Conventional and unconventional photon blockade effects in quantum optomechanics

Abstract

We theoretically study the photon blockade (PB) effect in a double-cavity optomechanical system with the two-photon driving. By analytical calculations and numerical simulations, the physical mechanisms of conventional photon blockade (CPB) and unconventional photon blockade (UPB) are discussed in detail. And then we obtain the optimal parameter conditions for PB. In our work, there exist both the CPB induced by strong nonlinear interaction and the UPB caused by quantum interference. In particular, we find that CPB and UPB can occur simultaneously under the same parameters. In addition, we also prove that the appropriate values of nonreciprocal coupling and two-photon driving are favorable to the improvement of PB. Our proposal provides an idea for simultaneously realizing CPB and UPB in the optomechanical system and offers a route for constructing high-quality single-photon sources.

Graphical abstract

Keywords

conventional photon blockade / unconventional photon blockade / two-photon driving

1 Introduction

Photon blockade (PB), an antibunching effect resulting from the excitation of the first photon blocking the injection of the second photon, is a prominent technique for generating single-photon sources. In general, there are two main schemes to realize photon antibunching, i.e., conventional photon blockade (CPB) [1−17] and unconventional photon blockade (UPB) [18−33], which are based on the anharmonic eigenenergy of the system and the destructive quantum interference in two-photon excitation processes, respectively. For CPB, strong nonlinearity (or strong coupling to nonlinear elements like atoms) is generally a prerequisite for realizing PB, which has been experimentally achieved in cavity quantum electrodynamics (QED) [1−6] and circuit QED systems [7, 8].

Compared with CPB, UPB can relax the requirement for strong nonlinearity, which offers a helpful way to achieve photon antibunching and reveal quantum correlation properties in weak-parameter devices. Usually, UPB originates from destructive quantum interference between distinct excitation pathways [18, 19]. Based on this mechanism, UPB has been studied theoretically and experimentally in different systems, such as superconducting resonators [20], second-order nonlinearity [21, 22], parametric interactions [23], non-Markovian reservoir [24], cavity QED system [25−27], optomechanical system [28−31], and magnons [32, 33].

Recently, some works have discovered that both CPB and UPB can occur in the same quantum system, and have analyzed the physical mechanisms behind CPB and UPB [34−40]. For example, they can be understood as the interference between squeezed and coherent components [40]. Generally, the single-photon source based on CPB has the advantage of high mean photon number (brightness), but relatively poor purity. By contrast, the one based on UPB has high purity but poor brightness. Further, under the same parameter conditions, two types of PB have been realized simultaneously in the cavity QED system [41−44] and coupled cavities with Kerr nonlinearity [45, 46].

Motivated by the above-mentioned works [34−46], we are going to explore CPB and UPB in the optomechanical system. Specifically, we consider the optomechanical system consisting of two mechanical modes and two optical modes in the presence of both one and two-photon driving. Then, we analyze the influences of the coupling between two cavities, two-photon driving, and optomechanical coupling on PB. The results demonstrate that both CPB and UPB are simultaneously achieved in the stronger coupling regime. To further relax the coupling strength, we choose the nonreciprocal coupling between two cavities. In this case, we also find that CPB and UPB can be simultaneously realized in the strong coupling regime. Thus, this approach can relax the couplings required for CPB and UPB from the stronger to strong coupling. Our proposal combines the advantages of CPB and UPB and provides a promising method to generate the single-photon source with high purity and brightness in optomechanics.

The remainder of this paper is organized as follows. In Section 2, we present the theoretical model and system Hamiltonian. Then, the validity of Hamiltonian and the effect of thermal noise on PB are discussed by the master equation and second-order correlation function. In Section 3, we calculate analytically the Schrödinger equation and derive the mean photon number and second-order correlation function. Subsequently, the physical mechanisms of CPB and UPB have been explained in detail. Most importantly, we combine CPB with UPB successfully. In Section 4, we also investigate the effects of nonreciprocal coupling and two-photon driving on PB, respectively. Finally, we provide the experimental feasibility and summarize the whole work in Section 5.

2 Theoretical model

As shown in Fig.1, we consider a double-cavity optomechanical system, which contains two mechanical oscillators and two optical cavities. Besides the optomechanical interaction coupling the mechanical oscillator to the optical cavity, these two optical cavities with the two-photon (i.e., parametric) driving [23−25, 47] are coupled by the nonreciprocal interaction [35, 36, 48−50]. In a frame rotating with the driving frequency {\omega }_l, the Hamiltonian of this system can be written as (setting \hslash =1)

(1)\begin{array}{rl}{H}_1=& \sum _{j=1}^2[-{\mathrm{\Delta }}_j{a}_j^{†}{a}_j+{\omega }_{m_j}{b}_j^{†}{b}_j-g_j{a}_j^{†}{a}_j(b_j^{†}+b_j)+\mathrm{i}{\lambda }_j{a}_j^{†2}\\ & -\mathrm{i}{\lambda }_j{a}_j^2]+E(a_1^{†}+a_1)+{\mu }_1{a}_1^{†}{a}_2+{\mu }_2{a}_2^{†}{a}_1,\end{array}

where a_j(a_j^{†}) is the annihilation (creation) operator of the jth optical cavity with the frequency {\omega }_j, and b_j(b_j^{†}) is the annihilation (creation) operator of the jth mechanical oscillator with the frequency {\omega }_{m_j}. {\mathrm{\Delta }}_j={\omega }_l- {\omega }_j is the detuning of the jth cavity from the laser drive. g_j is the coupling strength between the cavity a_j and the mechanical oscillator b_j. {\lambda }_j is the two-photon driving term. The amplitude of the driven laser field is described by \left|E\right|=\sqrt{2{\kappa }_{1}P/(\hslash {\omega }_l)} with the power P and corresponding cavity decay rate {\kappa }_1. {\mu }_j denotes the tunneling strength of photon hopping between cavities a_j. Here, this hopping is nonreciprocal (i.e., asymmetric, {\mu }_1\ne {\mu }_2), which can be achieved via non-Hermiticity [51−53], nanoparticles [54], quantum impurity [55], optical delay lines [56], reservoir engineering [57], and so on. For example, the nonreciprocal coupling is scaled by exp(\pm h) in opposite directions with h being the effect of imaginary vector potential [51−53].

In a displaced oscillator representation defined by the unitary operator U=\exp \left[{\sum }_{j=1}^2\frac{g_j}{{\omega }_{m_j}}{a}_j^{†}{a}_j\left(b_j^{†}-b_j\right)\right] [35, 36, 58−60], the transformed Hamiltonian can be derived as

(2)\begin{array}{rl}{H}_2=& \sum _{j=1}^2[-{\mathrm{\Delta }}_j{a}_j^{†}{a}_j+{\omega }_{m_j}{b}_j^{†}{b}_j-{\eta }_j{a}_j^{†}{a}_j{a}_j^{†}{a}_j+\mathrm{i}{\lambda }_j{a}_j^{†2}\\ & -\mathrm{i}{\lambda }_j{a}_j^2]+E(a_1^{†}+a_1)+{\mu }_1{a}_1^{†}{a}_2+{\mu }_2{a}_2^{†}{a}_1,\end{array}

with {\eta }_j=g_j^2/{\omega }_{m_j} (j=1,2, and g_j\ll {\omega }_{m_j}) being the Kerr-type nonlinear coupling. Here, the mechanical oscillators have been decoupled from the optical cavities. This indicates that the evolution of optical and mechanical parts is independent of each other. Thus, the mechanical part in Eq. (2) can be ignored safely when we study the optical properties of the system. The effective Hamiltonian can be given by

(3)\begin{array}{rl}{H}_3=& \sum _{j=1}^2[-{\mathrm{\Delta }}_j{a}_j^{†}{a}_j-{\eta }_j(a_j^{†}{a}_j{)}^2+\mathrm{i}{\lambda }_j{a}_j^{†2}-\mathrm{i}{\lambda }_j{a}_j^2]\\ & +E(a_1^{†}+a_1)+{\mu }_1{a}_1^{†}{a}_2+{\mu }_2{a}_2^{†}{a}_1,\end{array}

where this Hamiltonian describes a pure optical system consisting of two optical modes a_1 and a_2.

Here, we numerically study the quantum dynamics of the optomechanical or optical system and then employ the master equations for the system density operator \rho defined by

(4)\begin{array}{rl}\dot{\rho }=& -\mathrm{i}[H_1,\rho ]+\frac{{\kappa }_1}{2}L[a_1]\rho +\frac{{\kappa }_2}{2}L[a_2]\rho +\frac{{\gamma }_1}{2}(n_{th}+1)L[b_1]\rho \\ & +\frac{{\gamma }_2}{2}(n_{th}+1)L[b_2]\rho +\frac{{\gamma }_1}{2}{n}_{th}L[b_1^{†}]\rho +\frac{{\gamma }_2}{2}{n}_{th}L[b_2^{†}]\rho ,\\ \dot{\rho }=& -\mathrm{i}[H_3,\rho ]+\frac{{\kappa }_1}{2}L[a_1]\rho +\frac{{\kappa }_2}{2}L[a_2]\rho ,\end{array}

where L[o] \rho =2o\rho o^{†}-o^{†}o\rho -\rho o^{†}o (o=a_1, a_2,b_1, b_2) is the Liouvillian operator for the two optical cavities and two mechanical modes. n_{th}=1/\left[\exp \left(\hslash {\omega }_{m_j}/(k_{B}T)\right)-1\right] denotes the thermal phonon number of mechanical mode at the environmental temperature T with the Boltzmann constant k_B. {\kappa }_j and {\gamma }_j represent the decay rate of cavity and the damping rate of mechanical oscillator (j=1,2), respectively. When the present system is in the steady or transient state, the corresponding second-order correlation function in the a_1 mode can be evaluated by g_{\mathrm{s}\mathrm{s}}^{(2)}(0)={\mathrm{l}\mathrm{i}\mathrm{m}}_{t\to \mathrm{\infty }}⟨a_1^{†} a_1^{†}{a}_1{a}_1⟩(t)/⟨a_1^{†} a_1{⟩}^2(t) or g^{(2)}(0)= ⟨a_1^{†} a_1^{†}{a}_1{a}_1⟩(t)/⟨a_1^{†} a_1{⟩}^2(t) [61]. Here, the condition characterizes photon antibunching. Particularly, when g_{\mathrm{s}\mathrm{s}}^{(2)}(0) \to 0 [ g^{(2)}(0)\to 0], the perfect photon blockade (PB) will occur in the optical mode a_1.

In order to check the validity of this effective Hamiltonian in Eq. (3), we plot in Fig.2 the time evolution of g^{(2)}(0), which is calculated by the master equations, given in Eq. (4). In the numerical simulation, we can regulate the system parameters to satisfy the conditions for and g_j\ll {\omega }_{m_j}, and compute on the truncated Hilbert space of dimensions (N_{a_1}\times N_{a_2}\times N_{b_1}\times N_{b_2}{)}^2 or (N_{a_1}\times N_{a_2}{)}^2 in the optomechanical or optical system, where N_{a_1}=N_{a_2}=N_{b_1}=N_{b_2}=4. From Fig.2, it is clear to see that the approximate results obtained from effective Hamiltonian H_3 (dashed curve) are in good agreement with the exact numerical results of full Hamiltonian H_1 (solid curve). We also find that when \kappa t \approx 20, the correlation function g^{(2)}(0) approaches a steady value. For the dissipation rate \kappa =0.1 MHz, the relaxation time of the system is about 200 μs.

In the discussions above, we focus on the case of zero temperature (n_{th} =0). However, in the strong or stronger coupling regime, the mechanical thermal noise may affect the PB and cannot be neglected. Therefore, in Fig.3, we investigate the influence of thermal phonon number n_{th} on the second-order correlation function g_{\mathrm{s}\mathrm{s}}^{(2)}(0) based on Eq. (4) with H_1. For the system parameters \mathrm{\Delta } and \mu, we choose the optimal parameters {\mathrm{\Delta }}_{\mathrm{o}\mathrm{p}\mathrm{t}} and {\mu }_{\mathrm{o}\mathrm{p}\mathrm{t}}, as shown in Section 3. Specifically, in the strong coupling regime (\eta =10\kappa), the value of g_{\mathrm{s}\mathrm{s}}^{(2)}(0) sharply increases with the increase of thermal phonon number n_{th}. Eventually, the PB (or photon antibunching) can disappear for very small thermal phonon numbers, i.e., g_{\mathrm{s}\mathrm{s}}^{(2)}(0) >1. This suggests that the mechanical thermal noise has an undesirable effect on the PB. But in the stronger coupling regime (\eta = {10}^2\kappa), the PB can survive in the finite thermal phonon number.

3 Analytical calculation and photon blockade with {\mathit{\mu }}_{\mathbf{1}}\mathbf{=}{\mathit{\mu }}_{\mathbf{2}} and \mathit{\lambda }\mathbf{\ne }\mathbf{0}

To investigate the PB phenomenon in detail, we use the mean photon number and second-order correlation function, which can be obtained by analytically calculating the Schrödinger equation \mathrm{i}\frac{\mathrm{d}|\psi ⟩}{\mathrm{d}t}=H_{\mathrm{e}\mathrm{f}\mathrm{f}}|\psi ⟩. When the optical decay {\kappa }_j is phenomenologically introduced to H_3, the effective non-Hermitian Hamiltonian [36, 62−64] takes the form

(5)H_{\mathrm{e}\mathrm{f}\mathrm{f}}=H_3- {\displaystyle \frac{\mathrm{i}{\kappa }_1}{2}}{a}_1^{†} a_1-{\displaystyle \frac{\mathrm{i}{\kappa }_2}{2}}{a}_2^{†} a_2.

In the weak-driving regime (E\ll {\kappa }_j), only few photons (such as a single photon or two photons) in the cavity will be excited. Thus, the wave function of the system can be approximately expanded to the following form:

(6)\begin{array}{rl}|\psi ⟩=& C_{00}|00⟩+C_{01}|01⟩+C_{10}|10⟩+C_{11}|11⟩\\ & +C_{02}|02⟩+C_{20}|20⟩,\end{array}

with the probability amplitudes C_{N_{a_1}{N}_{a_2}} of the state |N_{a_1}{N}_{a_2}⟩. Here, | N_{a_1}⟩ and | N_{a_2}⟩ are the number states for the a_1 and a_2 modes, respectively.

Substituting the state | \psi ⟩ and the Hamiltonian H_{\mathrm{e}\mathrm{f}\mathrm{f}} into the Schrödinger equation, we can obtain a set of evolution equations:

(7)\begin{array}{rl}\mathrm{i}{\dot{C}}_{00}=& E{C}_{10}-\mathrm{i}\sqrt{2}{\lambda }_2{C}_{02}-\mathrm{i}\sqrt{2}{\lambda }_1{C}_{20},\\ \mathrm{i}{\dot{C}}_{01}=& {\mu }_2{C}_{10}+E{C}_{11}-{\mathrm{\Delta }}_2{C}_{01}-{\eta }_2{C}_{01}-\frac{\mathrm{i}{\kappa }_2}{2}{C}_{01},\\ \mathrm{i}{\dot{C}}_{10}=& {\mu }_1{C}_{01}+E{C}_{00}+\sqrt{2}E{C}_{20}-{\mathrm{\Delta }}_1{C}_{10}-{\eta }_1{C}_{10}\\ & -\frac{\mathrm{i}{\kappa }_1}{2}{C}_{10},\\ \mathrm{i}{\dot{C}}_{11}=& \sqrt{2}{\mu }_1{C}_{02}+\sqrt{2}{\mu }_2{C}_{20}+E{C}_{01}-{\mathrm{\Delta }}_1{C}_{11}-{\mathrm{\Delta }}_2{C}_{11}\\ & -{\eta }_1{C}_{11}-{\eta }_2{C}_{11}-\frac{\mathrm{i}{\kappa }_1}{2}{C}_{11}-\frac{\mathrm{i}{\kappa }_2}{2}{C}_{11},\\ \mathrm{i}{\dot{C}}_{02}=& \sqrt{2}{\mu }_2{C}_{11}+\mathrm{i}\sqrt{2}{\lambda }_2{C}_{00}-2{\mathrm{\Delta }}_2{C}_{02}-4{\eta }_2{C}_{02}\\ & -\mathrm{i}{\kappa }_2{C}_{02},\\ \mathrm{i}{\dot{C}}_{20}=& \sqrt{2}{\mu }_1{C}_{11}+\sqrt{2}E{C}_{10}+\mathrm{i}\sqrt{2}{\lambda }_1{C}_{00}-2{\mathrm{\Delta }}_1{C}_{20}\\ & -4{\eta }_1{C}_{20}-\mathrm{i}{\kappa }_1{C}_{20}.\end{array}

For mathematical simplicity, we set {\mathrm{\Delta }}_1= {\mathrm{\Delta }}_2= \mathrm{\Delta }, {\lambda }_1= {\lambda }_2=\lambda, {\eta }_1={\eta }_2= \eta, {\mu }_1={\mu }_2= \mu and {\kappa }_1={\kappa }_2=\kappa. Under the weak-driving condition, these probability amplitudes meet the relations of |C_{N_{a_1}{N}_{a_2}}|(N_{a_1}+N_{a_2}=2) \ll |C_{N_{a_1}{N}_{a_2}}|( N_{a_1}+N_{a_2}=1) \ll |C_{00}|\simeq 1. By neglecting the higher-order terms, the steady-state solutions (i.e., t\to \mathrm{\infty }) for the amplitudes C_{N_{a_1}{N}_{a_2}} are given as

(8)C_{01}={\displaystyle \frac{-E\mu }{-M^2+{\mu }^2}},

(9)C_{10}={\displaystyle \frac{-EM}{-M^2+{\mu }^2}},

(10)C_{11}={\displaystyle \frac{-\mu (2\mathrm{i}\lambda {\mu }^2-E^{2}M-2\mathrm{i}\lambda M^2-E^{2}N)}{B}},

(11)C_{02}={\displaystyle \frac{E^2{\mu }^2(M+N)-2\mathrm{i}\lambda {\mu }^{2}MN+2\mathrm{i}\lambda M^{3}N}{\sqrt{2}NB}} ,

(12)\begin{array}{rl}{C}_{20}=& -(E^2{\mu }^{2}M-E^2{\mu }^{2}N+2\mathrm{i}{\mu }^{2}MN-2E^2{M}^{2}N\\ & -2\mathrm{i}\lambda M^{3}N)/(\sqrt{2}NB),\end{array}

with M=\mathrm{\Delta }+ \frac{\mathrm{i}\kappa }{2}+\eta, N=\mathrm{\Delta }+\frac{\mathrm{i}\kappa }{2}+2\eta and B=2( M^2-{\mu }^2)(- {\mu }^2+MN). Here, the full analytical results for optimal conditions ({\lambda }_{\mathrm{o}\mathrm{p}\mathrm{t}} and {\mu }_{\mathrm{o}\mathrm{p}\mathrm{t}}) of C_{20}= 0 are too long to be given. However, it is possible to obtain the simple relation for \mathrm{\Delta } =0 as

(13){\lambda }_{\mathrm{o}\mathrm{p}\mathrm{t}}={\displaystyle \frac{E^2(160{\eta }^4+4{\eta }^2{\kappa }^2+3{\kappa }^4+\sqrt{G})}{4\kappa (64{\eta }^4+20{\eta }^2{\kappa }^2+{\kappa }^4)}},

where G=25600{\eta }^8+21760{\eta }^6{\kappa }^2+6352{\eta }^4{\kappa }^4 +24{\eta }^2{\kappa }^6-7{\kappa }^8. For weak or strong coupling ( or \eta >\kappa), we have the approximate expression: {\lambda }_{\mathrm{o}\mathrm{p}\mathrm{t}}\sim E^2/\kappa. For \mathrm{\Delta }\ne 0, we also have the similar results.

Then, the mean photon number of optical cavity a_1 can be analytically obtained via Eqs. (8)−(12) as

(14)\begin{array}{rl}⟨n⟩=& ⟨a_1^{†}{a}_1⟩\\ =& |C_{10}{|}^2+|C_{11}{|}^2+2|C_{20}{|}^2\\ \approx & |C_{10}{|}^2.\end{array}

And the second-order correlation function in the steady state is given by

(15)\begin{array}{rl}{g}_{\mathrm{s}\mathrm{s}}^{(2)}(0)=& \frac{⟨a_1^{†2}{a}_1^2⟩}{⟨a_1^{†}{a}_1{⟩}^2}\\ =& \frac{2|C_{20}{|}^2}{(|C_{10}{|}^2+|C_{11}{|}^2+2|C_{20}{|}^2{)}^2}\\ \approx & \frac{2|C_{20}{|}^2}{|C_{10}{|}^4}.\end{array}

We first discuss quantum interference-induced photon blockade (or more accurately unconventional photon blockade, UPB) in our system with {\mu }_1= {\mu }_2 and \lambda \ne 0. According to the expression for the second-order correlation function (15), if C_{20}= 0, we have g_{\mathrm{s}\mathrm{s}}^{(2)}(0) =0. In other word, the realization of UPB requires that the real and imaginary parts of C_{20} should be equal to zero at the same time. After some straightforward calculations, we obtain two sets of the optimal UPB conditions for {\mathrm{\Delta }}_{\mathrm{o}\mathrm{p}\mathrm{t}} and {\mu }_{\mathrm{o}\mathrm{p}\mathrm{t}}. However, these analytical results are too long to be presented here. Specifically, when the system parameters are considered as E=0.01\kappa and \lambda = 1\times {10}^{-6}\kappa, we get one set of optimal parameters: for the weak coupling (\eta =0.1\kappa), {\mathrm{\Delta }}_{\mathrm{o}\mathrm{p}\mathrm{t}}\approx 0.15\kappa and {\mu }_{\mathrm{o}\mathrm{p}\mathrm{t}}\approx 1.9\kappa; for the strong coupling (\eta =10\kappa), {\mathrm{\Delta }}_{\mathrm{o}\mathrm{p}\mathrm{t}}\approx -10\kappa and {\mu }_{\mathrm{o}\mathrm{p}\mathrm{t}}\approx 0.7\kappa; for the stronger coupling (\eta ={10}^2\kappa), {\mathrm{\Delta }}_{\mathrm{o}\mathrm{p}\mathrm{t}}\approx - {10}^2\kappa and {\mu }_{\mathrm{o}\mathrm{p}\mathrm{t}}\approx 0.7\kappa. Similarly, the other set of optimal parameters is obtained as follows: for \eta =0.1\kappa, {\mathrm{\Delta }}_{\mathrm{o}\mathrm{p}\mathrm{t}}\approx -2\kappa and {\mu }_{\mathrm{o}\mathrm{p}\mathrm{t}}\approx 12.8\kappa; for \eta = 10\kappa, {\mathrm{\Delta }}_{\mathrm{o}\mathrm{p}\mathrm{t}}\approx - 28\kappa and {\mu }_{\mathrm{o}\mathrm{p}\mathrm{t}}\approx 23\kappa; for \eta = {10}^2\kappa, {\mathrm{\Delta }}_{\mathrm{o}\mathrm{p}\mathrm{t}}\approx -251\kappa and {\mu }_{\mathrm{o}\mathrm{p}\mathrm{t}}\approx 151\kappa.

In Fig.4, we plot the second-order correlation function g_{\mathrm{s}\mathrm{s}}^{(2)}(0) as a function of the detuning \mathrm{\Delta }/\kappa in weak coupling regime (\eta =0.1\kappa). Fig.4 shows that except for the minimal value of g_{\mathrm{s}\mathrm{s}}^{(2)}(0), the analytical results (solid curves) in Eq. (15) agree well with the numerical results (circles) using the master Eq. (4) and H_3. This difference between two minimal values is mainly attributed to the Hilbert-space truncation. In the analytical derivation, we retain only the state |N_{a_1}{N}_{a_2}⟩ with N_{a_1}+N_{a_2}\le 2. But in the numerical simulation, the Hilbert space is truncated into finite dimensions (N_{a_1}\times N_{a_2}{)}^2, where N_{a_1}=N_{a_2}=10. Moreover, it clearly shows that the global minimum of g_{\mathrm{s}\mathrm{s}}^{(2)}(0) is located at the optimal value of {\mathrm{\Delta }}_{\mathrm{o}\mathrm{p}\mathrm{t}}\approx 0.15\kappa, -2\kappa. This feature (the dip D_1, g_{\mathrm{s}\mathrm{s}}^{(2)}(0) \ll 1) shows strong photon antibunching, which indicates the occurrence of UPB. In this situation, the physical mechanism behind UPB can be understood from quantum interference effect and is illustrated in more detail in Fig.5(a). The interference can happen between the four different transition pathways for two-photon excitation: two direct transitions | 10⟩\stackrel{E}{\to }|20⟩ and |00⟩\stackrel{\lambda }{\to }|20⟩, and two coupling-mediated transitions | 10⟩\stackrel{\mu }{\leftrightarrow }|01⟩\stackrel{E}{\to }(|11⟩\stackrel{\mu }{\leftrightarrow }|02⟩)\stackrel{\mu }{\leftrightarrow }|20⟩ and |00⟩\stackrel{\lambda }{\to }|02⟩\stackrel{\mu }{\leftrightarrow }|11⟩\stackrel{\mu }{\leftrightarrow }|20⟩. If the system parameters meet the optimal condition (i.e., C_{20}= 0), these transition pathways will destructively interfere, resulting in the complete suppression of two-photon excitation. We also find in Fig.4 that the second case can be able to produce stronger antibunching than the first case. Similarly, for \eta = 10\kappa, {10}^2\kappa, there are results like the case (\eta =0.1 \kappa ). For this, we mainly focus on studying the effect of the second case on PB in the following discussion.

Next, we consider the strong coupling regime, e.g., \eta = 10\kappa. In Fig.6(a), we depict the second-order correlation function g_{\mathrm{s}\mathrm{s}}^{(2)}(0) and the mean photon number ⟨n⟩ as functions of the detuning \mathrm{\Delta }/\kappa. For strong coupling, we clearly observe that there are three dips [D_1, D_2, and D_3 in Fig.6(a)] located at {\mathrm{\Delta }}_{\mathrm{o}\mathrm{p}\mathrm{t}}\approx - 28\kappa, -33 \kappa, and 13\kappa, respectively. For the dip D_1, there is a minimized value of g_{\mathrm{s}\mathrm{s}}^{(2)}(0). Here, the system parameters \mathrm{\Delta } and \mu can satisfy the optimal condition C_{20}= 0. Apparently, this feature belongs to the UPB. In addition, there also exist two same minima of g_{\mathrm{s}\mathrm{s}}^{(2)}(0) at two dips D_2 and D_3, corresponding to two peaks of the mean photon number ⟨n⟩ (shown by the orange solid curve). These behaviors are different from the UPB, which can be explained as follows. We begin by analyzing the low-energy level structure of the eigenstates of the present system [see Fig.5(b)]. Without considering the single- and two-photon drivings, we diagonalize the Hamiltonian H_{\mathrm{N}\mathrm{D}}= {\omega }_1{a}_1^{†} a_1+{\omega }_2{a}_2^{†}{a}_2-{\eta }_1(a_1^{†} a_1{)}^2- {\eta }_2(a_2^{†} a_2{)}^2+ {\mu }_1{a}_1^{†} a_2+{\mu }_2{a}_2^{†}{a}_1, in the few-photon subspace, yielding the energy eigenstates and eigenvalues. Here, we still assume {\omega }_1={\omega }_2= \omega, {\eta }_1={\eta }_2= \eta, and {\mu }_1={\mu }_2= \mu. In the zero-photon subspace, the corresponding eigenstate and eigenvalue are |0_0⟩=|00⟩ and 0. In the one-photon subspace, there are | 1_{\pm }⟩=1/\sqrt{2}( | 01⟩\pm |10⟩) and \omega - \eta \pm \mu. In the two-photon subspace, | 2_{1-}⟩=1/\sqrt{2}(|02⟩-|20⟩) and 2\omega -4\eta, |2_{2\pm }⟩= N_{\pm }[|02⟩+ | 20⟩+\frac{\eta \pm \sqrt{{\eta }^2+4{\mu }^2}}{\sqrt{2}\mu }|11⟩] and 2\omega -3 \eta \pm\sqrt{{\eta }^2+4{\mu }^2}, where N_{\pm }=1/\sqrt{2+(\frac{\eta \pm \sqrt{{\eta }^2+4{\mu }^2}}{\sqrt{2}\mu }{)}^2} is the normalization constant. In this case, the driving frequency is resonantly (i.e., \mathrm{\Delta } =-\eta - \mu) coupled to the |0_0⟩\to | 1_{-}⟩ transition, but the subsequent transition |1_{-}⟩\to | 2_{1-}⟩ or |2_{2+} ⟩ is suppressed due to large detuning 2(\mu -\eta ) or \sqrt{{\eta }^2+4{\mu }^2}+2\mu -\eta. Likewise, when \mathrm{\Delta }=- \eta +\mu, two-photon excitation process can also be suppressed. Thus, the large detuning can lead to the anharmonicity of the energy-level structure, which explains the essence of conventional photon blockade (CPB). More importantly, by comparing these dips D_1 and D_2 (or D_3) in the g_{\mathrm{s}\mathrm{s}}^{(2)}(0) curve, it is clear that the minimized value of g_{\mathrm{s}\mathrm{s}}^{(2)}(0) in UPB is much lower than in CPB. However, from the ⟨n⟩ curve, we find that the mean photon number in CPB is much higher than in UPB. That is to say, a single photon generated by UPB has high purity, but that by CPB has high brightness.

To achieve CPB and UPB simultaneously, we need to further increase the coupling strength. For example, we choose the stronger coupling with \eta ={10}^2\kappa. From Fig.6(b), we find that there are only two dips D_1 (D_2) and D_3 in the g_{\mathrm{s}\mathrm{s}}^{(2)}(0) and g_{\mathrm{s}\mathrm{s}}^{(3)}(0) curves. Specifically, the right one D_3 is the location of the CPB appearing, which corresponds to the local maximum value of ⟨n⟩. However, the left one D_1 (D_2) can satisfy the optimal conditions for UPB and CPB, i.e., \mu ={\mu }_{\mathrm{o}\mathrm{p}\mathrm{t}} and \mathrm{\Delta } ={\mathrm{\Delta }}_{\mathrm{o}\mathrm{p}\mathrm{t}}= -\eta -\mu. Under these conditions, the system has the strong antibunching (purity) and high mean photon number (brightness). The results indicate that the advantages of UPB and CPB can be brought together in our system. Furthermore, we also study the third-order correlation function in the steady state, i.e., g_{\mathrm{s}\mathrm{s}}^{(3)}(0) ={\mathrm{l}\mathrm{i}\mathrm{m}}_{t\to \mathrm{\infty }}⟨ a_1^{†}{a}_1^{†}{a}_1^{†} a_1{a}_1{a}_1 ⟩(t)/⟨a_1^{†} a_1{⟩}^3(t). As shown in Fig.6(b), both and at two dips D_1 (D_2) and D_3 mean that only the one-photon emission is allowed, but the emission of two or more photons is blocked. Hence, our work can provide a promising route to produce single-photon sources with high purity and brightness.

4 Photon blockade with {\mathit{\mu }}_{\mathbf{1}}\mathbf{\ne }{\mathit{\mu }}_{\mathbf{2}} and different \mathit{\lambda }

In the following, we will explore the performance of single-photon blockade (PB) with nonreciprocal coupling ({\mu }_1\ne {\mu }_2). For the weak coupling (\eta = 0.88\kappa), we plot in Fig.7 the second-order correlation function as a function of \mathrm{\Delta } /\kappa. We see that a strong PB is triggered at \mathrm{\Delta } \approx -0.76\kappa, which satisfies the optimal conditions ({\mathrm{\Delta }}_{\mathrm{o}\mathrm{p}\mathrm{t}} and {\mu }_1 = {\mu }_{\mathrm{o}\mathrm{p}\mathrm{t}}) obtained by calculating C_{20}=0. This PB belongs to unconventional photon blockade (UPB). However, the corresponding results are too cumbersome to present here. For the strong coupling (\eta = 10\kappa), we depict in Fig.8 the second-order correlation function g_{\mathrm{s}\mathrm{s}}^{(2)}(0) and the mean photon number ⟨n⟩ as functions of the detuning \mathrm{\Delta }/\kappa. According to the physical mechanism of conventional photon blockade (CPB), we have {\mathrm{\Delta }}_{+}=-\eta +\sqrt{{\mu }_1} \sqrt{{\mu }_2} and {\mathrm{\Delta }}_{-}=-\eta -\sqrt{{\mu }_1} \sqrt{{\mu }_2}. From Fig.8(a), there are two dips D_1 and D_2 located at {\mathrm{\Delta }}_{-}=-39.8 \kappa and {\mathrm{\Delta }}_{+}= 19.8\kappa, which correspond to the local maximum values of ⟨n⟩. In addition, the left one D_1 at \mathrm{\Delta }={\mathrm{\Delta }}_{\mathrm{o}\mathrm{p}\mathrm{t}}=-\eta -\sqrt{{\mu }_1}\sqrt{{\mu }_2} also satisfies the optimal condition for UPB. This means that CPB and UPB can be obtained simultaneously in the nonreciprocal and strong coupling conditions. Similarly, D_2 at \mathrm{\Delta }={\mathrm{\Delta }}_{\mathrm{o}\mathrm{p}\mathrm{t}}=-\eta +\sqrt{{\mu }_1} \sqrt{{\mu }_2} in Fig.8(b) also satisfies the optimal conditions for CPB and UPB. More importantly, the stronger coupling \eta = {10}^2\kappa mentioned before is not necessary to achieve CPB and UPB effects simultaneously. Therefore, our proposal is useful for relaxing the coupling condition of realizing CPB and UPB with nonreciprocal coupling.

To illustrate the influences of nonreciprocal coupling and two-photon driving on the PB effect more clearly, Fig.9 and Fig.10 show the second-order correlation function g_{\mathrm{s}\mathrm{s}}^{(2)}(0) for different values of {\mu }_1 and \lambda, respectively. From Fig.9(a), one can observe that a strong PB effect can be attained in the value of {\mu }_1=0.67{\mu }_2. This is because the intermediate value satisfies the optimal condition for the UPB. Under this condition, it is clear in Fig.10(a) that the PB effect can be realized without or with two-photon driving. Also, we find that when \lambda = 1\times {10}^{-6}\kappa, the PB is significantly enhanced compared to the \lambda =0 case. This phenomenon results from two-photon driving, i.e., the creation of the |00⟩\stackrel{\lambda }{\to }|20⟩ transition. However, with the further increase of \lambda, two-photon driving decreases the antibunching effect. For the strong coupling, \eta = 10\kappa [Fig.9(b) and Fig.10(b)], these results show behaviors similar to the weak-coupling case \eta =0.88\kappa [Fig.9(a) and Fig.10(a)]. Thus, the appropriate values of nonreciprocal coupling and two-photon driving are helpful to the enhancement of PB.

5 Experimental feasibility and conclusion

To realize a very strong PB effect under the combination of CPB and UPB, the parameter condition must be satisfied. Currently, optomechanical systems, such as microtoroid [65], photonic and phononic crystals [66−72], membrane [73], and nanoparticle [74], have the high-frequency mechanical mode (\sim GHz) and low-loss optics (or mechanics), so that the sideband resolution condition is met. Moreover, our work also needs to satisfy the single-photon strong-coupling condition . Experimentally, this strong coupling has been achieved in cold atoms [75−77]. Some theoretical works, including optical or mechanical parametric amplification [61, 78−80], have reported the exponentially-enhanced coupling. Therefore, our used parameters could be implemented in the currently available optomechanical systems.

In conclusion, we have investigated the CPB and the UPB in a double-cavity optomechanical system. After applying the unitary transformation and neglecting the mechanical part, we derive the effective Hamiltonian of the pure optical system. And then we verify its validity by the master equation. Specifically, we analyze the occurrence of UPB under the weak-coupling condition and the underlying physical mechanism. As the coupling strength is increasing, we find that CPB also emerges in the system. Interestingly, we discover that when the coupling between two optical cavities is reciprocal (or nonreciprocal), CPB and UPB can occur simultaneously in the stronger (or strong) coupling regime. The results imply that we successfully combine CPB and UPB together and utilize their respective advantages to allow one to prepare a single-photon source with high purity and high brightness. Finally, we illustrate the influences of nonreciprocal coupling and two-photon driving on the PB effect.

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