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Abstract
Herein, we propose a coupled Jaynes-Cummings model for the preparation of strong antibunched single photons and antibunched correlated photon pairs. Using the effective Hamiltonian method, we obtained the expression for the correlation function and then presented the optimal conditions for conventional/unconventional photon blockade. The results showed that on one hand, an intersection point exists between conventional photon blockade and unconventional photon blockade and that the performance of the single photon at the intersection point is better. On the other hand, under the condition of unconventional photon blockade, the photons produced by each Jaynes-Cummings system are strongly correlated with each other.
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1. Introduction
Non-classical light sources, such as single-photon sources and correlated(entangled) photon pairs, are widely applied in the fields of quantum information processing, quantum computation, quantum simulation, and such other fields [1–4]. To prepare these quantum states, many schemes have been proposed [5,6]; among them, the scheme based on photon blockade (PB) effect [7] has enormous potential. A single PB indicates that the photon that enters the cavity will block the next photon from entering the cavity, until this photon radiates out of the cavity, so that photons can be generated one by one. PB effect can be divided into two types. One is the use of input light to resonate with a specific energy level and suppress non-specific energy levels, which is known as the conventional photon blockade (CPB). The other, named unconventional photon blockade (UCPB), is the use of quantum interference effects between certain energy levels to suppress the population of specific energy levels [8,9]. The statistics of the light field induce an antibunching phenomenon under the effect of the PB mechanism. Therefore, PB effect can be used to generate an antibunched light source or a single-photon source. Usually, the single-photon source, generated based on CPB, has the advantage of high brightness but has low purity. Contrarily, the one based on UCPB has a high purity but a low brightness. Combining the advantages of both CPB and UCPB to generate single-photon source with higher purity and brightness is a pertinent topic for discussion [10].
In terms of theory and experiment, various systems are used to realize the PB, such as cavity QED systems, superconducting circuit systems, and optomechanical systems. [11–15]. In recent years, single-PB has been promoted in addition to multi-PB [16,17]; moreover, phonon blockade and nonreciprocal PB have been developed. Naturally, we can generalize the concept of a single-/multi-PB effect to a two-/multi-mode photon pair blockade.
When two photons show a strong bunching effect, they are considered a pair. Correlated photons pairs from parametric down-conversion [18–20] to four-wave mixing [21–23]are quantum light sources. It is due to the inherent nonlinear effect of the nonlinear medium that the output light can be converted into a pair of photons emitted at the same time. Although the photon pairs prepared by these schemes have strong correlation, the generated pairs are bunched. In addition, many other types of light sources provide different solutions, such as four-wave mixing in cold atom ensemble [24], biexciton-exciton radiative cascade of a single semiconductor quantum dot [25–27], superconducting circuit system [28], and Stokes-anti-Stokes process of Raman effect [29,30] achieved based on the PB effect in three-wave mixing [31]. Therefore, it is required to explore more quantum systems to realize two-/multi-mode photon pair blockade.
In this work, we studied the quantum statistics of two coupled Jaynes-Cummings (JC) models. An intersection point was determined between CPB and UCPB, where the advantages of the CPB and UCPB could be combined such that the generated photon source has high purity and high brightness simultaneously. Additionally, we found that the antibunched two-mode photon-pair state can be generated with this model. The photons from two independent JC models are independent; hence, the correlation between them obeys the Poisson distribution. When two JC models are coupled together through the tunneling between cavities using suitable parameters, we expect to obtain a strong correlation between the two cavity fields and antibunching for each of them.
The remainder of this paper is organized as follows. In Section 2, we present the effective Hamiltonian to describe the bi-driven coupled JC model system and the coefficients of the wave function. In Section 3, we study the physical mechanism of single-photon and two-mode photon-pair blockade using the wavefunction approximation approach and provide the optimal conditions for strong photon/photon-pair antibunching. Finally, we conclude this paper in Section 4.
2. Model
We studied a physical model composed of a double JC model, in which two cavities are coupled together, as shown in Fig. 1. For the sake of simplicity, we defined the parameters of the double JC model system to be the same. The resonance frequency of the cavity is $\omega _{a}$ and that of the two-level atom is $\omega _{0}$. In the frame rotating at the external driving laser frequency $\omega _{L}$, the system Hamiltonian can be described by $\left ( \hbar =1\right ) ,$
Fig. 1. Schematic for coupled identical JC models, driven by two external laser fields with the same frequencies and amplitudes, respectively. The tunneling strength between the two cavities is $J$.
To explore the quantum statistics of the photons with modes $a$ and $b$, we used the state vector method to calculate the state probability amplitude in the steady state and then derived the expressions for the correlation function and the average photon number of the cavity modes. More realistically, atomic spontaneous emission rate $\gamma$ and cavity leakage rate $\kappa$ were considered by the non-Hermitian Hamiltonian $\hat {H} _{non}=\hat {H}-\frac {i\kappa }{2}\left ( \hat {a}^{\dagger }\hat {a}+\hat {b} ^{\dagger }\hat {b}\right ) -\frac {i\gamma }{2}\left ( \hat {\sigma }_{1}^{+}\hat { \sigma }_{1}+\hat {\sigma }_{2}^{+}\hat {\sigma }_{2}\right )$. The state function of the system was set as $\left \vert \psi \right \rangle =\sum c_{nmjk}\left \vert nmjk\right \rangle$ with probability amplitude $c_{nmjk}$. Here $n$ and $m$ denote the number of photons occupied by cavities A and B, respectively; $j$ and $k$ represent the states of atoms $1$ and $2$, respectively, which can be ground state $\left \vert g\right \rangle$ or excited state $\left \vert e\right \rangle$. In the weak driving limit, i.e., $E\ll \left \{ \kappa,\gamma \right \}$, we expanded the wave function of the system in terms of the bare states up to two-photon excitation. After a few calculations, we determined the coefficients $c_{1,0,g,g}$, $c_{2,0,g,g}$ and $c_{1,1,g,g}$ as following:
3. Result and analysis
3.1 Antibunching of single photon
Usually, the equal-time second-order auto-correlation function $g^{(2)}(0)$ is employed to measure the quantum statistics of the cavity modes when the system is in a steady state. If $g^{(2)}(0)<1$, the photons in the cavities are antibunched. In the steady state (i.e., $t\rightarrow +\infty$), the equal-time auto-correlation function of the photons in cavities A can be expressed as $g_{a}^{\left ( 2\right ) }\left ( 0\right ) =\left \langle \hat {a} ^{\dagger }\hat {a}^{\dagger }\hat {a}\hat {a}\right \rangle /\left \langle \hat {a }^{\dagger }\hat {a}\right \rangle ^{2}$. Substituting the wave function into the auto-correlation functions expression, the expression of auto-correlation function approximately equals to
(1-1) $g^{2}-\delta ^{\prime }\left ( J+\Delta ^{\prime }\right ) =0$. This condition can be divided into a set of equations, according to the real and imaginary parts:
Obviously, the above two equations cannot be true simultaneously. However, in the strong coupling limit, $g\gg \{\kappa,\gamma \}$, only condition ( 6) can minimize the auto-correlation function and the corresponding value is much less than $1$. Meanwhile, we observe that the mean photon number(2-1) $g^{2}+\delta ^{\prime }\left ( J-\Delta ^{\prime }\right ) =0$. Comparing with the expression of $c_{20gg}$, we found that this condition can lead to the coefficient $c_{20gg}$ vanishing. We rewrite this condition according to the real and imaginary parts:
The above two equations cannot be true simultaneously. In the case of $g\gg \{\kappa,\gamma \}$, the auto-correlation function can take the minimized value less than $1$ when condition (9) is satisfied. However, this antibunched feature attributes to UCPB and exists only in the strong coupling condition.(2-2) $\left ( \delta ^{\prime }+\Delta ^{\prime }\right ) \left ( g^{2}+\delta ^{\prime }\left ( \delta ^{\prime }+\Delta ^{\prime }\right ) \right ) -\delta ^{\prime }J^{2}=0$. Similar to the previous condition, this condition also leads to the coefficients $c_{20gg}$ perfectly vanishing. Hence, the antibunching feature for this condition can be attributed to UCPB. Contrary to the above case, here the optimal condition can hold for both strong and weak coupling regimes. Additionally, if we neglect the dissipation, the optimal condition can reduce to
To verify the results discussed above, we numerically solved the quantum master equation and then calculated the correlation function and average photon number when the system was in the steady state. Figures 2(a) and (c) show the numerical results of the auto-correlation function and average photon number with the functions of atomic and cavity detuning. The coupling strength and tunneling strength are set as $g=10\gamma$ and $J=22\gamma$, respectively. The blue region in Fig. 2(a) indicates that the auto-correlation function is less than $1$, in which the cavity modes $a$($b$) show a sub-Poisson distribution or an antibunching behavior. The figure shows several deep blue structures corresponding to strong antibunching. To clarify the feature of these antibunching structures, we also plotted the optimal conditions [Eqs. (6,7,9–11)] with solid lines in Fig. 2(b). The optimal condition curves perfectly match the strong antibunching structures and can be attributed to CPB and UCPB. Interestingly, there is one intersection point (labeled with a thick dashed black circle) between the antibunching structures of CPB and UCPB, where the auto-correlation function is the smallest in the regions of the given parameters.
Fig. 2. (a) Auto-correlation functions $g_{a(b)}^{(2)}(0)$, (b) Optimal conditions for CPB and UCPB, and (c) average photon number $\langle N\rangle$ as a function of cavity detuning $\Delta$ and atomic detuning $\delta$. The atom-field coupling strength is $g=10\gamma$ and the cavity-cavity tunneling strength is $J=22\gamma$. We set the amplitude of the external driving field as $E=0.01\gamma$ and the cavity leakage is $\kappa =\gamma.$ In (b), the red and green dashed lines present the optimal conditions Eqs. (6), (9) and (11), respectively.
To analyze the quantum behavior of the cavity modes $a$($b$) at the intersection point in greater detail, we plotted the auto-correlation function and average photon number when the optimal conditions [Eqs. (6) and (11)] were true; the results are shown in Fig. 3 with solid lines. For comparison, we also plotted the auto-correlation function and average photon number of the single JC model, with the same parameters, in Fig. 3, which is indicated by circles. We can see that there is a sharp dip in the curve of the auto-correlation function at $\delta \simeq 3.3\gamma$ in the case of CPB. The value of the auto-correlation function at the dip in the double JC model is three orders of magnitude smaller than that of the single JC model. Meanwhile, the average photon number is the same for the single and double JC models. This result implies that the antibunching feature of CPB is greatly enhanced for the double JC model with no loss of big average photon number. In the UCPB case, as shown in Fig. 3(b), there is a wide and deep dip on the curve of the auto-correlation function at $\delta \simeq 3.3\gamma$, corresponding to the intersection point of CPB and UCPB. The minimal value of the auto-correlation function in the coupled JC model is smaller than that in the single JC model because the coupled JC model has more interference pathways. Especially, the average photon number is resonantly enhanced at the dip of the auto-correlation function curve in the coupled JC model compared to the single JC model. Thus, we can conclude that at the intersection point, the advantages of both CPB and UCPB can be achieved. In other words, if the system operates at the intersection point, the generated photon fields have stronger antibunching (purity) and a larger average photon number (brightness). This significant feature can help the PB effect to contribute to a new practical solution to the quantum light source.
Fig. 3. Auto-correlation functions $g^{(2)}_{a(b)}(0)$ and average photon number $\langle N\rangle$ as a function of atomic detuning $\delta$ with the optimal conditions for (a) CPB and (b) UCPB. The atom-field coupling strength is $g=10\gamma$ and the cavity-cavity tunneling strength is $J=22\gamma$. The figure shows the comparison of a single JC model and double JC model with the same parameters. Other parameters are the same as that in Fig. 2.
3.2 Antibunching of two-mode photon pair
In this section, we will discuss the quantum correlation between cavity modes $a$ and $b$ when they exhibit strong antibunching. To measure the correlation between photons in modes $a$ and $b$, we present the cross-correlation functions of the photons in two cavities: $g_{ab}^{\left ( 2\right ) }\left ( 0\right ) =\left \langle \hat {a}^{\dagger }\hat {b}^{\dagger } \hat {b}\hat {a}\right \rangle /\left ( \left \langle \hat {a}^{\dagger }\hat {a} \right \rangle \left \langle \hat {b}^{\dagger }\hat {b}\right \rangle \right )$. If $g_{ab}^{\left ( 2\right ) }\left ( 0\right ) >1$, we observe photons simultaneously in modes $a$ and $b$ with greater probability, vice versa. Substituting the wave function into the cross-correlation functions expression, we arrive at $g_{ab}^{\left ( 2\right ) }\left ( 0\right ) \simeq \left \vert c_{11gg}\right \vert ^{2}/\left \vert c_{10gg}c_{01gg}\right \vert ^{2}$.
First, we plotted the cross-correlation function when the CPB optimal condition [Eq. (6)] was true, as shown in Fig. 4(a). We can see that in the region of photon antibunching, the cross-correlation function is smaller than $1$. Hence, under the CPB optimal condition, the photons in modes $a$ and $b$ are not cross-correlated. In other words, the photons in cavity modes $a$ and $b$ cannot simultaneously exist or they tend to repel one another.
Fig. 4. Correlation function and average photon number as function of $\delta$. The blue solid lines represent the auto-correlation function. The red dashed lines are the cross-correlation function. The black dash-dotted lines are the average photon number. The coupling strengths and drive strength settings are consistent with the above figure. The cavity detuning satisfies different conditions Eq. (6) in (a), Eq. ( 9) in (b), Eq. (12) in (c) and Eq. ( 13) in (d). Other parameters are the same as that in Fig. 2.
Now we consider the case when the optimal conditions for UCPB are satisfied. Figure 4(b) shows that under the optimal condition [Eq. (9)], the cross-correlation function is greater than $1$ and smaller than $10$, while the auto-correlation function is considerably less than $1$ in the given parameter region; this indicates that antibunched photons $a$ and $b$ are weakly correlated in the given parameter region.
Next, we consider another optimal condition [Eq. (11)] for UCPB. To explicitly show the optimal condition between atomic and cavity detunings, we rewrite it as follows:
andFigures 4(c) and 4(d) illustrate the correlation function with the optimal conditions [Eq. (12)] and [Eq. (13)], respectively. Interestingly, we can see that at zero atomic detuning, the cross-correlation functions are significantly greater than $1$ with small auto-correlations for cavity modes $a$ and $b$, as shown in Figs. 4(c) and 4(d). Hence, antibunched strong correlated photon pair can be formed in this region. Comparing the auto-/cross-correlations under different PB conditions in Fig. 4, we conclude that the antibunched strong correlated photon pair can be realized in the second optimal condition of UCPB.
To clearly illustrate the effect of system parameters on antibunched correlated photon pairs under optimal condition (2-2) for UCPB, we present the explicit expressions of the coupling strength and tunneling strength as:
Fig. 5. Correlation functions and average photon number $\left \langle N\right \rangle$ as a function of atom detuning $\delta$ in (a) and cavity leakage $\kappa$ in (b). The tunneling strength $J$ and coupling strength $g$ satisfy the optimal condition (14) and (15), respectively. In (a) $\Delta =10\gamma$ and $\kappa =\gamma$, and in (b) $\Delta =\delta =10 \gamma$. Other parameters are the same as that in Fig. 2.
In the above discussion, we only analyzed the auto-/cross-correlation function behavior of cavity modes $a$ and $b$ when $\Delta \neq 0$ and the coupling strength obtained meets the optimal conditions Eq. (14) and Eq. (15). Now, we focus on the special case of $\{\Delta,\delta \}=0$, that is, the transition frequency of the two-level atom and the eigenfrequency of the cavity equal to that of the laser light field. Under this condition, the optimal condition $\left ( \delta ^{\prime }+\Delta ^{\prime }\right ) \left ( g^{2}+\delta ^{\prime }\left ( \delta ^{\prime }+\Delta ^{\prime }\right ) \right ) -\delta ^{\prime }J^{2}=0$ reduces to
Figure 6(a) shows the auto-/cross-correlation function as a function of $g$ while the tunneling strength $J$ satisfies the optimal condition [16 ]. We can see that the auto-/cross-correlation function monotonically increases with increasing coupling strength $g$ while the average photon number decreases. Hence, a large coupling strength is helpful to generate stronger and brighter correlated photon pairs. However, when $g>3.53\gamma$, the auto-correlation function is greater than $1$ and the photon pair is bunched. Therefore, generating a strong antibunched correlated photon pair requires a coupling strength smaller than $3.53\gamma$. For example, when $g=1.5\gamma$, $g_{a(b)}^{\left ( 2\right ) }\left ( 0\right ) =0.04$, $g_{ab}^{\left ( 2\right ) }\left ( 0\right ) =57$. Figure 6(b) presents the dependence of the auto-/cross-correlation function on the leakage rate of the cavity. We observe that both auto- and cross-correlation functions monotonically decrease with the increasing leakage rate of the cavity, while the average photon number is almost unchanged. This indicates that a relatively larger leakage rate contributes to a stronger correlation between photons $a$ and $b$ in a pair, at the cost the antibunching of the photon pair, which is different from the case of nonvanishing cavity detuning. When $\kappa >1.83\gamma$, the auto-correlation function is smaller than $1$ and the photon pair exhibits a strong antibunching and correlation. If we choose a leakage rate of $\kappa =15\gamma$, then $g_{a(b)}^{\left ( 2\right ) }\left ( 0\right ) =0.0032$, $g_{ab}^{\left ( 2\right ) }\left ( 0\right ) =95$.
Fig. 6. Correlation functions and average photon number $\left \langle N\right \rangle$ as function of atom-cavity coupling strength $g$ in (a) and of the cavity leakage $\kappa$ in (b) with the condition of the cavity and atom resonant with the driving light, i.e., $\Delta =\delta =0.$ In the plot, the tunneling strength $J$ is giving by Eq. (16). $g=5\gamma$ in (b). Other parameters are the same as that in Fig. 2.
4. Conclusion
We proposed a scheme of preparing a strong antibunched single-photon state and antibunched correlated two-mode photon states with a coupled J-C model based on PB effect. The results show that in the strong coupling regime $\left ( g,J\gg \gamma,\kappa \right )$, the auto-correlation function for both cavity modes exhibits both the CPB and UCPB effects and that these single-photon blockade conditions intersect. A comparison with the auto-correlation function and average photon number of the single JC model indicates that the coupled JC model has a lower auto-correlation function at the intersection point, that is, a single-photon source with higher purity and brightness can be realized. To prepare correlated two-mode photon pairs, we found that the strong cross-correlation function of $a$ and $b$ modes can be realized under UCPB condition. When the coupling strength and tunneling strength are sufficient, the photon pairs with high purity and correlation can be realized. In particular, when the cavity and atomic detuning are $0$, we found that the generation of photon pairs with high purity and correlation requires a relatively large cavity leakage.
Funding
National Natural Science Foundation of China (11664014, 11964014, 12064018); Major Discipline Academic and Technical Leaders Training Program of Jiangxi Province (20204BCJ23026); Natural Science Foundation of Jiangxi Province (20212BAB201018).
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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