Expand this Topic clickable element to expand a topic
Skip to content
Optica Publishing Group
Journal Home About Issues in Progress Current Issue All Issues Feature Issues

Strong antibunching effect under the combination of conventional and unconventional photon blockade

Hongyan Zhu, Xiaomiao Li, Zigeng Li, Fan Wang, and Xiaolan Zhong

Open Access Open Access

Abstract

Photon blockade (PB), an effective method of generating antibunching effect, is a critical way to construct a single photon source. The PB effect can be divided into conventional PB effect (CPB) and unconventional PB effect (UPB). Most studies focus on designing systems to successfully enhance CPB or UPB effect individually. However, CPB extremely depends on the nonlinearity strength of the Kerr materials to achieve strong antibunching effect while UPB relies on quantum interference beset with the high probability of the vacuum state. Here, we propose a method to utilize the relevance and complementarity of CPB and UPB to realize these two types simultaneously. We employ a hybrid Kerr nonlinearity two-cavity system. Because of the mutual assistance of two cavities, CPB and UPB can coexist in the system under certain states. In this way, for the same Kerr material, we reduce the value of the second-order correlation function due to CPB by three orders of magnitude without losing the mean photon number due to the presence of UPB, so the advantages of both PB effects are fully reflected in our system, which is a huge performance boost for single photons.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

High-quality and highly deterministic single photon source is an important prerequisite for realizing optical quantum computation and secure quantum teleportation [1–8]. To make a good single photon source, one has to ensure the excitation of the single-photon states while suppression the multi-photon states. Based on the above considerations, the photon blockade (PB) effect is fast becoming a key method, which is an effective way to produce an antibunching effect [9,10], because it prevents the production of the second photon. At the same time, the pure quantum effect that makes the Boson exhibit the exclusion properties similar to Fermion is of great significance in the field of fundamental physics and quantum science and technology [11–42].

There are two main schemes to realize a strong PB effect: conventional PB effect (CPB) [11] and unconventional [27] PB effect (UPB). What we called CPB is that strong nonlinear interaction between polaritons can lead to a quantum anharmonic ladder in the energy spectrum. And only one photon is allowed in the system due to the eigenenergy-level anharmonicity. A. Imamoḡlu et al. first present the PB effect with nonlinear material [11]. In the subsequent studies, most nonlinear materials used are Kerr materials. The main challenge faced by many experiments is the selection of materials. The stronger the nonlinearity of Kerr material is, the better the PB effect will be, which also means that the selection of materials is more demanding [33].

The physical mechanism of UPB owing to quantum interference between different paths will be realized mainly by introducing additional degrees of freedom [27], such as multi-cavity coupling [29], and multi-emitter coupling [32]. For the system, the UPB effect can inhibit the multi-photon state, but its ability to excite single-photon states is weak, and the probability of the vacuum state will increase, which is not favorable for the construction of a single photon source.

Both the CPB and the UPB have made great progress in theory and experiment [24–30], however, their shortcomings are still unavoidable. Inhibition of multi-photon states produced by the CPB effect and excitation of single-photon states produced by the UPB effect had been a largely under-explored domain.

In this paper, we propose a method to combine the two types of PB, so that we can make them complement each other. We theoretically provide a new physical model, which is a hybrid two-cavity system with Kerr nonlinearity. Under specific parameter conditions, we can achieve CPB and UPB effects simultaneously. By combining the two types of PB effects, we can get the minimum value of the second-order correlation function, due to the suppression of multi-photon states. This new method of combining CPB and UPB schemes exhibits a very strong PB effect, having three orders of magnitude reduction on second-order correlation function and yet retaining the high mean photon number, which is undoubtedly a significant improvement in the performance of a single photon.

The remainder of this paper is arranged in three parts as follows. In Sect. 2, we analytically solve the CPB and UPB effects of the system. In Sect. 3, we consider the effect of the Kerr strength ratio of the two cavities on the PB effect of cavity A. Finally, we conclude the whole work in the Sect. 4.

2. Model and analysis

Our model, shown in Fig. 1, is a hybrid two-cavity system formed by coupling two cavities filled with Kerr material. The frequency of cavity A (B) is ${\omega _a}$(${\omega _b}$), and the dissipation is ${\kappa _1}$(${\kappa _2}$). The frequency of the external driving field is ${\omega _{La}}$(${\omega _{Lb}}$) and the driving strength is ${\varepsilon _1}$(${\varepsilon _2}$). The parameter J denotes the strengths of the photon hopping interaction between two cavities. And note that our system is in a strong coupling regime. ($\kappa \ll J$)

 figure: Fig. 1.

Fig. 1. The hybrid two-cavity system. The dissipation of cavity A (B) is ${\kappa _1}$(${\kappa _2}$). The driving strength of the external driving field is ${\varepsilon _1}$(${\varepsilon _2}$). $\chi _A^{(3 )}$($\chi _B^{(3 )}$) is the Kerr nonlinear material filled in cavity A(B).

Download Full Size | PDF

Under the rotating frame with respect to ${U_L} = {\omega _{La}}{a^ + }a + {\omega _{Lb}}{b^ + }b$, the Hamiltonian of the system can be written as (setting $\hbar = 1$):

$$\begin{aligned} H = {\Delta _1}{a^ + }a + {\Delta _2}{b^ + }b + J({{a^ + }b + {b^ + }a} )+ {U_1}{a^ + }{a^ + }aa + {U_2}{b^ + }{b^ + }bb\\ + {\varepsilon _1}({{a^ + } + a} )+ {\varepsilon _2}({{b^ + } + b} ) \end{aligned}$$
where ${\Delta _1} = {\omega _a} - {\omega _{La}}$, ${\Delta _2} = {\omega _b} - {\omega _{Lb}}$. The parameter ${U_i}{\; }({i = 1,2} )$ represents the Kerr nonlinearity for the nonlinear cavity. ${U_i}$ is influenced by the effective volume of the cavity and the nonlinearity of the material filled in the cavity. For the same Kerr material, the smaller the effective volume of the cavity, the larger value of ${U_i}$; while for the same cavity, the strong nonlinearity of the material filling the cavity means a large value of ${U_i}$. The common Kerr materials such as SiO2 and Si are described in detail in Ref. [33]. The wave function of the system can be written as:
$$\psi = {C_{00}}|00\rangle + {C_{01}}|01\rangle + {C_{10}}|10\rangle + {C_{20}}|20\rangle + {C_{02}}|02\rangle + {C_{11}}|11\rangle$$

Considering the damping of this system, the effective Hamiltonian can be written as: ${H_{eff}} = H - \frac{i}{2}{\kappa _1}{a^ + }a - \frac{i}{2}{\kappa _2}{b^ + }b$.

Use the Schrödinger equation, we can get the equations:

$$i\dot{C_{01}} = \widetilde {{{\Delta }_2}}{C_{01}} + J{C_{10}} + {\varepsilon _1}{C_{11}} + {\varepsilon _2}\left( {\sqrt 2 {C_{02}} + {C_{00}}} \right)$$
$$i\dot {{C_{10}}} = \widetilde {{\Delta _1}}{C_{10}} + J{C_{01}} + {\varepsilon _2}{C_{11}} + {\varepsilon _1}\left( {\sqrt 2 {C_{20}} + {C_{00}}} \right)$$
$$i\dot {{C_{20}}} = 2\widetilde {{\Delta _1}}{C_{20}} + \sqrt 2 J{C_{11}} + 2{U_1}{C_{20}} + \sqrt 2 {\varepsilon _1}{C_{10}}$$
$$i\dot {{C_{02}}} = 2\widetilde {{\Delta _2}}{C_{02}} + \sqrt 2 J{C_{11}} + 2{U_2}{C_{02}} + \sqrt 2 {\varepsilon _2}{C_{01}}$$
$$i\dot {{C_{11}}} = ({\widetilde {{\Delta _1}} + \widetilde {{\Delta _2}}} ){C_{11}} + \sqrt 2 J({{C_{20}} + {C_{02}}} )+ {\varepsilon _1}{C_{01}} + {\varepsilon _2}{C_{10}}$$
where $\widetilde {{\Delta _1}} = {\Delta _1} - i\frac{{{\kappa _1}}}{2}$, $\widetilde {{\Delta _2}} = {\Delta _2} - i\frac{{{\kappa _2}}}{2}$.

Under the conditions of weak driving limit [43], the coefficients satisfy the relationship $|{{C_{00}}} |\approx 1 \gg |{{C_{10}}} |,|{{C_{01}}} |\gg |{{C_{20}}} |,|{{C_{02}}} |,|{{C_{11}}} |$. Therefore, Eq. (3a) and Eq. (3b) simplify to:

$$i\dot {{C_{01}}} = \widetilde {{\Delta _2}}{C_{01}} + J{C_{10}} + {\varepsilon _2}$$
$$i\dot {{C_{10}}} = \widetilde {{\Delta _1}}{C_{10}} + J{C_{01}} + {\varepsilon _1}$$

Since the difference between cavity A and cavity B is only the difference in parameters, we also study cavity B when we study cavity A. By solving the steady-state equations, the second-order correlation function of cavity A can be calculated as follows:

$$ g_a^{(2)}(0)=\frac{\left(-\widetilde{\Delta_1} \widetilde{\Delta_2}+J^2\right)^2 \times A}{\left(\widetilde{\Delta_2} \varepsilon_1-\varepsilon_2 J\right)^4 \times B} $$
where
$$A = {[({\widetilde {{\Delta _1}} + \widetilde {{\Delta _2}}} ){(\widetilde {{\Delta _2}}{\varepsilon _1} - {\varepsilon _2}J)^2} + ({\widetilde {{\Delta _2}}({\widetilde {{\Delta _1}} + \widetilde {{\Delta _2}}} )\varepsilon_1^2 - 2({\widetilde {{\Delta _1}} + \widetilde {{\Delta _2}}} ){\varepsilon_1}{\varepsilon_2}J + ({\varepsilon_1^2 + \varepsilon_2^2} ){J^2}} ){U_2}]^2},$$

$$B = {[ - \widetilde {{\Delta _1}}{J^2} + {\widetilde {{\Delta _2}}^2}{U_1} + {\widetilde {{\Delta _1}}^2}({\widetilde {{\Delta _2}} + {U_2}} )+ \widetilde {{\Delta _1}}({\widetilde {{\Delta _2}} + {U_1}} )({\widetilde {{\Delta _2}} + {U_2}} )- {J^2}({{U_1} + {U_2}} )+ \widetilde {{\Delta _2}}({ - {J^2} + {U_1}{U_2}} )]^2}.$$

The single PB will be observed when $g_a^{(2 )}(0 )< 1$ is satisfied. The closer $g_a^{(2 )}(0 )$ gets to 0, the better single photon performance will be. By setting $g_a^{(2 )}(0 )= 0$, we can easily achieve a series of equations as:

$${J^2} = \widetilde {{\Delta _1}}\widetilde {{\Delta _2}}$$
$$\frac{{({\widetilde {{\Delta _1}} + \widetilde {{\Delta _2}}} ){{(\widetilde {{\Delta _2}}{\varepsilon _1} - {\varepsilon _2}J)}^2} + ({\widetilde {{\Delta _2}}({\widetilde {{\Delta _1}} + \widetilde {{\Delta _2}}} )\varepsilon_1^2 - 2({\widetilde {{\Delta _1}} + \widetilde {{\Delta _2}}} ){\varepsilon_1}{\varepsilon_2}J + ({\varepsilon_1^2 + \varepsilon_2^2} ){J^2}} ){U_2}}}{{{{(\widetilde {{\Delta _2}}{\varepsilon _1} - {\varepsilon _2}J)}^2} + {\varepsilon _1}({\widetilde {{\Delta _2}}{\varepsilon_1} - 2{\varepsilon_2}J} ){U_2}}} = 0$$

In fact, the realization of $g_a^{(2 )}(0 )= 0$ requires to be satisfied the condition of Eq. (6a) or $A$ = 0. We neglect the invalid solution for $A$= 0 and keep only the valid solution (Eq. (6b)). In this way, the existence of two driving fields can be made practically meaningful, while other specific parameters can be made to satisfy the non-zero condition (i.e. the strength of the photon hopping interaction between the two cavities J), which can justify the existence of our hybrid two-cavity system.

2.1 Realized condition of CPB

According to Eq. (6a), the realization of $g_a^{(2 )}(0 )= 0$ requires the real and imaginary parts tend to zero at the same time, that is, the real part is ${\Delta _1}{\Delta _2} - {J^2} - {\kappa _1}{\kappa _2}/4 = 0$, and the imaginary part is ${\Delta _2}{\kappa _1} + {\Delta _1}{\kappa _2} = 0$. Obviously, the real and the imaginary part cannot be zero at the same time. Since our system is strongly coupled ($\kappa \ll J$), we try to lower the value of second-order correlation function by setting the real part to zero.

Considering the cases of ${\Delta _1} = {\Delta _2} = \Delta $, ${\kappa _1} = {\kappa _2} = \kappa $, Eq.  6(a) can be written as ${J^2} \approx {\Delta ^2}$. Under these conditions, the value of the second-order correlation function can be effectively reduced at $J ={\pm} \Delta $.

The mean number of photons in cavity A is:

$$\langle{a^ + }a\rangle \approx {|{{C_{10}}} |^2} = {(\frac{{ - \Delta {\varepsilon _1} + {\varepsilon _2}J}}{{ - {\Delta ^2} + {J^2}}})^2}$$

From Eq. (7), we can find that the mean photon number of cavity A $\langle{a^ + }a\rangle$ reach its local maximum value at $J \approx{\pm} \Delta $. As a result, the condition of $J \approx{\pm} \Delta $ plays an important role in the increase of the mean photon number while decreasing the value of the second-order correlation function at this point, which means that the enhancement of the PB effect is induced by increasing the excitation of single-photon states in cavity A. So, the above calculation results indicate that CPB occurs.

Since the mean number of photons at $J ={-} \Delta $ is always greater than at $J = \Delta $, for practical purposes, we can only consider the condition of $J ={-} \Delta $.

2.2 Realized condition of UPB

We expand the real and imaginary parts of Eq. (6b) and make the real part equal to 0, which will reduce the second-order correlation function value more effectively. By solving Eq. (6b), we can obtain the following conditions as:

$${U_2} ={-} \frac{{2\Delta {{({\Delta {\varepsilon_1} - {\varepsilon_2}J} )}^2}}}{{2{\Delta ^2}\varepsilon _1^2 - 4\Delta {\varepsilon _1}{\varepsilon _2}J + ({\varepsilon_1^2 + \varepsilon_2^2} ){J^2}}}$$
$${\varepsilon _1} = {\varepsilon _2},{\; }J = \Delta $$
$${\varepsilon _1} ={-} {\varepsilon _2},{\; }J = -\Delta $$

We ignore Eq. (8b) and Eq. (8c), even though they can effectively reduce the value of the second-order correlation function, the mean photon number at these conditions cannot be the global maximum value.

From Eq. (8a), we can see that the value of $g_a^{(2 )}(0 )$ can be reduced by ${U_2}$, but ${U_2}$ makes little influence on the increase of the mean photon number (Eq. (7) does not include ${U_2}$). This implies that the enhancement of the PB effect by ${U_2}$ is not by increasing the probability of single-photon states in cavity A, but by suppressing the excitation of multi-photon states in cavity A through the Kerr nonlinearity of cavity B. Therefore, Eq. (8a) corresponds to the occurrence of the UPB effect in the system at this time.

2.3 Condition of combining CPB and UPB

The CPB effect corresponds to improving the probability of a single-photon state in the system, while the UPB effect corresponds to reducing the probability of two- and multi-photon states. In principle, these two types can complement each other perfectly. In order to obtain a stronger PB effect, we combine the CPB effect with the UPB effect. The essence of the UPB effect is the interference between different quantum transition paths. The transition probability of our system depends on the drive ratio n of the two driving fields. According to Ref. [40], we know that the different drive ratios will affect the UPB effect. Therefore, we rewrite Eq. (8a) with drive ratio n ($n = {\varepsilon _1}/{\varepsilon _2}$) and total driving strength I (I=${\varepsilon _1} + {\varepsilon _2}$), as follows:

$${U_2} ={-} \frac{{2\Delta {{(n\Delta - J)}^2}}}{{{J^2} - 4\Delta nJ + ({{J^2} + 2{\Delta ^2}} ){n^2}}}$$

We also consider position $J ={-} \Delta $, then the condition will be simplified to:

$${U_2} \approx \frac{{2({1 + n} )}}{{1 + 3n}}J$$

According to Eq. (10), we can simultaneously produce CPB and UPB.

From Fig. 2, one can achieve this combining of these two different types of PB effect. Figure 2(a) only satisfies the solution of CPB, while Fig. 2(b) satisfies both CPB and UPB solutions at $\Delta ={-} J$. The value of the second-order correlation function generated by the combination of CPB and UPB effects is ${10^{ - 8.5}}$, which is two orders of magnitude lower than the value ${10^{ - 5.441}}$ that satisfies only the CPB condition. At the same time, the mean photon number of both is stabilized at 0.25, so the possibility of the vacuum state being boosted is not high. This is certainly a significant improvement in the single photon performance.

 figure: Fig. 2.

Fig. 2. Use the method of combining conventional and unconventional photon blockade effects to reduce the functional value of the second-order correlation of cavity A. (a) ${\varepsilon _1} = 0.125\kappa $, ${\varepsilon _2} = 0.375\kappa $, considering only the conventional photon blockade effect. (b) ${\varepsilon _1} = {\varepsilon _2} = 0.25\kappa $, use the unconventional photon blockade effect to continue to reduce the value of the second-order correlation function at the position $\Delta ={-} J$. The orange solid line represents the mean photon number in cavity A. The blue solid line represents the second-order correlation function for cavity A. The other parameters are chosen as $J = 10\kappa $, ${U_1} = 2.5\kappa $, ${U_2} = J$.

Download Full Size | PDF

In order to demonstrate the correctness of Eq. (10), we also plot the second-order correlation function and the mean photon number of cavity A with ${U_2}$ and n, as shown in Fig. 3. The white dashed line in Fig. 3(a) indicates curve ${U_2} = \frac{{2({1 + n} )}}{{1 + 3n}}J$ obtained by fitting the minimum of the second-order correlation function corresponding to different n, which is in full agreement with our analytical results. Meanwhile, for the same ${U_2} = J$, we can find a lower value of ${10^{ - 8.662}}$ for the second-order correlation function at n = 0.99, while its mean photon number remains stable at 0.25, as in Fig. 3(b), which also means that we further improve the performance of the single photon generated by this system.

 figure: Fig. 3.

Fig. 3. (a) The value of the second-order correlation function of cavity A with the change of the ${U_2}$ and the drive ratio n. (b)The mean photon number of cavity A varies with ${U_2}$ and the drive ratio n. $I = 0.5\kappa $, $\Delta ={-} J$, other parameters remain unchanged.

Download Full Size | PDF

3. Kerr strength ratio analysis

It is well known that the cavity filled with Kerr material will greatly affect the energy level of the system [33]. Therefore, the Kerr strength ${U_1}$ will affect the second-order correlation function of cavity A even though the simplified results do not include it. In this section, we investigate the effect of the ratio of the Kerr strength in the two cavities on the PB effect in cavity A. Here, we set ${U_1} = Z{U_2}$ and plot the second-order correlation function and the mean photon number as a function of Z, as shown in Fig. 4(a).

 figure: Fig. 4.

Fig. 4. (a)The change of the second-order correlation function and the mean photon number of cavities A and B with Z. The orange solid line (dashed line) is the mean photon number of cavity A (B), and the blue solid line (dashed line) is the second-order correlation function value of cavity A (B). Node 1 (2) is the intersection point of the second-order correlation function (mean photon number) of cavity A and B. n = 1 and other parameters remain unchanged. (b)Energy level diagram of the system. (c)Transition diagram of the system.

Download Full Size | PDF

It is normal that the value of the second-order correlation function of cavity A decreases with increasing Z (the CPB effect will become more pronounced due to the increase of the nonlinear strength). However, it is noteworthy that the value of the second-order correlation function increases with Z (in the part where Z < 1). The second-order correlation function and the mean photon number of cavity B are also plotted in Fig. 4(a). It is interesting to find that two nodes appear at Z = 1. To explain this abnormal phenomenon, we choose ${\omega _a} = {\omega _b}$, n = 1, I = 0.5$\kappa $ and ${U_2} = {U_1} = U = J ={-} \Delta $ to simplify the calculation and then draw the energy level diagram of the system. With these parameter settings, the system can still produce CPB and UPB effects.

Using equation $H|\Psi\rangle = E|\Psi\rangle $, we can obtain:

$${E_{1, \pm }} = {\omega _a} \pm J$$
$$|{1, + }\rangle = \frac{1}{{\sqrt 2 }}({|{10}\rangle + |{01}\rangle } )$$
$$|{1, - }\rangle = \frac{1}{{\sqrt 2 }}({ - |{10}\rangle + |{01}\rangle } )$$
$${E_{2, \pm }} = 2{\omega _a} + U \pm \sqrt {4{J^2} + {U^2}} $$
$${E_{2,0}} = 2{\omega _a} + 2U$$
$$|{2, + }\rangle = \frac{1}{{{C_1}}}\left( {|{20}\rangle +|02\rangle - \frac{{\sqrt 2 U - \sqrt 2 \sqrt {4{J^2} + {U^2}} }}{{2J}}|{11}\rangle } \right)$$
$$|{2,0}\rangle = \frac{1}{{\sqrt 2 }}({ - |{20}\rangle + |{02}\rangle } )$$
$$|{2, - }\rangle = \frac{1}{{{C_2}}}\left( {|{20}\rangle +|02\rangle - \frac{{\sqrt 2 U + \sqrt 2 \sqrt {4{J^2} + {U^2}} }}{{2J}}|{11}\rangle } \right)$$
where ${C_1}$ and ${C_2}$ are the normalizing constants.

When the frequency of the external driving field is $\omega + J$ (i.e. $J ={-} \Delta $), the energy level transition of the system is shown in Fig. 4(b), and this figure also shows the mechanism of the CPB effect in the system. The $|{2,0}\rangle $ state does not contain the $|{11}\rangle $ state, leading to the destruction of the quantum transition path in Fig. 4(c) (i.e. UPB effect), and due to the presence of ${\varepsilon _1} = {\varepsilon _2}$ (i.e., n = 1), the system jumps to the $|{01}\rangle $, $|{10}\rangle $, $|{02}\rangle $, $|{20}\rangle $ states with the same transition probability, forming the two nodes in Fig. 4(a). We can learn that the process of increasing Z from 0 to 1 is the process in which the UPB effect becomes weaker and weaker. As shown in Fig. 4(a), the main function of ${U_1}$ at this time is to enhance the PB effect of cavity B, which can be seen from the decrease of the second-order correlation function of cavity B.

If we choose the optimal Kerr parameter ${U_1}$ and drive ratio n (Z = 0 and n = 0.99), we can reduce the value of the second-order correlation function to ${10^{ - 9.233}}$ while keeping the mean photon number at 0.25, which is certainly a great improvement in the PB effect of this system, as shown in Fig. 5. Note that we do not need to consider the part of Z > 1, because in this range, the decrease in the value of the second-order correlation function is due to the enhancement of the nonlinearity of the Kerr material.

 figure: Fig. 5.

Fig. 5. (a) The value of the second-order correlation function of cavity A with the change of the Z and the drive ratio n. (b)The mean photon number of cavity A varies with Z and the drive ratio n. Other parameters remain unchanged.

Download Full Size | PDF

In Table 1, we calculate the second-order correlation function values under the two models to demonstrate the superiority of our results. The value of the second-order correlation function for a single cavity filled with Kerr material is ${10^{ - 5.941}}$, and our system is ${10^{ - 9.233}}$. Clearly, the combination of CPB and UPB effects can reduce the second-order correlation function of the system more effectively. The value of the second-order correlation function for the same Kerr material can be reduced by three orders of magnitude.

Tables Icon

Table 1. Comparison of conventional and unconventional photon blockade effects

View Table

4. Conclusion

In conclusion, we construct a hybrid two-cavity system in which one cavity is coupled to another cavity filled with Kerr material. Through analytical calculations, we use this hybrid two-cavity system to generate both CPB and UPB effects at the same time. The combination of these two PB effects complements each other so that cavity A achieves a more obvious antibunching effect by simultaneously exciting the single-photon state and suppressing the vacuum and multi-photon states without the need for stronger Kerr strength material, which better reduces the difficulty of Kerr selection in the experiment. By adjusting the drive ratio n and the Kerr strength ratio Z of the hybrid two-cavity system, we determined the optimal parameters n = 0.99 and Z = 0 (i.e., the cavity A is not filled with Kerr material), resulting in a second-order correlation function value for the cavity A that is three orders of magnitude lower than the value brought by the Kerr material alone, with no loss in the mean photon number. There is no doubt that this is an excellent scheme for achieving a high-quality single photon source. Benefiting from the improvement of the microcavity manufacturing technology, our single photon source construction method can be better applied to the study of quantum information processing. It will also have important applications in the field of quantum communication as well as quantum computing.

Funding

Beijing Municipal Natural Science Foundation (4212051); National Natural Science Foundation of China (11804018, 62075004).

Acknowledgments

X. Zhong thanks the National Natural Science Foundation of China and the Beijing Natural Science Foundation for help identifying collaborators for this work.

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 may be obtained from the authors upon reasonable request.

References

1. I. Aharonovich, D. Englund, and M. Toth, “Solid-state single-photon emitters,” Nat. Photonics 10(10), 631–641 (2016). [CrossRef]  

2. Y. M. He, G. Clark, J. R. Schaibley, Y. He, M. C. Chen, Y. J. Wei, X. Ding, Q. Zhang, W. Yao, X. Xu, C. Y. Lu, and J. W. Pan, “Single quantum emitters in monolayer semiconductors,” Nat. Nanotechnol. 10(6), 497–502 (2015). [CrossRef]  

3. T. T. Tran, K. Bray, M. J. Ford, M. Toth, and I. Aharonovich, “Quantum emission from hexagonal boron nitride monolayers,” Nat. Nanotechnol. 11(1), 37–41 (2016). [CrossRef]  

4. E. G. Melo, W. Eshbaugh, E. B. Flagg, and M. Davanco, “Multiobjective Inverse Design of Solid-State Quantum Emitter Single-Photon Sources,” ACS Photonics 10(4), 959–967 (2022). [CrossRef]  

5. P. Senellart, G. Solomon, and A. White, “High-performance semiconductor quantum-dot single-photon sources,” Nat. Nanotechnol. 12(11), 1026–1039 (2017). [CrossRef]  

6. M. Pelton, C. Santori, J. Vučković, B. Zhang, G. S. Solomon, J. Plant, and Y. Yamamoto, “Efficient Source of Single Photons: A Single Quantum Dot in a Micropost Microcavity,” Phys. Rev. Lett. 89(23), 233602 (2002). [CrossRef]  

7. P. Jiang, N. Ma, P. Liu, W. Wu, and K. Zhang, “An easy-implemented on-chip waveguide coupled single photon source based on self-assembled quantum dots membrane,” Appl. Sci. 11(2), 695 (2021). [CrossRef]  

8. N. Luo, M. Zhang, A. Hu, and X. Guo, “Study of single photon source efficiency based on cavity quantum electrodynamics,” Phys. B 612(March), 412945 (2021). [CrossRef]  

9. H. Jabri and H. Eleuch, “Bunching and antibunching in cavity QED,” Commun. Theor. Phys. 56(1), 134–138 (2011). [CrossRef]  

10. H. Eleuch, “Photon statistics of light in semiconductor microcavities,” J. Phys. B: At., Mol. Opt. Phys. 41(5), 055502 (2008). [CrossRef]  

11. A. Imamoglu, H. Schmidt, G. Woods, and M. Deutsch, “Strongly interacting photons in a nonlinear cavity,” Phys. Rev. Lett. 79(8), 1467–1470 (1997). [CrossRef]  

12. R. Huang, K. Özdemir, J. Q. Liao, F. Minganti, L. M. Kuang, F. Nori, and H. Jing, “Exceptional Photon Blockade: Engineering Photon Blockade with Chiral Exceptional Points,” Laser Photonics Rev. 16(7), 2100430 (2022). [CrossRef]  

13. R. Huang, A. Miranowicz, J. Q. Liao, F. Nori, and H. Jing, “Nonreciprocal Photon Blockade,” Phys. Rev. Lett. 121(15), 153601 (2018). [CrossRef]  

14. P. Rabl, “Photon blockade effect in optomechanical systems,” Phys. Rev. Lett. 107(6), 063601 (2011). [CrossRef]  

15. B. Li, R. Huang, X. Xu, A. Miranowicz, and H. Jing, “Nonreciprocal unconventional photon blockade in a spinning optomechanical system,” Photonics Res. 7(6), 630 (2019). [CrossRef]  

16. L. J. Feng, L. Yan, and S. Q. Gong, “Unconventional photon blockade induced by the self-Kerr and cross-Kerr nonlinearities,” Front. Phys. 18(1), 12304 (2023). [CrossRef]  

17. M. Li, Y. L. Zhang, S. H. Wu, C. H. Dong, X. B. Zou, G. C. Guo, and C. L. Zou, “Single-Mode Photon Blockade Enhanced by Bi-Tone Drive,” Phys. Rev. Lett. 129(4), 043601 (2022). [CrossRef]  

18. X. W. Xu, Y. Li, B. Li, H. Jing, and A. X. Chen, “Nonreciprocity via Nonlinearity and Synthetic Magnetism,” Phys. Rev. Appl. 13(4), 044070 (2020). [CrossRef]  

19. H. Z. Shen, Y. H. Zhou, and X. X. Yi, “Tunable photon blockade in coupled semiconductor cavities,” Phys. Rev. A 91(6), 063808 (2015). [CrossRef]  

20. J. Li, R. Yu, and Y. Wu, “Proposal for enhanced photon blockade in parity-time-symmetric coupled microcavities,” Phys. Rev. A 92(5), 053837 (2015). [CrossRef]  

21. J. Li, J. Li, Q. Xiao, and Y. Wu, “Giant enhancement of optical high-order sideband generation and their control in a dimer of two cavities with gain and loss,” Phys. Rev. A 93(6), 063814 (2016). [CrossRef]  

22. Y. Dong, X. Zheng, D. Wang, and J. Ding, “Fluctuation-enhanced Kerr nonlinearity in an atom-assisted optomechanical system with atom-cavity interactions,” Opt. Express 29(4), 5367 (2021). [CrossRef]  

23. Y. F. Jiao, T. X. Lu, and H. Jing, “Optomechanical second-order sidebands and group delays in a Kerr resonator,” Phys. Rev. A 97(1), 013843 (2018). [CrossRef]  

24. K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, “Photon blockade in an optical cavity with one trapped atom,” Nature 436(7047), 87–90 (2005). [CrossRef]  

25. A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vučković, “Coherent generation of non-classical light on a chip via photon-induced tunnelling and blockade,” Nat. Phys. 4(11), 859–863 (2008). [CrossRef]  

26. T. C. H. Liew and V. Savona, “Single photons from coupled quantum modes,” Phys. Rev. Lett. 104(18), 183601 (2010). [CrossRef]  

27. M. Bamba, A. Imamoǧlu, I. Carusotto, and C. Ciuti, “Origin of strong photon antibunching in weakly nonlinear photonic molecules,” Phys. Rev. A 83(2), 021802 (2011). [CrossRef]  

28. O. Kyriienko and T. C. H. Liew, “Triggered single-photon emitters based on stimulated parametric scattering in weakly nonlinear systems,” Phys. Rev. A 90(6), 063805 (2014). [CrossRef]  

29. M. Wang, T.-S. Yin, Z.-Y. Sun, H.-G. Cheng, B.-F. Zhan, and L.-L. Zheng, “Unconventional phonon blockade via atom-photon-phonon interaction in hybrid optomechanical systems,” Opt. Express 30(7), 10251 (2022). [CrossRef]  

30. G. Q. Zhang, Y. P. Wang, and J. Q. You, “Dispersive readout of a weakly coupled qubit via the parity-time-symmetric phase transition,” Phys. Rev. A 99(5), 052341 (2019). [CrossRef]  

31. H. J. Snijders, J. A. Frey, J. Norman, H. Flayac, V. Savona, A. C. Gossard, J. E. Bowers, M. P. Van Exter, D. Bouwmeester, and W. Löffler, “Observation of the Unconventional Photon Blockade,” Phys. Rev. Lett. 121(4), 043601 (2018). [CrossRef]  

32. R. Trivedi, M. Radulaski, K. A. Fischer, S. Fan, and J. Vučković, “Photon Blockade in Weakly Driven Cavity Quantum Electrodynamics Systems with Many Emitters,” Phys. Rev. Lett. 122(24), 243602 (2019). [CrossRef]  

33. S. Ferretti and D. Gerace, “Single-photon nonlinear optics with Kerr-type nanostructured materials,” Phys. Rev. B 85(3), 033303 (2012). [CrossRef]  

34. H. Xie, L. W. He, X. Shang, G. W. Lin, and X. M. Lin, “Nonreciprocal photon blockade in cavity optomagnonics,” Phys. Rev. A 106(5), 053707 (2022). [CrossRef]  

35. H. Jabri and H. Eleuch, “Enhanced unconventional photon-blockade effect in one- and two-qubit cavities interacting with nonclassical light,” Phys. Rev. A 106(2), 023704 (2022). [CrossRef]  

36. Y. T. Guo, F. Zou, J. F. Huang, and J. Q. Liao, “Retrieval of photon blockade effect in the dispersive Jaynes-Cummings model,” Phys. Rev. A 105(1), 013705 (2022). [CrossRef]  

37. L. J. Feng and S. Q. Gong, “Two-photon blockade generated and enhanced by mechanical squeezing,” Phys. Rev. A 103(4), 043509 (2021). [CrossRef]  

38. X. Su, J. S. Tang, and K. Xia, “Nonlinear dissipation-induced photon blockade,” Phys. Rev. A 106(6), 063707 (2022). [CrossRef]  

39. Z. Li, X. Li, and X. Zhong, “Strong photon blockade in an all-fiber emitter-cavity quantum electrodynamics system,” Phys. Rev. A 103(4), 043724 (2021). [CrossRef]  

40. H. Zhu, X. Li, Z. Li, and X. Zhong, “A robust and flexible high-order photon blocking effect based on drive ratio analysis,” Eur. Phys. J. Spec. Top. 231(4), 735–742 (2022). [CrossRef]  

41. H. Eleuch, “Quantum trajectories and autocorrelation function in semiconductor microcavity,” Appl. Math. Inf. Sci. 3(2), 185–196 (2010).

42. Z. G. Li, X. M. Li, G. F. Zhang, and X. L. Zhong, “Realizing strong photon blockade at exceptional points in the weak coupling regime,” Frontiers in Physics 11 (2023).

43. H. J. Carmichael, R. J. Brecha, and P. R. Rice, “Quantum interference and collapse of the wavefunction in cavity QED,” Opt. Commun. 82(1-2), 73–79 (1991). [CrossRef]  

Data availability

Data underlying the results presented in this paper are not publicly available at this time may be obtained from the authors upon reasonable request.

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)
Fig. 1.
Fig. 1. The hybrid two-cavity system. The dissipation of cavity A (B) is ${\kappa _1}$(${\kappa _2}$). The driving strength of the external driving field is ${\varepsilon _1}$(${\varepsilon _2}$). $\chi _A^{(3 )}$($\chi _B^{(3 )}$) is the Kerr nonlinear material filled in cavity A(B).

View in Article | Download Full Size | PDF

Fig. 2.
Fig. 2. Use the method of combining conventional and unconventional photon blockade effects to reduce the functional value of the second-order correlation of cavity A. (a) ${\varepsilon _1} = 0.125\kappa $, ${\varepsilon _2} = 0.375\kappa $, considering only the conventional photon blockade effect. (b) ${\varepsilon _1} = {\varepsilon _2} = 0.25\kappa $, use the unconventional photon blockade effect to continue to reduce the value of the second-order correlation function at the position $\Delta ={-} J$. The orange solid line represents the mean photon number in cavity A. The blue solid line represents the second-order correlation function for cavity A. The other parameters are chosen as $J = 10\kappa $, ${U_1} = 2.5\kappa $, ${U_2} = J$.

View in Article | Download Full Size | PDF

Fig. 3.
Fig. 3. (a) The value of the second-order correlation function of cavity A with the change of the ${U_2}$ and the drive ratio n. (b)The mean photon number of cavity A varies with ${U_2}$ and the drive ratio n. $I = 0.5\kappa $, $\Delta ={-} J$, other parameters remain unchanged.

View in Article | Download Full Size | PDF

Fig. 4.
Fig. 4. (a)The change of the second-order correlation function and the mean photon number of cavities A and B with Z. The orange solid line (dashed line) is the mean photon number of cavity A (B), and the blue solid line (dashed line) is the second-order correlation function value of cavity A (B). Node 1 (2) is the intersection point of the second-order correlation function (mean photon number) of cavity A and B. n = 1 and other parameters remain unchanged. (b)Energy level diagram of the system. (c)Transition diagram of the system.

View in Article | Download Full Size | PDF

Fig. 5.
Fig. 5. (a) The value of the second-order correlation function of cavity A with the change of the Z and the drive ratio n. (b)The mean photon number of cavity A varies with Z and the drive ratio n. Other parameters remain unchanged.

View in Article | Download Full Size | PDF

Tables (1)

Tables Icon

Table 1. Comparison of conventional and unconventional photon blockade effects

View Table

Equations (28)

Equations on this page are rendered with MathJax. Learn more.

H = Δ 1 a + a + Δ 2 b + b + J ( a + b + b + a ) + U 1 a + a + a a + U 2 b + b + b b + ε 1 ( a + + a ) + ε 2 ( b + + b )
ψ = C 00 | 00 ⟩ + C 01 | 01 ⟩ + C 10 | 10 ⟩ + C 20 | 20 ⟩ + C 02 | 02 ⟩ + C 11 | 11 ⟩
i C 01 ˙ = Δ 2 ~ C 01 + J C 10 + ε 1 C 11 + ε 2 ( 2 C 02 + C 00 )
i C 10 ˙ = Δ 1 ~ C 10 + J C 01 + ε 2 C 11 + ε 1 ( 2 C 20 + C 00 )
i C 20 ˙ = 2 Δ 1 ~ C 20 + 2 J C 11 + 2 U 1 C 20 + 2 ε 1 C 10
i C 02 ˙ = 2 Δ 2 ~ C 02 + 2 J C 11 + 2 U 2 C 02 + 2 ε 2 C 01
i C 11 ˙ = ( Δ 1 ~ + Δ 2 ~ ) C 11 + 2 J ( C 20 + C 02 ) + ε 1 C 01 + ε 2 C 10
i C 01 ˙ = Δ 2 ~ C 01 + J C 10 + ε 2
i C 10 ˙ = Δ 1 ~ C 10 + J C 01 + ε 1
g a ( 2 ) ( 0 ) = ( − Δ 1 ~ Δ 2 ~ + J 2 ) 2 × A ( Δ 2 ~ ε 1 − ε 2 J ) 4 × B
A = [ ( Δ 1 ~ + Δ 2 ~ ) ( Δ 2 ~ ε 1 − ε 2 J ) 2 + ( Δ 2 ~ ( Δ 1 ~ + Δ 2 ~ ) ε 1 2 − 2 ( Δ 1 ~ + Δ 2 ~ ) ε 1 ε 2 J + ( ε 1 2 + ε 2 2 ) J 2 ) U 2 ] 2 ,
B = [ − Δ 1 ~ J 2 + Δ 2 ~ 2 U 1 + Δ 1 ~ 2 ( Δ 2 ~ + U 2 ) + Δ 1 ~ ( Δ 2 ~ + U 1 ) ( Δ 2 ~ + U 2 ) − J 2 ( U 1 + U 2 ) + Δ 2 ~ ( − J 2 + U 1 U 2 ) ] 2 .
J 2 = Δ 1 ~ Δ 2 ~
( Δ 1 ~ + Δ 2 ~ ) ( Δ 2 ~ ε 1 − ε 2 J ) 2 + ( Δ 2 ~ ( Δ 1 ~ + Δ 2 ~ ) ε 1 2 − 2 ( Δ 1 ~ + Δ 2 ~ ) ε 1 ε 2 J + ( ε 1 2 + ε 2 2 ) J 2 ) U 2 ( Δ 2 ~ ε 1 − ε 2 J ) 2 + ε 1 ( Δ 2 ~ ε 1 − 2 ε 2 J ) U 2 = 0
⟨ a + a ⟩ ≈ | C 10 | 2 = ( − Δ ε 1 + ε 2 J − Δ 2 + J 2 ) 2
U 2 = − 2 Δ ( Δ ε 1 − ε 2 J ) 2 2 Δ 2 ε 1 2 − 4 Δ ε 1 ε 2 J + ( ε 1 2 + ε 2 2 ) J 2
ε 1 = ε 2 , J = Δ
ε 1 = − ε 2 , J = − Δ
U 2 = − 2 Δ ( n Δ − J ) 2 J 2 − 4 Δ n J + ( J 2 + 2 Δ 2 ) n 2
U 2 ≈ 2 ( 1 + n ) 1 + 3 n J
E 1 , ± = ω a ± J
| 1 , + ⟩ = 1 2 ( | 10 ⟩ + | 01 ⟩ )
| 1 , − ⟩ = 1 2 ( − | 10 ⟩ + | 01 ⟩ )
E 2 , ± = 2 ω a + U ± 4 J 2 + U 2
E 2 , 0 = 2 ω a + 2 U
| 2 , + ⟩ = 1 C 1 ( | 20 ⟩ + | 02 ⟩ − 2 U − 2 4 J 2 + U 2 2 J | 11 ⟩ )
| 2 , 0 ⟩ = 1 2 ( − | 20 ⟩ + | 02 ⟩ )
| 2 , − ⟩ = 1 C 2 ( | 20 ⟩ + | 02 ⟩ − 2 U + 2 4 J 2 + U 2 2 J | 11 ⟩ )
Select as filters
Select Topics Cancel
Top
       
© Copyright 2024 | Optica Publishing Group. All rights reserved, including rights for text and data mining and training of artificial technologies or similar technologies.
Privacy | Terms of Use