1. Introduction

With the rapid development of micro- and nano- processing technologies, the waveguide quantum electrodynamics (QED), in which the atoms are coupled to a one-dimensional (1D) propagation field, has become one of the important physical platform for realizing quantum information processing [1–6] and quantum simulation [7–11]. The optical modes coupling to the atoms near band edges can be well understood by mapping to the Jaynes-Cummings model [12,13]. This leads to many interesting phenomena and applications such as the generation of long-distance entanglement [14,15], correlated photon scattering [16,17], simulation of topological states [18,19].

Topological insulators, because of many interesting features such as robustness to local decoherence processes and potential applications in quantum information, has attracted a great of interests and attentions in quantum physics [20–23]. The Su-Schrieffer-Heeger (SSH) model originally used to describe the transport properties of conducting polyacetylene [24] has also received increasing attention in recent years [4,18,19,25–37]. As one of the simplest topological insulator models with its simple structure and abundant physical images, the SSH model has been implemented in several quantum systems including photonic crystals [29,30], spin-phonon crystals [18,19], cavity optomechanical system [31–34], circuit QED [3,4,35] , trapped ions [36], plasmonic system [37] et al. Based on these platforms, a lot of interesting phenomena and potential applications have been done for example, topological phase transitions [38–40], quantum state transfer [41–43], non-Hermitian [44] et al. Recently, M. D’Angelis et al. proposed a faster and higher-fidelity state transfer scheme by considering next-nearest neighbor interactions, which significantly impove of the transfer speed compared to conventional adiabatic processes [45]. Furthermore, Wanjura et al. proposed a method using dynamic matrix topological property to predict and to describe the directional-amplifying transmission in driven-dissipative cavity arrays [46,47], which establishes a connection between directional transmission and topological insulator theory and greatly enhance the links between quantum physical phenomena and topological physics.

In this paper, we propose a scheme to simulate the SSH model by designing the positions of the atoms in the coupled cavity arrays. After eliminated the photon modes, the long-range coherent and correlated dissipative interactions between atoms can be established. When the detuning between atoms and cavity fields lies in the band gap, the dissipative coupling approaches zero, so the dynamics of the system is completely dominated by the coherent interaction. Under this condition, we designed three atomic arrays with different geometries. By considering the positions of atoms in coupled cavity arrays, we can achieve topologically trivial and non-trivial phases of atomic arrays. Moreover, the topological phase transition can be induced by tuning the drive parameters when we introduce periodic atomic drives. Finally, we shows that when photon-induced next-nearest-neighbor interactions were taken into account, the degenerate bandgap states are broken, and a topological state transfer channel is established.

2. Effective coherent and dissipative coupling in atomic array

We consider $2N_a$ two-level atoms coupling to a one-dimensional waveguide as shown in Fig. ( 1). Each atom interacts with the cavity through the local Jaynes-Cummings Hamiltonian with coupling strength $g$. The Hamiltonian can be written as

 

Fig. 1. Schematic diagram of the $2N_{a}$ two-level atoms coupled into an optical waveguide. The coupling strength and distance between adjacent cavities are $J$ and $x_0$.

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Next, we consider the weak coupling or broadband limit $g/J\ll 1$. In this case, the photonic waveguide modes simply act as a collective reservoir of the atoms and can be eliminated by using the Born-Markov approximation, which can be guaranteed by the condition that the relaxation time of the environment is much faster than the evolution time scale of the states of the atomic array. The master equation for the reduced density operator of the atoms is as [48]

In Fig. (2), we plot the correlated decay rates $\Gamma _{i,j}$ and coherent interactions $U_{i,j}$ versus $\delta$. The distance between atoms $|x_{i}-x_{j}|=0, x_0, 2x_0, 3x_0$ corresponds to the blue ($U_0, \Gamma _0$), red ($U_1, \Gamma _1$), purple ($U_2, \Gamma _2$), and cyan ($U_3, \Gamma _3$) curves, respectively. We take $x_0=1$ in the numerical calculation. The presence of dissipative coupling is harmful for the topological study of the system, so it is necessary to suppress it. As for the atomic detuning range in the $-2J<\delta <2J$, the dissipative couplings cannot be zero simultaneously for different distance between atoms as shown in Fig. 2(b). When the atomic frequency lies in the bandgap region ($\delta >2J$ or $\delta < -2J$), it can be seen that the coherent and dissipative couplings exhibit the same asymptotic behavior and converge to zero with increasing or decreasing $\delta$ for different distance between atoms $|x_{i}-x_{j}|$. This asymptotic behavior offers us the possibility to consider only several-times coupling in terms of $|x_{i}-x_{j}|$, which is obviously different from [38,50] where the strength of the coupling between the atoms caused by the waveguide exhibits periodic oscillations with distance between the atoms. For the fixed detuning ($\delta =2.2J$), the coherent interactions $|U_{i,j}|$ gradually decrease with the atomic distance increases. Under the parameters in Fig. 2(a), for enough value $\delta$ ($\delta >2.8J$), $|U_{i,j}|$ with $|x_{i}-x_{j}|=3x_0$ is smaller than that with $|x_{i}-x_{j}|=0, x_0, 2x_0$ so that it can be ignored. Most importantly, $U_{i,j}$ and $\Gamma _{i,j}$ for $|x_i-x_j|>3x_0$ are all near zero under the group of parameters in Fig. 2, because of the complex $K$ making $A_{ij}$ exponentially decreasing with the increasing of $|x_i-x_j|$. Thus $U_{i,j}$ can be approximated as follows

 

Fig. 2. Correlated decay rates $\Gamma _{i, j}$ and coherent dipole-dipole interactions $U_{i, j}$ versus $\delta$ for different distance between atoms $|x_{i}-x_{j}|=0, x_0, 2x_0, 3x_0$ corresponds to the blue ($U_0, \Gamma _0$), red ($U_1, \Gamma _1$), purple ($U_2, \Gamma _2$), and cyan ($U_3, \Gamma _3$), respectively. The other parameters are $\gamma _{a}=0.25J$ and $g=0.3J$.

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Therefore, the dynamics of the system is governed by the Hamiltonian Eq. (11). Because the effective coupling strengths $U_i$ ($i=0,1,2$) depends on the distance between the atoms. In next section, we will design the special position of the atoms in the coupled cavity arrays to simulate the SSH model with different topological phases. Moreover, the on-site potential induced by photons is the same for every atom. This leads to a drift in the energy spectrum and does not affect the quantization of the zak phase as detailed in Appendix A. Therefore, in the following, we neglected this item.

3. Simulated SSH model by designing atomic position

In the previous section, we simulate the SSH model by changing the $2N_{a}$ atoms arrangement inside the waveguide with three different atom arrays. As shown in Fig. (3), we designed atomic arrays with three different geometries. For the A-Type atomic arrangement structure shown in Fig. 3(a), the $2N_a$ atoms are adjacent to each other in the coupled cavity arrays and the intra-cell atomic distance $x_0$ equals to the extra-cell atomic distance. The Hamiltonian of this structure can be written as Eq. (1). For the B-Type atomic arrangement structure in Fig. 3(b), each cell contains two atom-cavity sites, and an empty cavity connects the cells. So the intra-cell atomic distance is $x_0$, and the extra-cell atomic distance is $2x_0$. The interaction Hamiltonian can be written as

 

Fig. 3. (a) (b) (c) Three different arrangements of atomic array in coupled cavity arrays. We label the odd number of atoms as $A_{n}$ and the even number of atoms as $B_{n}$, $n=1,2,\ldots,N_{a}$.

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No matter what kinds of structures, the Fourier transformation for the Hamiltonian Eq. (3) is exactly applicable, and the method of eliminating the waveguide is the same as that achieving Eq. (11). The only difference is the position of the atoms, which is exhibited by the effective coupling $U_j (j=1,2)$ under $\delta >2.8J$. For convenience, here we label the odd number of atoms as $A_{n}$ (gray ball) and the even number of atoms as $B_{n}$ ( blue ball) , $n=1,2,\ldots,N_{a}$. Thus, the same subscript is in one cell. For A-Type atomic array, since the atoms are equally spaced, according to Eq. (11), there will be next-nearest neighbor interactions between the atoms. In terms of $A_n$ and $B_n$, the Hamiltonian of A-Type atomic array can be rewritten as

It describes a one-dimensional chain of ordinary atoms, and $U_{1}$ as well as $U_{2}$ represents the nearest-neighbor and next-nearest-neighbor interactions between atoms. As for the atomic array with the structure of Fig. 3(b), due to the presence of empty cavity, the next-nearest neighbor interactions between atoms $U_3$ can be neglected according to the conclusions in Fig. (2). Meanwhile $U_{1}$ and $U_{2}$ are both nearest-neighbor interactions. Then, the Hamiltonian B-Type atomic array can be written as

 

Fig. 4. (a)(b)(c) Energy spectra corresponding to A-Type, B-Type and C-Type atomic array, respectively. (d) The probability distribution of the zero energy state at the lattice point corresponds to (c). The parameters are set to $\delta =2.9J$, $U_{1}=-0.017J$, $U_{2}=0.006J$ and $N_{a}=30$.

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From the above results, it can be found that the topologically trivial phase and non-trivial phase can be achieved by designing the position of the atoms in the coupled cavity array. However, it is noted that the coupling coefficient $U_{i,j}$ decreases with the increasing of atomic distance when $\delta >2.2J$. Therefore, it is impossible to adjust the other parameters of the system to change the intra-cell coupling strength from greater than the extra-cell coupling strength to less than the extra-cell coupling strength. This means that the system does not undergo the topological phase transition by changing in the parameters of the current system.

4. Topological phase transition induced by atomic periodic driving

As mentioned above, the topological phase can be simulated by designing different geometry structures of the atomic arrays, but the topological phase transition can not be achieved due to the fixed intra-cell and extra-cell coupling strength. Fortunately, if we introduce periodic modulation of the atoms, then the topological phase transition can be induced by adjusting the external field parameters.

For atomic periodic driving [36,52], it can be read as

Now, we perform a rotating transformation $U^{\dagger }HU-iU^{\dagger }\dot {U}$ with $U=\exp \{i\sum\limits _{j=1}^{2N_{a}}\frac {\eta }{2}$ $\sin {\omega _{d}t} \cos {(\Delta kx_{j}+\phi )}\sigma _{j}^{z}\}$. After that, the Hamiltonian reads

For A-Type atomic arrays, since the distance between atoms is homogeneous, so we can fix $\Delta k=\frac {\pi }{2x_0}$ to achieve the periodic couplings. According to the expression of Bessel function, we find that $\varXi _{j,i}$ has the following relation

Combining the above relationship and Eq. (14), then the Hamiltonian of the A-Type atomic arrays can be written in the following form

In Fig. (5), we plot the variation of coupling coefficient $v_{a}$, $w_{a}$, $T_{a}$, $T_{b}$ with the drive parameters $\phi$ and $\eta$. As shown in Fig. 5(a), (b), (c) and (d), we can see that the periods of all coupling coefficients are almost $\pi$, while the yellow area of the coupling coefficient become narrow with the increasing of $\eta$. We also plot the above coupling coefficients varying with the phase $\phi$ when $\eta =2$ in Fig. 5(e) and (f), it is clearly seen that the coupling coefficients $v_{a} (T_b)$ and $w_{a} (T_b)$ both exhibit a sinusoidal function with period $\pi$ but opposite phase. This means that we can periodically control the strength of the intra-cell and extra-cell coupling by adjusting the drive parameters. For B-Type and C-Type atomic arrays, the same conclusion is shown in Fig. 5(g) and (h). In Fig. 6(a), (e) and (h), we plot the energy spectrum as functions of phase $\phi$ for A-Type (a), B-Type (e) and C-Type atomic arrays (h). For A-Type atomic array, we can see that the energy spectrum is no longer symmetrical, due to the presence of next-nearest-neighbor interactions between atoms that break the chiral symmetry. Moreover, the system has a pair of bandgap states are separated from each other in $\phi \in [0.5\pi, \pi ] \cup [1.5\pi, 2\pi ]$, depicted by red lines in the Fig. 6(a). It corresponds to the $N_a$th , ($N_a+1$)th eigenvalue of the A-Type atomic array, respectively. We use $\Psi _L$ and $\Psi _R$ to denote these two bandgap states. We also draw the probability distribution of the bandgap states ($\Psi _L$, $\Psi _R$) and find that different driving parameters $\phi$ affects the existence of edge states. When we select the appropriate drive parameter $\phi =0.8\pi$, these two band gap states are respectively localized at the end of the atomic chain depicted in Fig. 6(b). The presence of edge states implies that the atomic array is in topologically nontrivial phase. Furthermore, when we increase the driving parameter $\phi =1.3\pi$, we see that this state is not localized at the end of the atomic array, which indicates that the system lies in a topologically trivial phase as shown in Fig. 6(c). Eventually when $\phi =1.7\pi$, this state is localized at the right end shown in Fig. 6(d). Thus, we found that by slowly and adiabatically adjusting the driving parameters, the topological phase transition can be induced in the atomic chains.

 

Fig. 5. The variation of coupling coefficient with the drive parameters $\phi$ and $\eta$. (a) $v_{a}$, (b) $w_{a}$, (c) $T_{a}$, (d) $T_{b}$. (e) (f) (g) (h) $v_{a}, w_{a}, T_{a}, T_{b}$, $v_{b}, w_{b}, v_{c}, w_{c}$ as a function of drive parameters $\phi$. The parameters are $\delta =2.9J$, $U_{1}=-0.017J$ and $U_{2}=0.006J$, $\eta =2$ for (e) (f), $\eta =2.7$ for (g) and $\eta =4.5$ for (h).

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Fig. 6. The energy spectrum for A-Type (a), B-Type (e) and C-Type atomic arrays (h) varying with $\phi$. The probability distribution of the bandgap states ($\Psi _L$, $\Psi _R$) when $\phi =0.8\pi$ (b) , $\phi =1.3\pi$ (c), $\phi =1.7\pi$ (d) and $\eta =2$ for A-Type atomic arrays. The winding number as functions of phase $\phi$ (e) and the probability distribution of the bandgap states $\Psi _L$ (f) when $\phi =\pi$ and $\eta =2.6$ for B-Type atomic array. (i) The winding number varying with $\phi$ and (j) the probability distribution of the bandgap states $\Psi _L$ when $\phi =\pi$ and $\eta =4.5$ for C-Type atomic array. The parameters are $\delta =2.9J$, $U_{1}=-0.017J$, $U_{2}=0.006J$, and $N_{a}=30$.

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For B-Type and C-Type atomic arrays, we also plot the energy spectrum and the winding number as functions of phase $\phi$, where the definition and the calculation of winding number is given in Appendix A. We can see the symmetry of the energy spectrum, while as $\phi$ changes, zero energy states appear in the bandgap shown in Fig. 6(e) and (h). By calculating the topological invariants of the bulk, such as the winding number $\nu$, we can predict the existence of edge states at the boundary, or vice versa, which is known as the bulk-edge correspondence [20]. In Fig. 6(f) and (i), it can be seen that for both types of atomic arrays, as $\phi$ varies, the winding number in the system changes between 0 and -1, corresponding to the transition from topologically trivial to non-trivial phase. In fact, for SSH chains with an even number of sites, there will be edge states at the end of the long chains in the topologically non-trivial phase. But for finite chains, the two edge modes are hybridized as shown in Fig. 6(g) and (j), where $\Psi _L$ corresponding to the $N_a$th eigenvalue of the B-Type and C-Type atomic array, respectively. Furthermore, for A-Type atomic arrays, $H_{a}^{\prime }$ no longer has a chiral symmetry due to the presence of atomic next-neighbour interactions, in which case the winding number can no longer be used as a topological invariant to describe the topological properties of the system. One can employ the Chen number [20] to indicate the topological properties.

Therefore, by introducing atomic periodic drives, we find that the topological phase transitions can be induced in the three structures of atomic arrays by slowly and adiabatically tuning the drive parameters. Furthermore, we note that the difference between the A-Type, B-Type and C-Type atomic arrays is the presence or absence of next-nearest neighbor interactions between atoms. In conjunction with Fig. 6(b), (g) and (j), it can be seen that the degenerate bandgap state is broken and localized at one end of the chain by considering the next-nearest-neighbor interaction of the atom.

5. Topological state transfer induced by next-nearest-neighbor interactions

In this section, we will show that the introduction of coupling in the next-nearest-neighbor leads to topological channels that allow quantum state transfer. We consider A-Type atomic array with $N_a=15$. For the SSH chain with an odd number of sites, there is only one edge state, which is positioned at the left or right edge of the chain, depending on the ratio of intra-cell to extra-cell. That is, if we can adiabatically change the coupling strength between inside and outside the cell from less than 1 to large than 1, then it is possible to pump the local state from one edge to the other thereby achieving quantum state transfer [53–56].

In Fig. 7(a) and (c), we plot the energy spectrum as functions of $\phi$ within the range $[0.75\pi, 1.75\pi ]$ for two defferent detuning $\delta =3.2J$ (a), $2.9J$ (c) corresponding to the absence and presence of next-nearest neighbor interactions, respectively. As shown in Fig. 7(a), the system has a pair of degenerated zero-energy states in the bandgap $\phi \in [0.75\pi, \pi ]\cup [1.5\pi, 1.75\pi ]$. The zero-energy states are distributed with equal probability at two ends of the atomic array depicted in Fig. 7(b). This is because the edge states are not separated but mixed with each other due to the size effect, so it is difficult to achieve the topological quantum state transfer if one wants to use this bandgap state.

 

Fig. 7. (a) and (c) are the energy spectrum with the variation of drive parameter $\phi$, corresponding to $\delta =3.2J$, $2.9J$, respectively. (b) and (d) are their distribution of the bandgap states with the variation of drive parameter $\phi$. The parameters are set to $\eta =1.5$, and $N_{a}=30$.

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However, if we appropriately reduce the detuning $\delta$ and consider the next-nearest-neighbor interactions, then the transfer state problem mentioned above will be well resolved. In Fig. 7(c), it can be seen that the previously degenerated bandgap states gradually separate and merge into the bulk state. The probability distribution of bandgap states is given in Fig. 7(b), it can be seen that at $\phi \in [0.75\pi, 1\pi ]$, the bandgap states are mainly distributed on the left side of the atomic array and gradually integrate into the bulk state with the slowly and adiabatically change of $\phi$. When $\phi \in [1.5\pi, 1.75\pi ]$, the bandgap states concentrate in the rightmost. From the above results, it can be seen that topological state transfer channels can be induced by considering the next-nearest-neighbor interactions, and thus state transfer between left-edge states and right-edge states can be achieved by slowly and adiabatically tuning the phase $\phi$. The state transfer channel mentioned above is similar to Rice-Mele pumping [25], where the topological pumping is achieved through periodically modulated on-site potentials.

6. Discussion and conclusion

In this section we discuss the experimental feasibility analysis. In recent years, a large number of systems can be designed with some degree of waveguide QED physics such as solid-state emitters in nanophotonic structures [2,7,57], circuit QED [3,9,58], cold atoms in state-dependent optical lattices [59] et al. As we all know, two counter-propagating lasers can form a standing wave field and the atoms can be trapped in the optical lattice, therefore the equally spaced atomic array can be trapped [60,61]. Meanwhile, each atom is coupled to an additional optical cavity via dipole interactions. Hence, for the A-Type atomic arrays, it will be implemented using the standing-wave mechanism. However, for B-Type and C-Type atomic arrays, controlling inhomogeneous arrays of atoms using optical lattice or optical waveguides will be a challenge. In addition, our proposed scheme maybe implemented in the superconducting circuit, where the superconducting quantum qubit acts as a two-level atom, the cavity and the waveguide can be realized by using coupled superconducting transmission line resonators [4,62–65]. Thus, the position of the atoms and cavities is fixed, and the transition frequency of the qubit is modulated by two periodic driving fields [52]. Experimentally, the coupling of adjacent resonator $J$ can be achieved by coupling an auxiliary capacitor, and the coupling strengths of $2\pi \times 31$ MHz can be obtained, which also can be significantly improved by increasing the capacitance [65]. Similarly, the coupling strength $g$ between resonator and qubit can be increased by boosting the impedance of the resonator [62]. It is therefore possible to adjust the capacitance of the auxiliary capacitor, and the inductance of the oscillator to achieve $g\ll J$. Moreover, the strong coupling between atoms and cavities ($g>\gamma _a, \gamma _s$) has been achieved experimentally in superconducting circuit [64]. Additionally, in Eq. (8), we ignore the decoherence rate ($\gamma _s$) of the atoms, which requires atoms to have long lifetimes. State-of-the-art superconducting qubits have coherence times in excess of 100 $\mu s$ (decoherence rates below 10 kHz). Considering the value of $U_2=0.006 \times 2\pi \times 31$ MHz, the dynamical evolution time is 0.85 $\mu s$ which is much shorter than 100 $\mu s$. Therefore, it should be reasonable to ignore the atomic inherent losses.

In conclusion, we propose a simple and feasible scheme to simulate the SSH model in the waveguide QED system. The photon bound state occurs when the atomic transition frequency happens to be in the photon bandgap, which establishes the coherent and dissipative coupling between atoms. Then, the topologically trivial and non-trivial phases can be achieved by considering atomic arrays with different distance structures. Furthermore, by introducing atomic periodic drives the topological phase transitions can be induced in the system by slowly and adiabatically tuning the drive parameters. Finally, for an even number of chains, we find that the introduction of next-nearest-neighbor interactions breaks the degenerated zero-energy state and establishes the topological state transport channel.

A. Appendix A: The bulk Hamiltonian and the topological invariants

To investigate the topological properties of the atomic arrays such as the winding number, topological phases and the existence of edge states, as in [18,25], we consider $N_a \to \infty$ and write the Hamiltonian Eq. (14) in momentum space. For A-Type atomic array, the bulk Hamiltonian as $H_{a}=\sum\limits _{k}\Psi _{k}^{\dagger }H_{a}(k)\Psi _{k}$, where $\Psi _{k}=(A_{k},B_{k})$ [66]. Introducing the vector of Pauli matrices $\vec{\sigma}$ and the identity matrix $I$, we obtain

(25)$$H_{a}(k)=h_{I}(k)I+\vec{h}_a(k)\cdot \vec{\sigma},$$

with $h_{I}(k)=2U_{2}\cos {(kx_{0})}$, $\vec {h}_{a}(k)=(U_{1}+U_{1}\cos { (kx_{0})},U_{1}\sin {(kx_{0})})$. We can derive the eigenvalues of $H_{a}(k)$ as well as the eigenvectors as

(26)$$E_{a}(k,\pm )=2U_{2}\cos {kx_{0}}\pm \sqrt{2U_{1}^{2}\cos {(kx_{0})}},\quad |a({\pm} k)\rangle =\frac{1}{\sqrt{2}}(e^{{-}i\phi _{a}(k)},\pm 1)^{T},$$

where $\phi _{a}(k)=\arctan {\frac {\sin (kx_{0})}{1+1\cos (kx_{0})}}$ (no relation with $U_{1}$ and $U_{2}$ ). The same method also can be used for both B, C-Types atomic chains. In ($h_x, h_y$) space, the direction of the vector $\vec {h}(k)$ represents an eigenstate, and the magnitude of the vector will give its eigenvalue. Due to the periodicity of the Brillouin zone, the vector $\vec {h}(k)$ must form a closed loop. As shown in Fig. ( 8), for A-Type and B-Type atomic arrays, the curve $\vec {h}(k)$ does not enclose the origin, and $\vert \phi (k)\vert <\pi /2$ for all $k$. However, for C-Type atomic array, the vector $\vec {h}(k)$ rotates around the origin and the phase $\phi (k)$ can take any values. The two-dimensional vector $\vec {h}(k)$ defines a trajectory in ($h_x, h_y$) space that is linked to the appearance of topological invariants. To describe the topological properties of the system, we calculated the Zak phase [67], defined as

(27)$$Z=i\oint dk\langle k\vert\frac{d}{dk}\vert k\rangle=\frac{1}{2 }\oint dk\frac{d\phi}{dk},$$

where $\vert k\rangle$ is the eigenvector of the system in momentum space. Additionally, the winding number $\nu$ associated with the Zak phase can be found

(28)$$\nu=\frac{1}{2\pi }\oint dk\frac{d\phi}{dk}=\frac{\Delta\phi}{2\pi},$$

where $\Delta \phi$ is the change in $\phi (k)$ with $k$ varying throughout the Brillouin zone. Therefore the Zak phase is $\pi$ times the winding number of the curve $\vec {h}(k)$ around the origin. So,

(29)$$v=\left\{ \begin{array}{rl} 0, &{\rm A,\ B\ Type} \\ -1. & {\rm C\ Type} \end{array} \right.$$

where the minus sign refers to the clockwise rotation of the vector $\vec {h}(k)$ around the origin. Therefore, the winding number $\nu$ can be found for two different structures of the atomic arrays A, B-Type ($\nu =0$) and C-Type ($\nu =-1$), corresponding to the topological trivial phase and topological non-trivial phases, respectively. Moreover, we note that the first term of the Eq. (25) not breaks the chiral symmetry so that the energy spectrum is shifted but has no effect on the eigenstate Eq. (26) of the system, so the Zak phase is still quantized to multiples of $\pi$. We therefore ignore the on-site potential induced by photons in Eq. (14), Eq. (15) and Eq. (16).

 

Fig. 8. Trajectories of $\vec {h}(k)$ over the Brillouin zone in the ($h_x, h_y$) space for different atomic arrays (a) A-Type atomic array. (b) B-Type atomic array. (c) C-Type atomic array. The red arrow represents the vector $\vec {h}(k)$, whose direction represents an eigenstate and whose mode indicates its corresponding eigenvalue. The blue circles represents the trajectory of vector $\vec {h}(k)$ in $\vec {k}$ space. The parameters are set to $U_{1}=-0.017J$ and $U_{2}=0.006J$.

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Funding

Key Technologies Research and Development Program (2021YFE0193500); National Natural Science Foundation of China (12274053, 11874099).

Disclosures

The authors declare no conflict of interest.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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