1. Introduction

The study of photon-atom interaction is the basis of quantum optics. Theoretical and experimental studies [1–6] have been substantially done on the interaction between local quantum emitters and photons propagating in waveguide. Although photons confined in a 1D waveguide are easily achieved currently [7–13], the direction interactions among them are weak, so nonlinear media are required, e.g., a two-level system (2LS) [14–17], two two-level atoms [18–20], a V-type three-level atom (3LA) [21], or a small atomic ensemble [22], in addition, a cavity containing a two-level atom [23,24], or a single-mode nonlinear cavity [25–28].

The ability to tailor the interaction between photons is one of the important requirements for quantum information processing at the few-photon level. The literatures [15,16] theoretically showed that a strong antibunching effect can be observed in the reflected photons. The strong correlation between photons in the waveguide is generated by the local interaction since the atom prevents multiple occupancy of photons. A single two-level atom (2LA) is highly saturated by a single photon; thus, it creates a strong optical nonlinearity for multiple incident photons. In addition, compared to a two-level system, a multilevel system provides more controllable parameters. So, it is interesting if we introduce a cascade three-level atom which is controlled by a cavity. We will discuss how to induce either an effective repulsion or attraction between two photons by the intracavity photon count.

The paper is organized as follows: In Sec. 2, we present the Hamiltonian of our system, and use the parity operator to divide it into the scatter-free subspace and the controllable subspace. Then, we show that the system with cascade configuration can be solved exactly in the two-photon case. The result contains rich physics, previously, people mostly focused on phenomena such as single photon’s quantum routing at the single photon level. Here, we show how to control the number of photon-photon bound states. These two-photon bound states can exhibit bunching or anti-bunching statistics. Moreover, the local spatial range of bound states are strongly dependent on the cavity field. In Sec. 3, our results show how to induce effective attractive or repulsive interactions for transmitted photons by adjusting the cavity. Additionally, the quantum switch can be achieved by only one photon in the cavity. Finally, in Sec. 4, we conclude with a brief summary of the results.

2. Hamiltonian and two-photons scattering eigenstate

The system under study is depicted in Fig. 1(a): the system consists of a 1D waveguide evanescently coupled to a three-level atom with a cascade configuration. The Hamiltonian of the system is given by $\left ( \hbar =1\right )$

The first term $H_{w}$, describing the free propagation of the photons, is given by

Here, we assume that the group velocity of the waveguide is 1, and $R^{\dagger }\left ( x\right ) \left [ L^{\dagger }\left ( x\right ) \right ]$ is a bosonic operator creating a right-going (left-going) photon in the waveguide at position $x$. The cavity is represented by the annihilation operators $a$ with eigenfrequencies $\omega _{a}$, its Hamiltonian is

The third term $H_{A}$ gave the three-level system’s Hamiltonian, atomic states $\left \vert 1\right \rangle$, $\left \vert 2\right \rangle$, $\left \vert 3\right \rangle$ and their corresponding frequency $\omega _{1},\omega _{2},\omega _{3}$ are representations of three-level system. We select the ground state energy as the zero energy surface, i.e., $\omega _{1}=0.$ Besides, $\sigma _{mn}=\left \vert m\right \rangle \left \langle n\right \vert \left ( m,n=1,2,3\right )$ represents the atomic energy-level population operators.

The transition from level $\left \vert 2\right \rangle$ to $\left \vert 3\right \rangle$ is driven by the cavity field with driving strength $\lambda$ and $\omega _{32}=\omega _{3}-\omega _{2}$ is far from the cutoff frequency of the waveguide. Moreover the frequency difference $\omega _{32}$ satisfies the constraint conditions, i.e., $\left \vert \omega _{32}-\omega _{a}\right \vert \ll \left \vert \omega _{32}+\omega _{a}\right \vert$. And yet, the frequency difference between the levels $\left \vert 2\right \rangle$ and $\left \vert 1\right \rangle$ is assumed below the cutoff frequency of the waveguide. Hence the guided photon is coupled to the transition $\left \vert 1\right \rangle \leftrightarrow \left \vert 2\right \rangle$ with coupling strength $\gamma _{g}$. In other words, the dispersion relation for the 1D waveguide associated wave vector $k$ is approximately linear around the transition frequency $\omega _{2}$ as $\omega _{k}=\omega _{2}+v(k-k_{0})$, where $v$ is the photon group velocity, and it is set to $1$. Consequently, the interaction part mainly considers the interaction between the transition $\left \vert 1\right \rangle \leftrightarrow \left \vert 2\right \rangle$ and the waveguide, the transition $\left \vert 2\right \rangle \leftrightarrow \left \vert 3\right \rangle$ and the cavity, ignoring the weak interaction between the transition $\left \vert 2\right \rangle \leftrightarrow \left \vert 3\right \rangle$ (the cavity field) and the waveguide. Overall, the interaction Hamiltonian $H_{int}$ has the form

 

Fig. 1. (a) Schematic representation of a three-level system with cascade configuration interacts with a 1D waveguide. The blue line represents the chiral coupling between the level transitions $\left \vert 1\right \rangle \leftrightarrow \left \vert 2\right \rangle$ and the left and right going photons of the bottom waveguide, and the coupling rates are $ \gamma _{r}$ and $ \gamma _{l}$, respectively. While the brown line represents the coupling between the level transitions $\left \vert 2\right \rangle \leftrightarrow \left \vert 3\right \rangle$ and the cavity with strength $ \lambda$. (b) An equivalent few-photon router by introducing the scatter-free channel and the controllable channel: the three-level system only interacts with the e-mode with a strength determined $ \gamma$.

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As a first step, we introduce the parity operators $c _{e}^{\dagger }\left ( x\right ) =\sqrt {\frac {\gamma _{r}}{\gamma }}R^{\dagger }\left ( x\right ) + \sqrt {\frac {\gamma _{l}}{\gamma }}L^{\dagger }\left ( -x\right ) ,c _{o}^{\dagger }\left ( x\right ) =\sqrt {\frac {\gamma _{l}}{\gamma }}R^{\dagger }\left ( x\right ) -\sqrt {\frac {\gamma _{r}}{\gamma }}L^{\dagger }\left ( -x\right )$ with $\gamma =\gamma _{r}+\gamma _{l}$. By employing the above transformation, the original Hamiltonian is transformed into two decoupled Hamiltonians, i.e., $H=H_{e}+H_{o}$, where

The decomposition of the Hamiltonian $H$ in Eq. (1) into two parts leads to the discussion of the problem from a one-mode interacting model Hamiltonian $H_{e}$ Eq. (6), which greatly simplifies the calculations. The system described by the Hamiltonians $H_{e}$ is referred to as the chiral system with unidirectional propagation of photons. A two-photon interacting eigenstate $\left \vert \Phi _{ee}\right \rangle$ of the chiral system described by $H_{e}$ satisfies the Schrödinger equation $H_{e}\left \vert \Phi _{ee}\right \rangle =E_{k_{1},k_{2}}\left \vert \Phi _{ee}\right \rangle$ and $\left \vert \Phi _{ee}\right \rangle$ has the following general form:

So, the wavefunction of the two-photon out-state is

Here, $\eta,$ $C_{\pm }$ are coefficients, and their expressions are

Which the scattering factor $U_{k_{i}}$ and the detuning $\Delta _{k_{i}}$ are the single photon physical parameters, they are defined as $U_{k_{i}}\equiv t_{k_{i}}^{e}-1$, $\Delta _{k_{i}}=vk_{i}-\omega _{2}$ $\left ( i=1,2\right )$. And $\alpha _{\pm }=\Delta _{k_{1},k_{2}}^{n}+\frac {1}{2}\left [ \left ( \Delta _{a}+\mathrm {i}\frac {\gamma }{2}\right ) \pm \eta \right ]$. In the vacuum cavity field scenario ($n = 0$), we recover the results given in Ref. [15], and it is obvious that there is only one two-photon bound state, that is, $\alpha =\Delta _{k_{1},k_{2}}^{n}+\mathrm {i}\gamma /2$ and the two-photon bound-state term is

However, for the $n\neq 0$ situation under our consideration, the two bound states are generated, while the atom can only absorb one photon at one time [34].

To sum up, in our scheme, whether the cavity contains photons determines the effective configuration of the emitter. When the control cavity field is in the vacuum state $\left ( n=0\right )$, the emitter can be considered as a two-level system, these photon’s scattering results coincide with those already published [15]. When the number of photons in the control cavity is $n$ $\left ( n\neq 0\right )$, our scheme becomes a 1D waveguide coupled to a V-type three-level system in the dressed-state representation. The expression of the dressed states can be obtained as

The effective transition energies of $\left \vert 1,n\right \rangle \leftrightarrow \left \vert \phi _{\pm }^{n}\right \rangle$ can be derived as $E_{\pm }^{n}=\omega _{2}+\frac {1}{2}[ -\Delta _{a}\pm \sqrt {\Delta _{a} ^{2}+4n\lambda ^{2}}]$. When the incoming photons satisfy the single-photon coupled-resonator-induced transparency [CRIT] condition, i.e., $\Delta _{k_{i}}+\Delta _{a}=0$. The contribution of bound states at $x_{1}=x_{2}$ vanishes since the parameters $C_{\pm }=0$. This result coincides with those already published in Ref.[35].

3. Transmitted photon-photon interaction

Having solved the two-photon scattering state in subspace $ee$, we now proceed to construct the two-photon eigenstate of the Hamiltonian in Eq. (1). In order to study the two-photon scattering, we restrict the problem to two-photon incoming from an arbitrarily selected port, for example, from port 1. The solutions corresponding to the input situation through port 2 are similar. Therefore, in the future, we will discuss in detail the case of two photons incoming from port 1. First of all, we take into account that there are two incident photons in the waveguide from port 1 with momenta $k_{1}$ and $k_{2}$, respectively. Besides, the initial state of the 3LS being $\left \vert 1\right \rangle$, we will follow a similar procedure as in Ref. [36,37], so the scattering output state can be represented as

The magnitude of out-state wave functions can be measured in Ref. [15,16]. Because two photons cannot be simultaneously emitted by the atom, it simply indicates the antibunching in $\rho _{mm}$ $\left ( m\neq 2\right )$ at $x_{1}=x_{2}$ for all $k_{1},k_{2}$. Next, we focus on the transmitted photons by $\rho _{22}\left ( x_{1},x_{2}\right ) ,$ the expressions of the probability amplitude $\rho _{22}\left ( x_{1},x_{2}\right )$ as

Here $t_{k_{i}}=\frac {2\left ( \Delta _{k_{i}}\left ( \Delta _{k_{i}}+\Delta _{a}\right ) -n\lambda ^{2}\right ) +\mathrm {i}\left ( \gamma _{l}-\gamma _{r}\right ) \left ( \Delta _{k_{i}}+\Delta _{a}\right ) }{2\left ( \Delta _{k_{i}}\left ( \Delta _{k_{i}}+\Delta _{a}\right ) -n\lambda ^{2}\right ) +\mathrm {i}\gamma \left ( \Delta _{k_{i}}+\Delta _{a}\right ) }$ is the single-photon with incident frequency $k_{i}$ transmission amplitude. When $vk=E_{\pm }^{n}=\omega _{2}+\frac {1}{2}\left [ -\Delta _{a}\pm \sqrt { \Delta _{a}^{2}+4n\lambda ^{2}}\right ] ,$ the transmission amplitude $t_{k}$ is found to have a minimum value. The wave function $\rho _{22}\left ( x_{1},x_{2}\right )$ includes plane-wave term $\left [ t_{k_{1}}t_{k_{2}}P_{k_{1},k_{2}}\left ( x_{1},x_{2}\right ) \right ]$, and a bound-state term, $\left [ \left ( \frac {\gamma _{r}}{\gamma }\right ) ^{2}B_{k_{1},k_{2}}\left ( x_{1},x_{2}\right ) \right ]$. Then, we have defined $x_{c}\equiv \left ( x_{1}+x_{2}\right ) /2$ and $x\equiv \left ( x_{1}-x_{2}\right )$ for the center-of-mass and relative coordinates, respectively. And $\delta _{k_{1},k_{2}}^{n}\leftrightarrow E$ is the total energy of the photon pair, $\Delta \equiv \left ( k_{1}-k_{2}\right ) /2$ measures the energy difference between two photons. So the plane-wave and the bound-wave express as

We note that the Eqs. (21a)–(21b) are a product of the center-of-mass wave function $e^{ \mathrm {i}Ex_{c}}$ and the relative wave function $e^{\mathrm {i}\kappa \left \vert x\right \vert }$, so $\left \vert \rho _{22}\left ( x_{c},x\right ) \right \vert ^{2}$ is even function of $\left \vert x\right \vert.$ The out-wave function $\rho _{22}$ is

Before going into detail, we mention some general properties of the above analytic expressions. The second term of $\left \vert \rho _{22}\left ( x\right ) \right \vert ^{2}$ is a bound function which is a superposition of two bound states with different localized width $\gamma /2 +$Im $\eta$ and $\gamma /2 -$Im $\eta$. As a result, the probability $\left \vert \rho _{22}\left ( x\right ) \right \vert ^{2}$ oscillates. We now discuss the effects of varying both $k_{1}$ and $k_{2}$ as well as the coupling strength $\gamma _{r}$, the joint probability of photons with a separation $x=0$ is

If $\left \vert \rho _{22}\left (0\right ) \right \vert ^{2}$ reduces from its finite value to zero. The transmitted photons thus change from bunching to antibunching. That is to say, the atom can induce either an effective repulsion or attraction between two photons in Refs. [15,16,18,22,25]. Based on this, we will discuss the effects of the two bound states. First, we take one of the two incident photons on resonance with the dressed states, i.e., $vk_{1}=E_{+}^{n},$ or $vk_{1}=E_{-}^{n}$, and the waveguide symmetrical coupled with the atom, i.e., $\frac {\gamma _{r}}{\gamma }=0.5,$ so that the plane-wave can disappear. For simplicity, all the parameters but $n$ are in units of $\gamma$ and we consider the case $\Delta _{a}=0$ in the following text. We assume that the two incident photons are both resonant with the dressed states, i.e., $vk_{1}=E_{+}^{n},vk_{2}=E_{-}^{n}$ or $vk_{1}=E_{-}^{n},vk_{2}=E_{+}^{n}$ [Fig. 2(a)], and $vk_{1}=vk_{2}=E_{+}^{n}$ or $vk_{1}=vk_{2}=E_{-}^{n}$ [Fig. 2(b)]. The effect of the bound states makes two photons being bound together, i.e., $\left \vert \rho _{22}\left ( 0\right ) \right \vert ^{2}\neq 0$. It is obvious that regardless of the value of the parameter $\sqrt {n}\lambda$, the binding function will locally within a certain space. We note that the oscillation of the function is related to the coherence between the two bound states.

 

Fig. 2. The joint probability of transmitted photons with a separation $x$, $\left \vert \rho _{22}\left ( x\right ) \right \vert ^{2}$, as a function of scaled relative coordinate $ \gamma x$, for various values of the coupling strength parameter $ \sqrt {n} \lambda$ of the atom and the cavity. The energies of the two incident photons are (a) $vk_{1}=E_{+}^{n},vk_{2}=E_{-}^{n}$ or $vk_{1}=E_{-}^{n},vk_{2}=E_{+}^{n}$, and (b) $vk_{1}=vk_{2}=E_{+}^{n}$ or $vk_{1}=vk_{2}=E_{-}^{n}$, as well as (c) $vk_{1}=E_{+}^{n},vk_{2}=\omega _{2}$ or $vk_{1}=E_{-}^{n},vk_{2}=\omega _{2}$. Here, we chose $ \sqrt {n} \lambda$=0,0.1,0.5,1, expressed as the black solid, green solid, red dashed, and blue dotdashed, lines in panels, respectively. The other parameters are $ \omega _{2}=10$, $\Delta _{a}=0$, $ \gamma _{r}/ \gamma =0.5$, all the parameters but $n$ are in units of $ \gamma$.

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When $n\neq 0$ and $\sqrt {n}\lambda$ less than $\gamma /4,$ there is no significant difference between our emitter system and a two-level atom emitter while taking $t_{k_{i}}=0 (i=1,2)$. And it has a local area with a width of $\gamma$. But it is obvious that a transmission path has been opened, leading to a valley at $x=0,$ in Fig. 2(c)[ $vk_{1}=E_{+}^{n},vk_{2}=\omega _{2}$ or $vk_{1}=E_{-}^{n},vk_{2}=\omega _{2}$]. In this case, it manifests the significant difference between our system and the two-level system. When $\sqrt {n}\lambda$ greater than $\gamma /4,$ the oscillation becomes more and more obvious as $\sqrt {n}\lambda$ increases, in addition, the localized width around $x = 0$ of the joint probability of transmitted photons $\left \vert \rho _{22}\left ( x\right ) \right \vert ^{2}$ becomes smaller, indicating that the photon pairs are becoming increasingly compact.

Reducing the joint probability of having both transmitted photons at the same position $\left \vert \rho _{22}\left ( 0\right ) \right \vert ^{2}$ to zero corresponds to photon antibunching in Refs. [15,16,18,22,25]. When $vk_{1}=E_{+}^{n},vk_{2}=\omega _{2}$ or $vk_{1}=E_{-}^{n},vk_{2}=\omega _{2}$, $\left \vert \rho _{22}\left ( 0\right ) \right \vert ^{2}$ exhibits a antibunching behavior with a global minimum $\left \vert \rho _{22}\left ( 0\right ) \right \vert ^{2}=0$, as shown in Fig. 2(c).

Then, we show in Fig. 3(a) how to modulate $\left \vert \rho _{22}\left ( 0\right ) \right \vert ^{2}$ through changing $vk_{2}$ for the selected $vk_{1}=E_{\pm }^{n}$, $\gamma _{r}/\gamma =0.5$. A particularly interesting aspect of the results is that the evolution of $\left \vert \rho _{22}\left ( 0\right ) \right \vert ^{2}$ with $vk_{2}$ [Fig. 3(a)], which is in contrast to the transmission spectrum [Fig. 3(b)]. Obviously, when $n\neq 0$, $\left \vert \rho _{22}\left ( 0\right ) \right \vert ^{2}$ will disappear in $\Delta _{k_{2}}=0$. And it is not affected by the number of the cavity photons. Therefore, the single-photon switch to control the single-photon source can be achieved by only one photon in the cavity.

 

Fig. 3. The joint probability of transmitted photons at the same position $\left \vert \rho _{22}\left ( 0\right ) \right \vert ^{2}$ and the single-photon transmission $T _{k_{2}}$, as a function of $vk_{2}$, for various values of the coupling strength parameter $ \sqrt {n} \lambda$, we choosed $vk_{1}=E_{\pm }^{n}$. The values of other parameters are consistent with those in the Fig. 2.

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Besides, when the energy of the other photon $vk_{2}$ also resonates with the dressed state, there is no way to achieve antibunching. So, we try to break the symmetry of waveguide coupling in Fig. 4. The situation is achieved by injecting two photons with $vk_{1},vk_{2}=E_{\pm }^{n}$. In this case the two photons are mainly reflected, but if the interference between the transmitted plane wave and the bound wave is destructive, i.e., $\gamma _{r}/\gamma =0.25,$ the two transmitted photons have effective repulsion.

 

Fig. 4. The joint probability of transmitted photons at the same position $\left \vert \rho _{22}\left ( 0\right ) \right \vert ^{2}$ as a function of $ \gamma _{r}/ \gamma$, for $vk_{1}=E_{+}^{n},vk_{2}=E_{-}^{n}$ or $vk_{1}=E_{-}^{n},vk_{2}=E_{+}^{n}$ and $vk_{1}=vk_{2}=E_{\pm }^{n}$, respectively.

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To sum up, we might as well discuss the effects of varying both the total energy $E$ and the energy difference $\Delta$ under certain coupling conditions. For clarity, we still consider the case $\Delta _{a}=0$. The joint probability density plot of having both transmitted photons at the same position $\left \vert \rho _{22}\left ( 0\right ) \right \vert ^{2}$ show in Figs. 5(a)–6(f). In Fig. 5, for $\gamma _{r}/\gamma =1/2$, if the total energy is determined, the number of occurrences with zero joint probability is two at $n=0$ in Fig. 5(a), in contrast, in the case of $n\neq 0,$ the number increased to twice [Fig. 5(d)]. Especially when the energy of the photon pair remains twice the bare state energy, i.e., $E=2\omega _{2}$, the antibunching phenomenon emerges on $\Delta =\pm \gamma /2$ when $n=0$, in other words, it is located on both sides of the half height of the single photon transmission spectrum, i.e., $vk_{1}=\omega _{2} + \gamma /2$, $vk_{2}=\omega _{2} - \gamma /2$ or $vk_{1}=\omega _{2} - \gamma /2$, $vk_{2}=\omega _{2} + \gamma /2$. And yet when $\sqrt {n}\lambda /\gamma =1$, the transmission spectrum split resulting in four positions at half height, $\Delta _{k}=\mu _{\pm }^{n},\nu _{\pm }^{n}$, here

Therefore, the antibunching effect appears in $E=2\omega _{2}$, $\Delta =\pm \frac {1}{2}\left [ \gamma _{r}\pm \sqrt {\left ( \gamma _{r}\right ) ^{2}+4n\lambda ^{2}}\right ] .$ There are four situations in total [Fig. 5(d)]. Here, we can find that the cavity initially injects photons, and the frequency channel of antibunching adds. The Rabi splitting, which is induced by the coupling of the cavity with the level transition between $\left \vert 2\right \rangle \leftrightarrow \left \vert 3\right \rangle$, grows the transmission valleys and shifts. We can also control the energy difference $\Delta$ by changing the total energy $E$ to achieve antibunching of photon pairs, as shown in Fig. 5(f), when $\left \vert \Delta \right \vert =\gamma _{r}/2$, then the total energy is $E=2\omega _{2} + \sqrt {\left ( \gamma _{r}\right ) ^{2}+4n\lambda ^{2}}$ or $E=2\omega _{2} - \sqrt {\left ( \gamma _{r}\right ) ^{2}+4n\lambda ^{2}}$.

 

Fig. 5. (a) and (d) The joint probability $\left \vert \rho _{22}\left ( 0\right ) \right \vert ^{2}$ versus the total energy $E$ and the energy difference $\Delta$, we set $ \omega _{2}=10$, $\Delta _{a}=0$, $ \gamma _{r}/ \gamma =0.5$, and $ \sqrt {n} \lambda$= 0, 1, respectively. (b) and (e) $\left \vert \rho _{22}\left ( 0\right ) \right \vert ^{2}$ versus the energy difference $\Delta$, for the total energy $E=2 \omega _{2}$. (c) and (f) $\left \vert \rho _{22}\left ( 0\right ) \right \vert ^{2}$ versus the total energy $E$, for the energy difference $\Delta =\pm \gamma _{r} /2$.

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Fig. 6. (a) and (d) The joint probability $\left \vert \rho _{22}\left ( 0\right ) \right \vert ^{2}$ versus the total energy $E$ and the energy difference $\Delta$, we set $ \omega _{2}=10$, $\Delta _{a}=0$, $ \gamma _{r}/ \gamma =0.25$, and $ \sqrt {n} \lambda$= 0, 1, respectively. (b) and (e) $\left \vert \rho _{22}\left ( 0\right ) \right \vert ^{2}$ versus the energy difference $\Delta$, for the total energy $E=2 \omega _{2}$. (c) and (f) $\left \vert \rho _{22}\left ( 0\right ) \right \vert ^{2}$ versus the total energy $E$, for the energy difference $\Delta =0$.

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In the case of chiral coupling, let’s consider $\gamma _{r}/\gamma =1/4$, in Fig. 6. It is very interesting that there are several intersections in the density map, and their details are shown on the right side. It is obvious that the double relationship still holds. In addition, behind these intersections is the low valley of the transmission spectrum.

4. Conclusion

In this article, we have presented an accurate analytic solution of two-photon scattering in real space. The system is composed of a 1D waveguide coupling a three-level atom with a cascade configuration, which provides a scheme to induce photon-photon correlations. The bound states appear due to the photon-photon interaction induced by the atom, and influence the correlations. Here, the coupling between the cavity and the atom’s transition $\left \vert 2\right \rangle \leftrightarrow \left \vert 3\right \rangle$ decides the number of bound states. When the cavity is in the vacuum state, i.e., $n=0$, only one bound state appears. When the cavity is in the fock state $\left \vert n\right \rangle$ and $n\neq 0$, there are two bound states here, and their coherent superposition causes oscillation in space. In the relative coordinate of photon pairs, we have shown that the coupling strength $\sqrt {n}\lambda$ impacts the adhesive width of photon pairs. And, as $\sqrt {n}\lambda$ increases, the photon pairs adhere more tightly, and the interference between bound states is more and more obvious, leading to faster wave function oscillation.

When selecting one of the two incident photons to make it perfectly reflecting, there is only a pure bound wave. If another photon is fully transmitted, a trough at $x=0$ in $\left \vert \rho _{22}\left ( x\right ) \right \vert ^{2}$ reduces from finite value to zero[minimum]. It signifies that the joint probability of having both transmitted photons at the same position is zero, i.e., the transmitted photons are antibunching. Hence, the atom can induce an effective repulsion between two photons. In addition, if the other photon is also fully reflected, the pure bound wave cannot cause the antibunching of the photon pair. In order to achieve the effective repulsion between transmitted photons, it is necessary to eliminate the interference between plane waves and bound waves by breaking the symmetry of waveguide coupling.

Finally, by adjusting the total energy of photon pairs and the energy difference between photons, it was found that the cavity field, which contains photons at the initial moment, increases the way to attain the antibunching effect of photon pairs. We hope that our proposal can provide a feasible and robust approach to induce either an effective repulsion or attraction between two photons.