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Abstract
We propose a scheme to generate a single-photon source based on photon blockade in the Jaynes-Cummings (J-C) model with a two-photon dissipation (TPD) process. We present the optimal conditions for conventional/unconventional photon blockade via the wave function method with an effective Hamiltonian involving TPD. The results show that the second-order correlation function for the J-C model with TPD is considerably less than that of the J-C model with single-photon dissipation. Additionally, the average photon number can reach 0.5 in the large atomic detuning regime. This feature makes the J-C model with TPD a high-quality single photon source.
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1. Introduction
The preparation of a single photon source with high purity and brightness has been of great interest in the fields of quantum optics, information, and computing [1–5]. Currently, one method involves examining the quantum statistics of the light field via the photon blockade (PB) effect [6–8], which corresponds to sub-Poissonian distribution and strong antibunching.
As a typical photon nonlinear phenomenon, the PB effect implies that a photon can block the transmission of subsequent photons. The PB effect can be primarily divided into conventional PB (CPB) and unconventional PB (UCPB) [9,10], based on different physical mechanisms. The former is caused by the nonlinearity of the energy spectrum of a system, while the latter is induced by the destructive quantum interference between different transition paths. From then on, photon blockade has been extended to multiphoton regime [11–14], or even with multi-photon dissipation [14]. As an analog of photon blockade, phonon blockade has also been proposed and studied in various scenarios [15–18]. Recently, a hybrid photon-phonon blockade was observed in an optomechanical systems [19].
There are several quantum systems that can offer the strong nonlinearity required for CPB, including the Kerr-type nonlinearity [6,7,20–22], strong optomechanical coupling [23–29], a two-level emitter (e.g., atom [30–44] or superconducting qubit systems [23,45–47]) strongly coupled to an optical cavity [48,49], nonreciprocal systems [50–53] and non-Hermitian systems [54]. Conversely, UCPB effect has been illustrated in quantum optics systems, e.g., coupled Kerr-type nonlinear cavities systems [9,10,55–59], cavity-QED systems [60–62], coupled optomechanical systems [63,64], and other quantum composite systems [65–69]. CPB was first observed experimentally in a system with a trapped atom in an optical cavity [31], subsequently demonstrated via a quantum dot in a photonic crystal cavity [32,34,38], and a single superconducting artificial atom coupled to a microwave transmission-line resonator [45,46]. The UCPB effect was experimentally demonstrated in an optical microcavity coupled to a single semiconductor quantum dot [70] and in a superconducting circuit comprising two coupled resonators [71].
Previous studies on PB have mainly aimed at a single-photon dissipation (SPD) system. However, a single photon source can also be prepared via a two-photon dissipation (TPD) process [6,72,73]. TPD is a nonlinear optical phenomenon wherein two overlapping photons are simultaneously absorbed, while the absorption of the individual photons is restricted, that is, it occurs with a much lower efficiency. TPD has been studied and successfully demonstrated in some physical systems [74–77]. The master equation for the density operator with a TPD system was proposed via quantum-state diffusion and the quantum-jump model [78]. In Ref. [79], a practical and efficient scheme was proposed to develop a single-photon source via the four-wave mixing process in a cavity, which is effectively equivalent to the TPD process. Quantum Zeno blockade via two-photon absorption (TPA) was also proposed, which can produce high-fidelity entangled photons on-demand at MHz rates or higher [80]. The non-Hermitian Hamiltonian and master equation of a quantum van der Pol (vdP) oscillator system with both SPD and TPD processes was proposed, and the results showed that phonon antibunching depends significantly on the nonlinear two-phonon loss of the vdP oscillator [81]. Additionally, Ref. [82] introduced the PB effect in a Kerr nonlinear resonator with TPD. Therefore, an interesting question arises: Is the PB effect of a system with TPD significantly better than that of the SPD system?
Therefore, in this work, we investigated the PB effect of the J-C model with TPD. In fact, the researchers have investigated the initial photon states on the PB effect for a Kerr-type nonlinearity derived from the J-C model with two-photon driving and dissipation under large detuning and strong nonlinearity [83]. Here, We shall investigates PB effect of J-C model with single-photon driving and two-photon dissipation and discuss in detail the CPB and UCPB effect based on the wave function and density matrix methods respectively. We found that the antibunching of the J-C model with TPD is stronger than that of the J-C model with SPD. More importantly, without lowering the purity, the average photon number of the J-C model with TPD can reach $0.5$ for a large atomic detuning when the system operates at an optimal condition for CPB. This intriguing feature makes the J-C model with TPD a good candidate for a high-quality single photon source.
The remainder of this paper is organized as follows. In Sec. 2, we present a physical model of a two-level atom with an optical cavity mode with TPD and analytically study the PB effect. In Sec. 3, by using the wave-function method, the optimal parameter conditions are proposed for strong photon antibunching. Subsequently, the density matrix method is adopted to derive the analytical expression for the second-order correlation function in the large atomic regime. Finally, we conclude the work in Sec. 4.
2. Model
First, we introduce our model of the system. We consider a simple J-C model, driven by a weak light field, wherein a two-level atom (with the transition frequency $\omega _{0}$) is coupled to an optical cavity (with the fundamental mode frequency $\omega _{a}$) in Fig. 1. In the frame rotating with the driving laser frequency $\omega _{d}$, the system Hamiltonian is given by (taking $\hbar =1$)
Fig. 1. Schematic of the two-photon decay J-C model. A quantum dot is coupled to a microcavity with two orthogonally polarized cavity modes.
We further assume that our system is subject to two dissipation processes: one is a TPD of rate $\kappa$, implying that two photons are destroyed for each dissipative event, and the other is atomic spontaneous emission with the rate $\gamma$. In fact, we can place nonlinear materials in the cavity to quickly absorb two photons, but it does not work when there is a single photon in the cavity. To study the quantum statistics of the photons in the cavity, we provide a non-Hermitian effective Hamiltonian $H_{\text {eff}}$ for our system, followed by calculating the probability amplitude in the steady state by using the wave function method, and then providing the correlation function expression and the average photon number of the cavity modes, which characterize the purity and brightness of the single-photon source, respectively. By using the quantum-jump method presented in Ref. [78] and phenomenologically introducing TPD and the atom spontaneous emission term, the non-Hermitian effective Hamiltonian is described by
Herein, the effective Hamiltonian form of TPD for PB is proposed in the literature for the first time, and the analytical results are provided subsequently.
3. Results and analysis
In this section, we find the optimal conditions for the CPB and UCPB of the J-C model with TPD.
3.1 Photon blockade-based wave function method
Currently, two major methods are used to generate PB effect. One is induced by strong nonlinear interaction between polaritons, which is called the CPB effect and the other is induced by quantum interference, which is called UCPB. In Ref. [48], the CPB effect of the J-C model driven by a weak driving field has been discussed; thus, we will not discuss it here. In this section, we use the wave function method to present a unified expression for both CPB and UCPB.
First, we assume that the largest photon excitation number is $n=2$ in the weak driving limit (i.e., $E\ll \left \{ \kappa,\gamma \right \}$), as the higher-photon-excitation states have considerably low population during a PB. Therefore, we can discuss PBs involving only transitions between the bare states $\left \vert 0,g\right \rangle,\left \vert 0,e\right \rangle,\left \vert 1,g\right \rangle,\left \vert 1,e\right \rangle$ and $\left \vert 2,g\right \rangle$, where $\left \vert g\right \rangle$ and $\left \vert e\right \rangle$ are the ground and excited states of an atom, respectively.
According to the approach presented in Ref. [9], in the weak driving limit, we expand the wave function of the system in terms of the bare states up to two-photon excitation:
After substituting Eqs. (2) and (3) into the Schrödinger equation $i\frac {d\left \vert \Psi \right \rangle }{dt}=H_{\text {eff} }\left \vert \Psi \right \rangle$, we can obtain a set of equations regarding the coefficients. By using a perturbation method, that is, discarding the higher-order terms in the equations for lower-order variables and setting $\left \{ \dot {C}_{n,g},\dot {C}_{n,e}\right \} =0$ and $C_{0,g}\simeq 1$, we finally obtain the equations of the coefficients as follows:
with $\delta ^{\prime }=\delta -i\gamma /2$. Notably, we ignore the equation $EC_{1,g}=0$ according to the assumption here, which is very important for the subsequent analysis. The expressions (6)–(9) also directly illustrate the energy levels and the links between the steady states $\left \vert n,g\right \rangle$ (or $\left \vert n,e\right \rangle$) under the weak driving field. After simple calculations, we obtain the coefficients $C_{1,g}$ and $C_{2,g}$ as follows:Both CPB and UCPB share the same quantum statistics of the photons, that is, sub-Poissonian, although their mechanisms differ. Based on Eq. (5), we obtain a general expression of the second-order correlation function for the photons in the cavity as:
Here, we have assumed $\kappa =\gamma$ for simplicity.
In Eq. (12), in the strong-coupling regime, both $g^{2}+\delta \left ( \delta +\Delta\right ) -\gamma ^{2}/4=0$ and $g^{2}-\delta \Delta =0$ can make $g^{\left ( 2\right ) }\left ( 0\right )$ assume local minimal values smaller than $1$. As discussed above, $g^{2}-\delta \Delta =0$ [See Eq. (6) in Ref. [48] for details] is the optimal condition for CPB, while $g^{2}+\delta \left ( \delta +\Delta\right ) =0$ is an approximated expression of the optimal condition for UCPB in the strong-coupling limit.
We first confirm the optimal analytical conditions obtained above via numerical simulation. We present the contour plots of $g^{\left ( 2\right ) }\left ( 0\right )$ as a function of atomic $\delta$ and cavity detunings $\Delta$ in Fig. 2(a). There are four strong antibunching regions located in two different parameter regions. In the first region, the atomic and cavity detunings have the same sign, that is, $\delta \Delta >0$, and the antibunching regions are located in the first and third quadrants of the $(\delta,\Delta )$ plane. In this region, the two antibunching regions are symmetric with respect to the origin $(0,0)$. The corresponding CPB is induced by a strong nonlinear interaction between the polaritons. Figure 2(b) shows the CPB structure with red lines based on the optimal condition $g^{2}=\delta \Delta$. In the second region, the atomic and cavity detunings have opposite signs, that is, $\delta \Delta <0$, and the antibunching regions are located in the second and fourth quadrants of the $(\delta,\Delta )$ plane. The UCPB, in this case, arises from quantum interference. The corresponding optimal condition is $g^{2}+\delta \left ( \delta +\Delta\right ) =0$, plotted in green dashed lines in Fig. 2(b). Figure 2(c) and (d) present high-order correlation functions with the variance of atomic $\delta$ and cavity detunings $\Delta$, thus satisfying the optimal conditions for CPB and UCPB, respectively. Notably, $1>g^{\left ( 2\right ) }\left ( 0\right ) >g^{\left ( 3\right ) }\left ( 0\right ) >g^{\left ( 4\right ) }\left ( 0\right )$ for CPB while $g^{\left ( 3\right ) }\left ( 0\right ) >g^{\left ( 4\right ) }\left ( 0\right ) >1>g^{\left ( 2\right ) }\left ( 0\right )$ for UCPB. This result verifies our classification regarding the antibunching structure displayed in Fig. 2(a).
Fig. 2. (a) The second-order correlation function $\log _{10}g^{(2)}(0)$ as functions of the atomic detuning $ \delta$ and cavity detuning $\Delta$. (b) Antibunching structure for CPB and UCPB with the same parameters used in (a). (c) Correlation functions as a function of atomic detuning $ \delta$ with cavity detuning $\Delta =g^2/ \delta$. (d) Correlation functions as a function of atomic detuning $ \delta$ with cavity detuning $\Delta =-g^{2}/ \delta - \delta$. The other parameters are considered as $ \kappa = \gamma,g=10 \gamma$ and $E=0.001 \gamma$.
The correlation function characterizes the purity of a single photon source. The smaller the correlation function, the stronger the PB effect and the higher the purity of the single photon source becomes. This implies that the single-photon source prepared in this manner demonstrates superior performance. For a better demonstration of the behavior of the second-order correlation function with the optimal conditions satisfied, we plot $g^{\left ( 2\right ) }\left ( 0\right )$ as a function of atomic detuning $\delta$ with $\Delta =g^{2}/\delta$ in Fig. 3(a) and $\Delta =-g^{2}/\delta -\delta$ in Fig. 3(b) based on numerical calculations and analytical results. In addition, to compare the antibunching effect of the J-C model with SPD and that with TPD with the same parameters, we also provide $g^{\left ( 2\right ) }\left ( 0\right )$ of the J-C model with SPD by numerical calculations in Fig. 3. From Fig. 3(a) and (b), we can see that $g^{\left ( 2\right ) }\left ( 0\right ) \ll 1$ even at as low as $10^{-5}$, for the J-C model with TPD in the CPB $g^{2}=\delta \Delta$ and the UCPB optimal conditions $g^{2}+\delta \left ( \delta +\Delta\right ) =0$. This means that the J-C system with TPD shows an excellent antibunching effect, with CPB or UCPB. This can be understood that, with two-photon dissipation, the population of even photon state is more suppressed comparing to odd photon state, which lead to the enhancement of CPB and UCPB with TPD.
Fig. 3. The second-order correlation function $\log _{10}g^{(2)}(0)$ as a function of atomic detuning $ \delta$ with (a) $\Delta =g^{2}/ \delta$ and (b) $\Delta =-g^{2}/ \delta - \delta$ under the conditions of SPD and TPD, respectively. The remaining parameters are taken as $ \kappa = \gamma,g=10 \gamma$, and $E=0.001 \gamma$.
In Fig. 3(a), for minor atomic detuning $\delta$, the second-order correlation function $g^{\left ( 2\right ) }\left ( 0\right )$ for TPD is almost the same as that for SPD, implying that they have the same antibunching effect. However, for the large $\delta$, $g^{\left ( 2\right ) }\left ( 0\right )$ for TPD is considerably smaller than that of SPD, and the gap between them increases with the continual increase of $\delta$, which means that the J-C system with TPD shows a stronger antibunching effect than that with SPD. In addition, the second-order correlation function for SPD reaches the minimum $g_{\min }^{\left ( 2\right ) }\left ( 0\right ) \approx 7\times 10^{-2}$ at $\delta \simeq 6.8\gamma$ and then approaches $1$, indicating that the antibunching effect disappears, and a stable single photon source can no longer be obtained. The function for TPD gradually decreases with the increase in atomic detuning $\delta$, reaches the minimum $g_{\min }^{\left ( 2\right ) }\left ( 0\right ) \approx 10^{-5}$ at $\delta \simeq 240\gamma$ and does not return to $1$ with the increase in $\delta$, but constantly maintains this value. This indicates that we can obtain a continuous and stable high-purity single-photon source. Therefore, we can conclude that the antibunching effect can be obtained in a larger atomic detuning range for the CPB optimal condition $g^{2}=\delta \Delta$ of the J-C model with TPD, which is highly important in the development of single-photon sources.
In Fig. 3(b), the case with TPD is superior to that with SPD, that is, $g_{\min,\,\text {SPD}}^{\left ( 2\right ) }\left ( 0\right ) <g_{\min,\,\text { TPD}}^{\left ( 2\right ) }\left ( 0\right )$, which denotes that the system with TPD still displays a stronger antibunching effect than that with SPD under the UCPB optimal condition. In addition, the variation trend of the second-order correlation function for TPD in the UCPB optimal condition $g^{2}+\delta \left ( \delta +\Delta\right ) =0$ is different from that in the CPB optimal condition $g^{2}=\delta \Delta$; that is, it no longer approaches a constant value with the increase in $\delta$, but returns to $1$, which is similar to the case with SPD.
Notably, the analytical results deviate from numerical results under the optimal condition of CPB for large atomic detuning $\delta$. This shows that the probability amplitude method of the wave function is invalid in the large atomic detuning regime, that is, $\delta \gg \gamma$, under the optimal condition of CPB for the J-C model with TPD. The reason for this problem is that the wave function method used here is a semiclassical description, which ignores completely the effect of quantum jumps, as explained in the context of non-Hermitian quantum mechanics in [84]. To address the problem, in the next section, we use the density matrix method to provide the approximated expression of the second-order correlation function in the large atomic detuning regime, which is based on the master equation and is fully quantum and includes the effect of quantum jumps on blockade. The differences between these two approaches can be understood and manipulated via the hybrid-Liouvillian formalism [85].
3.2 Photon blockade-based density matrix method
In this section, we employ the master equation method to study the quantum statistics for the density matrix $\rho$ of the system. The master equation for the J-C model with TPD is given by
In the weak-driving limit, we expand the density matrix of the system in terms of bare states up to two-photon excitation in the Hilbert subspace, that is, $\hat {\rho }=\sum _{{}}^{{}}\rho _{mn}\left \vert m\right \rangle \left \langle n\right \vert$ ($m,n\leq 4$) based on the excitation number states $\left \vert 0\right \rangle =\left \vert 0,g\right \rangle$, $\left \vert 1\right \rangle =\left \vert 0,e\right \rangle$, $\left \vert 2\right \rangle =\left \vert 1,g\right \rangle$, $\left \vert 3\right \rangle =\left \vert 1,e\right \rangle$, and $\left \vert 4\right \rangle =\left \vert 2,g\right \rangle$. We assume that the system is in a steady state, that is, $\dot {\rho }=0$. Then, we obtain the coupled equation set of the matrix elements (see the detailed equation sets in the appendix). If we find the exact expression for all the matrix elements, the equal-time second-order correlation function can be approximately evaluated as $g^{\left ( 2\right ) }\left ( 0\right ) \simeq 2\rho _{44}/\rho _{22}^{2}$.
It is difficult to present a simple explicit expression for the density elements. Here, we consider the case of large atomic detuning. After long and straightforward calculations, we obtain the asymptotic expressions for density matrix elements:
Finally, we obtain the second-order correlation function as
To show the dependence of $g^{\left ( 2\right ) }\left ( 0\right )$ on atomic detuning $\delta$ more clearly, we plot $g^{\left ( 2\right ) }\left ( 0\right )$ as a function of $\delta$ with $\Delta =g^{2}/\delta$ by Eq. (16) in Fig. 4(a). Evidently, the analytical results (red circle symbol ’o’) are in good agreement with the numerical results for large $\delta$, as expected. The differences between the wave function method and the density matrix method are clearly seen for small values of $\delta /\gamma$. Thus, by a proper postselection of quantum trajectories, the reported semiclassical predictions could, in principle, be confirmed experimentally even in the above-mentioned parameter range in Fig. 4(a). In Eq. (16), in the large atomic detuning limit $\delta \rightarrow \infty$, $g^{\left ( 2\right ) }\left ( 0\right ) \simeq 8E^{2}/\kappa ^{2}$. The result shows that the limit value of the second-order correlation function is only dependent on the driving strength $E$ and the two-photon decay rate $\kappa$ of the cavity field, but independent of the coupling strength $g$. In Fig. 4(b) and (c), we plot $g^{\left ( 2\right ) }\left ( 0\right )$ as functions of the driving strength $E$ and the coupling strength $g$ with $\delta =500\gamma$, while other parameters satisfy the optimal condition for CPB. Notably, $g^{\left ( 2\right ) }\left ( 0\right )$ quadratically increases with the increasing driving strength $E$. While $g^{\left ( 2\right ) }\left ( 0\right )$ first dramatically decreases and then stabilizes at a constant value $8\times 10^{-6}$ by increasing the coupling strength $g$. This result shows that the second-order correlation function is independent of $g$ when the coupling strength is in the strong coupling regime, that is, $g\gg \gamma$.
Fig. 4. (a) Second-order correlation function and average photon number as a function of atomic detuning $ \delta$ with cavity detuning $\Delta =g^2/ \delta$ for the J-C model with TPD. Second-order correlation function and average photon number as a function of (b) the driving strength $E$ and (c) the coupling strength with atomic detuning $ \delta =g^2/\Delta$. The cycle symbol ’o’ denotes the analytical result [Eq. (16)]. The other parameters are considered as $\Delta =0.2 \gamma, \delta =500 \gamma, \kappa = \gamma,g=10 \gamma$, and $E=0.001 \gamma$.
In addition to the purity, brightness is an important index to characterize the quality of a single-photon source. Generally, we use the average photon number of the system to characterize brightness. The average photon number $\left \langle N\right \rangle \simeq$ $\rho _{22}$ in this work. Notably, the average photon number monotonically increases and finally approaches a constant value $0.5$ with the increase of atomic detuning, as shown in Fig. 4(a). This result is consistent with the analytical one $\left \langle N\right \rangle _{\delta \rightarrow \infty }=0.5$. Furthermore, the dependence of the average photon number on the driving strength $E$ and the coupling strength $g$ is shown in Fig. 4(b) and (c), respectively. The average photon number first rapid increases and then approaches a constant value $0.5$ with the increasing driving strength $E$. Conversely, the behavior of the average photon number in terms of the coupling strength $g$ is similar to that for the driving strength. This phenomenon implies that high brightness requires strong driving and coupling strengths.
Evidently, when $g\rightarrow 0$, the J-C model with TPD studied here reduces to a model comprising an empty cavity with TPD, which has been investigated in Ref. [81]. Here, we compare the J-C model with TPD with the model of empty cavity with TPD. Figure 5 shows a considerably narrow and deep dip on the curve of the second-order correlation function for the empty cavity with TPD. The minimum value of $g^{\left ( 2\right ) }\left ( 0\right )$ reaches $8\times 10^{-6}$ at $\Delta =0$. While there is a wide and deep dip on the curve of $g^{\left ( 2\right ) }\left ( 0\right )$ for the J-C model with TPD. The minimum value is the same as that of the empty cavity with TPD. This means that the antibunching effect of the J-C model with TPD is more robust than that of the empty cavity with TPD when the detuning changes. This feature makes the J-C model with TPD more suitable as a single-photon source with high quality.
Fig. 5. Second-order correlation function $g^{\left ( 2\right ) }\left ( 0\right )$ as a function of cavity detuning $\Delta$ for an empty cavity model (red line) and the J-C model (blue dashed line) with TPD and $ \delta =g^2/\Delta$. The other parameters are considered as $g=10 \gamma$ and $E=0.001 \gamma$.
Finally, we study the time-dependent photon number probability. The zero-photon probability $P_{0}\simeq \rho _{00}$ and single-photon probability $P_{1}\simeq \rho _{22}$ as a function of time are shown in Fig. 6. The probability oscillates with time, and the changes of the two probabilities are opposite, but the sum of the two probabilities is approximately $1$. Here, we pay more attention to the change of the single-photon probability $P_{1}$ with time. In Fig. 6, in one cycle, the single-photon probability starts from $0$, increases to the maximum value of $0.996$ at time $Et=\pi$, and then decreases to $0$ at time $Et=2\pi$. Thus, we can implement a $\pi$-pulsed laser exciting the system to prepare the single-photon source.
Fig. 6. The evolution of the zero-photon probability $P_{0}$ and one-photon probability $P_{1}$ with time. The other parameters are taken as $ \kappa = \gamma,g=10 \gamma,E=0.001 \gamma, \delta =2000 \gamma$ and $\Delta =0.05 \gamma$.
4. Conclusion
In this work, we investigated the quantum statistics of the photon field in a J-C model with TPD. With the wave function method and the density matrix method, we present the asymptotic analytical expressions of the second-order correlation function for small and large atomic detuning regimes, respectively. Compared with the J-C model with SPD, the second-order correlation function in the J-C model with TPD can reach a smaller value with several orders of magnitude when the system operates at the optimal condition for CPB. Especially, the average photon number can reach $0.5$, when the second-order correlation function assumes the minimum value. This feature makes the J-C model with TPD a good candidate for high-quality single photon sources.
A. Appendix
In the weak-driving limit, we expand the density matrix of the system in terms of bare states up to two-photon excitation in the Hilbert subspace; thus, the density operator $\rho$ can be expressed as
Here, we assume that $\left \vert 0\right \rangle$ corresponds to the state $\left \vert 0,g\right \rangle$, $\left \vert 1\right \rangle$ corresponds to the state $\left \vert 0,e\right \rangle$, $\left \vert 2\right \rangle$ corresponds to the state $\left \vert 1,g\right \rangle$, $\left \vert 3\right \rangle$ corresponds to the state $\left \vert 1,e\right \rangle$ and $\left \vert 4\right \rangle$ corresponds to the state $\left \vert 2,g\right \rangle$. Then, the density matrix element is expressed as $\rho _{mn}=\left \langle m\left \vert \rho \right \vert n\right \rangle$, e.g., $\rho _{0,0}=\left \langle 0\left \vert \rho \right \vert 0\right \rangle =\left \langle 0,g\left \vert \rho \right \vert 0,g\right \rangle$, $\rho _{0,1}=\left \langle 0\left \vert \rho \right \vert 1\right \rangle =\left \langle 0,g\left \vert \rho \right \vert 0,g\right \rangle$. By using a perturbation method, that is, discarding the higher-order terms in the equations for the lower-order variables and assuming $\left \{ \dot {\rho } _{m,n}\right \} =0$ and $\left \{ \rho _{00},\rho _{22}\right \} \gg \left \{ \rho _{11},\rho _{44}\right \} \gg \left \{ \rho _{33}\right \}$ under the weak-driving limit, we finally obtain the coupled equations of the density matrix elements as follows:
If we solve the coupled equations directly, the expression will become highly complex. Considering that our purpose is to find the asymptotic expressions of the second-order correlation function in the large atomic detuning regime under the CPB optimal condition $g^{2}=\delta \Delta$, we only present the asymptotic analytical solution of the density matrix elements in the case of large atomic detuning. Therefore, after an approximate treatment, we obtain the asymptotic solution in the large atomic detuning regime under the CPB optimal condition $g^{2}=\delta \Delta$:
Funding
Major Discipline Academic and Technical Leaders Training Program of Jiangxi Province (20204BCJ23026); Natural Science Foundation of Jiangxi Province (20212BAB201018); National Natural Science Foundation of China (11664014, 11964014, 12064018).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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