## Abstract

By solving the master equation in the steady-state limit and calculating the zero-delay-time second-order correlation function, a strong photon antibunching is found in two coupled cavities with weak nonlinearity *χ*^{(2)}. An optimal antibunching condition is derived that is complementary to the result in a recent publication [Phys. Rev. A **89**, 031803(R) (2014)]. Numerical simulations confirm the optimal condition. These results are extended from the case with a single drive to the case with two drives. We find that the parameter added to the system due to the second drive can lift the restriction on the detuning for the antibunching to occur. This may open a new door towards the application of the antibunching into single photon sources.

© 2016 Optical Society of America

## 1. Introduction

There is a crucial requirement for the generation and manipulation of the single photon in information and communication technology. A single-photon source typically relies on a system which can produce sub-Poissonian light when it is driven by a classical light field. The basic physical principle for the generation of single photon is the photon blockade (PB) effect, which requires strong optical nonlinearity in most proposals published in the literature. This topic has become a research focus in modern quantum optics for the important applications.

The PB, also known as conventional phonon blockade, is a quantum optical effect preventing the injection of more than one photon into a nonlinear cavity mode [1, 2]. The PB origins from the strong photon-photon interaction which requires the nonlinear strength much greater than the decay rate of the system. It is generally considered that the actualization of PB demands a strong third-order (Kerr) nonlinearity for a photonic mode, which was first observed in an optical cavity coupled to a single trapped atom [3]. Subsequently, a sequence of experimental groups observed the strong antibunching behaviors in different systems [4–6]. Afterward it was proposed that a single-photon blockade can be achieved in second-order (*χ*^{(2)}) nonlinear system [7, 8]. The potential applications of PB cover the realization of single-photon transistor [9], interferometers [10] and quantum optical diode with semiconductor microcavities [8].

Recently, a new mechanism for the generation of sub-Poissonian light was put forward by Liew and Savona, where a strong photon antibunching can be obtaind with nonlinearity smaller than the decay rate of the cavity mode [11]. This mechanism is termed unconventional photon blockade (UPB), which only requires a much smaller optical nonlinearity than its conventional counterpart [12, 13]. Different from the conventional mechanism, the blockade is enforced by quantum interference among multiple excitation pathways. The typical system for UPB consists of two coupled resonators, a very weak third-order nonlinearity characterizes the two resonators [14–18]. Up to present, many other systems are proposed to realize the UPB, such as coupled optomechanical systems [19], bimodal optical cavity with a quantum dot [20, 21], and second-order nonlinearity systems [22–25].

In [23], the UPB can be realized in microcavities with second-order nonlinearity, where the optimal condition that leads to strong antibunching is obtained. In this paper, we explore the strong antibunching under the condition complementary to that in [23]. We first derive a Hamiltonian to describe two coupled cavities with second-order nonlinearity, which has the same form as that in [23]. Then we investigate the optimal condition for strong antibunching. Although the model is same, the method and conclusion in ours are different from that in [23]. The strong antibunching conditions in [23] are obtained by transforming the second-order nonlinearity term in the three-mode Hamiltonian into a two-mode Kerr-type nonlinear Hamiltonian under the condition Δ* _{c}* ≫ 2Δ

*, where Δ*

_{b}*represents the detuning between the driving laser and the corresponding cavity-mode*

_{i}*i*. Whereas in our scheme, we shall investigate the photon antibunching condition directly by using the second-order nonlinear Hamiltonian for the three-mode system. To get the strong antibunching condition with our method, the detuning Δ

*= 0 is needed. So we analyze the case of Δ*

_{i}*= 0 with a single drive that is different from that in [23]. In addition, when two drives are applied to the system, we find that the strong photon antibunching can also be obtained when Δ*

_{i}*= 0. Different from the case of single drive, the UPB with two drives has the following features: (i) The optimal conditions for strong antibunching exist and can be found when Δ*

_{i}*≠ 0. (ii) The UPB can be obtained even if both the*

_{i}*χ*

^{(2)}nonlinearity and the coupling strength are small. This study is quite different from our previous work of the UPB based on a two-mode (by contrast, three modes are considered in this paper) system with second-order nonlinearity [22], where the coupling

*J*between the modes was not taken into account, and the antibunching cannot happen with a single drive since the quantum interference pathways cannot be formed. Besides, the effect of cavity decay on the UPB is different for the two-mode and three-mode system.

## 2. Model

Let us start with the nonlinear response of dielectric material to an electric field described by

*x*,

*y*and

*z*. Assuming that the dielectric is inhomogeneous but isotropic, i.e.,

*ε*(

_{ij}**r**) →

*ε*(

**r**) [7], and restricting ourself to consider only the second-order nonlinearity

*χ*

^{(2)}(

**r**), i.e., ${\chi}_{ijkm}^{(n)}(\mathbf{r})=0$,

*n*≥ 3, with the electromagnetic field for the three-mode system,

*ĵ*(

*j*=

*a*,

*b*,

*c*) is the destruction operator of mode

*j*, and

*B̂*(

**r**) = −∇ ×

*Ê*(

**r**)/

*ω*

_{0}, we can derive a Hamiltonian for the system.

Now we go to the details. To derive the Hamiltonian with second-order nonlinearity, we set *ω _{c}* = 2

*ω*= 2

_{a}*ω*. The three dimensional cavity field for the three modes

_{b}*ϕ*(

_{i}**r**) can be normalized by ∫ |

*ϕ⃗*(

_{i}**r**)|

^{2}

*d*

^{3}

**r**= 1 (

*i*=

*a*,

*b*,

*c*). The classical expression of the energy density is given by ${H}_{\mathit{em}}=\frac{1}{2}\int \left[\widehat{E}(\mathbf{r},t)\widehat{D}(\mathbf{r},t)+\widehat{H}(\mathbf{r},t)\widehat{B}(\mathbf{r},t)\right]{d}^{3}\mathbf{r}$, where

*Ĥ*=

*B̂/μ*

_{0}. Then the Hamiltonian for this system can be calculated as

*a*is in the first cavity, mode

*b*and the second-harmonic mode

*c*are in the second cavity, as shown in Fig. 1. Since the modes in different cavities are coupled, we only keep the leading order in Hamiltonian (3) and drop the nonlinear interaction coefficients

*v*and

*u*by assuming that the overlapped of these wave functions is very small. With these considerations, the Hamiltonian Eq. (3) reduces to

*a*mode is essential. The driving frequency and driving strength are denoted by

*ω*and

_{L}*F*, respectively. The Hamiltonian for the system with such a drive reads:

*Û*(

*t*) =exp[

*iω*(

_{L}t*â*

^{†}

*â*+

*b̂*

^{†}

*b̂*+ 2

*ĉ*

^{†}

*ĉ*)], which leads to a rotated Hamiltonian by

*Ĥ*=

_{d}*ÛĤÛ*

^{†}−

*ih̄ÛdU*

^{†}/

*dt*,

*=*

_{i}*ω*−

_{i}*ω*(

_{L}*i*=

*a*,

*b*) and Δ

*=*

_{c}*ω*− 2

_{c}*ω*denote the detunings of the

_{L}*a*,

*b*and

*c*modes from the driving laser, respectively. The derivation of the Hamiltonian Eq. (3) strictly relies on the relation

*ω*= 2

_{c}*ω*= 2

_{a}*ω*= 2

_{b}*ω*, or equivalently, Δ

*= 2Δ*

_{c}*= 2Δ*

_{a}*. Hereafter, we only analyze the resonant case (Δ*

_{b}*= 2Δ*

_{c}*= 2Δ*

_{a}*= 2Δ = 0) because our method cannot obtain a condition for antibunching with non-zero Δ (for details, see the APPENDIX).*

_{b}Taken the cavity decay into account, the evolution of the system can be described by a master equation in the Lindblad form

*κ*,

_{a}*κ*and

_{b}*κ*being the damping rates of

_{c}*a*mode,

*b*mode and

*c*mode, respectively. For the case of photonic crystal cavities, the higher order mode has always a lower quality factor

*Q*(and thus a higher

*κ*). The most optimistic assumptions in literature take

*Q*=

_{a}*Q*= 2

_{b}*Q*[7], which resulting in

_{c}*κ*= 4

_{c}*κ*= 4

_{a}*κ*= 4

_{b}*κ*. We will use these parameters in this paper. To discuss the UPB, we need to calculate the second-order correlation function defined by

*τ*is the time delay between different detectors. In the following, we consider the zero-time delay correlation function in steady state for mode

*a*

*ρ*is the steady-state density matrix calculated from Eq. (8). The second-order correlation function

_{s}*g*

^{(2)}(0) < 1 corresponds to sub-poissonian statistics, which indicates photon antibunching.

## 3. The comparison of the optimal condition with numerical simulations for single drive

In the weak driving limit, the zero-time delay correlation function *g*^{(2)}(0) can be given approximately by *g*^{(2)}(0) ≃ 2|*C*_{200}|^{2}/|*C*_{100}|^{4}. This suggests that the conditions for *g*^{(2)}(0) → 0 are equivalent to *C*_{200} = 0. So the optimal condition of photon antibunching can be obtained by solving the steady-state solution of the system and setting *C*_{200} = 0.

We can expand the wave function of the system on a Fock-state basis |*n _{a}n_{b}n_{c}*〉, where

*n*,

_{a}*n*and

_{b}*n*denote the photon number in mode

_{c}*a*,

*b*and

*c*, respectively. The system is truncated up to at most two photons, i.e.,

*n*+

_{a}*n*+ 2

_{b}*n*≤ 2. In this circumstances, the state of the system can be expanded as

_{c}*κ*= 4

_{c}*κ*= 4

_{a}*κ*= 4

_{b}*κ*. Substituting the state Eq. (11) and Hamiltonian

*H̃*into Schrödinger’s equation

*ih̄∂*|

_{t}*ψ*〉 =

*H̃*|

*ψ*〉, we get a set of equations for the coefficients

*F*≪

*κ*, we have |

*C*

_{000}| ≫ {|

*C*

_{100}|, |

*C*

_{010}|} ≫ {|

*C*

_{200}|, |

*C*

_{110}|, |

*C*

_{020}|, |

*C*

_{001}|}. Thus the occupation number of the vacuum state is approximately equal to one, i.e.,

*C*

_{000}= 1. For one-photon states, the steady-state coefficients are determined by

*a*, we set

*C*

_{200}= 0, and then we get the condition for optimal photon antibunching as When the above optimal condition for strong antibunching is satisfied, the UPB would occur.

In order to describe the UPB effect, we plot the second-order correlation function *g*^{(2)}(0) as a function of the parameters in the system, this can be done by substituting Hamiltonian (Eq. (7)) into the master equation and solving Eq. (8) numerically. The Hilbert spaces of the system are truncated to five dimensions for cavity modes *a*, *b* and two dimensions for mode *c*. We rescale all parameters by the dissipation rate *κ* of the cavity modes in this paper.

In the derivation of the optimal photon antibunching condition, we have assumed that the detunings satisfy Δ* _{c}* = 2Δ

*= 2Δ*

_{a}*= 2Δ = 0. In Fig. 2, we plot the numerical result for the zero-time second-order correlation functions*

_{b}*g*

^{(2)}(0) versus

*g/κ*. We find that

*g*

^{(2)}(0) ≪ 1 appears at

*g/κ*≃ ±0.1, which agrees with the analytic solution calculated by Eq. (16). And the position of

*g/κ*where the correlation function arrives at its minimum would not be affected obviously with the change of Δ/

*κ*. However, the value of

*g*

^{(2)}(0) gets higher with the increase of Δ/

*κ*. There is no obvious UPB phenomenon with Δ

*= 0.1. Although the antibunching effect is the strongest in the resonance case, the strict resonance condition is not necessary. The optimal analytical condition of Eq. (16) can be written in the form of*

_{a}/κ*J/κ*. We can also learn from optimal condition Eq. (17) that real

*g*and

*κ*require $J/\kappa \in \left(-\infty ,-1/\sqrt{2}\right)\cup \left(1/\sqrt{2},\infty \right)$. Thus there will be no photon antibunching in the range $J/\kappa \in \left[-1/\sqrt{2},1/\sqrt{2}\right]$ even if the optimal condition is satisfied.

In [23], the dependence of the photon antibunching is checked against the quality factor *Q _{c}*, where the second-harmonic mode loss rate ranges from

*κ*≃

_{c}*κ*up to

*κ*≃ 10

_{c}^{2}

*κ*(

*κ*=

_{a}*κ*=

_{b}*κ*), see Fig. 3(a). It is shown that the photon antibunching can be observed for both

*Q*>

_{c}*Q*and

*Q*<

_{c}*Q*(

*Q*=

_{a}*Q*=

_{b}*Q*). The reason is as follows: in their scheme, the Hamiltonian in Eq. (7) is transformed into a Kerr-type nonlinear Hamiltonian as

*Ĥ*=

_{d}*h̄*Δ

*â*

^{†}

*â*+

*h̄*Δ

*b̂*

^{†}

*b̂*+

*h̄J*(

*â*

^{†}

*b̂*+

*b̂*

^{†}

*â*) +

*h̄g*[

_{eff}*b̂*

^{†}

*b̂*+

*b̂*

^{†}

*b̂*

^{†}

*b̂b̂*] +

*h̄F*(

*â*

^{†}+

*â*), and the antibunching conditions are obtained based on this Hamiltonian. The mode

*c*is decoupled from mode

*a*and mode

*b*, thus the Hamiltonian is equivalent to a two mode system Hamiltonian, which makes the influence of

*κ*on

_{c}*g*

^{(2)}(0) smaller and

*g*

^{(2)}(0) ≪ 1 can be obtained even if

*κ*changes on a large scale. In our scheme,

_{c}*g*

^{(2)}(0) as a function of the quality factor

*Q*is also plotted, which is shown in Fig. 3(b). Different from the result in Fig. 3(a), the optimal value of photon antibunching appears at a fixed point

_{c}*Q*= 1/2. In fact, if we do not consider the condition

_{C}/Q*κ*= 4

_{c}*κ*, a more general antibunching conditions can be obtained as 8

*g*

^{2}

*J*

^{2}− 4

*g*

^{2}

*κ*

^{2}−

*κ*

^{3}

*κ*= 0, which is different from the antibunching conditions in [23], because

_{c}*κ*arise in the condition. The antibunching condition of Eq. (16) is satisfied only when

_{c}*κ*= 4

_{c}*κ*, so the optimum value of photon antibunching appeares at a fixed point

*Q*= 1/2.

_{C}/QIt is worth evaluating the actual experimental possibilities of the strong photon antibunching effect with weak nonlinearity. Here we choose the parameters in [23]. The present scheme could be realized with state-of-the-art technology employing the main III–V materials, such as GaAs [26, 27], GaN, and AIN [28], where bulk nonlinear susceptibility can be of the order of 10–200 pm/V in optoelectronics. A realistic order of magnitude estimates for the second-order nonlinearity is *h̄g* ≈ 1*μ* eV with engineered photonic crystal cavities, which is the same with that in [23]. Working in the typical telecommunication band, *h̄ω* ∼ 0.8eV, the loss rate is about *h̄κ* = 10*μ* eV corresponding to a fundamental mode *Q* factor *Q* ∼ 80000 [23], the *Q _{c}* for the second harmonic mode is

*Q*∼ 40000. These values could be routinely achieved in photonic crystal cavities made of III–V semiconductor materials [29]. The coupling strength is

_{c}*h̄J*≈ 71

*μ*eV, which is smaller than that in [23], and it could be further reduced for larger

*g/κ*according to Eq. (17). The time interval of antibunching occurs for UPB is limited by

*π/J*[23].

## 4. The case with two drives

In this section, we investigate the photon antibunching with two drives. The Hamiltonian for such a system is *Ĥ*_{d1} = *h̄*Δ_{a}â^{†}*â* + *h̄*Δ_{b}b̂^{†}*b̂* + *h̄*Δ_{c}ĉ^{†}*ĉ* + *h̄J*(*â*^{†}*b̂* + *b̂*^{†}*â*) + *h̄g*[(*b̂*^{†})^{2}*ĉ* + *ĉ*^{†}*b̂*^{2})] + *h̄F*_{1}(*â*^{†} + *â*) + *h̄F*_{2}(*ĉ*^{†} + *ĉ*), where Δ* _{c}* = 2Δ

*= 2Δ*

_{a}*= 2Δ, the driving strengths for mode*

_{b}*a*and mode

*c*are denoted by

*F*

_{1}and

*F*

_{2}, respectively. In the following discussion, we will set

*F*

_{1}=

*F*,

*F*

_{2}/

*F*

_{1}=

*n*, where

*n*is called strength ratio. The sign of

*n*(minus or plus) can be tuned by adjusting the corresponding phases of the driving fields [14, 31].

Different from the case with single drive, the photon antibunching can be obtained in the mode *a* even if the detuning exists. The optimal strong photon antibunching conditions can be calculated as:

*= 2Δ*

_{c}*= 2Δ*

_{a}*= 2Δ. The photon antibunching would take place at the location where the requirements in Eqs. (18) meet.*

_{b}For the sake of convenience, we rewrite Eqs. (18) as

*g*

_{1},

*n*

_{1}) and (

*g*

_{2},

*n*

_{2}) are equivalent with Eqs. (18). The zero-time second-order correlation functions

*g*

^{(2)}(0) are plotted in Fig. 4. In Fig. 4(a), the dips in the plot of

*g*

^{(2)}(0) ≪ 1 correspond to two rings. This numerical condition agrees well with the analytical result given by

*g*and

*κ*, we must require Δ/

*κ*in (−2.9382, −0.447545) ∪ (0.447545, 2.9382). Then the nonlinear interaction strength

*g/κ*can only take values in [−2.16, 0) ∪ (0, 2.16]. In Fig. 4(b), the two symmetrical-quadratic-function-like analytical solution can be calculated as

*κ*to take values in (−2.9382, −0.447545) ∪ (0.447545, 2.9382).

So far, we have investigated the photon antibunching driven by two fields with non-zero detunings Δ/*κ*, and we find the numerical simulation is in excellent agreement with the analytical solution. Next we will consider the case of resonance, i.e., the detunings are zero. In this case, the first equation of Eqs. (18) vanishes and Eqs. (18) reduce to

*J/κ*. For the single drive, there will be no photon antibunching in the range $J/\kappa \in \left[-1/\sqrt{2},1/\sqrt{2}\right]$. But for two drives, we will show that the UPB can be obtained even with weak nonlinearity and weak coupling strength, which is shown in Fig. 5. In addition, although the antibunching conditions are complicated for two drives compared with the single drive, both the driving strengths and the strength ratio

*n*are included in the antibunching conditions and can be tuned precisely in experiments [14, 31], which makes our scheme available for tunable single photon sources.

The optimal analytical condition for strong photon antibunching depends on the strength ratio *n*

*J/κ*, which is shown in Fig. 6. In Fig. 6(a), by setting

*J/κ*= 2, two valleys corresponding to

*g*

^{(2)}(0) ≪ 1 appear in the Logarithmic plot of the zero-time second-order correlation functions. It is shown that the nonlinear interaction strength

*g/κ*changes continuously with the ratio

*n*in the photon antibunching region. The value of the nonlinear interaction strength is

*g/κ*≃ ±0.378 when the ratio is

*n*= 0. When

*J/κ*= 0.2, the shapes of two valleys change which is shown in Fig. 6(b).

*g/κ*changes discontinuously with the ratio

*n*for the photon antibunching region. And

*n*can only take

*n*⊂ (−∞, −0.082] ∪ [0.082, ∞). We also compare the current scheme with that based on Kerr nonlinearity which is also with two drives [14]. For Kerr nonlinearity system, in order to make sure that the strong antibunching occurs in mode

*a*in the weak nonlinear regime, the strength ratio

*n*should be larger than one, but second-order nonlinearity system is not restricted by this condition. As the strength ratio

*n*is not too large, e.g.,

*n*< 10, the antibunching effect is strong. With the increase of

*n*, the strength of antibunching becomes weaker. But in second-order nonlinearity system, the influence of

*n*to the value of

*g*

^{(2)}(0) seems very small.

The system with two coupled cavities (with coupling strength *J*) for the case that both the two cavity modes are driven coherently have been demonstrated in different systems [14, 18, 21, 30], such as a quantum-dot-bimodal-cavity system [21], Kerr nonlinear system [14, 18, 30] and second-order nonlinearity system [25]. Here we also adopt photonic crystal cavities, and the second-order nonlinearity might take *h̄g* ≈ 1*μ* eV, the loss rate is about *h̄κ* = 10*μ* eV, *h̄F* = 5*μ*eV, *n* = 0.329447 and *h̄J* = 3*μ* eV. The strength ratio *n* can be tuned precisely in experiments [14,31]. The coupling strength between the cavity modes depends exponentially upon the space of the cavities, and only a small *J* is needed to obtain antibunching. There is a range for *J/κ* to obtain antibunching when the others parameters are fixed. The range is called tolerance. When *g*^{(2)}(0) < 0.5, antibunching can be obtained, the range of *J/κ* is about 0.1*J/κ* (the tolerance is about 0.1*J/κ*), which is shown in Fig. 5. The range depends on the effective nonlinearity *g*, which is the same with that in [23]. The tolerance is small due to a small *J/κ*, we can get a big tolerance when we take a big *J/κ*. To pump the two resonators with two distinct lasers, we might drive the mode *a* from the left side of the first cavity with frequency *ω* and drive the mode *c* from the right side of the second cavity with frequency 2*ω*, this scheme was used in [18]. In [25], second-order nonlinear system for two drives can be realized in a double quantum well embedded in a semiconductor microcavity, which might be a candidate to realize our scheme.

The physical principle of the photon antibunching is the quantum interference among multiple excitation pathways, which is shown in Fig. 7, where we show the energy-level diagram and the transition paths. There are three paths for two-photon excitation : (i) Directly input two photons in mode *a*, i.e.,
$|000\u3009\stackrel{{F}_{1}}{\to}|100\u3009\stackrel{{F}_{1}}{\to}|200\u3009$. (ii) Input one photon in mode *a* and swap it to mode *b*, input the second photon in mode *a* and the photon in mode *b* coming back to mode *a*, i.e.,
$|000\u3009\stackrel{{F}_{1}}{\to}|100\u3009\stackrel{J}{\to}|010\u3009\stackrel{{F}_{1}}{\to}|110\u3009\stackrel{J}{\to}|200\u3009$. (iii) Input one photon in mode *c*, transfer the photon to mode *b* and mode *a* by the second-order nonlinearity and the linear process, i.e.
$|000\u3009\stackrel{{F}_{2}}{\to}|001\u3009\stackrel{g}{\to}|020\u3009\stackrel{J}{\to}|110\u3009\stackrel{J}{\to}|200\u3009$. The photons coming from the three pathways take destructive interference when the antibunching condition Eqs. (18) are satisfied. The interference of the three excitation pathways leads to zero population of photon in the state |200〉, i.e., the photons can not occupy the state |200〉. This mechanism is different from the photon antibunching based on *χ*^{(3)}, in which the level structure has been changed.

## 5. Conclusion

In conclusion, we have studied the photon blockade caused by weak *χ*^{(2)} nonlinearity in two coupled cavities. By solving the master equation in the steady-state limit and computing zero-delay-time second-order correlation function, strong photon antibunching is predicted in the first cavity. The exact numerical simulation for optimal antibunching agrees well with the analytical results. When both two cavities are driven, the ratio of the first driving strength to the second one becomes an additional parameter to modulate the strong photon antibunching. Thus we can always obtain the optimal antibunching independent of the detuning. We believe that our results may find new applications in generating single-photon sources due to the range of validity of the scheme.

## APPENDIX: Non-zero Δ results in no photon antibunching conditions

In this section, we will prove that there are no antibunching conditions when the detuning is not zero. In the presence of Δ, the optimal strong photon antibunching conditions can be derived as

*= 2Δ*

_{c}*= 2Δ*

_{a}*= 2Δ. This condition is necessary, as mentioned in the maintext. Now we solve the equations and find 4 solutions for*

_{b}*g*and Δ,

*g*, we must have By solving this inequality, we find that the condition of

_{i}*g*have a real solution is

_{i}*κ*≤ 0. Thus in the case with single drive, we only consider the resonant case, i.e., Δ

*= Δ*

_{a}*= Δ*

_{b}*= 0. Real*

_{c}*g*is necessary, otherwise the Hamiltonian is not hermitian(see the maintext).

## Funding

This work is supported by the National Natural Science Foundation of China (NSFC) under Grants Nos. 11534002, 61475033 and 11204028.

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