1. Introduction

Generating and manipulating single-photon sources [1] have attracted extensive attention in the past several decades because of their important applications in quantum information and communication technology [2,3], quantum simulation [4], and quantum metrology [5,6]. One of the effective methods to obtain such sources is via the photon blockade (PB) effect (or, more accurately, single-photon blockade). In general, this PB effect based on two different physical mechanisms can be divided into conventional photon blockade (CPB) [7–10] and unconventional photon blockade (UPB) [11–21]. The former results from the quantum anharmonicity ladder of energy-level structure, where the strong nonlinearity is required. Under this mechanism, the resonant excitation of a single photon blocks the second or subsequent photons from simultaneous excitation. As a consequence, one can observe a one by one orderly photon output on the side of the cavity. This phenomenon is usually characterized by using higher-order correlation functions at zero-time $g^{(n)}(0)$ with $n\geq 2$. The value of $g^{(n)}(0)<1$ corresponds to sub-Poissonian photon-number statistics, which is referred to as photon antibunching effect, and the limit $g^{(2)}(0)\rightarrow 0$ indicates that the system is in the CPB regime. It should be noted that single photon generated via CPB has high brightness (i.e., large mean photon number) but poor purity.

Different from the strong nonlinearity-induced CPB, UPB arises from the destructive quantum interference between distinct excitation pathways and can exist in a weakly nonlinear system. Specifically, this mechanism requires at least an additional component [11–21], such as an atom, an auxiliary cavity, an auxiliary driving field and so on, which is coupled to the original system for constructing two or more transition pathways. When this system parameters satisfy the optimal conditions, namely, the probability of detecting two photons is almost zero, different excitation pathways will destructively interfere, resulting in the complete suppression of two-photon excitation. Correspondingly, it can give rise to optimally sub-Poissonian statistics and nearly perfect antibunching in the second-order correlation function $g^{(2)}(0)\approx 0$. Unlike the CPB, the higher-order correlation function of UPB meets the condition $g^{(n)}(0)>1$ for $n>2$. This feature means that two-photon emission is suppressed only but the emission of single photon and multiple photons is allowed. Therefore, single photon produced by UPB may be not a good single-photon source.

On the other hand, the cavity quantum electrodynamics (QED) system provides an excellent platform for the implementation of PB due to the light-matter interaction. As a fundamental model in the QED system, the Jaynes-Cummings (J-C) model can be used to investigate the interaction between an optical cavity and a two-level system, which is formed by either an atom, an ion, a quantum dot (QD), a superconducting qubit, or other two-level systems. In this model, some works about conventional and unconventional photon blockades have been proposed theoretically and observed experimentally [10,13,18,22–35]. However, it is worth noting that at the total resonance, CPB or UPB cannot take place in the case of the atom (or two-level emitter) drive [30,32]. This brings about an interesting question: how to revive the PB at the total resonance in atom-driven case by using other quantum manipulation?

Recently, two works [36,37] were devoted to the study of single-photon blockade via two-photon absorption in the resonant case. This blockade is also referred to as environmentally induced photon blockade (EPB). In contrast to CPB and UPB, EPB refers to a process in which two photons can leak out from the cavity simultaneously. Motivated by the above-mentioned works [36,37], we are going to explore the PB effect induced by the two-photon absorption in the atom-driven J-C model. The analytical and numerical results show that in the resonant case, strong antibunching of photons can be achieved without requiring strong nonlinearity or auxiliary component, and this physical mechanism can be understood by analyzing the two-photon absorption process. It should be stressed that our proposal is essentially different from a very recent work [35]. Here we list out these differences: Firstly, Ref. [35] has studied the PB based on the anharmonic energy-level structure and the destructive quantum interference. While in our work, the realization of PB is only due to the two-photon absorption. Secondly, we consider the atom-driven case rather than the cavity-driven case [35]. And thirdly, the strong photon antibunching in Ref. [35] occurs at $\Delta =g^2/\delta$ and $\Delta =-g^2/\delta -\delta$ in the strong-coupling regime, where $\Delta$ and $\delta$ are the cavity and atomic detunings, and g is the coupling strength. However for our work, the strong antibunching behavior appears at the detuning $\Delta _{a}=\Delta _{c}=0$ in the weak-coupling regime. Moreover, we find that owing to the two-photon absorption, the photon blockade can also be realized in the Tavis-Cummings (T-C) model. Thus, this work could provide a reliable method for producing the PB in both the J-C and T-C models.

2. Single-atom-cavity system

First, we consider a single-atom-cavity system [see Fig. 1] which is composed of a two-level atom coupled to a single-mode cavity. We assume that the atom is driven by a coherent laser at frequency $\omega _{L}$ with strength $\eta$. In a frame rotating at the laser frequency $\omega _{L}$ defined by the transformation $U=\mathrm {exp}[-i\omega _{L}t(a^{\dagger }a+ \sigma ^{\dagger }\sigma ^{-})]$, the Hamiltonian describing this system can be written as (setting $\hbar =1$)

 

Fig. 1. A schematic illustration of a two-level atom (orange, transition frequency $ \omega _{a}$, ground state $|g\rangle$ and excited state $|e\rangle$) located in a single-mode cavity (blue, frequency $ \omega _{c}$). The green arrow depicts the atom drive with driving frequency $ \omega _{L}$ and strength $ \eta$. $ \kappa$ and $ \gamma$ are the cavity decay rate and the atomic spontaneous emission rate, respectively. $ \kappa _{2}$ is the two-photon decay rate. $g$ is the coupling strength between cavity and atom.

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The vast majority of previous studies focused on the PB effect induced by the anharmonicity of the energy-level structure or the destructive quantum interference between distinct excitation pathways. Only a few studies, including Refs. [27,35–37], were analyzing PB induced by two-photon absorption. The physics behind this mechanism is that when there are two photons in the cavity, they can get away from the cavity simultaneously. Here, we will take into account the PB in the atom-cavity system coupled to a thermal reservoir via a two-photon absorption process. Correspondingly, they are described by the interaction Hamiltonian $H_{I}=\sum _{k}g_{k}(o_{k} a^{\dagger 2}+ a^{2}o_{k}^{\dagger })$, where $g_{k}$ is the system-reservoir coupling strength and $o_{k}^{\dagger }$ ($o_{k}$) is the reservoir creation (annihilation) operator. Under the Markov approximation, the two-photon absorption process is treated by a master equation approach [38–42], resulting in the following master equation of the density matrix $\rho$ for the whole system

To induce the PB, we need a large two-photon decay rate. Currently, several physical systems such as an optical cavity with N-type atoms [36], a driven single-mode cavity [37], and a Kerr resonator [40], have considered the large rate of two-photon absorption. These include $\kappa _{2}\approx 28\kappa$, $\kappa _{2}=10\kappa$, $\kappa _{2}=\kappa$, respectively. Further, some studies [27,35,41–48] suggest that the two-photon decay rate can be dominantly larger than the cavity decay rate ($\kappa _{2}\gg \kappa$ and even $\kappa =0$). Several experiments have also reported the enhanced two-photon absorption. From Refs. [49–51], the large two-photon absorption has been achieved in tapered optical fibers, hollow-core photonic-band-gap fibers, and two superconducting cavities coupled through a Josephson junction.

Numerically solving the above Eq. (2) based on Quantum Toolbox in MATLAB and Python [52–54], we can obtain the steady-state density matrix $\rho _{\text {ss}}$. Then, we can calculate the mean photon number and equal-time second-order correlation function in the steady state, respectively,

Below, we will analytically calculate the mean photon number and second-order correlation function. For simplicity, we phenomenologically introduce an anti-Hermitian term to Hamiltonian in Eq. (1) to describe the dissipation parts of cavity and atom [35,55–58]. This effective non-Hermitian Hamiltonian can be given as

Under the assumption that the two-level atom is initially in the ground state. In the weak-driving regime, the excitation number of the whole system will not exceed two. Hence, the state of this system can be approximately expressed as

By using the perturbation method and steady-state assumption [59], we have the approximate analytical solutions of the coefficients $C_{\alpha,n}$, as follows

In the limit $\eta \rightarrow 0$, the whole system remains in the ground state $|g,0\rangle$. In the weak-driving limit, we have the approximate scales $C_{g,0}\sim 1$, $\left \{C_{e,0},C_{g,1}\right \}\sim \eta$, and $\left \{C_{e,1},C_{g,2}\right \}\sim \eta ^{2}$, i.e., $|C_{g,0}|^2\gg |C_{e,0}|^2,|C_{g,1}|^2\gg |C_{e,1}|^2,|C_{g,2}|^2$. Using the above relationship and combining with Eqs. (3), (4), and (6), the mean photon number and second-order correlation function in the steady state are approximated as, respectively,

In the resonant case ($\Delta _{a}=\Delta _{c}=0$), the steady-state solutions of mean photon number and correlation function become

Note that Eq. (12) is directly related to the inherent properties of cavity and atom ($g$, $\kappa$, $\kappa _{2}$, and $\gamma$), but is independent of the driving strength $ \eta$. To check the validity of the approximate analytical solution (12), in Fig. 2 we plot the second-order correlation function $g^{(2)}(0)$ as a function of the atom-cavity coupling strength $g$ without and with the two-photon absorption. Here, the circles are based on the numerical solution of Eqs. (2) and (4), while the solid curves are plotted using the analytical solution in Eq. (12). From Fig. 2, we see clearly that the analytical solution is in excellent agreement with the numerical solution. This means that the analytical solution obtained using the effective Hamiltonian $H'_{1}$ is reliable in the weak-driving case.

 

Fig. 2. The second-order correlation function $g^{(2)}(0)$ versus the atom-cavity coupling strength $g$ with the total resonance case ($\Delta _{a}=\Delta _{c}=0$) for $\gamma =\kappa$ and $\eta =0.01\kappa$. These system parameters are the same as in Ref. [30] except for the two-photon absorption. The circles (the solid curves) represent the numerical solution (the analytical solution) using Eqs. (2) and (4) [Eq. (12)], respectively.

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Specifically, we choose the system parameters $\Delta _{a}=\Delta _{c}=0$, $\gamma =\kappa$, and $\eta =0.01\kappa$, which are the same as those used in Ref. [30] except for the two-photon absorption. In the absence of two-photon absorption ($\kappa _{2}=0$), the correlation function, given in Eq. (12), is reduced to $g^{(2)}(0)=\frac {[4(g/\kappa )^2+1]^2}{[4(g/\kappa )^2+2]^2}$. For the weak-coupling regime ($g\ll \kappa$), we have $g^{(2)}(0)\approx 0.25$, while for the strong-coupling regime ($g\gg \kappa$), we have $g^{(2)}(0)\approx 1$. These analytical results are well consistent with the blue circles and curves in Fig. 2. Correspondingly, we discuss these phenomena in both the weak- and strong-coupling regimes. In the case of $g\ll \kappa$, we see $g^{(2)}(0)<1$, indicating photon antibunching. This is because the high-order states cannot be excited under the weak driving condition (see Fig. 3). However in the case of $g\gg \kappa$, we see $g^{(2)}(0)\rightarrow 1$, indicating neither photon antibunching nor photon bunching. This is due to strong atom-cavity coupling strength which causes a splitting of the two eigenenergies on resonance ($\Delta _{a}=\Delta _{c}=0$), and can be understood by considering the energy-level structure of the eigenstates (see the right side of Fig. 3). The interaction between atom and cavity in Eq. (1) yields energy eigenstates that form an anharmonic ladder of doublets $|n_{\pm }\rangle =(|g,n\rangle \pm |e,n-1\rangle )/\sqrt {2}$, which denote the higher- (lower-) energy level of the $n$th excited state with the energy eigenvalues $E_{n_{\pm }}=\pm \sqrt {n}g$, $n=1,2$. With increasing the coupling strength $g$, the anharmonic energy-level splitting $E_{n_{+}}-E_{n_{-}}=2\sqrt {n}g$ becomes the dominant factor so that the driving laser is far off-resonance with all states. More importantly, we can observe that there is no photon-blockade effect in the system. We try to explain the reason leading to this phenomenon by using the energy-level diagram for atom-driven scheme on the left side of Fig. 3. Obviously, there only exists one transition pathway for this system to reach the two-photon state, i.e., $|g,0\rangle {\rightarrow }|e,0\rangle {\rightarrow }|g,1\rangle {\rightarrow }|e,1\rangle {\rightarrow }|g,2\rangle$. Thus, the PB induced by quantum interference does not exist in the atom-driven system.

 

Fig. 3. Energy-level diagram of the atom-driven system at the total resonance ($\Delta _{a}=\Delta _{c}=0$). States are labeled $|\alpha,n\rangle =|\alpha \rangle \otimes |n\rangle$, where $|\alpha \rangle$ $(\alpha =e,g)$ denotes the excited (ground) state of the atom, and $|n\rangle$ $(n=0,1,2)$ is corresponding to the zero-photon, one-photon, and two-photon states. $|n_{+}\rangle$ ($|n_{-}\rangle$) represents the higher- (lower-) energy eigenstate with $n=1,2$ excitations. Symbols: driving strength $\eta$, coupling strength $g$, cavity decay rate $\kappa$, atomic spontaneous emission rate $\gamma$, two-photon decay rate $\kappa _{2}$.

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In order to produce and enhance the photon-blockade effect, in Fig. 2, we consider that two-photon decay rate is large ($\kappa _{2}=10\kappa$), and plot the correlation function $g^{(2)}(0)$ against the coupling strength $g$. Clearly, the plot shows that the PB appears near the weak-coupling strength, exhibiting strong photon antibunching $g^{(2)}(0)\ll 1$, as shown by the red circles and curves of Fig. 2. This interesting phenomenon results from the two-photon absorption, which will be interpreted in the following. As the coupling strength $g$ increases, the photon antibunching decreases progressively and then vanishes. Indeed, in the resonant case ($\Delta _{a}=\Delta _{c}=0$), the strong coupling strength is unfavorable to produce the photon-blockade effect.

Now we explore the influence of two-photon absorption on the photon-blockade effect in the weak-coupling regime. To this end, in Fig. 4(a), we plot the correlation function $g^{(2)}(0)$ and the probability of detecting state $|C_{g,2}|^2$ as a function of the two-photon decay rate $\kappa _{2}$. It can be observed from Fig. 4(a) that the probability of detecting state $|C_{g,2}|^2$ decreases greatly with the increase of the two-photon decay rate $\kappa _{2}$. This result will give rise to an important phenomenon, i.e., the photon-blockade effect $g^{(2)}(0)\rightarrow 0$, which is well illustrated by the analytical solutions derived from Eqs. (8), (10), and (12). Therefore, the key role of two-photon absorption is the reduction of two-photon probability $|C_{g,2}|^2$. Physically, once there exist two photons in the cavity, they will leak out of the cavity simultaneously (see the transition from $|g,2\rangle {\rightarrow }|g,0\rangle$ in Fig. 3). In Fig. 4(b), we plot the time evolution of mean photon number ($\langle n\rangle =\langle a^{\dagger }a\rangle$) for different driving strengths $\eta$. We can see that the mean photon number gradually increases with increasing $\eta$. Regardless of the value of $\eta$, the mean photon number first increases and then reaches a steady-state value. For example, the steady-state value is of the order of $10^{-7}$ when the driving strength $\eta =0.01\kappa$, and the corresponding count rate is close to kHz, which can be detectable in cavity QED experiments [60,61]. For the weak driving strength, e.g., $\eta =0.01\kappa$ (blue circles) or $\eta =0.2\kappa$ (red circles), the numerical results based on Eqs. (2) and (3) are almost the same as the analytical results based on Eq. (11). However, for the strong driving strength, e.g., $\eta =2\kappa$ (black circles), the analytical results significantly deviate from the numerical results. This is because the approximate relationship is only valid in the weak-driving case. That is to say, the analytical results are valid (invalid) for the weak (strong) driving strength. In addition, we notice that by comparing Figs. 4(a) and 4(b) with $\kappa _{2}=10\kappa$, the occupying probability of state $|C_{g,2}|^2$ (about $10^{-16}$) is considerably smaller than that of state $|C_{g,1}|^2$ (about $10^{-7}$). This means that with the large two-photon decay rate, the state $|g,2\rangle$ is more suppressed comparing to the state $|g,1\rangle$, and it can lead to the generation of PB.

 

Fig. 4. (a) The second-order correlation function $g^{(2)}(0)$ (red curve) given in Eq. (12) and the probability of detecting state $|C_{g,2}|^2$ (blue curve) given by Eq. (8) versus the two-photon decay rate $\kappa _{2}$. (b) The mean photon number $\langle n\rangle$ based on Eqs. (2) and (3) versus the time $\kappa t$ with different driving strengths $\eta$. The inset shows the $\eta =0.01\kappa$ case. The parameters are the same as those in Fig. 2 except for $g=0.02\kappa$.

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To further avoid confusion of such a type of PB and conventional (or unconventional) PB, we analyze the differences and similarities between these types of PBs. Generally, all these types of PBs satisfy the condition $g^{(2)}(0)\rightarrow 0$. However, the steady-state third-order correlation function, namely,

 

Fig. 5. The $k$th-order equal-time correlation functions $g^{(k)}(0) =\langle a^{\dagger k}a^{k}\rangle /\langle a^{\dagger }a\rangle ^{k}$ ($k=2,3$) versus the coupling strength $g$. (a) $\eta =0.01\kappa$ and (b) $\eta =2\kappa$. The circles are plotted using the numerical solutions based on the master Eqs. (2), (4), and (13). The parameters used here are the same as those in Fig. 2.

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3. Two-atom-cavity system

Next, we consider a two-atom-cavity system as shown in Fig. 6, where two identical two-level atoms driven by a coherent laser are trapped in a single mode cavity. When two atoms are separated by much larger than the cavity wavelength, we do not need to consider the dipole-dipole interaction between two atoms [62,63]. In a frame rotating with frequency $\omega _{L}$, the Hamiltonian of the system can be expressed as (setting $\hbar =1$)

 

Fig. 6. A schematic illustration of two identical two-level atoms located in a single-mode cavity. $g_{j}$ ($j=1,2$) indicates the position-dependent coupling strength between cavity and $j$th atom. Other symbols are the same as those in Fig. 1.

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Then, the master equation describing the system dynamics is given by

Here, the Liouvillian superoperator $\mathcal {L}^{(j)}_{\gamma }\rho$ refers to the spontaneous decay of $j$th atom with the decay rate $\gamma$, which is given by $\mathcal {L}_{\gamma }^{(j)}\rho =\gamma /2(2\sigma ^{-}_{j}\rho \sigma ^{\dagger }_{j}-\sigma ^{\dagger }_{j}\sigma ^{-}_{j}\rho -\rho \sigma ^{\dagger }_{j}\sigma ^{-}_{j})$. Numerically solving Eq. (15), we can examine the collective radiance behavior of two atoms. This behavior can be described by the radiance witness [63],

To clearly show the influence of two-photon absorption on the collective radiance effect of two atoms, we plot the radiance witness $R$ as a function of the interatomic phase $\phi$ for different values $\kappa _{2}$ and $g$ at the total resonance $(\Delta _{a}=\Delta _{c}=0)$ in Fig. 7. As can be seen from Fig. 7(a), the two-photon absorption can significantly influence the radiance effect of atoms in the weak-coupling regime. In the ranges $0<\phi <\pi /2$ and $3\pi /2<\phi <2\pi$, the increase of two-photon decay rate $\kappa _{2}$ will suppress the collective radiance effect $R$. In the range $\pi /2<\phi <3\pi /2$, the two-photon absorption has little influence on the collective radiance. However in the strong-coupling regime, the collective radiance is not sensitive to the change of two-photon absorption [see Fig. 7(b)]. Physically, these phenomena can be understood by using the collective states of two atoms $\left \{|gg\rangle,|\pm \rangle,|ee\rangle \right \}$ as the basis, where the states $|\pm \rangle =(|eg\rangle \pm |ge\rangle )/\sqrt {2}$ are the symmetric and antisymmetric Dicke states. In this representation, the atom-cavity and laser-atom interactions in Eq. (14) can be rewritten in terms of the collective operators $S_{\pm }=(\sigma ^{-}_{1}\pm \sigma ^{-}_{2})/\sqrt {2},$ yielding $g_{\pm }(aS_{\pm }^{\dagger }+a^{\dagger }S_{\pm })/\sqrt {2}+\sqrt {2}\eta (S_{+}^{\dagger }+S_{+})$ with $g_{\pm }=g[1\pm \cos (\phi )]$ [69]. More specifically, we choose the interatomic phase $\phi =0$ or $\pi$ as two examples. For the case of $\phi =0$ and $g_{-}=0$, i.e., in-phase radiation ($g_{1}=g_{2}=g$), the antisymmetric Dicke state $|-\rangle$ is uncoupled from the system dynamics [shown in Fig. 8(a)]. For the case of $\phi =\pi$ and $g_{+}=0$, i.e., out-of-phase radiation ($g_{1}=-g_{2}=g$), the cavity only couples to the atoms via the state $|-\rangle$ [shown in Fig. 8(b)]. Obviously in the weak-coupling regime and without the two-photon absorption ($\kappa _{2}=0$), the collective radiation is enhanced ($0<R<1$) at the in-phase radiation ($\phi =0$), see Fig. 7(a). This is due to the constructive interference for the cavity coupled two atoms. But for atoms radiating out-of-phase ($\phi =\pi$), this system exhibits the subradiant behavior ($R<0$). In the strong-coupling regime, the radiation of two atoms in-phase ($\phi =0$) is suppressed, whereas at $\phi =\pi$, we can observe hyperradiance ($R>1$), see Fig. 7(b), which can be explained by the cavity backaction [63]. Comparing the case of $\kappa _{2}=0$, it is obvious in Fig. 7(a) that the two-photon absorption $\kappa _{2}=10\kappa$ can change the radiant character drastically. However, it has no influence on the radiant behavior in Fig. 7(b). Therefore, in the following, we will discuss the effect of two-photon absorption on the PB in the case of $\phi =0$ and $g<\kappa$.

 

Fig. 7. The radiance witness $R$ versus the interatomic phase $\phi$ with $\eta =0.5\kappa$. (a) $g=0.5\kappa$ and (b) $g=5\kappa$. The rest of the system parameters are the same as those used in Fig. 2.

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Fig. 8. Energy-level diagram of the two-atom-driven system at the total resonance ($\Delta _{a}=\Delta _{c}=0$). (a) in-phase radiation $\phi =0$ and (b) out-of-phase radiation $\phi =\pi$. Here, the interaction term $g_{+}$ ($g_{-}$) couples the cavity to $|+\rangle$ ($|-\rangle$) depending on the different value $\phi$. Other symbols are the same as in Fig. 3.

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Analogous to the single-atom-driven case, the non-Hermitian Hamiltonian and the state of this system are respectively,

After some straightforward calculations, the second-order correlation function in the case of steady state can be found as

Figure 9 displays the second-order correlation function $g^{(2)}(0)$ as a function of the atom-cavity coupling strength $g$ at $\phi =0$. Clearly, the analytical solution (solid blue or red curve) based on Eq. (20) is in conformity with the numerical one (blue or red circles) given in Eqs. (4) and (15), which illustrates that the analytical expression for $g^{(2)}(0)$ is accurate in the region of the system parameters. According to Eq. (20), when $\Delta _{a}=\Delta _{c}=0$, we obtain $g^{(2)}(0)= \frac {(\kappa +2\gamma )^2(8g^2+\kappa \gamma )^2}{4[4g^2(\kappa +\kappa _{2}+2\gamma )+\gamma (\kappa +\gamma )(\kappa +\kappa _{2})]^2}$. Considering the condition $\gamma =\kappa$ and $\kappa _{2}=0$, the above expression can be further simplified as $g^{(2)}(0)=\frac {9[8(g/\kappa )^2+1]^2}{16[6(g/\kappa )^2+1]^2}$. In the weak-coupling case ($g\ll \kappa$), we have $g^{(2)}(0)\approx 0.56$, i.e., there is no PB effect. This result is consistent with the blue circles and curves of Fig. 9. To understand the physical mechanism of this phenomenon, we examine the energy-level diagram for in-phase radiation of two atoms (Fig. 8(a)). As shown in Fig. 8(a), there exist two different transition pathways for two-photon state, corresponding to $|gg,0\rangle {\rightarrow }|+,0\rangle {\rightarrow }|gg,1\rangle {\rightarrow }|+,1\rangle {\rightarrow }|gg,2\rangle$ and $|gg,0\rangle {\rightarrow }|+,0\rangle {\rightarrow }|ee,0\rangle {\rightarrow }|+,1\rangle {\rightarrow }|gg,2\rangle$. Here, we notice that these two transition paths are symmetric and indistinguishable. This will cause the constructive interference between two paths, which results in two-photon excitation. In contrast, we consider the case of $\kappa _{2}\ne 0$. From Fig. 9, we find that with increase in the two-photon decay rate, $g^{(2)}(0)$ decreases in weak-coupling regime. More importantly, the presence of large two-photon absorption ($\kappa _{2}=10\kappa$) can produce and even improve the PB effect. Hence, this proves that two-photon absorption has an important role in achieving PB.

 

Fig. 9. The second-order correlation function $g^{(2)}(0)$ versus the atom-cavity coupling strength $g$ with $\phi =0$. The remaining parameters are the same as those in Fig. 2. The circles (the solid curves) represent the numerical solution (the analytical solution) using Eqs. (4) and (15) [Eq. (20)], respectively.

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4. Conclusion

In this work, we have proposed a feasible scheme to realize the photon blockade (PB) in the Jaynes-Cummings (J-C) and Tavis-Cummings (T-C) models by introducing the two-photon absorption. Through analytically solving the Schrödinger equation and numerically simulating the master equation, we obtain the equal-time second-order (or third-order) correlation function describing the PB effect, and then find a perfect agreement between the analytical and numerical solutions. Due to the two-photon absorption, the strong PB can be achieved in the atom-driven case and at the total resonance. In contrast to the conventional and unconventional PBs, the strong nonlinearity and the auxiliary component are not required for our work, allowing us to achieve $g^{(2)}(0)\rightarrow 0$ even in the weak-coupling regime. In addition, we also discuss the collective radiance characteristics like subradiance, superradiance, and hyperradiance for the T-C model. It is observed that for in-phase radiation and in the weak-coupling regime, the two-photon absorption can significantly influence the collective radiance effect. Therefore, we believe that our proposal could be helpful for investigating the PB and radiance effects in a variety of cavity quantum electrodynamics (QED) systems.