Omega _{\mathrm {P}}$ on the corner site and measure the reflected photon currents from the corner sites and their neighbors. Based on the input-output formalism [82], the reflected signal can be expressed in terms of the Green function $\left [ \Omega _{\mathrm {P}}-\mathcal {H}_\mathrm {T} \right ]^{-1}$ of the lattice. Therefore we can extract the spatial and spectral distribution of the ZECMs from the reflected signals. This scheme has also been used in recent circuit QED experiments [83]. In the following, we perform numerical simulations following the first scheme: The coherent monochromatic pumping can be described by
Suppose $(\mathbf {r}_{\mathrm {c}},\beta )$ is a corner site, we can measure the SSPN $n_{\mathbf {r}_\mathrm {c},\beta } = \left \langle \beta _{r_{c}}^{\dagger } \beta _{r_{c}}\right \rangle$ at exactly this site, and the summation of SSPN at $(\mathbf {r}_{\mathrm {c}},\beta )$ and its few neighbors, which we denote as $N_{\mathbf {r}_\mathrm {c},\beta }$. Then the concentration factor $R_{\mathbf {r}_\mathrm {c},\beta }= n_{\mathbf {r}_\mathrm {c},\beta }/N_{\mathbf {r}_\mathrm {c},\beta }$ serves as a good index of the existence of the ZECMs. A significant high $R_{\mathbf {r}_\mathrm {c},\beta }$ can be obtained only if the following three requirements are met simultaneously: 1. the ZECMs do exist, i.e. the lattice is in its topological non-trivial phase. 2. The site $(\mathbf {r}_{\mathrm {c}},\beta )$ is pumped, i.e. we pump a corner site (or a bunch of corner sites). 3. $\Omega _{\mathrm {P}}\approx 0$. Only in this situation can the ZECMs be effectively excited, and the injected photons will prefer locating on the corner site $(\mathbf {r}_\mathrm {c},\beta )$. Otherwise, we will have very low $R_{\mathbf {r}_\mathrm {c},\beta }$: If the ZECMs do not exist, we can only excite the spatially extended bulk or edge modes if $\Omega _{\mathrm {P}}$ is in the bulk or edge bands, leading to the dilution of the weight of SSPN on $(\mathbf {r}_\mathrm {c},\beta )$, or even can not effectively excite the lattice if $\Omega _{\mathrm {P}}$ is in the gap; if the ZECMs exist but we either do not pump the corner sites or pump the corner sites with $\Omega _{\mathrm {P}} \neq 0$, our pumping is not compatible with the ZECMs and thus can not effectively excite the ZECMs.
We perform our numerical simulation based on several recently reported experiments [84–86] and show our results in Fig. 4(a)–(c). Here we choose $\kappa =0.03$ and set $\gamma =0.5$, $\lambda _i=1$ following our previous analysis in Sec. 3. (see also Fig. 2(i)). To detect the ZECMs, we set $\Omega _{\mathrm {P}}=0$ and choose $\mathcal {P}^{\dagger }\mathbf {a}=A_{1,1}+B_{1,8}+C_{8,1}+D_{8,8}$ (here we simulate the four-site pumping for better visualization, that is, the pump is added on the four corner sites $(\mathbf {r}_{\mathrm {c}},\beta ) \in \{ [(1,1),A],[(1,8),B],[(8,1),C],[(8,8),D] \}$ . This choice does not affect our discussion and conclusion because the four single-site pumpings can be regarded as independent due to their distant spacing. Consequently, the definition of the concentration ratio is modified as $R= {\sum _{_{\mathbf {r}_\mathrm {c},\beta }}n_{\mathbf {r}_\mathrm {c},\beta }}/{\sum _{_{\mathbf {r}_\mathrm {c},\beta }}N_{\mathbf {r}_\mathrm {c},\beta }}$ ). We observe that the concentration ratio $R$ jumps dramatically at $\pi /2$ as shown in Fig. 4(a). This jump is a clear evidence of the predicted topological phase transition induced by varying $\phi$ at $\phi =\pi /2$. In addition, our numerical calculation implies that $R\gt 0.7$ can serve as for the existence of the ZECMs. To better visualize the influence of the magnetic field $\phi$, we plot the SSPN of the whole lattice with topological non-trivial $\phi =2\pi /3$ and topological trivial $\phi =2\pi /10$ in Fig. 4(b) and (c), respectively. The localization of the SSPN at $\phi =2\pi /3$ and its diffusion at $\phi =2\pi /10$ can be clearly identified.
Fig. 4. Numerically simulated SSPN distributions versus magnetic field $\phi$ and dissipation factor $\kappa$. (a) depicts the concentration factor $R$ at $\kappa =0.03$. Here the summation $\sum _{{\mathbf {r}_\mathrm {c},\beta }}N_{\mathbf {r}_\mathrm {c},\beta }$ is performed at the corner sites and their nearest six neighbors indicated by the dashed triangle in Fig. 1(a). The red vertical dashed lines denote the topological phase transition points $\phi =\pi /2$ and $\phi =3\pi /2$, and the black horizontal dotted line labels the empirical critical value $R= 0.7$. The SSPNs distribution of the whole lattice at $\phi = 2\pi /3$ and $\phi = 2\pi /10$ are sketched in (b) and (c), respectively. (d–f) and (g–I) show the same calculations at $\kappa =0.01$ and $\kappa =0.005$, respectively.
Here we offer a brief remark on the influence of the dissipation factor $\kappa$ on the localization of the photons. As the dissipation can also hinder the diffusion of the photons, we should discriminate whether the observed localization is caused by the appearance of ZECMs or by the dissipation of the lattice. Therefore, we consider the behavior of $R$ and the SSPN under different $\kappa$. If their behavior is basically unchanged, we can conclude that the localization of the SSPNs is indeed caused by the appearance of ZECMs. We then perform the same numerical simulation with $\kappa =0.01$ and $\kappa =0.005$, and show the corresponding results in Figs. 4(d)–(f) and 4(g)–(i), respectively. With decreasing $\kappa$, the obtained results are largely unchanged, indicating that the observed localization is indeed caused by the appearance of ZECMs. Moreover, the jump of $R$ at $\phi = \pi /2$ becomes increasingly sharp, and the SSPN at the topological trivial becomes more diffused. Therefore, smaller dissipation can help us to better identify the topological trivial/non-trivial regions. Meanwhile, as indicated in Fig. 4(a), a moderate, currently achievable $\kappa =0.03$ is already sufficient for the discrimination the phase boundary.
Our proposed architecture can also be used to verify the previously-studied anisotropy-induced HOTI predicted in Ref. [66], where the magnetic field $\phi$ equals $0$ or $\pi$. Here we set $\phi =\pi,\gamma =1$, and $\lambda _2=\lambda _3=3$. In this situation, the discrete translation symmetry and consequently the concept of Bloch bands is preserved. Therefore, the vector Zak phase can be exploited to characterize the higher-order topology of the considered lattice [80], the component of which has the form
On the other hand, in the Berry phase theory of polarization, the charge polarization of insulators is expressed as
The HOTI phase diagram of the lattice versus $\lambda _1$ and $\lambda _4$ is shown in Fig. 5(a). In particular, the yellow, green and blue regions label different polarization vectors [1] $\boldsymbol {P} = (0,0), \boldsymbol {P} = (0,1/2)$ (upper left) or $\boldsymbol {P} = (1/2,0)$ (lower right), and $\boldsymbol {P} = (1/2,1/2)$, respectively. These different polarization vectors in turn correspond to different numbers and locations of the ZECMs. The energy spectra of the lattice at several representative points are calculated and shown in Fig. 5(b)–(d): In the yellow region the lattice has four ZECMs locating at the four corners, while in the green region the lattice has two ZECMs, the location of which depending on $\lambda _1/\lambda _4$; However, the lattice has only extended bulk and edge modes but no isolated ZECMs in the blue region.
Fig. 5. (a) HOTI phase diagram of the 2D SSH lattice with $\phi =\pi$, $\gamma =1$ and $\lambda _2=\lambda _3=3$. The phases are discriminated by the number and location of the corner modes and labelled by different colors. In the blue, green, and yellow regions, the lattice has polarization vectors $\boldsymbol {P} = (0,0), \boldsymbol {P} = (0,1/2)$ (upper left) or $\boldsymbol {P} = (1/2,0)$ (lower right), and $\boldsymbol {P} = (1/2,1/2)$ and consequently zero, two, and four ZECMs, respectively. The spectra of the lattice at several representative points, labelled by triangle: $\lambda _1=\lambda _4=3$, star: $\lambda _1=3$, $\lambda _4=0.5$, hollowed star: $\lambda _1=0.5$, $\lambda _4=3$ and round: $\lambda _1=\lambda _4=0.5$, are shown in (b–d), respectively. The spectrum behavior around the zero energy are detailed in the insets. The corresponding SSPN distributions under $\kappa =0.03$, $\Omega _{\mathrm {P}}=0$, and $\mathcal {P}^{\dagger }\mathbf {a}=A_{1,1}+B_{1,8}+C_{8,1}+D_{8,8}$ are shown in (e–h), respectively.
We further numerically calculate the SSPN distribution of the lattice at these four representative points with $\kappa =0.03$, $\Omega _{\mathrm {P}}=0$, and $\mathcal {P}^{\dagger }\mathbf {a}=A_{1,1}+B_{1,8}+C_{8,1}+D_{8,8}$. In Fig. 5(e), we choose $\lambda _1=\lambda _4=3$. We find that the SSPN is localized at four corners, indicating that there exist four ZECMs locating at the four corners. This is consistent with the spectrum shown in Fig. 5(b). Meanwhile, in Fig. 5(f) with $\lambda _1=3$ and $\lambda _4=0.5$, the resulting SSPN is localized only at two adjacent corner sites along the horizontal direction at the bottom. The SSPN distribution implies there exists only two ZECMs, which is consistent with Fig. 5(f). In Fig. 5(g), with inversely chosen $\lambda _1=0.5$ and $\lambda _4=3$, the SSPN is localized at two adjacent corner sites along vertical direction. The comparison between Fig. 5(f) and 5(g) manifests their different polarization vectors. Finally, in Fig. 5(h) we choose $\lambda _1= \lambda _4=0.5$. Now the SSPN is no longer localized at the corner sites, implying that there is no ZECMs. These calculated SSPN distributions thus confirm the validity of our method, that is, we can extract the spatial and spectral information of the ZECMs by pumping the corner sites and detecting the corner sites and their few neighbors. Here we notice that our result is consistent with the previous theoretical work which states that the higher-order topological property is protected by the chiral symmetry and is not dependent on any spatial symmetry [66].
5. Conclusion and outlook
In conclusion, we have shown that it is not only possible, but also advantageous to implement and detect in SQC system the HOTI phase transition of 2D SSH lattice induced by continuous magnetic field. Meanwhile, it is our feeling that we are still in the beginning of this research direction. Due to the flexibility of the proposed SQC architecture, our further direction should be the extension to other HOTI lattice configurations, the incorporation of other mechanisms including strong correlation, disorder, and non-Hermicity, and the interplay of these mechanisms, which will bring us into the realm of rich but less-explored physics.
Funding
National Natural Science Foundation of China (11774114, 11874156, 11874160).
Acknowledgments
We thank Y. H. Wu for helpful discussion.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes, “Quantized electric multipole insulators,” Science 357(6346), 61–66 (2017). [CrossRef]
2. F. Schindler, A. M. Cook, M. G. Vergniory, Z. Wang, S. S. P. Parkin, B. A. Bernevig, and T. Neupert, “Higher-order topological insulators,” Sci. Adv. 4(6), 1 (2018). [CrossRef]
3. Z. Song, Z. Fang, and C. Fang, “(d−2)-dimensional edge states of rotation symmetry protected topological states,” Phys. Rev. Lett. 119(24), 246402 (2017). [CrossRef]
4. W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes, “Electric multipole moments, topological multipole moment pumping, and chiral hinge states in crystalline insulators,” Phys. Rev. B 96(24), 245115 (2017). [CrossRef]
5. T. Liu, J. J. He, and F. Nori, “Majorana corner states in a two-dimensional magnetic topological insulator on a high-temperature superconductor,” Phys. Rev. B 98(24), 245413 (2018). [CrossRef]
6. S. Franca, J. van den Brink, and I. C. Fulga, “An anomalous higher-order topological insulator,” Phys. Rev. B 98(20), 201114 (2018). [CrossRef]
7. F. Liu and K. Wakabayashi, “Novel topological phase with a zero Berry curvature,” Phys. Rev. Lett. 118(7), 076803 (2017). [CrossRef]
8. M. Kim and J. Rho, “Topological edge and corner states in a two-dimensional photonic Su-Schrieffer-Heeger lattice,” Nanophotonics 9(10), 3227–3234 (2020). [CrossRef]
9. Y. Che, C. Gneiting, T. Liu, and F. Nori, “Topological quantum phase transitions retrieved through unsupervised machine learning,” Phys. Rev. B 102(13), 134213 (2020). [CrossRef]
10. W. Nie, M. Antezza, Y.-x. Liu, and F. Nori, “Dissipative topological phase transition with strong system-environment coupling,” Phys. Rev. Lett. 127(25), 250402 (2021). [CrossRef]
11. X.-X. Li, B. Li, and P.-B. Li, “Simulation of topological phases with color center arrays in phononic crystals,” Phys. Rev. Res. 2(1), 013121 (2020). [CrossRef]
12. Y. Meng, G. Chen, and S. Jia, “Second-order topological insulator in a coinless discrete-time quantum walk,” Phys. Rev. A 102(1), 012203 (2020). [CrossRef]
13. Y. Otaki and T. Fukui, “Higher-order topological insulators in a magnetic field,” Phys. Rev. B 100(24), 245108 (2019). [CrossRef]
14. J. Herzog-Arbeitman, Z.-D. Song, N. Regnault, and B. A. Bernevig, “Hofstadter topology: Noncrystalline topological materials at high flux,” Phys. Rev. Lett. 125(23), 236804 (2020). [CrossRef]
15. M. Rodriguez-Vega, A. Kumar, and B. Seradjeh, “Higher-order Floquet topological phases with corner and bulk bound states,” Phys. Rev. B 100(8), 085138 (2019). [CrossRef]
16. E. Khalaf, W. A. Benalcazar, T. L. Hughes, and R. Queiroz, “Boundary-obstructed topological phases,” Phys. Rev. Res. 3(1), 013239 (2021). [CrossRef]
17. S. Mittal, V. V. Orre, G. Zhu, M. A. Gorlach, A. Poddubny, and M. Hafezi, “Photonic quadrupole topological phases,” Nat. Photonics 13(10), 692–696 (2019). [CrossRef]
18. Z.-W. Zuo, W. A. Benalcazar, Y. Liu, and C.-X. Liu, “Topological phases of the dimerized Hofstadter butterfly,” J. Phys. D: Appl. Phys. 54(41), 414004 (2021). [CrossRef]
19. K. Asaga and T. Fukui, “Boundary-obstructed topological phases of a massive Dirac fermion in a magnetic field,” Phys. Rev. B 102(15), 155102 (2020). [CrossRef]
20. I. M. Georgescu, S. Ashhab, and F. Nori, “Quantum simulation,” Rev. Mod. Phys. 86(1), 153–185 (2014). [CrossRef]
21. T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. 91(1), 015006 (2019). [CrossRef]
22. L. Lu, J. D. Joannopoulos, and M. Soljacic, “Topological photonics,” Nat. Photonics 8(11), 821–829 (2014). [CrossRef]
23. M. Ezawa, “Non-Hermitian higher-order topological states in nonreciprocal and reciprocal systems with their electric-circuit realization,” Phys. Rev. B 99(20), 201411 (2019). [CrossRef]
24. B.-Y. Xie, G.-X. Su, H.-F. Wang, H. Su, X.-P. Shen, P. Zhan, M.-H. Lu, Z.-L. Wang, and Y.-F. Chen, “Visualization of higher-order topological insulating phases in two-dimensional dielectric photonic crystals,” Phys. Rev. Lett. 122(23), 233903 (2019). [CrossRef]
25. X.-D. Chen, W.-M. Deng, F.-L. Shi, F.-L. Zhao, M. Chen, and J.-W. Dong, “Direct observation of corner states in second-order topological photonic crystal slabs,” Phys. Rev. Lett. 122(23), 233902 (2019). [CrossRef]
26. B.-Y. Xie, H.-F. Wang, H.-X. Wang, X.-Y. Zhu, J.-H. Jiang, M.-H. Lu, and Y.-F. Chen, “Second-order photonic topological insulator with corner states,” Phys. Rev. B 98(20), 205147 (2018). [CrossRef]
27. M. Serra-Garcia, V. Peri, R. Susstrunk, O. R. Bilal, T. Larsen, L. G. Villanueva, and S. D. Huber, “Observation of a phononic quadrupole topological insulator,” Nature 555(7696), 342–345 (2018). [CrossRef]
28. C. W. Peterson, W. A. Benalcazar, T. L. Hughes, and G. Bahl, “A quantized microwave quadrupole insulator with topologically protected corner states,” Nature 555(7696), 346–350 (2018). [CrossRef]
29. S. Imhof, C. Berger, F. Bayer, J. Brehm, L. W. Molenkamp, T. Kiessling, F. Schindler, C. H. Lee, M. Greiter, T. Neupert, and R. Thomale, “Topolectrical-circuit realization of topological corner modes,” Nat. Phys. 14(9), 925–929 (2018). [CrossRef]
30. M. Serra-Garcia, R. Süsstrunk, and S. D. Huber, “Observation of quadrupole transitions and edge mode topology in an LC circuit network,” Phys. Rev. B 99(2), 020304 (2019). [CrossRef]
31. M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photonics 7(12), 1001–1005 (2013). [CrossRef]
32. S. Mukherjee, M. Di Liberto, P. Öhberg, R. R. Thomson, and N. Goldman, “Experimental observation of Aharonov-Bohm cages in photonic lattices,” Phys. Rev. Lett. 121(7), 075502 (2018). [CrossRef]
33. I. Carusotto, A. A. Houck, A. J. Kollár, P. Roushan, D. I. Schuster, and J. Simon, “Photonic materials in circuit quantum electrodynamics,” Nat. Phys. 16(3), 268–279 (2020). [CrossRef]
34. S. Schmidt and J. Koch, “Circuit QED lattices: Towards quantum simulation with superconducting circuits,” Ann. Phys. 525(6), 395–412 (2013). [CrossRef]
35. X.-F. Shi, Y. Yu, J. Q. You, and F. Nori, “Topological quantum phase transition in the extended Kitaev spin model,” Phys. Rev. B 79(13), 134431 (2009). [CrossRef]
36. J. Q. You, Z. D. Wang, W. Zhang, and F. Nori, “Encoding a qubit with Majorana modes in superconducting circuits,” Sci. Rep. 4(1), 5535 (2014). [CrossRef]
37. A. Blais, A. L. Grimsmo, S. M. Girvin, and A. Wallraff, “Circuit quantum electrodynamics,” Rev. Mod. Phys. 93(2), 025005 (2021). [CrossRef]
38. J. M. Martinis, M. H. Devoret, and J. Clarke, “Quantum Josephson junction circuits and the dawn of artificial atoms,” Nat. Phys. 16(3), 234–237 (2020). [CrossRef]
39. W. Nie, Z. H. Peng, F. Nori, and Y.-x. Liu, “Topologically protected quantum coherence in a superatom,” Phys. Rev. Lett. 124(2), 023603 (2020). [CrossRef]
40. Y. Chen, C. Neill, P. Roushan, N. Leung, M. Fang, R. Barends, J. Kelly, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, A. Megrant, J. Y. Mutus, P. J. J. O’Malley, C. M. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, M. R. Geller, A. N. Cleland, and J. M. Martinis, “Qubit architecture with high coherence and fast tunable coupling,” Phys. Rev. Lett. 113(22), 220502 (2014). [CrossRef]
41. J. Stehlik, D. M. Zajac, D. L. Underwood, T. Phung, J. Blair, S. Carnevale, D. Klaus, G. A. Keefe, A. Carniol, M. Kumph, M. Steffen, and O. E. Dial, “Tunable coupling architecture for fixed-frequency transmon superconducting qubits,” Phys. Rev. Lett. 127(8), 080505 (2021). [CrossRef]
42. P. Roushan, C. Neill, A. Megrant, Y. Chen, R. Babbush, R. Barends, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, A. Fowler, E. Jeffrey, J. Kelly, E. Lucero, J. Mutus, P. J. J. O’Malley, M. Neeley, C. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. White, E. Kapit, H. Neven, and J. Martinis, “Chiral ground-state currents of interacting photons in a synthetic magnetic field,” Nat. Phys. 13(2), 146–151 (2017). [CrossRef]
43. P. Roushan, C. Neill, J. Tangpanitanon, V. M. Bastidas, A. Megrant, R. Barends, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, A. Fowler, B. Foxen, M. Giustina, E. Jeffrey, J. Kelly, E. Lucero, J. Mutus, M. Neeley, C. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. White, H. Neven, D. G. Angelakis, and J. Martinis, “Spectroscopic signatures of localization with interacting photons in superconducting qubits,” Science 358(6367), 1175–1179 (2017). [CrossRef]
44. X. Gu, A. F. Kockum, A. Miranowicz, Y.-X. Liu, and F. Nori, “Microwave photonics with superconducting quantum circuits,” Phys. Rep. 718-719, 1–102 (2017). [CrossRef]
45. M. J. Hartmann, F. G. S. L. Brand ao, and M. B. Plenio, “Strongly interacting polaritons in coupled arrays of cavities,” Nat. Phys. 2(12), 849–855 (2006). [CrossRef]
46. A. D. Greentree, C. Tahan, J. H. Cole, and L. C. L. Hollenberg, “Quantum phase transitions of light,” Nat. Phys. 2(12), 856–861 (2006). [CrossRef]
47. I. Carusotto, D. Gerace, H. E. Tureci, S. De Liberato, C. Ciuti, and A. Imamoǧlu, “Fermionized photons in an array of driven dissipative nonlinear cavities,” Phys. Rev. Lett. 103(3), 033601 (2009). [CrossRef]
48. I. Carusotto and C. Ciuti, “Quantum fluids of light,” Rev. Mod. Phys. 85(1), 299–366 (2013). [CrossRef]
49. R. Ma, B. Saxberg, C. Owens, N. Leung, Y. Lu, J. Simon, and D. I. Schuster, “A dissipatively stabilized Mott insulator of photons,” Nature 566(7742), 51–57 (2019). [CrossRef]
50. K. Fang, Z. Yu, and S. Fan, “Realizing effective magnetic field for photons by controlling the phase of dynamic modulation,” Nat. Photonics 6(11), 782–787 (2012). [CrossRef]
51. E. Zakka-Bajjani, F. Nguyen, M. Lee, L. R. Vale, R. W. Simmonds, and J. Aumentado, “Quantum superposition of a single microwave photon in two different ‘colour’ states,” Nat. Phys. 7(8), 599–603 (2011). [CrossRef]
52. Y.-P. Wang, W.-L. Yang, Y. Hu, Z.-Y. Xue, and Y. Wu, “Detecting topological phases of microwave photons in a circuit quantum electrodynamics lattice,” npj Quantum Inf 2(1), 16015 (2016). [CrossRef]
53. Z.-H. Yang, Y.-P. Wang, Z.-Y. Xue, W.-L. Yang, Y. Hu, J.-H. Gao, and Y. Wu, “Circuit quantum electrodynamics simulator of flat band physics in a Lieb lattice,” Phys. Rev. A 93(6), 062319 (2016). [CrossRef]
54. F. Liu, H.-Y. Deng, and K. Wakabayashi, “Helical topological edge states in a quadrupole phase,” Phys. Rev. Lett. 122(8), 086804 (2019). [CrossRef]
55. M. Ezawa, “Higher-order topological insulators and semimetals on the breathing kagome and pyrochlore lattices,” Phys. Rev. Lett. 120(2), 026801 (2018). [CrossRef]
56. E. J. Bergholtz, J. C. Budich, and F. K. Kunst, “Exceptional topology of non-Hermitian systems,” Rev. Mod. Phys. 93(1), 015005 (2021). [CrossRef]
57. C. Leefmans, A. Dutt, J. Williams, L. Yuan, M. Parto, F. Nori, S. Fan, and A. Marandi, “Topological dissipation in a time-multiplexed photonic resonator network,” Nat. Phys. 18(4), 442–449 (2022). [CrossRef]
58. D. L. Underwood, W. E. Shanks, J. Koch, and A. A. Houck, “Low-disorder microwave cavity lattices for quantum simulation with photons,” Phys. Rev. A 86(2), 023837 (2012). [CrossRef]
59. A. L. C. Hayward, A. M. Martin, and A. D. Greentree, “Fractional quantum Hall physics in Jaynes-Cummings-Hubbard lattices,” Phys. Rev. Lett. 108(22), 223602 (2012). [CrossRef]
60. F. Nguyen, E. Zakka-Bajjani, R. W. Simmonds, and J. Aumentado, “Quantum interference between two single photons of different microwave frequencies,” Phys. Rev. Lett. 108(16), 163602 (2012). [CrossRef]
61. J. S. C. Hung, J. H. Busnaina, C. W. S. Chang, A. M. Vadiraj, I. Nsanzineza, E. Solano, H. Alaeian, E. Rico, and C. M. Wilson, “Quantum simulation of the Bosonic Creutz ladder with a parametric cavity,” Phys. Rev. Lett. 127(10), 100503 (2021). [CrossRef]
62. C. M. Wilson, G. Johansson, A. Pourkabirian, M. Simoen, J. R. Johansson, T. Duty, F. Nori, and P. Delsing, “Observation of the dynamical Casimir effect in a superconducting circuit,” Nature 479(7373), 376–379 (2011). [CrossRef]
63. J. M. Martinis and M. R. Geller, “Fast adiabatic qubit gates using only ${\sigma }_{z}$ control,” Phys. Rev. A 90(2), 022307 (2014). [CrossRef]
64. N. Goldman, G. Juzeliūnas, P. Ohberg, and I. B. Spielman, “Light-induced gauge fields for ultracold atoms,” Rep. Prog. Phys. 77(12), 126401 (2014). [CrossRef]
65. E. Paladino, Y. M. Galperin, G. Falci, and B. L. Altshuler, “1/f noise: Implications for solid-state quantum information,” Rev. Mod. Phys. 86(2), 361–418 (2014). [CrossRef]
66. L. Li, M. Umer, and J. Gong, “Direct prediction of corner state configurations from edge winding numbers in two- and three-dimensional chiral-symmetric lattice systems,” Phys. Rev. B 98(20), 205422 (2018). [CrossRef]
67. A. J. Kollár, M. Fitzpatrick, and A. A. Houck, “Hyperbolic lattices in circuit quantum electrodynamics,” Nature 571(7763), 45–50 (2019). [CrossRef]
68. J. Koch, A. A. Houck, K. L. Hur, and S. M. Girvin, “Time-reversal-symmetry breaking in circuit-QED-based photon lattices,” Phys. Rev. A 82(4), 043811 (2010). [CrossRef]
69. C.-A. Li, S.-B. Zhang, J. C. Budich, and B. Trauzettel, “Random flux driven metal to higher-order topological insulator transition,” (2021).
70. J. Bourassa, F. Beaudoin, J. M. Gambetta, and A. Blais, “Josephson-junction-embedded transmission-line resonators: From Kerr medium to in-line transmon,” Phys. Rev. A 86(1), 013814 (2012). [CrossRef]
71. M. Leib, F. Deppe, A. Marx, R. Gross, and M. Hartmann, “Networks of nonlinear superconducting transmission line resonators,” New J. Phys. 14(7), 075024 (2012). [CrossRef]
72. M. Naghiloo, M. Abbasi, Y. N. Joglekar, and K. W. Murch, “Quantum state tomography across the exceptional point in a single dissipative qubit,” Nat. Phys. 15(12), 1232–1236 (2019). [CrossRef]
73. T. Liu, Y.-R. Zhang, Q. Ai, Z. Gong, K. Kawabata, M. Ueda, and F. Nori, “Second-order topological phases in non-Hermitian systems,” Phys. Rev. Lett. 122(7), 076801 (2019). [CrossRef]
74. X.-W. Luo and C. Zhang, “Higher-order topological corner states induced by gain and loss,” Phys. Rev. Lett. 123(7), 073601 (2019). [CrossRef]
75. A. B. Bernevig and T. L. Hughes, Topological Insulators and Topological Superconductor (Princeton University Press, Princeton and Oxford, 2013).
76. A. Cerjan, M. Jürgensen, W. A. Benalcazar, S. Mukherjee, and M. C. Rechtsman, “Observation of a higher-order topological bound state in the continuum,” Phys. Rev. Lett. 125(21), 213901 (2020). [CrossRef]
77. W. A. Benalcazar and A. Cerjan, “Bound states in the continuum of higher-order topological insulators,” Phys. Rev. B 101(16), 161116 (2020). [CrossRef]
78. C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljacic, “Bound states in the continuum,” Nat. Rev. Mater. 1(9), 16048 (2016). [CrossRef]
79. D. Vanderbilt, Berry phases in electronic structure theory: electric polarization, orbital magnetization and topological insulators (Cambridge University Press, 2018).
80. D. Obana, F. Liu, and K. Wakabayashi, “Topological edge states in the Su-Schrieffer-Heeger model,” Phys. Rev. B 100(7), 075437 (2019). [CrossRef]
81. A. J. Sirois, M. A. Castellanos-Beltran, M. P. DeFeo, L. Ranzani, F. Lecocq, R. W. Simmonds, J. D. Teufel, and J. Aumentado, “Coherent-state storage and retrieval between superconducting cavities using parametric frequency conversion,” Appl. Phys. Lett. 106(17), 172603 (2015). [CrossRef]
82. A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, “Introduction to quantum noise, measurement, and amplification,” Rev. Mod. Phys. 82(2), 1155–1208 (2010). [CrossRef]
83. N. M. Sundaresan, Y. Liu, D. Sadri, L. J. Szőcs, D. L. Underwood, M. Malekakhlagh, H. E. Türeci, and A. A. Houck, “Beyond strong coupling in a multimode cavity,” Phys. Rev. X 5(2), 021035 (2015). [CrossRef]
84. Z. Wang, Y. Wu, Z. Bao, Y. Li, C. Ma, H. Wang, Y. Song, H. Zhang, and L. Duan, “Experimental realization of a deterministic quantum router with superconducting quantum circuits,” Phys. Rev. Appl. 15(1), 014049 (2021). [CrossRef]
85. M. Mergenthaler, A. Nersisyan, A. Patterson, M. Esposito, A. Baumgartner, C. Schönenberger, G. A. D. Briggs, E. A. Laird, and P. J. Leek, “Circuit quantum electrodynamics with carbon-nanotube-based superconducting quantum circuits,” Phys. Rev. Appl. 15(6), 064050 (2021). [CrossRef]
86. X. Tan, Y. X. Zhao, Q. Liu, G. Xue, H.-F. Yu, Z. D. Wang, and Y. Yu, “Simulation and manipulation of tunable Weyl-semimetal bands using superconducting quantum circuits,” Phys. Rev. Lett. 122(1), 010501 (2019). [CrossRef]
87. J. Zak, “Berry’s phase for energy bands in solids,” Phys. Rev. Lett. 62(23), 2747–2750 (1989). [CrossRef]
