## Abstract

We theoretically study the electromagnetic field controlled photon correlation and radiation properties in a two-atom-cavity quantum electrodynamic (QED) system with different atom-cavity coupling strengths. When two three-level atoms are coherently driven by a probe field and a strong control field, forming a ladder type electromagnetically induced transparency configuration, there exist four different excitation pathways via the two photon processes, yielding four different frequencies for realizing two photon blockades. Carefully choosing the intensity of the control field, resonant single-photon and two-photon excitation can be achieved at the same probe field frequency. Thus, two photon blockade phenomenon can be significantly improved and the collective radiation behavior becomes superradiance or even hyperradiance. Our system presented here can be considered to be a converter of a weak coherent light to two correlated photons.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Nonclassical light plays a major role in many quantum information technologies. The ability to generate high quality nonclassical light efficiently is essential for advances in optical quantum technologies [1]. The photon number state (i.e., the Fock state) is a fundamental nonclassical light. In particular, for a single photon state, the second-order correlation function $g^{(2)}(0)\rightarrow 0$ implies that two coincident photons cannot be detected [2,3], which is well known as the photon antibunching [4,5]. In past few decades, various approaches have been proposed to generate antibunched photons, such as spontaneous parametric down conversion [6,7] and quantum emitter in the cavity [8,9].

Single photon blockade (PB), as a method for realizing photon antibunching and characterized by vanishing the second order correlation function of cavity photons, has been intensively studied. To date, there are two well-known methods to obtain strong PB effects. One is called as the conventional photon blockade (CPB), and the other is recognized as the unconventional photon blockade (UCPB). The former rely on the energy anharmonicity in cavity QED systems [10], but the later is based on the destructive quantum interference between two different quantum excitation pathways from the ground state to a doubly excited state [11–14]. Both methods have been studied theoretically and experimentally in various quantum systems [5,15–22].

Recently, the concept of photon blockade has been extended to multiphoton excitation [23–26]. For two photon blockade, two photons leak from the cavity simultaneously, resulting in the steady-state third-order correlation function $g^{(3)}(0)<1$, but the second-order correlation function $g^{(2)}(0)>1$. This phenomenon has been observed in a single atom cavity QED system [27]. Although many advanced techniques have been used to improve the quality of the single photon blockade, including the quantum interference [28], dipole-dipole interaction [29] and Rydberg interaction [30,31], the research of two photon blockade is still on its early stage. How to realize strong two photon blockade is challenging both in the theory and experiment.

In this paper, we study the photon correlation and radiation properties in a two atoms cavity QED system under the two photon blockade regime. Here, two identical three-level ladder-type atoms are embedded in a single mode cavity with different atom-cavity coupling strengths, and coherently driven by a weak probe field and a strong control field, forming a typical Ladder type electromagnetically induced transparency (EIT) configuration [32]. Although the EIT technique has been used to manipulate the photon blockades [33] many other interesting features have not yet been studied. Using this technique, we show that the control field will induce two additional excitation pathways via the two photon processes. Therefore, there exist four frequencies to implement the two photon blockade. By carefully choosing the control field intensity, one can obtain a specific probe field frequency at which both states in one-photon and two-photon spaces can be resonantly excited. As a result, the two photon blockade can be significantly improved with an enhanced mean photon number. We show that the steady-state third-order correlation function $g^{(3)}(0)$ can be reduced to $10^{-4}$, while the second-order correlation function is larger than unity. Moreover, the collective radiation property of two atoms becomes superradiance or hyperradiance with a strong two photon blockade.

## 2. Model

The schematic diagram of the system considered here is shown in Fig. 1(a). Two identical three-level ladder-type atoms with energy levels $|g\rangle$, $|m\rangle$ and $|e\rangle$ are embedded in a single mode cavity with resonant frequency $\omega _\textrm {cav}$. The $|g\rangle \leftrightarrow |m\rangle$ transition is driven by a probe field with Rabi frequency $\eta$ and interacts with the cavity photons simultaneously. Another strong control field with Rabi frequency $\Omega _c$ couples with the $|m\rangle \leftrightarrow |e\rangle$ transition, forming a ladder type EIT configuration.

Under the rotating wave approximation, the Hamiltonian of the system can be written as

where $H_0=\sum _{i=1,2}(\Delta _\textrm {cav}a^{\dagger} a+\Delta _e\sigma _{ee}^{(i)}+\Delta _m\sigma _{mm}^{(i)})$ with the i-th (i=1,2) atom’s operator $\sigma _{\alpha \beta }^{(i)}=|\alpha \rangle _{i}\langle \beta |$ ($\alpha,\beta =g,m,e$) and $a^{\dagger}\ (a)$ being the photon creation (annihilation) operator. The cavity detuning is defined as $\Delta _\textrm {cav}=\omega _\textrm {cav}-\omega _p$. The atomic one-photon and two photon detunings are given by $\Delta _m=\omega _{me}-\omega _p$ and $\Delta _e=\omega _{eg}-\omega _p-\omega _C$ where $\omega _{\alpha \beta }=\omega _\alpha -\omega _\beta \ (\alpha =g,m,e)$ with $\omega _{p(C)}$ being the probe (control) field angular frequency and $\hbar \omega _\alpha$ being the energy of state $|\alpha \rangle$. Here, we set $\Delta _e=\Delta _m=\Delta _\textrm {cav}=\Delta$ for mathematical simplicity. The atom-cavity interaction is described as $H_I=\sum _{i=1,2}[g_i(a^{\dagger}\sigma _{gm}^{(i)}+a\sigma _{mg}^{(i)})]$ where $g_i=2\pi z_i/\lambda _\textrm {cav}$ is the position-dependent coupling strength with $z_i$ and $\lambda _\textrm {cav}$ being the position of the i-th atom and the wavelength of the cavity mode, respectively. Carefully choosing the positions of two atoms, we set $g_1=-g_2=g_0$ in this work. The driving term is $H_D=\sum _{i=1,2}[\eta (\sigma _{gm}^{(i)}+\sigma _{mg}^{(i)})+\Omega _c(\sigma _{me}^{(i)}+\sigma _{em}^{(i)})]$.Generally, the dynamical behavior of the whole system can be described by the master equation, i.e.,

## 3. Perturbation theory

To show the physical mechanism, we first use the perturbation theory to analysis the eigenvalues and eigenstates of the system by neglecting the detunings and the weak probe field. Defining the collective states $|GG,n\rangle =|gg,n\rangle$, $|MG\pm,n-1\rangle =(|mg,n-1\rangle \pm |gm,n-1\rangle )/\sqrt {2}$, $|EG\pm,n-1\rangle =(|eg,n-1\rangle \pm |ge,n-1\rangle )/\sqrt {2}$, $|MM,n-2\rangle =|mm,n-2\rangle$, $|EM\pm,n-2\rangle =(|em,n-2\rangle \pm |me,n-2\rangle )/\sqrt {2}$ and $|EE,n-2\rangle =|ee,n-2\rangle$ with $|n\rangle$ being the Fock state of the cavity field, one can easily obtain the eigenstates in one photon space:

Likewise, the eigenstates in two photon space are given by

For $\Omega _c=0$, as shown in Fig. 1(b), the eigenstates $\Psi _{1\pm }^{(1)}$ are absent, and the single photon excitation $\Psi ^{(0)}{\overset{1\textrm{ph}}{\rightarrow}}\Psi ^{(1)}_{2\pm }$ (blue dashed arrows labeled by A) are not allowed. Thus, the two photon excitation $\Psi ^{(0)}{\overset{2\textrm{ph}}{\rightarrow}}\Psi _{3\pm }^{(2)}$ (orange arrows labeled by B) are predominant, and then two photon blockade can be observed by increasing the probe field intensity. Consequently, the second order correlation function is larger than unity, but the third order correlation function $g^{(3)}(0)<1$ [see Fig. 1(c)]. In such a system, the collective radiation behavior of two atoms can be quantified by calculating the radiance witness $R=(\langle a^{\dagger}a\rangle _2-2\langle a^{\dagger}a\rangle _1)/(2\langle a^{\dagger}a\rangle _1)$, where $\langle a^{\dagger}a\rangle _{i}$ indicates the mean photon number with $i=1,2$ atoms in the cavity. Here, $R=0$ reveals an uncorrelated scattering. $R<0$ indicates that the radiation is suppressed, corresponding to a subradiation, while $R>0$ means the radiation is enhanced. Particularly, $R=1$ implies that the radiation scales with the square of the number of atoms, i.e., the superradiance [34]. $R>1$ denotes the domain of hyperradiance [35]. In the case of $\Omega _c=0$, obviously, the witness $R<1$, i.e., the collective radiation behavior is subradiance when the two photon blockade occurs [see orange curve in Fig. 1(c)].

For the case of $\Omega _c\neq 0$, more excitation pathways appear in one- and two-photon spaces, e.g., $\Psi ^{(0)}{\overset{1\textrm{ph}}{\rightarrow}}\Psi _{1\pm }^{(1)}$ (blue solid arrows labeled by C) and $\Psi ^{(0)}{\overset{2\textrm{ph}}{\rightarrow}}\Psi _{2\pm }^{(2)}$ (red arrows labeled by D). The former will not cause any photon in the cavity, while the later results in two photon blockade due to the energy level anharmonicity.

To verify these analysis, we calculate the mean photon number $\langle a^{\dagger} a\rangle$ (blue solid curve) and the value of $\langle a^{\dagger} a^{\dagger} aa\rangle$ (orange dashed curve), characterizing the two photon process, as a function of the detuning $\Delta$. Here, we choose $\eta /\kappa =1$ and the control field is chosen as (a) $\Omega _c/\kappa =0$ and (b) $10$, respectively. When $\Omega _c/\kappa =0$, as shown in Fig. 2(a), there exist three peaks in the cavity excitation spectrum, corresponding to the detunings $\Delta =0$ and $\Delta =\pm \sqrt {6}g/2$, respectively. Although the two photon blockade with $g^{(3)}<1$ and $g^{(2)}>1$ can be achieved at $\Delta =\pm \sqrt {6}g/2$, the mean photon number is too small to be detected. For $\Omega _c/\kappa =10$, as shown in Fig. 2(b), there exist four peaks in the cavity excitation spectrum with detunings $\Delta =\pm \sqrt {(\alpha -\sqrt {\beta })/8}$ and $\Delta =\pm \sqrt {(\alpha +\sqrt {\beta })/8}$, corresponding to arrows D and B respectively. Obviously, the two photon processes are still dominant at these four detunings. Particularly, for the detunig $\Delta =\pm \sqrt {(\alpha -\sqrt {\beta })/8}$ (see arrows D), the mean photon number can be significantly enhanced, resulting in strong two photon blockade with reasonable photon number.

## 4. Two photon blockade with superradiance or hyperradiance

To show the quality of two photon blockade, we plot the third order correlation function of cavity photons in Fig. 3(a). The second order correlation function is always larger than unity. As shown in panel (a), $g^{(3)}(0)<1$ near the detunings $\Delta =\pm \sqrt {(\alpha +\sqrt {\beta })/8}$ and $\Delta =\pm \sqrt {(\alpha -\sqrt {\beta })/8}$, indicated by (B) the white dash-dotted and (D) white dashed curves respectively. At the detuning of $\Delta =\pm \sqrt {(\alpha +\sqrt {\beta })/8}$, the energy level anharmonicity is enhanced by the control field, and the value of $g^{(3)}(0)$ decreases significantly with the increase of $\Omega _c$, yielding a very strong improvement of the two photon blockade (see the dash-dotted curves). With these system parameters, the third order correlation function can be reduced to $10^{-5}$.

Likewise, near the detuning $\Delta =\pm \sqrt {(\alpha -\sqrt {\beta })/8}$, the two photon blockade can also be improved by choosing a suitable control field $\Omega _c$. In particular, as shown in Fig. 3(a), there exists two crossing points, where resonant two photon excitation (i.e., $\Delta =\pm \sqrt {(\alpha -\sqrt {\beta })/8}$) and single photon excitation (i.e., $\Delta =\pm \Omega _c$) can be achieved simultaneously. Thus, one can easily obtain the condition, i.e.,

to realizing strong two photon blockade with enhanced collective radiation behavior. To show this point, we plot the radiance witness $R$ in Fig. 3(b). Clearly, the system exhibits a subradiant behavior (i.e., $R<0$) near $\Delta =\pm \sqrt {(\alpha +\sqrt {\beta })/8}$, but it becomes superradiance or hyperradiance ($R\geq 1$) near the frequencies of $\Delta =\pm \sqrt {(\alpha -\sqrt {\beta })/8}$ (see white dashed curves).Finally, let’s discuss the influence of the pump field intensity to the two photon blockade and radiation behavior by choosing the detuning near the crossing point $\Delta =\sqrt {2}g/2$. In Fig. 4, the third-order correlation function $g^{(3)}(0)$ [panel (a)] and the witness parameter $R$ [panel (b)] are plotted as functions of the pump field intensity and the control field Rabi frequency, respectively. Clearly, the two photon blockade with hyperradiant radiation can be achieved near the control field Rabi frequency $\Delta =\sqrt {2}g/2$. By choosing a suitable pump field intensity, the magnitude of the third-order correlation function can be less than $10^{-3}$, while the magnitude of the mean photon number is above $0.01$ [see the black curves in panel (a)]. It is worth to point out that this value of the mean photon number is similar to the recent experimental report in Ref. [27], but the value of the third order correlation function is more than two times order smaller than that in single atom cavity QED system. With this strongly improved two photon blockade, the hyperradiant behavior remains unchanged as shown in Fig. 4(b). Further increasing the pump field intensity, the two photon blockade will be lifted since the states in multiphoton spaces will be excited. It is noted that such a system can be implemented in various quantum systems, such as the Rydberg cavity QED system [36–38] and superconducting circuit QED system [39–41].

## 5. Conclusion

In conclusion, we have shown that a system of two three-level atoms in a single mode cavity with different atom-cavity coupling strengths can lead to stronger two-photon blockade with superradiance or hyperradiance behavior when these two atoms are driven by a probe field and a control field. In the presence of the control field, the system dressed states are significantly changed and additional excitation pathways appear in one and two photon spaces, respectively. Since the asymmetric states are decoupled from the probe field and the control field induced single photon excitation will not result in any cavity photons, two photon transitions are predominant in such a system. Carefully choosing the control field intensity, i.e., $\Omega _c=\Delta =\pm \sqrt {2}g/2$, resonant single-photon and two-photon excitation can be achieved at the same frequency of the probe field. As a result, the third-order correlation function $g^{(3)}(0)$ can be reduced to $10^{-4}$, yielding strong two photon blockade phenomenon. At the same time, significantly enhanced mean photon number and superradiance or hyperradiance behavior of two atoms can be achieved in such a system, which can be used as a converter of a weak coherent field to two correlated photons.

## Funding

National Natural Science Foundation of China (11774262, 11874287, 61975154).

## Disclosures

The authors declare no conflicts of interest.

## Data availability

No data was generated in this study.

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