1. Introduction

Cavity Quantum Electrodynamics is an important platform for quantum optics which focused on the interaction of matter and quantized electromagnetic field. In 1963, Jaynes and Cummings proposed the Jaynes-Cummings model to describe the interaction between a two-level atom and a single-mode cavity, which is the simplest but widely used model in quantum optics [1]. In 1970, G. S. Agarwal established the quantum master equation method suitable for dissipative systems [2]. In 1985, C. W. Gardiner and M. J. Collett developed the quantum Langevin equations to calculate the input-output relations of the system [3]. These works were based on the interaction of a single atom and cavity. Although atomic gas was used in the experiment, the experimental results were still consistent with the results of the single-atom model [4,5]. With the development of atomic cooling technology, atoms can be accurately located in the cavity [6]. And then people began to study the interaction of several separated atoms and the cavity. The atoms can interact with each other through the cavity field; therefore, there is no direct interaction among atoms when van der Waals interaction can be ignored [7,8]. With the development of superconducting circuit technology, a new type of artificial atom called SQUID appears. Especially, the direct coherent coupling between SQUID can be realized through the superconducting circuit [9,10], making it possible to study the influence of coherent atom chain on the input and output relationship of the cavity. The atom chain with direct adjacent coupling is called the coherent atom chain here.

Recently, topological photonics has become a hot spot, and a series of one-dimensional and two-dimensional topological optical structures have been proposed and realized [11–21]. The topological structures satisfy the bulk-edge correspondence, namely the nontrivial topological finite structure possesses topological protected edge state, which received widespread attention, and put forward a series of application scenarios, such as transport, and quantum information processing [11–21]. For example, Majorana zero-mode becomes a candidate for quantum information processing due to its robustness to local disturbance and disorder [11]. In 2013, M. C. Rechtsman et al. studied the photon transport in 2D optical topological structure and confirmed that the edge state can affect the photon transport characteristics [14]. The edge state of topological optical structure can also improve the interaction between light and matter. Our group recently studied the dynamic evolution of two atoms separated by a one-dimensional Su-Schrieffer-Heeger (SSH) coupling cavity array, and found that the edge mode of the SSH coupling cavity chain can switch the effective coupling between atoms from coherent coupling to complete decoupling [22]; When the two-level atom is located in 2D optical topological optical coupling cavity lattice, it was found that the atom can excite the edge mode, but the efficiency of this excitation depends on the coupling strength between the atom and the element cavity [23].

Here, we do not pay attention to the topological optical structure but focus on the topological coherent coupling atom chain and its interaction with cavity. In 2020, W. Nie et al studied the superradiation generated by the edge state of topological SQUID array in microwave cavity based on superconducting circuits [9,10]. Here we continue their research and discuss the transmission spectrum of cavity containing SSH-type coherent coupling atom chains. In particular, we should make clear the difference between atomic chain and atomic gas, that is, the role of the topological property of atomic chain, i.e., edge state, on the transmission spectrum.

The paper is arranged as follows. Section 2 introduces the theoretical models; Section 3 calculates and explains the transmission spectrum of a cavity containing several coherent atoms. Section 4 extends to SSH-type atom chain, and discusses the influence of topological phase of the atom chain on the transmission spectrum, and interprets the results analytically. Section 5 summarizes the work.

2. Theoretical models

Here we adopt the experimental platform in Ref. [9,29], as shown in Fig. 1(a). The SQUIDs is a kind of artificial atoms which are made of three-dimensional superconducting circuits. They are connected one by one through the LC resonators, and the frequency of the LC resonator is greatly detuned from the SQUID, thus the interaction between SQUIDs is realized without any real photons transferred through the LC resonator [9,10]. With the Schrieffer-Wolff transformation, the Hamiltonian of the system can be equivalent to the SQUID chain with the direct near-neighbor coupling [9,10]. In this paper, SQUID is called atom, SQUID chain is called atom chain, and microwave cavity in the experiment is called cavity for short.

 

Fig. 1. (a) 3D circuit QED schematic diagram [9]; (b) schematic diagram of SSH artificial atom chain-single-mode cavity coupling system with input-output ports.

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The theoretical model based on the above is shown in Fig. 1(b). 2N identical atoms (circles) with transition frequency ${\mathrm{\omega }_a}$ are placed in the cavity with resonant frequency ${\mathrm{\omega }_c}$. These atoms form atom chains through nearest-neighbor interactions (thick and thin lines). In order to form an Su-Schrieffer-Heeger (SSH) chain, these neighbor interactions is alternately modulated, namely ${t_\textrm{A}}$ and ${t_\textrm{B}}$. So there are N cells, and there are two atoms in each cell, a total of 2N atoms. ${t_\textrm{A}}$ is the intracellular coupling strength, while ${t_\textrm{B}}$ is the intercellular coupling strength.

All atoms interact with the cavity mode with the same coupling strength $\mathrm{\Omega }$. Besides, the cavity mode is coupled to the electromagnetic bath on the left and right sides through the cavity wall. The Hamiltonian ($\hbar $=1) of the system can be expressed as

In Eq. (1), the first row represents the Hamiltonian of the atom chain, in which the first term refers to free atoms, and the second and third terms refer to the near-neighbor interaction between atoms with the intracellular coupling coefficient ${\textrm{t}_\textrm{A}}$ and the intercellular coupling coefficient ${\textrm{t}_\textrm{B}}$, respectively. The first term of the second row is the Hamiltonian of the cavity mode, and the second term describes the interaction between the cavity mode and atoms with the coupling coefficient $\mathrm{\Omega }$. The third row represent in turn the Hamiltonian of the left and right electromagnetic bath and the interaction between the cavity modes and the electromagnetic baths with coupling coefficients ${g_k}$ and ${g_l}$.

Here we discuss the transmission spectrum of the cavity containing the coherently coupled atom chain under a weak incident field with frequency ${\omega _L}$, shown in Fig. 1(b). By introducing the atomic spontaneous decay rates $\gamma $ and the leak rate $\mathrm{\kappa }$ of the cavity, and taking the mean-field approximation (${s_i} = \langle S_i\rangle^ - $, $a = \langle a\rangle$), we get the simultaneous equations as

By solving the steady-state solution of the above equations, we obtain the expression of transmission coefficient

3. Transmission spectrum of the cavity containing several atoms

Atom can affect the cavity’s transmission spectrum. It is known that the empty cavity has a Lorentz-type transmission spectrum. If a resonant atom is placed in the cavity, Rabi splitting will occur in the transmission spectrum. The frequency difference between the two splitting transmission peaks is proportional to the coupling coefficient $\mathrm{\Omega }$ between the atom and the cavity. If the atomic gas composed of N identical atoms is placed in the cavity, Rabi splitting will still occur, and the frequency difference between the two transmission peaks is proportional to $\sqrt N \mathrm{\Omega }$. Here, we will explore what happens when 2N coherent coupling atoms are placed in the cavity.

In the following, we set ${\mathrm{\Omega }_0}$ as the unit. The atomic spontaneous decay rate $\mathrm{\gamma }$ and leak rate $\mathrm{\kappa }$ of the cavity are set as $\mathrm{\gamma } = 0.0001{\mathrm{\Omega }_0}$ and $\mathrm{\kappa } = 0.01{\mathrm{\Omega }_0}$.

3.1 Two-atoms in the cavity

We first consider two atoms with coupling coefficient ${\textrm{t}_\textrm{A}}$ in the cavity. Setting $\mathrm{\Omega } = {\mathrm{\Omega }_0}$, we plot the transmission spectrum as the function of ${\textrm{t}_\textrm{A}}$ in Fig. 2. When ${\textrm{t}_\textrm{A}} = 0$, two atoms are equivalent to a larger atom which has the equivalent coupling coefficient $\sqrt 2 {\mathrm{\Omega }_0}$ with cavity mode, so the transmission spectrum shows the typical Rabi splitting at ${\textrm{t}_\textrm{A}} = 0$. Gradually increasing ${\textrm{t}_\textrm{A}}$, the transmission spectrum gradually becomes an asymmetric double peak, and the blue-shifted peak ($\mathrm{\delta } > 0$) is weaker than that of the redshifted peak ($\mathrm{\delta } < 0$), while the center frequency of the redshifted peak gradually approaches $\mathrm{\delta } = 0$. When ${\textrm{t}_\textrm{A}} > 15{\mathrm{\Omega }_0}$, the blueshifted peaks almost disappeared, leaving only the redshifted peak. Continuing to increase ${\textrm{t}_\textrm{A}}$, the transmissivity T of red-shifted single-peak gradually increases to 1, and the central frequency of it tends to the resonance frequency of the cavity. This can be understood as follows. When ${\textrm{t}_\textrm{A}} \gg {\mathrm{\Omega }_0}$, two strongly coupled atoms form two dressed states whose eigenvalues greatly detuned with the cavity, therefore the system is equivalent to the empty cavity, resulting in a transmission peak at the cavity frequency.

 

Fig. 2. The transmissivity as function of $\mathrm{\delta }$ and ${\textrm{t}_\textrm{A}}$. The coupling coefficient between atom and cavity mode is $\mathrm{\Omega } = {\mathrm{\Omega }_0}$. Other parameters are N = 1, $\mathrm{\gamma } = 0.0001{\mathrm{\Omega }_0}$ and $\mathrm{\kappa } = 0.01{\mathrm{\Omega }_0}$.

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3.2 Six atoms in the cavity

We then consider the case of six atoms in the cavity, i.e., N = 3. Fixing ${t_A} = {\mathrm{\Omega }_0}$ and $\mathrm{\Omega } = {\mathrm{\Omega }_0}$, the transmission spectrum as the function of ${\textrm{t}_\textrm{B}}$ is shown in Fig. 3(a). When ${\textrm{t}_\textrm{B}} = 0$, it is just the case of two correlated atoms in the cavity, corresponding to the bimodal transmission spectrum at ${t_A} = {\mathrm{\Omega }_0}$ in Fig. 2. With the increase of ${\textrm{t}_\textrm{B}}$, the redshifted transmission peak ($\mathrm{\delta } < 0$) is basically unchanged, and the corresponding frequency is slightly biased to the resonance frequency of the cavity, while the blue-shifted transmission peak ($\mathrm{\delta } > 0$) gradually decreases and moves away from the resonance frequency. The red-shifted transmission peak shows an slight split near ${t_B} = 1.5{\mathrm{\Omega }_0}$, as shown in the red circle. When ${t_B} > 0.5{\mathrm{\Omega }_0}$, a new central transmission peak appears.

 

Fig. 3. (a) Transmission spectrum as function of the interatomic coupling coefficient ${\textrm{t}_\textrm{B}}$ by fixing ${\textrm{t}_\textrm{A}} = {\mathrm{\Omega }_0}$ and $\mathrm{\Omega } = {\mathrm{\Omega }_0}$; (b) transmission spectrum as function of the atom-cavity coupling coefficient $\mathrm{\Omega }$ by fixing ${\textrm{t}_\textrm{A}} = {\mathrm{\Omega }_0}$ and ${\textrm{t}_\textrm{B}} = 3{\mathrm{\Omega }_0}$. Common parameters are N = 3, $\mathrm{\gamma } = 0.0001{\mathrm{\Omega }_0}$ and $\mathrm{\kappa } = 0.01{\mathrm{\Omega }_0}$.

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Next we fixed ${\textrm{t}_\textrm{A}} = {\mathrm{\Omega }_0}$ and ${\textrm{t}_\textrm{B}} = 3{\mathrm{\Omega }_0}$ and plotted the transmission spectrum as function of the atom-cavity coupling coefficient $\mathrm{\Omega }$ in Fig. 3(b). When $\mathrm{\Omega }$ increases, the detuning of the blue-shifted peak with cavity and the detuning of the red-shifted peak with cavity increases linearly with Ω. Besides, the red-shifted peak shows a slight split near $\mathrm{\Omega } = 1.5{\mathrm{\Omega }_0}$, marked by the red circle.

3.3 Explain the transmission spectrum by dressed state theory

From Figs. 2 and 3, we find that the maximum number of transmission peaks is three in the cases of two atoms or six atoms in the cavity if the slight splitting can be ignored. In this section, we explain the transmission spectrum according to the dressed states theory.

The input and output problem discussed here belongs to the linear region. It can be converted into the full quantum theory by ignoring the leakage of the cavity and the atomic spontaneous decay rate. Therefore, we just consider the system including only the cavity mode and the atom chain, and calculate its eigenstates and eigenvalues. The Hilbert space with a single exciton is sufficient to explain the transmission problem. When 2N atoms are in the cavity of Fig. 1(b), the single excitation space just involves 2N + 1 basis vectors, i.e. $|{g_1},{g_2}, \ldots ,{g_{2N}},1\rangle$, and $S_i^ + |{g_1},{g_2}, \ldots ,{g_{2N}},0\rangle$ (i = 1,…..,2N). The first basis vector indicates that there is one photon in the cavity, but all atoms are in the ground state. The second one means that there is no photon in the cavity, and only the ith-atom is in the excited state. The eigenstates can be obtained by diagonalization, and are superposition of $|{g_1},{g_2}, \ldots ,{g_{2N}},1\rangle$, and $S_i^ + |{g_1},{g_2}, \ldots ,{g_{2N}},0\rangle$. The eigenstate can be excited if the frequency of the incident field agrees with it’s eigenvalue. Though there are 2N + 1 eigenstates, not all eigenstates correspond to transmission peaks. Only those eigenstates with non-zero mean photon numbers $\langle{a^ + }a\rangle$ correspond to transmission peaks according to Eq. (2f). The larger the average photon number $\langle{a^ + }a\rangle$, the higher the transmissivity.

We analyzed in detail the case of Fig. 3(a), in which six atoms are in the cavity with ${t_A} = {\mathrm{\Omega }_0}$ and $\mathrm{\Omega } = {\mathrm{\Omega }_0}$. Such a system has 7 eigenstates. The eigenvalue of these 7 dressed states as functions of ${t_B}$ is shown in Fig. 4(a). We sort the eigenstates according to the size of the eigenvalues. Different colors represent different eigenstates. The mean photon numbers $\langle{a^ + }a\rangle $ of corresponding eigenstates are shown in Fig. 4(b).

 

Fig. 4. (a) Eigenvalue and (b) mean photon number $\langle{a^ + }a\rangle $ of dressstates of the system including six coupled atom in the cavity. Common parameters are N = 3, ${\textrm{t}_\textrm{A}} = {\mathrm{\Omega }_0}$ and $\mathrm{\Omega } = {\mathrm{\Omega }_0}.$

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Next we explain the transmission peaks in Fig. 3 with the help of Fig. 4. The frequency of the transmission peaks in Fig. 3(a) actually corresponds to the eigenvalues of some eigenstates in Fig. 4(a).

From Fig. 4 (b), the mean photon numbers $\langle{a^ + }a\rangle $ of the 4^th and 6^th eigenstates are 0, so the 4^th and 6^th eigenstates do not contribute to the transmission peak in Fig. 3(a).

From Fig. 4(a), the eigenvalue of the 1^st and 2^nd eigenstates crosses near ${\textrm{t}_\textrm{B}} = 1.1{\mathrm{\Omega }_0}$. We check $\langle{a^ + }a\rangle$ of 1^st and 2^nd eigenstates in Fig. 4 (b), and found the mean photon number $\langle{a^ + }a\rangle$ of 1^st and 2^nd eigenstates exchanges abruptly near ${t_B} = 1.1{\mathrm{\Omega }_0}$.This explains that the red-shifted peak is slightly splitting near ${t_B} = 1.1{\mathrm{\Omega }_0}$, shown in the circle in Fig. 3(a). Furthermore, with the increase of ${t_B}$, $\langle{a^ + }a\rangle$ shifts from 1^st to 2^nd and finally to 3^rd eigenstate, meanwhile the eigenvalues of these three eigenstates with significant $\langle{a^ + }a\rangle$ are almost equal, which constitutes the redshift peak in Fig. 3(a).

$\langle{a^ + }a\rangle$ of 5^th eigenstate increases with ${t_B}$, while the eigenvalue of the 5^th eigenstate is near 0. This explains the appearance of central transmission peak when ${t_B} > 0.5{\mathrm{\Omega }_0}$ in Fig. 3(a).

The blue-shifted peak in Fig. 3(a) can also be explained by the eigenvalue and average photon number of the 7th eigenstate.

Above analysis explains the transmission spectrum calculated by Eqs. (3) and (4). It shows that the number of transmission peaks is less than 2N + 1, and only those eigenstates with non-zero mean photon numbers contribute to the transmission peaks.

4. SSH-type atom chain in the cavity

Su-Schrieffer-Heeger (SSH) model is the most representative 1D topological model [24]. Recently, the optical correspondence of SSH model has become a research hotspot. Optical SSH models had been implemented in in different electromagnetic systems, such as microwave photonic crystals [16,25], coupled waveguide arrays [20], pure dielectric cavities [26], waveguide ring resonators [27], plasmonic superlattices [28], and so on. However, we are interested in the influence of SSH-type atomic chain on the cavity transmissivity.

Let's first introduce the properties of SSH-type atom chains. The first line in Eq.(1) is the Hamiltonian of SSH type atomic chain, that is

Such 2N atoms can be diagonalized with the basis vector $S_i^ + |{g_1},{g_2}, \ldots ,{g_{2N}}\rangle$ to obtain 2N eigenstates $|{\Psi _m}\rangle$ and the corresponding eigenvalue ${\omega _m}$ (m = 1,2,….,2N). It is known that the atom chain is topological trivial when ${t_A} > {t_B}$, while is topological non-trivial when ${t_A} < {t_B}$. As an example, we set N = 10, fix ${t_A} + {t_B}$=2${\Omega _0}$. We choose the ground state as zero energy point, and then plot the eigenvalue ${\omega _m} - {\omega _a}$ as the function of ${t_A}/{t_B}$ in Fig. 5.

 

Fig. 5. (a) The eigenvalue ${\mathrm{\omega }_\textrm{m}} - {\mathrm{\omega }_\textrm{a}}$ of the atom chain as function of ${\textrm{t}_\textrm{A}}/{\textrm{t}_\textrm{B}}$. N = 10, ${\textrm{t}_\textrm{A}} + {\textrm{t}_\textrm{B}}$=2${\mathrm{\Omega }_0}$. (b) The wavefunction C_m,i of two edge states as N = 10; (c) wavefunction C_i of two edge states as N = 50. The common parameters of (b) and (c) are ${\textrm{t}_\textrm{A}} = 0.2{\mathrm{\Omega }_0}$ and ${\textrm{t}_\textrm{B}} = 1.8{\mathrm{\Omega }_0}$.

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It is clear that when ${t_A}/{t_B} > 1$, these eigenstates fall in two bands, i.e. ${\omega _\textrm{m}} - {\omega _a} \in [ - ({{t_A} + {t_B}} ),$$- ({{t_A} - {t_B}} ) ]\cup [{({{t_A} - {t_B}} ),({{t_A} + {t_B}} )} ]$, they are all bulk states. When ${t_A}/{t_B} < 1$, the eigenstates can be divided into two categories. 2N-2 eigenstates refer to bulk states whose eigenvalues fall in ${\omega _\textrm{m}} - {\omega _a} \in [{ - ({{t_A} + {t_B}} ), - ({{t_B} - {t_A}} )} ]\cup [{({{t_B} - {t_A}} ),({{t_A} + {t_B}} )} ]$. The other 2 eigenstates are the edge states, whose eigenvalues are close to 0. The two edge states are the symbols of topological non-trivial phase.

The eigenstates have the general expression that

Here C_mi refers to the probability amplitude when i-th atom is excited. C_mi as function of i is just the wavefunction of m-th eigenstate. We plot the wavefunction C_i of edge states in Fig. 5(b). It is clear that the edge states connect the atoms on both sides in the form of symmetry and anti-symmetry, as shown at the black and red dots in Fig. 5(b). The edge states of the long SSH-type atom chain are different from that of the short SSH-type atom chain. For example, when the number of atoms is relatively large, i.e., N = 50, the edge states are either localized to the left atom or to the right atom, shown in Fig. 5(c).

Our work is to find out the influence of the edge state of the atomic chain on the cavity transmission spectrum. In other words, we want to make clear the role of topological phase of atom chain on it. Next, we use Eq. (3) and (4) to calculate the transmission spectrum of the cavity containing SSH atom chain at first. The common parameters are $\mathrm{\gamma } = 0.0001{\mathrm{\Omega }_0}$ and $\mathrm{\kappa } = 0.01{\mathrm{\Omega }_0}$.

4.1 Atom-cavity coupling coefficient $\Omega $ is of the same order as ${t_A} + {t_B}$

We start by calculating the transmission spectrum when the atom-cavity coupling coefficient $\mathrm{\Omega }$ has the same order of magnitude as the coupling coefficients ${t_A} + {t_B}$.

Setting N = 10, ${\textrm{t}_\textrm{A}}$=${\mathrm{\Omega }_0}$,and $\mathrm{\Omega } = {\mathrm{\Omega }_0}$, we plot the transmission spectrum as function of ${\textrm{t}_\textrm{B}}$ in Fig. 6(a). We can see that when ${\textrm{t}_\textrm{B}} < {t_A}$, the transmission spectrum has two peaks; when ${\textrm{t}_\textrm{B}}$>${\textrm{t}_\textrm{A}}$, a new central peak appears and the transmission spectrum has three peaks. In detail, with the increase of ${\textrm{t}_\textrm{B}}$, the transmissivity of the blue-shifted peak gradually decreases, while that of the central peak gradually enhances, and that of the red-shifted peak is nearly unchanged.

 

Fig. 6. (a) Transmission spectrum as function of the interatomic coupling coefficient ${t_B}$ by fixing ${t_A} = {\mathrm{\Omega }_0}$ and $\mathrm{\Omega } = {\mathrm{\Omega }_0}\; $; (b) transmission spectrum as function of the atom-cavity coupling coefficient $\mathrm{\Omega }$ by fixing ${t_A} = {\mathrm{\Omega }_0}$ and ${t_B} = 3{\mathrm{\Omega }_0}$. Common parameters are N = 10, $\mathrm{\gamma } = 0.0001{\mathrm{\Omega }_0}$ and $\mathrm{\kappa } = 0.01{\mathrm{\Omega }_0}$.

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Then we fixed ${\textrm{t}_\textrm{A}}$=${\mathrm{\Omega }_0}$ and ${\textrm{t}_\textrm{B}}$=3${\mathrm{\Omega }_0}$ and plotted the transmission spectrum as the function of $\mathrm{\Omega }$ in Fig. 6 (b). The spectrum is similar to the case of six atoms, i.e., Figure 3(b). Compared Fig. 6 and Fig. 3, we found there is nearly no difference between 6 atoms and 20 atoms except that the more slight splitings in redshifted transmission peak in Fig. 6.

From the previous analysis, we know that ${t_B} = {t_A}$ is the critical value, ${t_B}/{t_A} > 1$ corresponds to the existence of the edge state of the atom chain, while ${t_B}/{t_A} < 1$ means that there are no edge states. By looking at Fig. 6(a), we find that the conclusion related to this critical value is that ${t_B}/{t_A} < 1$ corresponds to two peaks in the transmission spectrum, while ${t_B}/{t_A} > 1$ corresponds to three peaks in the spectrum. From Figs. 6(a) and (b), we can draw a conclusion that even if there are 20 atoms in the cavity, there are only 3 peaks at most in the transmission spectrum.

4.2 Atom-cavity coupling coefficient $\Omega $ is much weaker than ${t_A} + {t_B}$

We check the transmission spectrum when $\mathrm{\Omega }$ is much smaller than ${t_A}$+${t_B}$. Since the wavefunction of edge states depends on the number of atoms, shown in Figs. 5(b) and (c), we selected two typical cases of N = 10 and N = 50, corresponding to 20 and 100 atoms respectively. Fixing ${t_A} + {t_B} = 2{\Omega _0}$ and $\mathrm{\Omega }$=0.01, we plot the transmitivity spectrum as function of ${t_A}$ for N = 10 and N = 50 in Figs. 7(a) and (b), respectively.

In Fig. 7(a), the case of 20 atoms, when ${\textrm{t}_\textrm{A}} < 0.8{\mathrm{\Omega }_0}$, the transmission spectrum shows two symmetric peaks, just like the Rabi splitting in the case of the cavity containing a resonant atom. When ${\textrm{t}_\textrm{A}} > 0.9{\mathrm{\Omega }_0}$, the transmission spectrum shows a central peak, just like the case of the empty cavity.

 

Fig. 7. (a) Transmission spectrum as function of ${t_\textrm{A}}$ when N = 10; (b) transmission spectrum as function of ${t_\textrm{A}}$ when N = 50. Common parameters are ${t_B} = 2{\varOmega _0} - {t_A}$, $\mathrm{\Omega }$=0.01${\mathrm{\Omega }_0}$, k = 0.01${\mathrm{\Omega }_0}$, and $\gamma $=0.0001${\mathrm{\Omega }_0}$.

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In Fig. 7(b), in the case of 100 atoms, with the increase of ${\textrm{t}_\textrm{A}}$, the transition from the Rabi splitting type double transmission peak to the empty’s central peak is very clear. The critical value is just ${t_\textrm{A}} = {\mathrm{\Omega }_0} = {t_B}$, which is also the critical value to distinguish topological non-trivial from topological trivial of atom chain.

It can be seen from Fig. 7 that when $\mathrm{\Omega }$ is much smaller than ${t_A}$+${t_B}$, there are only two peaks in the transmission spectrum at most. It is important to note that whether the transmission spectrum is Rabi-splitting-like double peaks or single peak is related to whether the atom chain is topologically non-trivial or trivial.

4.3 SSH atom chain is equivalent to three-level atom

In principle, we can use the method in Section 3.3 to diagonalize the system containing atoms and cavities, and point out that the transmission peaks correspond to dressed states containing non-zero mean photon numbers. However, this does not explain the phenomenon that even if the number of atoms continues to increase, there are only three transmission peaks at most.

According to Fig. 5(a), the SSH-type atom chain with N = 10 can be regarded as an atom with 20 transition levels. Thus, the interaction of the atom chain and the cavity can be transformed into the interaction between a multi-level atom and cavity. The Hamiltonian ${H_{atom\; chain}}$ in Eq. (5) can be rewritten as

And then the Hamiltonian of system including atom chain and cavity can be rewritten as

Here, $\xi _m^z = |{\Psi _m}\rangle\langle{\Psi _m}|$, $\xi _m^ +{=} |{\Psi _m}\rangle \langle{g}|$ and ${\xi _m} = |g\rangle\langle{\Psi _m}|$ are operators for equivalent multi-level atoms, and $|g\rangle$=$|{g_1},{g_2}, \ldots ,{g_{2N}}\rangle$ represents that all the atoms are in the ground state. We choose the ground state as the energy zero point. ${\mathrm{\Omega }_{{\Psi _m}}}$ is the effective coupling coefficient between the m^th eigenstate and the cavity, it can be obtained by the transformation in [10]. Following Hamiltonian Eq. (8), the transmission coefficient of the cavity containing atom chain can be rewritten as

Here ${\Delta _\textrm{m}} = {\omega _L} - {\omega _m}$ represents the detuning of the m-th eigenstate of the atom chain with the incident field. With ${\omega _m}$ and ${\mathrm{\Omega }_{{\Psi _m}}}$ in hand, we can still calculate the transmission spectrum.

For the atom chain with N = 10, we calculate the effective coupling coefficient ${\mathrm{\Omega }_{{\Psi _m}}}$ in the case of topological nontrivial and trivial in Figs. 8(a) and (b), respectively. In Fig. 8(a), the parameters are ${t_\textrm{A}}$=0.2${\mathrm{\Omega }_0}$ and ${t_\textrm{B}}$=$1.8{\mathrm{\Omega }_0}$, which corresponds to a topological nontrivial atom chain. The eigenstates of m = 10 and 11 are edge states of atom chain. From Fig. 8, it is clear ${\mathrm{\Omega }_{{\Psi _{10}}}} = 1.27\mathrm{\Omega } \approx \sqrt 2 \mathrm{\Omega }$ and ${\mathrm{\Omega }_{{\Psi _{11}}}} = 0$. Besides, ${\mathrm{\Omega }_{{\Psi _m}}}$ with odd m is zero. When m < 10, ${\mathrm{\Omega }_{{\Psi _m}}}$ can be ignored. When m > 12 ${\mathrm{\Omega }_{{\Psi _m}}}$ with even m increases with the increase of m. So ${\mathrm{\Omega }_{{\Psi _{20}}}} = 4.1\mathrm{\Omega } \approx \sqrt {20} \mathrm{\Omega }$ is the largest.

 

Fig. 8. The relative effective coupling coefficient ${\mathrm{\Omega }_{{\Psi _m}}}/\mathrm{\Omega }$ as function of m. (a) N = 10, ${\textrm{t}_\textrm{A}}$=0.2${\mathrm{\Omega }_0}$ and ${\textrm{t}_\textrm{B}}$=$1.8{\mathrm{\Omega }_0}$. The inset show the wavefunction of $|{\Psi _1}\rangle$ (black) and $|{\Psi _{20}}\rangle$ (red). (b) N = 10, ${\textrm{t}_\textrm{A}}$=1.8${\mathrm{\Omega }_0}$ and ${\textrm{t}_\textrm{B}}$=0$.2{\mathrm{\Omega }_0}$; (c) N = 50, ${\textrm{t}_\textrm{A}}$=0.2${\mathrm{\Omega }_0}$ and ${\textrm{t}_\textrm{B}}$=$1.8{\mathrm{\Omega }_0}$. m is sorted by the size of the eigenvalue ${\omega _m}$.

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Since each eigenstate $|{\Psi _m}\rangle$ is equivalent to an atom level with frequency ${\omega _\textrm{m}}$, and the corresponding equivalent dipole momentum d_m,eff is proportional to the sum of the probability amplitudes, i.e. ${d_{m,eff}} \propto \mathop \sum \nolimits_{i = 1}^{2N} {C_{mi}}$, we can explain the above properties of ${\mathrm{\Omega }_{{\Psi _m}}}$ through the wave function of eigenstate $|{\Psi _m}\rangle$ . For the non-trivial atom chain with N = 10, $|{\Psi _{10}}\rangle$ is the symmetric edge state, whose wavefunction is shown at the black dots in Fig. 5(b). Since the probability amplitude has the same maximum value on the two outermost atoms, i.e. C₁= C₂₀=-0.67, such edge state is equivalent to two atoms with the same dipole, so we get ${\mathrm{\Omega }_{{\Psi _{10}}}} = 1.27\mathrm{\Omega }$. Meanwhile, $|{\Psi _{11}}\rangle$ is the anti-symmetric edge state, whose wavefunction is shown at the red dots in Fig. 5(b). Since the probability amplitude has the opposite maximum value on the two outermost atoms, i.e. C₁=-C₂₀=-0.67, such edge state is equivalent to two atoms with opposite dipoles which can be cancel to each other, so we get ${\mathrm{\Omega }_{{\Psi _{11}}}} = 0$. For the eigenstates with odd m, their wave functions are odd relative to the center point, shown at the black curve in the inset of Fig. 8(a). Therefore $\mathop \sum \nolimits_{i = 1}^{2N} {C_{mi}}$ as well as the equivalent dipole momentum d_m,eff are zero, and we get ${\mathrm{\Omega }_{{\Psi _m}}} = 0$. For the eigenstates with even m, although their wave function is an even function, the probability amplitude C_m,i still oscillates with the atomic position i, and the wavelength decreases with the increase of m. Therefore, $\mathop \sum \nolimits_{i = 1}^{2N} {C_{mi}}$ ${\approx} 0$ when m < 10. Only when m is large, especially when the wavelength is smaller than the atomic interval, all probability amplitudes are in phase, shown at the red curve in the inset of Fig. 8(a), we get ${\mathrm{\Omega }_{{\Psi _{20}}}} = 4.1\mathrm{\Omega }$.

In Fig. 8(b), the parameters are ${\textrm{t}_\textrm{A}}$=1.8${\mathrm{\Omega }_0}$ and ${\textrm{t}_\textrm{B}}$=$0.2{\mathrm{\Omega }_0}$, which corresponds to topological trivial atom chain, and there are no edge states. Except that there is no edge state, ${\mathrm{\Omega }_{{\Psi _m}}}$ has similar properties to these in Fig. 8(a).

Note that ${\mathrm{\Omega }_{{\Psi _m}}}/\mathrm{\Omega }$ is independent of the size of $\mathrm{\Omega }$.

Therefore, for the topological non-trivial atom chain consisting of 2N atoms, although it has 2N equivalent diodes, only two equivalent diodes interact with the cavity, one is the symmetric edge state of atom chain with ${\mathrm{\Omega }_{{\Psi _{edge}}}} \approx \sqrt 2 \mathrm{\Omega }$ and ${\omega _{edge}} \approx {\omega _a}$, the other is the high-energy bulk state of atom chain with ${\mathrm{\Omega }_{{\Psi _{bulk}}}} \approx \sqrt {2N} \mathrm{\Omega }$ and ${\omega _{bulk}} \approx {\omega _a} + {t_A} + {t_B}$.

For the topological trivial atom chain, there is only one equivalent diode interacting with the cavity, which is the high-energy bulk state of the atom chain with ${\omega _{bulk}} \approx {\omega _a} + {t_A} + {t_B}$.

It should be noticed that although several bulk states of atom chain have a large coupling coefficient ${\mathrm{\Omega }_{{\Psi _m}}}$, the eigenvalues ${\omega _m}$ of these bulk states are very close to ${\omega _{bulk}} \approx {\omega _a} + {t_A} + {t_B}$, shown in Fig. 5(a), so they can be integrated into one bulk dipole.

When there are many atoms, the above conclusion is still true. We set N = 50, ${\textrm{t}_\textrm{A}}$=0.2${\mathrm{\Omega }_0}$ and ${\textrm{t}_\textrm{B}}$=$1.8{\mathrm{\Omega }_0}$, and plot the effective coupling coefficient ${\mathrm{\Omega }_{{\Psi _m}}}$ in Fig. 8(c). It refers to topological nontrivial atom chain. The eigenstates of m = 50 and 51 are edge states. From Fig. 8(c), ${\mathrm{\Omega }_{{\Psi _{50}}}} = \,{\mathrm{\Omega }_{{\Psi _{51}}}} = 0.91\mathrm{\Omega } \approx \mathrm{\Omega }$, which reflects the wavefunction of edge states are localized in the left or right atom when N is large, shown in Fig. 5(c). Although both edge states can interact with the cavity mode, their eigenvalues are highly degenerate, they can still be regarded as one edge dipole with ${\mathrm{\Omega }_{{\Psi _{edge}}}} \approx \sqrt 2 \mathrm{\Omega }$.

Therefore, the whole system approximately contains only three entities. One is the cavity mode with ${\omega _c}$, the second is the bulk dipole with ${\omega _{bulk}} \approx {\omega _a} + {t_A} + {t_B}$,and the third is the symmetry edge dipole with ${\omega _{edge}} \approx {\omega _a}$. The other states have almost no interaction with photons, but contribute to the slight splitting of transmission peak, shown in the red circle in Fig. 3. Clearly, there is no interaction between bulk dipole and symmetry edge dipole, but bulk dipole and symmetry edge dipole can interact with cavity mode.

Under such three entities approximation, the accumulation in the denominator of Eq. (9) can be simplified to the accumulation of two items, one is the symmetric edge states, and the other is high energy bulk states. Then Eq. (9) can be approximated into

The first fraction in the denominator represents the contribution of the bulk dipole, in which $\sqrt {2N} \mathrm{\Omega }$ is the coupling strength of uninteracting 2N atoms gas with cavity. The second fraction represents the contribution of the symmetry edge dipole. Notice that the edge dipole only works for topological non-trivial atom chain, i.e. t_A< t_B, otherwise it does not work. So the peak in the transmission spectrum is the result of the superposition of single photon state, symmetry edge state, and high-energy bulk states.

To confirm our analysis, we repeat the result in Fig. 6(a) through Eq. (10). The approximated transmission spectrum is shown in Fig. 9. Figure 9 is almost the same as Fig. 6(a), except that some slight splitting in Fig. 6(a) disappear in Fig. 9.

 

Fig. 9. The approximated transmission spectrum as function of ${\textrm{t}_\textrm{B}}$ calculated through Eq. (10). Other parameters are N = 10, ${t_A} = {\mathrm{\Omega }_0}$ and $\mathrm{\Omega } = {\mathrm{\Omega }_0}$, $\mathrm{\gamma } = 0.0001{\mathrm{\Omega }_0}$ and $\mathrm{\kappa } = 0.01{\mathrm{\Omega }_0}$.

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Finally, we make a summary. For the interaction between atomic chain and cavity, the atom chain can be seen as the equivalent three-level atom with two dipoles. Therefore, three transmission peaks can appear at most. If only the symmetry edge dipole interacts with the cavity, it will lead to symmetrical Rabi splitting type double peaks in transmission spectrum. If only the bulk dipole interacts with the cavity, as it is greatly detuned with cavity, the transmission spectrum will have asymmetric double peaks. The coupling strength of bulk dipole ${\mathrm{\Omega }_{{\Psi _{bulk}}}} \approx \sqrt {2N} \mathrm{\Omega}$ increases with the atom number 2N, but the detuning ${\omega _{bulk}} - {\omega _a} \approx {t_A} + {t_B}$ is fixed. Therefore, when coupling $\sqrt {2N} \mathrm{\Omega }$ is far less than detuning ${\omega _{bulk}} - {\omega _a}$, that is $\mathrm{\Omega } \ll {t_A}\; + \; {t_B}$, the bulk dipole has no contribution to the transmission spectrum, shown in Fig. 7. According to the shape of the transmission spectrum.

We can get three criteria:

5. Conclusion

In this paper, we study transmission spectrum of the cavity containing SSH-type atom chain, and explain the spectrum through quantum theory in the low excitation limit. When the atom chain is topological non-trivial, we found that the whole system actually contains only three entities. One is the cavity mode with frequency ${\mathrm{\omega }_\textrm{c}}$, the second is the bulk dipole with ${\mathrm{\Omega }_{{\mathrm{\Psi }_{\textrm{bulk}}}}} \approx \sqrt {2\textrm{N}} \mathrm{\Omega \,and\,}{\mathrm{\omega }_{\textrm{bulk}}} \approx {\mathrm{\omega }_\textrm{a}} + {\textrm{t}_\textrm{A}} + {\textrm{t}_\textrm{B}}$ ($2N$ is the number of atoms), the third is the edge dipole with ${\mathrm{\Omega }_{{\Psi _{edge}}}} \approx \sqrt 2 \mathrm{\Omega \,and\,}{\omega _{edge}} = {\omega _a}$. This is the reason why the spectrum is much different from that of the cavity containing single atom or atom gas.

It is necessary to point out the difference between the high-energy bulk state of atom chain and the atom gas. Although their effective coupling strengths are almost the same, i.e. $\sqrt {2\textrm{N}} \mathrm{\Omega }$, the high-energy bulk state is greatly detuned with cavity, while the atomic gas is resonant with cavity, if ${\omega _a} = {\omega _c}$ is satisfied.

Besides, the atoms in the atomic gas are lack of correlation, while the atoms in the atom chain have strong coherence, so the cavity containing the atomic chain can be used to prepare the multiparty entangled state. As mentioned, when the atomic chain is topologically non-trivial, the transmission peak is the result of the superposition of single photon state, symmetry edge state, and high energy bulk state of atom chain. We can prepare Many-body entangled states by inputting CW laser with a certain frequency in this system. Because the edge state and high-energy bulk states are the collective state of atoms, our system is also conducive to the preparation of Schrodinger cat state. We will discuss this issue in subsequent articles.