Simultaneously enhanced photon blockades in two microwave cavities via driving a giant atom

We propose a scheme to enhance photon blockades simultaneously in two microwave cavities connected by a giant atom. In the case that only one cavity is weakly driven, we observe that the enhanced photon blockades occur in both cavities at the same time, which is attributed to the anharmonic eigenenergy spectrum constructed by the resonantly coupled giant atom. When the cavity and the atom are simultaneously driven, the stronger photon blockades in two cavities can be successfully achieved by the destructive quantum interference of two-photon excitation. Interestingly, we find that high single-photon occupations can be obtained in both cases. Moreover, we give the optimal conditions for conventional and unconventional photon blockades through analytical calculations, which are in good agreement with numerical results. Our scheme opens a prospective path to achieve simultaneous photon blockades in two indirectly coupled cavities, and provides a promising method to generate the high-quality and brightness single photon source.


Introduction
In recent years, single photon sources are extensively applied in the fields of quantum optics and quantum information technologies [1][2][3]. To generate the single photon source, many schemes based on photon blockade (PB) effect have been proposed by using different physical systems, including cavity quantum electrodynamics (QED) system [4][5][6][7][8], superconducting circuit system [9], optomechanical system [10][11][12][13][14][15], and so on. The PB, also known as the conventional PB (CPB), results from the anharmonicity of energy level in the system [16][17][18], where photons in the nonlinear cavity will block the transmission of subsequent photons. However, the realization of CPB requires strong nonlinearity in the system, which is difficult to achieve in experiment. Therefore, the PB generated from destructive quantum interference, called unconventional photon blockade (UPB), was proposed and thoroughly investigated [19][20][21][22][23]. Generally, there are two methods to open additional transition pathways for realizing the destructive quantum interference effect. One is by adding an auxiliary cavity between two cavity modes [24], and the other is to add an auxiliary driving field so that the cavity and atoms are simultaneously driven [6]. Based on this feature, many schemes have been proposed to implement UPB, such as two coupled single-mode cavities with nonlinearity [24][25][26][27], cavity embedded with a quantum dot [28][29][30], degenerate optical parametric amplifier system [31][32][33], and two coupled superconducting resonators system [34][35][36][37].
Both UPB and CPB can be characterized by the second-order correlation function g (2) (0) [6]. It is well known that the single photon source generated by UPB has very low g (2) (0) due to interference, but relatively poor brightness, whereas that by CPB has high mean photon number, but relatively poor purity [7]. Hence UPB and CPB have different advantages. Although the mechanisms of CPB and UPB are different, they can coexist in the generation of PB [38,39], which can combine their individual advantage. In [39], the strong PB has been realized by utilizing the two-level emitter to enhance energy-spectrum anharmonicity and construct the destructive quantum interference for two-photon excitation in atom-cavity system. Recently, there have been many related studies with regard to the giant atom, such as decoherence-free interaction between giant atoms [40], tunable chiral bound states with giant atoms [41], giant atoms in a synthetic frequency dimension [42], and coherent single-photon scattering spectra [43]. The giant atom can be coupled to two or more desirable resonators, providing us another method to control the resonance frequency, the dipole moment, and the states of the photons.
In this paper, we propose an atom-cavity model to enhance the PBs simultaneously in two indirectly coupled single-mode microwave cavities, which are connected by a giant atom [43,44]. The atom acts as the nonlinear medium that mediates the conversion of the photons between the two cavities. Different from [39], owing to the destructive quantum interference, not only the two-photon excitation can be totally canceled, but also the single-photon excitation is maximized in our scheme. This indicates that the generated photon fields have stronger antibunching (purity) and a larger mean photon number (brightness). By analytically and numerically calculating the second-order correlation function of photons, two methods are proposed to simultaneously enhance PBs in both cavities. First, when one of the cavities is driven, we observe that the enhanced PB effect can occur in both cavities due to the anharmonic eigenenergy spectrum constructed by the resonant coupled giant atom. Second, for the case of simultaneously driving the cavity and the atom, the driving field acting on the giant atom further promotes the photon antibunching via improving the destructive quantum interference. Moreover, we also find that high single-photon occupation can be obtained in both cases. By analytically calculating the eigenvalues of the system Hamiltonian and the steady-state Schrödinger equation, we obtain the optimal conditions for CPB and UPB. Comparing the optimal conditions with the numerical results, it is found that they are in great agreement, which confirms that our analytical results is reasonable. Our scheme provides an alternative method for the experimental implementation of PBs in two indirectly coupled cavities.
The paper is arranged as follows. In section 2, we introduce the physical model and present the Hamiltonian of the atom-cavity system. In section 3, we first investigate the simultaneously enhanced single PBs in two cavities by the energy-level anharmonicity when one cavity is driven. Then, the enhancement of the PBs based on quantum interference is investigated when the atom and cavity are driven at the same time. Finally, we conclude this paper in section 4.

System and Hamiltonian
The atom-cavity system we consider is shown in figure 1, where a superconducting giant atom is coupled with two single-mode cavities simultaneously. In a frame rotating at driving frequency ω d defined by , the Hamiltonian of the system is written as (ℏ = 1) [44] where a (a † ) and b (b † ) denote respectively the annihilation (creation) operators of cavity modes with frequencies ω a and ω b , ∆ j = ω j − ω d (j = a, b) and ∆ 0 = ω 0 − ω d are the detunings of the cavity and atom from the driving field, respectively, σ ± are the raising and lowering operators of atom with resonant transition frequency ω 0 , and g denotes the atom-cavity coupling strength. Here H d is the driving term, depending on whether the cavity or the atom is driven. For the case of the atom (cavity) is driven by a driving field, Taking the dissipations of two cavities and atom into account, the dynamics of the system is governed by the master equationρ where κ a , κ b , and γ are the dissipation rates of mode a, mode b, and atom, respectively. The Lindblad superoperator L[o] acts as for o = a, b, σ − . In order to observe the PB, we are concerned with the statistical properties of photons, which can be described by the equal-time second-order correlation function in the steady state Here, ρ s is the steady-state density operator given byρ = 0. The second-order correlation function g (2) (0) < 1 (g (2) (0) > 1) is referred to as the sub-Poissonian (super-Poissonian) statistics, and correspondingly photons have antibunching (bunching) behavior. In the following we will discuss this phenomenon in detail.

Conventional photon blockade
First, we consider that only cavity a is driven by a coherent field. The basis of the system can be denoted by |mng(e)⟩, in which |m⟩ and |n⟩ are the number states of cavity modes a and b, |g⟩ (|e⟩) represents the atomic state, respectively. In order to have an analytical description of the system, we will study two subspaces of the system, single-photon excitation and two-photon excitation cases, separately. In the weak driving limit, we restrict the system to contain single-photon excitation for mode a. Therefore, the Hamiltonian can be expanded with the basis |10g⟩, |01g⟩, and |00e⟩. Then, the system Hamiltonian can be expressed as a matrix [45] where the weak driving term has been neglected. For simplicity, ω a = ω b = ω 0 is considered [46,47]. Thus, the three eigenfrequencies of the system Hamiltonian can be obtained by diagonalizing the above matrix where the subscript 1 represents the single-photon excitation for mode a. The corresponding eigenstates are For the single-excitation resonance condition for CPB, using the frequencies given in (6) as the drive frequency, the drive detuning must be In general, when one of the single excitation resonance conditions in equation (8) is satisfied, the system will occupy an eigenstate in equation (7), which can trigger the occurrence of the CPB. However, when ∆ a = 0 is satisfied, the CPB does not appear, which will be discussed in detail in the following. Similarly, in the two-photon excitation subspace {|20g⟩, |02g⟩, |11g⟩, |10e⟩, |01e⟩}, the matrix form of the system Hamiltonian is written as Five eigenfrequencies can be obtained by diagonalizing H 2 in equation equation (9), which can be described as The subscript 2 represents the two-photon excitation number of mode a. The corresponding eigenstates are The energy-level diagram of the system is shown in figure 2, where ω j 1 (j = 1, 2, 3) denoted by equation (6) and ω k 2 (k = 1, 2, 3, 4, 5) are the energy levels of the system with single-photon excitation and two-photon excitation of mode a, respectively. When ∆ a = 0, there is a two-photon resonance with the transition 0 → ω 3 2 ; thus the absorption of the first photon favors that of the second photon, resulting in photon-induced tunneling. However, when the driving field is on resonance with the 0 → ω 1,3 1 transition, i.e. ∆ a = ± √ 2g, the ω 1,3 1 → ω k 2 transition is suppressed due to large detuning 2( √ 2 − 1)g. This leads to a sharp increase in the single-photon occupation probability P 1 . However, the two-photon occupation probability P 2 does not change significantly due to detunings at the same time. Under the weak driving condition, the equal-time correlation function can be approximated as g (2) (0) ≈ 2P 2 /P 2 1 [30,45]. The significant increase of single-photon probability P 1 leads to the correlation function g (2) (0) tending to zero, which makes the strong PB triggered. According to equation (7), both modes a and b occupy the single-photon state, which indicates that single photon antibunching can be simultaneously realized in two cavities.
Next, we numerically study the CPB effect based on the master equation (2), and compare the results with the conditions shown in equation (8). The logarithmic plot of equal-time second-order correlation functions g (2) (0) of modes a and b versus g and ∆ a is respectively shown in figures 3(a) and (b). For simplicity, we assume ∆ a = ∆ b = ∆ 0 and κ a = κ a = κ. Other parameters are set as Ω/κ = 0.1, and γ/κ = 0.1. The white dashed lines passing through the origin of the coordinate indicate the single excitation resonance conditions ∆ a = ± √ 2g, which are in great agreement with the numerical results. Obviously, the photon antibunching g (2) (0) < 1 appears at detuning ∆ a = ± √ 2g, which indicates that CPB can simultaneously occur in two modes. This can be understood from the anharmonicity of the energy level spacing of the system, i.e. if a single-photon transition 0 → ω 1,3 1 appears, the subsequent transition ω 1,3 1 → ω k is suppressed due to large detunings, as shown in figure 2. In this situation, the first excited photon will block the second one being excited. The photon bunching g (2) (0) > 1 can be observed at detuning ∆ a = 0. The photon bunching means that the number of photons is greater than 1 in the cavity, which comes from the two-photon transition 0 → ω 3 2 . The mean photon number N = Tr(j † jρ s ) represents brightness, which can be obtained by solving the master equation, as shown in figures 3(c) and (d). Combined with figures 3(a) and (b), the region with strong photon antibunching corresponds to a large mean photon number, which is caused by the first level of resonance excitation. Different from the previous scheme, the advantage of the system is that the photon antibunching can be realized in two indirectly coupled cavities at the same time.
To analyze the statistic behavior of the cavity modes a and b in greater detail, we respectively show the dependence of the correlation function g (2) (0) and the photon-number occupation on detuning ∆ a /κ in figure 4. The photon-number distributions of modes a and b can be respectively expressed as P m = Tr[|m⟩⟨m|ρ s ] and P n = Tr[|n⟩⟨n|ρ s ]. In figures 4(a) and (b), we can observe that there are two sharp dips in the curve of the correlation function at ∆ a = ±25κ. The value of g (2) a (0) is close to 10 −2 , whereas the g (2) b (0) can reach 10 −3 . Due to the direct interaction between the cavities and the atom, the single-photon occupation probability of two modes are approximately equal as shown in figure 4(c). Also, the two-photon state relationship of the two modes is as follows: |20g⟩ → |10e⟩ → |11g⟩ → |01e⟩ → |02g⟩, which results in a lower probability of mode b than mode a, as shown in figure 4(d). This leads to a better PB in mode b compared to mode a, mainly due to the lower values of the two-photon occupation probability of mode b. Moreover, we also find that the single-photon occupation probability of two modes are maximized at ∆ a = ±25κ.

Unconventional photon blockade
Next, we consider that cavity a and atom are driven at the same time. We propose to enhance the PB effect via the destructive interference between the different transition pathways. The wave function of the system can be written as |ψ⟩ = C 00g |00g⟩ + C 10g |10g⟩ + C 01g |01g⟩ with probability amplitude C mng(e) . The dissipations of two cavities and atom can be considered by the non-Hermitian HamiltonianH where H is given in equation (1). Substituting the wave function |ψ⟩ and HamiltonianH into the Schrödinger equation i∂ t |ψ⟩ =H|ψ⟩, a set of equations for the probability amplitudes can be obtained here In the weak driving regime, C 00g ≫ {C 10g , C 01g , C 00e } ≫ {C 20g , C 02g , C 11g , C 10e , C 01e } and C 00g ≈ 1, we can obtain the probability amplitudes C 10g , C 01g , C 20g , and C 02g in the steady state, where . Thus the second-order correlation functions of two cavities can be expressed as Obviously, if 2g 2 − ∆ ′ a ∆ ′ 0 = 0, the second-order correlation functions of the two cavities will be 0. In the strong coupling limit, g ≫ {κ, γ}, the optimal condition for strong photon antibunching can be obtained The optimal driving strength ε for C 20g = 0 is which are associated with the destructive quantum interference between different transition pathways. At the same time, we can obtain the driving strength ε for C 02g = 0 as √ 2Ω/2. In figure 5, there exist three transition pathways to the state |20g⟩ of mode a: (i) |10g⟩ → |20g⟩, (ii) |10g⟩ → |10e⟩ → |20g⟩, and (iii) |00e⟩ → |10e⟩ → |20g⟩. Indeed, when the contributions of |10g⟩ and |10e⟩ to |20g⟩ cancel each other, the destructive interference occurs. To realize the PB of mode b, the excitation of the state |01e⟩ is usually required to be prohibited by the destructive quantum interference so that the two-photon state |02g⟩ remains unexcited. Clearly, we can observe that there are three different transition pathways to the state |01e⟩: (i) |10e⟩ → |11g⟩ → |01e⟩, (ii) |01g⟩ → |11g⟩ → |01e⟩, and (iii) |01g⟩ →→ |01e⟩. When the optimal conditions given by equations (17) and (18) are satisfied, the total transitions result in a destructive quantum interference and vanishing occupations on |20g⟩ and |02g⟩.
In order to verify the above analysis, based on equations (2) and (4), we plot the equal-time second-order correlation function g (2) (0), single-photon and two-photon probability amplitude for modes a and b, as a function of ε/Ω under the optimal conditions in figure 6. Clearly, there are two sharp dips in the curve of g (2) a (0) at ε = √ 2Ω/2 and 5 √ 2Ω/2 in figure 6(a), which is in great consistence with the driving strength as displayed in equation (18). However, when ε = √ 2Ω/2, the photon antibunching phenomenon of mode a is much weaker compared with ε = 5 √ 2Ω/2. This is because of the fact that there are two pathways to reach the single-photon state |10g⟩ as shown in figure (5): (i) the direct pathway, i.e. |00g⟩ → |10g⟩, and (ii) the pathway through atom, i.e. |00g⟩ → |00e⟩ → |10g⟩, which makes the destructive interference between the two pathways happen when the driving strength ε = √ 2Ω/2 is satisfied. In addition, we can find that the value of |C 10g | 2 is approximately 0 at ε = √ 2Ω/2, while the value of |C 10g | 2 is approximately 0.05 for ε = 5 √ 2Ω/2 as shown in figure 6(c). In figure 6(d), the value of |C 20g | 2 at both points is close to 0. This will lead to a larger value of g (2) a (0), mainly due to the lower values of |C 10g | 2 . In figure 6(b), we can observe that there is a sharp  peak in the curve of g (2) b (0) at ε = √ 2Ω/2. This phenomenon attributes to the destructive interference between the two transition pathways reaching state |01g⟩, i.e. (i) |00g⟩ → |00e⟩ → |01g⟩, and (ii) |00g⟩ → |10g⟩ → |00e⟩ → |01g⟩, which can minimize the probability amplitude C 01g as shown in figure 6(e). In figure 6(f), it can be observed that the two-photon state is suppressed due to the destructive quantum interference, which means that the value of C 02g will be approximately 0. Figure 6 also shows that the antibunching phenomenon of the two modes is enhanced when the atom is driven.
Next, we mainly discuss the situation of ε = 5 √ 2Ω/2. In figure 7, we plot the logarithmic value of g (2) (0) as a function of ∆ a /κ and γ/κ, and the mean photon number N a ≃ |C 10g | 2 (N b ≃ |C 01g | 2 ) as a function of versus ∆ a /κ, respectively. In figures 7(a) and (b), we can observe that there are two sharp dip in the curve of g (2) (0) at ∆ a = ±25κ, as predicted in equation (17), which verifies the validity of the optimal conditions obtained analytically. The smallest second-order correlation function values of two modes are respectively g (2) a (0) ≈ 10 −6 and g (2) b (0) ≈ 10 −3 . Comparing figures 7(c) and (d) with 7(a) and (b), we find that the locations with strong photon antibunching correspond to the largest mean photon number. In addition, comparing with figures 4(c) and (d), we find that the single-photon occupation is enhanced with the quantum interference mechanism, which means that high-brightness single photon sources can be obtained. Figures 7(e) and (f) show that the value of the g (2) (0) gradually increases with the increase of γ/κ, which means that the photon antibunching effect becomes weaker. However, it is worth emphasizing that the PB effect still occurs for smaller or larger dissipation γ/κ.

Conclusions
In conclusion, we have presented a promising scheme to enhance the photon antibunching simultaneously in two cavities connected by a giant atom, which not only acts as the nonlinear medium that mediates the conversion of the photons between the two cavities, but also provides a way to enhance the PBs effect by adding atom driving. Through analyzing the CPB and UPB mechanisms, we show that the enhanced PB effect can be ascribed to two aspects: (i) the resonant coupled giant atom constructs the anharmonic eigenenergy spectrum; (ii) the driving field acting on the giant atom further promotes the destructive quantum interference for two-photon excitation. By numerically calculating the equal-time second-order correlation function and the mean photon number, we study the statistical characteristics of photons and find that the strong photon antibunching with high single-photon occupations of two indirectly coupled cavities can be obtained under the two mechanisms, which is different from the previous scheme. Moreover, the optimal parameter conditions are given to optimize the PB (g (2) a (0) ≈ 10 −6 and g (2) b (0) ≈ 10 −3 ) and maximize the single-photon occupation (N a ≈ 10 −1 and N b ≈ 10 −1 ) at the same time. Therefore, our scheme may have potential applications in the generation of single photon sources with high-quality and brightness of two indirectly coupled cavities.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).