Unconventional photon blockade induced by the self-Kerr and cross-Kerr nonlinearities

Abstract

We study the use of the self-Kerr and cross-Kerr nonlinearities to realize strong photon blockade in a weakly driven, four-mode optomechanical system. According to the Born−Oppenheimer approximation, we obtain the cavity self-Kerr coupling and the inter-cavity cross-Kerr coupling, adiabatically separated from the mechanical oscillator. Through minimizing the second-order correlation function, we find out the optimal parameter conditions for the unconventional photon blockade. Under the optimal conditions, the strong photon blockade can appear in the strong or weak nonlinearities.

Graphical abstract

Keywords

unconventional photon blockade / cross-Kerr nonlinearity / self-Kerr nonlinearity / optomechanical system

1 Introduction

Cavity optomechanics, based on the radiation-pressure coupling between light and mechanical oscillator, has attracted extensive attention in recent years [1–3]. Thanks to this inherent nonlinear coupling, the optomechanical system provides a powerful platform to investigate the photon blockade. So far, there have been two methods for realizing the photon blockade, including the conventional photon blockade (CPB) and the unconventional photon blockade (UPB). The former arises due to the anharmonicity of the energy-level structure [4, 5]. Specifically, the resonant excitation of a single photon prevents the second or subsequent photons from simultaneous excitation. This phenomenon can be observed by evaluating the correlation function, which requires and . However, to produce the sufficient anharmonicity, this scheme requires the strong nonlinear coupling between the cavity and mechanical modes. Therefore, some theoretical schemes have been proposed to significantly enhance the inherent nonlinear coupling, such as the postselected weak measurement [6], the delayed quantum feedback [7], the optical coalescence [8], the Josephson effect [9–11], and the squeezing effects of the cavity or mechanical mode [12–16].

In contrast, the realization of UPB only needs the weak Kerr nonlinearity in the system consisting of two coupled quantum boxes [17]. This arises from the destructive quantum interference between distinct excitation pathways [18], and satisfies the conditions and g^{(3)}(0) >1. Based on this idea, the phenomenon of UPB has been observed in two coupled superconducting resonators [19] and quantum dot cavity-QED system [20]. Afterwards, it was studied in the three-mode optomechanical system, which contains two coupled optical modes and one mechanical mode [21–23].

We should note that most studies of CPB and UPB to date are based on the Kerr or Kerr-like nonlinearity. Different from these studies, Liao et al. [24, 25] used the cross-Kerr coupling between the cavity and mechanical modes to achieve the CPB effect. Also, Wang et al. [26] presented a scheme for implementing the UPB effect via the cross-Kerr coupling between two resonators. Recently, Yang et al. [27] found that the strong self-Kerr and cross-Kerr nonlinearities can significantly strengthen the CPB effect. Inspired by the above-mentioned works, in this paper, we are going to investigate the UPB with the self-Kerr and cross-Kerr nonlinearities in the multimode optomechanical system. This system can be considered as a four-mode system [28], in which two optical cavities are coupled to a mechanical oscillator and an auxiliary optical cavity via radiation pressure and photon tunneling, respectively. Under the Born−Oppenheimer (BO) approximation [28–31], two optical cavities are adiabatically separated from the mechanical oscillator, and the optomechanical parts simplify to the cavity self-Kerr and the inter-cavity cross-Kerr nonlinearities. Comparing with the previous work [27], the main novelty of our work is the realization of strong photon blockade in the condition of weak nonlinearity. Specifically, we calculate the equal-time second-order and third-order correlation functions, and then find that the UPB can be achieved for the strong or weak nonlinearities. Moreover, we briefly discuss the effect of pure dephasing on the UPB. Therefore, we hope that these results obtained will provide guidance for the implementation of strong photon blockades in the optomechanical systems.

2 Model and Hamiltonian

We consider a four-mode optomechanical system [28] as shown in Fig.1(a). In this system, two optical modes a_1 and a_2 with frequencies {\omega }_1 and {\omega }_2 can couple with an auxiliary optical mode c with frequency {\omega }_c via photon tunneling, and they are also coupled to one mechanical mode b with frequency {\omega }_m by radiation pressure. To probe the photon blockade effect, we can drive the optical mode c using a monochromatic field with {\omega }_d and \mathrm{\Omega } being the driving frequency and amplitude. In a rotating frame defined by U(t) =\exp [-\mathrm{i}{\omega }_{d}t(a_1^{†}{a}_1+ a_2^{†}{a}_2+c^{†}c)], the system Hamiltonian is given by (setting \hslash =1)

(1)\begin{array}{rl}H=& H_m+{\mathrm{\Delta }}_1{a}_1^{†}{a}_1+{\mathrm{\Delta }}_2{a}_2^{†}{a}_2+{\mathrm{\Delta }}_c{c}^{†}c+\mathrm{\Omega }(c^{†}+c)\\ & +J_1(a_1^{†}c+c^{†}{a}_1)+J_2(a_2^{†}c+c^{†}{a}_2),\end{array}

where H_m={\omega }_m{b}^{†}b+g_1{a}_1^{†}{a}_1(b^{†}+b) +g_2{a}_2^{†} a_2(b^{†} +b) describes the mechanical part. a_1 (a_1^{†}), a_2 (a_2^{†}), and c (c^{†}) represent the photon annihilation (creation) operators for the optical modes. Similarly, b (b^{†}) denotes the phonon annihilation (creation) operator of the mechanical mode. The last two terms in Eq. (1) correspond to the photon tunneling interaction between the two optical and the auxiliary optical modes with the tunneling coupling strengths J_1 and J_2. The second and third terms in H_m are the radiation pressure interaction between the optical and mechanical modes with the optomechanical coupling strengths g_1 and g_2 [32]. {\mathrm{\Delta }}_k={\omega }_k-{\omega }_d (k=1,2,c) is the detuning of the optical field from the driving field. Here, by adjusting the driving frequency {\omega }_d, the detuning {\mathrm{\Delta }}_k can be much smaller than the mechanical frequency {\omega }_m. With this condition, the optical and mechanical mode can be considered as the slow and fast variables, respectively, and the BO approximation can be applied to H_m [28–31].

Introducing the mechanical position and momentum operators with m_{\mathrm{e}\mathrm{f}\mathrm{f}} being the effective mass, x= \sqrt{\frac{1}{2m_{\mathrm{e}\mathrm{f}\mathrm{f}}{\omega }_m}}( b^{†}+b) and p=\mathrm{i}\sqrt{\frac{m_{\mathrm{e}\mathrm{f}\mathrm{f}}{\omega }_m}{2}}(b^{†}-b), H_m can be rewritten as

(2)\begin{array}{rl}{H}_m=& \frac{p^2}{2m_{\mathrm{e}\mathrm{f}\mathrm{f}}}+\frac{m_{\mathrm{e}\mathrm{f}\mathrm{f}}{\omega }_m^2{x}^2}{2}+\sqrt{2m_{\mathrm{e}\mathrm{f}\mathrm{f}}{\omega }_m}(g_1{a}_1^{†}{a}_1+g_2{a}_2^{†}{a}_2)x\\ =& \frac{p^2}{2m_{\mathrm{e}\mathrm{f}\mathrm{f}}}+\frac{m_{\mathrm{e}\mathrm{f}\mathrm{f}}{\omega }_m^2}{2}{\left[x+\sqrt{\frac{2}{m_{\mathrm{e}\mathrm{f}\mathrm{f}}{\omega }_m^3}}(g_1{a}_1^{†}{a}_1+g_2{a}_2^{†}{a}_2)\right]}^2\\ & -\frac{(g_1{a}_1^{†}{a}_1+g_2{a}_2^{†}{a}_2{)}^2}{{\omega }_m}\\ =& \frac{p^2}{2m_{\mathrm{e}\mathrm{f}\mathrm{f}}}+\frac{m_{\mathrm{e}\mathrm{f}\mathrm{f}}{\omega }_m^2{X}^2}{2}-\frac{(g_1{a}_1^{†}{a}_1+g_2{a}_2^{†}{a}_2{)}^2}{{\omega }_m}\\ =& {\omega }_m{A}^{†}A-U_1{a}_1^{†}{a}_1{a}_1^{†}{a}_1-U_2{a}_2^{†}{a}_2{a}_2^{†}{a}_2-U{a}_1^{†}{a}_1{a}_2^{†}{a}_2,\end{array}

where we have used the BO adiabatic separation between the the optical and mechanical modes. A=\sqrt{\frac{m_{\mathrm{e}\mathrm{f}\mathrm{f}}{\omega }_m}{2}}\left(X+\frac{\mathrm{i}p}{m_{\mathrm{e}\mathrm{f}\mathrm{f}}{\omega }_m}\right) and X=x+\sqrt{\frac{2}{m_{\mathrm{e}\mathrm{f}\mathrm{f}}{\omega }_m^3}}( g_1{a}_1^{†} a_1+ g_2{a}_2^{†}{a}_2) are the annihilation and displacement operators. U_k=\frac{g_k^2}{{\omega }_m} and U=\frac{2g_1{g}_2}{{\omega }_m} are the self- and cross-Kerr nonlinear coefficients. By tracing the Hamiltonian (1) over the mechanical space, we obtain the reduced Hamiltonian for the pure optical system

(3)\begin{array}{rl}{H}_{\mathrm{e}\mathrm{f}\mathrm{f}}=& \mathrm{T}\mathrm{r}[H|m⟩⟨m|]\\ =& {\mathrm{\Delta }}_1{a}_1^{†}{a}_1+{\mathrm{\Delta }}_2{a}_2^{†}{a}_2+{\mathrm{\Delta }}_c{c}^{†}c+\mathrm{\Omega }(c^{†}+c)\\ & +J_1(a_1^{†}c+c^{†}{a}_1)+J_2(a_2^{†}c+c^{†}{a}_2)\\ & -U_1{a}_1^{†}{a}_1{a}_1^{†}{a}_1-U_2{a}_2^{†}{a}_2{a}_2^{†}{a}_2-U{a}_1^{†}{a}_1{a}_2^{†}{a}_2,\end{array}

where the mechanical energy m{\omega }_m has been removed by resetting the ground energy, and |m⟩ is the mechanical eigenvector. From the above Hamiltonian, our model can be considered as a three-coupled-cavity system [see Fig.1(b)] and also indicates that the self-Kerr nonlinearity of the cavity mode and the cross-Kerr nonlinearity between the optical modes are induced by the BO adiabatic separation between the optical and mechanical modes.

3 Results and discussion

In the following, we will investigate the impact of the self- and cross-Kerr nonlinearities on the unconventional photon blockade (UPB) in the mode c, and check the validity of the effective Hamiltonian H_{\mathrm{e}\mathrm{f}\mathrm{f}}. Usually, the UPB effect can be studied by analyzing the equal-time second-order and third-order correlation functions in the steady state, i.e., g_c^{(\mu )}(0) ={\mathrm{l}\mathrm{i}\mathrm{m}}_{t\to \mathrm{\infty }}\frac{⟨c^{†\mu }{c}^{\mu }⟩(t)}{⟨c^{†}c{⟩}^{\mu }(t)} for \mu =2,3. The values of and g_c^{(3)}(0) >1, manifesting two-photon antibunching and three-photon bunching, indicates that UPB only blocks the emission of two photons, but simultaneously allows the emission of three photons.

Specifically, the equal-time second-order correlation function g_c^{(2)}(0) can be analytically calculated from the non-Hermitian Schrödinger equation. Here, we phenomenologically introduce the optical decay to H_{\mathrm{e}\mathrm{f}\mathrm{f}}, and then obtain the non-Hermitian Hamiltonian

(4)H_{\mathrm{n}\mathrm{o}\mathrm{n}}=H_{\mathrm{e}\mathrm{f}\mathrm{f}}- {\displaystyle \frac{\mathrm{i}{\kappa }_1}{2}}{a}_1^{†} a_1-{\displaystyle \frac{\mathrm{i}{\kappa }_2}{2}}{a}_2^{†} a_2-{\displaystyle \frac{\mathrm{i}{\kappa }_c}{2}}{c}^{†}c,

where {\kappa }_k (k=1,2,c) is the decay rate. In the weak-driving regime, i.e., \mathrm{\Omega } \ll {\kappa }_c, by truncating the optical space to n_{a_1}+n_{a_2}+n_c\le 2, the state of this optical system can be written as

(5)\begin{array}{rl}|\psi ⟩=& C_{000}|000⟩+C_{001}|001⟩+C_{100}|100⟩+C_{010}|010⟩\\ & +C_{002}|002⟩+C_{101}|101⟩+C_{011}|011⟩+C_{110}|110⟩\\ & +C_{200}|200⟩+C_{020}|020⟩,\end{array}

with the probability amplitudes C_{n_{a_1}{n}_{a_2}{n}_c}. Then we substitute the non-Hermitian Hamiltonian (4) and the state (5) into the Schrödinger equation \mathrm{i}\frac{\mathrm{d}|\psi ⟩}{\mathrm{d}t}=H_{\mathrm{n}\mathrm{o}\mathrm{n}}|\psi ⟩ to obtain the evolution equations of the coefficients C_{n_{a_1}{n}_{a_2}{n}_c}, which reads

(6)\begin{array}{rl}\mathrm{i}{\dot{C}}_{000}=& \mathrm{\Omega }{C}_{001},\\ \mathrm{i}{\dot{C}}_{001}=& \mathrm{\Omega }{C}_{000}+\sqrt{2}\mathrm{\Omega }{C}_{002}+J(C_{100}+C_{010})\\ & +(\mathrm{\Delta }-\mathrm{i}\kappa /2)C_{001},\\ \mathrm{i}{\dot{C}}_{100}=& \mathrm{\Omega }{C}_{101}+J{C}_{001}+(\mathrm{\Delta }-\mathrm{i}\kappa /2-U/2)C_{100},\\ \mathrm{i}{\dot{C}}_{010}=& \mathrm{\Omega }{C}_{011}+J{C}_{001}+(\mathrm{\Delta }-\mathrm{i}\kappa /2-U/2)C_{010},\\ \mathrm{i}{\dot{C}}_{002}=& \sqrt{2}\mathrm{\Omega }{C}_{001}+\sqrt{2}J(C_{101}+C_{011})+2(\mathrm{\Delta }-\mathrm{i}\kappa /2)C_{002},\\ \mathrm{i}{\dot{C}}_{101}=& \mathrm{\Omega }{C}_{100}+\sqrt{2}J(C_{002}+C_{200})+J{C}_{110}\\ & +2(\mathrm{\Delta }-\mathrm{i}\kappa /2-U/4)C_{101},\\ \mathrm{i}{\dot{C}}_{011}=& \mathrm{\Omega }{C}_{010}+\sqrt{2}J(C_{002}+C_{020})+J{C}_{110}\\ & +2(\mathrm{\Delta }-\mathrm{i}\kappa /2-U/4)C_{011},\\ \mathrm{i}{\dot{C}}_{110}=& J(C_{011}+C_{101})+2(\mathrm{\Delta }-\mathrm{i}\kappa /2-U)C_{110},\\ \mathrm{i}{\dot{C}}_{200}=& \sqrt{2}J{C}_{101}+2(\mathrm{\Delta }-\mathrm{i}\kappa /2-U)C_{200},\\ \mathrm{i}{\dot{C}}_{020}=& \sqrt{2}J{C}_{011}+2(\mathrm{\Delta }-\mathrm{i}\kappa /2-U)C_{020},\end{array}

where we simply consider {\mathrm{\Delta }}_k=\mathrm{\Delta }, {\kappa }_k=\kappa, J_1= J_2=J, and g_1=g_2=g. By neglecting higher-order term in each equation, we can approximately solve these equations in Eq. (6), and then obtain the steady-state solutions of the coefficients C_{n_{a_1}{n}_{a_2}{n}_c}:

(7)\begin{array}{rl}{C}_{000}=& 1,\\ C_{001}=& -2(U-2\mathrm{\Delta }+\mathrm{i}\kappa )\mathrm{\Omega }/M,\\ C_{100}=& C_{010}=-4J\mathrm{\Omega }/M,\\ C_{002}=& 2\sqrt{2}[8J^{2}U+(U-2\mathrm{\Delta }+\mathrm{i}\kappa )(2U-2\mathrm{\Delta }+\mathrm{i}\kappa )\\ & \cdot (U-4\mathrm{\Delta }+2\mathrm{i}\kappa )]{\mathrm{\Omega }}^2/N,\\ C_{101}=& C_{011}=8J(2U-2\mathrm{\Delta }+\mathrm{i}\kappa )(U-4\mathrm{\Delta }+2\mathrm{i}\kappa ){\mathrm{\Omega }}^2/N,\\ C_{110}=& 16J^2(U-4\mathrm{\Delta }+2\mathrm{i}\kappa ){\mathrm{\Omega }}^2/N,\\ C_{200}=& C_{020}=C_{110}/\sqrt{2},\end{array}

with

(8)\begin{array}{rl}M=& 8J^2+(2\mathrm{\Delta }-\mathrm{i}\kappa )(U-2\mathrm{\Delta }+\mathrm{i}\kappa ),\\ N=& M[16J^2(U-2\mathrm{\Delta }+\mathrm{i}\kappa )+(2\mathrm{\Delta }-\mathrm{i}\kappa )(2U-2\mathrm{\Delta }+\mathrm{i}\kappa )\\ & \cdot (U-4\mathrm{\Delta }+2\mathrm{i}\kappa )].\end{array}

For the weak-driving case, we have the approximate solutions, \left\{C_{002},C_{101},C_{011},C_{110},C_{200},C_{020}\right\}\sim {\mathrm{\Omega }}^2 and \left\{C_{001},C_{100},C_{010}\right\}\sim \mathrm{\Omega }. Hence, the second-order correlation function is given by

(9)\begin{array}{rl}{g}_c^{(2)}(0)& =\frac{2|C_{002}{|}^2}{(|C_{001}{|}^2+|C_{101}{|}^2+|C_{011}{|}^2+2|C_{002}{|}^2{)}^2}\\ & \approx \frac{2|C_{002}{|}^2}{|C_{001}{|}^4}.\end{array}

Following the similar method as above, the third-order correlation function is

(10)g_c^{(3)}(0) \approx {\displaystyle \frac{6|C_{003}{|}^2}{|C_{001}{|}^6}},

where

(11)\begin{array}{rl}{C}_{003}=& -512\sqrt{2/3}{J}^4[18U^2-8(2\mathrm{\Delta }-\mathrm{i}\kappa {)}^2+U(-6\mathrm{\Delta }+3\mathrm{i}\kappa )]\\ & \cdot (3U-2\mathrm{\Delta }+\mathrm{i}\kappa ){\mathrm{\Omega }}^3/G+32\sqrt{2/3}{J}^{2}U[104U^3\\ & +304U(2\mathrm{\Delta }-\mathrm{i}\kappa {)}^2-71(2\mathrm{\Delta }-\mathrm{i}\kappa {)}^3\\ & +361U^2(-2\mathrm{\Delta }+\mathrm{i}\kappa )](4U-6\mathrm{\Delta }+3\mathrm{i}\kappa ){\mathrm{\Omega }}^3/G\\ & +36\sqrt{2/3}(U-2\mathrm{\Delta }+\mathrm{i}\kappa )(2U-2\mathrm{\Delta }+\mathrm{i}\kappa )\\ & \cdot (3U-2\mathrm{\Delta }+\mathrm{i}\kappa )(U-4\mathrm{\Delta }+2\mathrm{i}\kappa )(U-6\mathrm{\Delta }+3\mathrm{i}\kappa )\\ & \cdot (4U-6\mathrm{\Delta }+3\mathrm{i}\kappa {)}^2{\mathrm{\Omega }}^3/G\\ & +2048\sqrt{2/3}{J}^6(U+4\mathrm{\Delta }-2\mathrm{i}\kappa ){\mathrm{\Omega }}^3/G,\end{array}

with

(12)\begin{array}{rl}G=& -8J^{2}N[36U^2+27(2\mathrm{\Delta }-\mathrm{i}\kappa {)}^2+77U(-2\mathrm{\Delta }+\mathrm{i}\kappa )]\\ & \cdot (4U-6\mathrm{\Delta }+3\mathrm{i}\kappa )-3N(2\mathrm{\Delta }-\mathrm{i}\kappa )(3U-2\mathrm{\Delta }+\mathrm{i}\kappa )\\ & \cdot (U-6\mathrm{\Delta }+3\mathrm{i}\kappa )(4U-6\mathrm{\Delta }+3\mathrm{i}\kappa {)}^2\\ & +512N{J}^4(U-2\mathrm{\Delta }+\mathrm{i}\kappa ).\end{array}

According to the analytical expression for the second-order correlation function (9), if C_{002}=0, then g_c^{(2)}(0) \to 0. In this case, the photons will exhibit the nearly perfect antibunching, indicating the appearance of complete photon blockade. Therefore, we obtain the optimal conditions for the unconventional photon blockade (UPB) in the mode c as

(13)\begin{array}{rl}0=& 8J^{2}U+2U^3-14U^2\mathrm{\Delta }+28U{\mathrm{\Delta }}^2-16{\mathrm{\Delta }}^3\\ & -7U{\kappa }^2+12\mathrm{\Delta }{\kappa }^2,\\ 0=& 7U^2\kappa -28U\mathrm{\Delta }\kappa +24{\mathrm{\Delta }}^2\kappa -2{\kappa }^3.\end{array}

For fixed J and \kappa, the optimal parameters {\mathrm{\Delta }}_{\mathrm{o}\mathrm{p}\mathrm{t}} and U_{\mathrm{o}\mathrm{p}\mathrm{t}} can be obtained from the above equations. However, these results are too cumbersome to be given here. Fig.2 displays the optimal parameters {\mathrm{\Delta }}_{\mathrm{o}\mathrm{p}\mathrm{t}}/\kappa and U_{\mathrm{o}\mathrm{p}\mathrm{t}}/\kappa as functions of normalized photon tunneling J/\kappa. Obviously, there are two sets of optimal parameters, and they are valid for J/\kappa >0.68. In Fig.2(a), the optimal nonlinearity U_{\mathrm{o}\mathrm{p}\mathrm{t}}/\kappa (i.e., the optimal self- or cross-Kerr nonlinearity) increases with increasing photon tunneling J /\kappa, but in Fig.2(b), the optimal nonlinearity U_{\mathrm{o}\mathrm{p}\mathrm{t}}/\kappa decreases with increasing J /\kappa. This implies that the strong or weak nonlinearity could lead to the complete photon blockade.

Qualitatively, the origin of complete photon blockade can be understood from quantum interference effect, as shown in Fig.3. The interference can happen between the three different paths for two-photon excitation: the direct excitation from |001⟩\stackrel{\mathrm{\Omega }}{\to }|002⟩, and the two tunnel-coupling-mediated transitions |011⟩\stackrel{J}{\to }|002⟩ and |101⟩\stackrel{J}{\to }|002⟩. If the parameters \mathrm{\Delta } and U can satisfy the optimal conditions (13), the three different excitation paths will destructively interfere, resulting in the complete suppression of two-photon excitation. Apparently, these remarkable features are attributed to the UPB effect.

To further verify the analytical results g_c^{(2)}(0) and g_c^{(3)}(0), we numerically study the quantum dynamics of the present system. We introduce the Lindblad master equation of the density operator \rho for the driven-dissipative system,

(14)\begin{array}{rl}\dot{\rho }=& -\mathrm{i}[H,\rho ]+\frac{{\kappa }_1}{2}L[a_1]\rho +\frac{{\kappa }_2}{2}L[a_2]\rho +\frac{{\kappa }_c}{2}L[c]\rho \\ & +\frac{\gamma }{2}{N}_{\mathrm{t}\mathrm{h}}L[b^{†}]\rho +\frac{\gamma }{2}(N_{\mathrm{t}\mathrm{h}}+1)L[b]\rho ,\end{array}

where the Hamiltonian H is given by Eq. (1) or Eq. (3), L[o]\rho = 2o\rho o^{†} -o^{†}o\rho -\rho o^{†}o (o=a_1,a_2,c, b,b^{†}) is the Lindblad superoperator for the three optical modes and the mechanical mode, {\kappa }_k and \gamma denote the decay rate of the cavity fields (k=1,2,c) and the damping rate of the mechanical oscillator, respectively. N_{\mathrm{t}\mathrm{h}}=[\mathrm{e}\mathrm{x}\mathrm{p}(\hslash {\omega }_m/ (k_{B}T ))-1{]}^{-1} is the thermal phonon number in the mechanical mode with the environmental temperature T and the Boltzmann constant k_B. Here, the steady-state value of g_c^{(2)}(0) and g_c^{(3)}(0) can be obtained by numerically simulating the Lindblad master equation (14) and from the density matrix operator in the steady state as g_c^{(\mu )}(0) =\frac{\mathrm{T}\mathrm{r}(\rho {\mathrm{c}}^{†\mu }{\mathrm{c}}^{\mu })}{[\mathrm{T}\mathrm{r}(\rho {\mathrm{c}}^{†}{\mathrm{c}}^{}){]}^{\mu }} with \mu =2,3.

Next, we numerically check the validity of the approximation from the full Hamiltonian (1) to the effective Hamiltonian (3). In Fig.4, we show the evolution of the second-order correlation function g_c^{(2)}(0) with the Hamiltonians H and H_{\mathrm{e}\mathrm{f}\mathrm{f}} as a function of the scaled time \kappa t. The values of \mathrm{\Delta } and U are considered to be the optimal parameters {\mathrm{\Delta }}_{\mathrm{o}\mathrm{p}\mathrm{t}} and U_{\mathrm{o}\mathrm{p}\mathrm{t}}. When the photon tunneling J=2\kappa, {\mathrm{\Delta }}_{\mathrm{o}\mathrm{p}\mathrm{t}} and U_{\mathrm{o}\mathrm{p}\mathrm{t}} from Eq. (13) are calculated as follows: {\mathrm{\Delta }}_{\mathrm{o}\mathrm{p}\mathrm{t}}\approx 4.934\kappa and U_{\mathrm{o}\mathrm{p}\mathrm{t}}\approx 13.637\kappa; {\mathrm{\Delta }}_{\mathrm{o}\mathrm{p}\mathrm{t}}\approx -0.233\kappa and U_{\mathrm{o}\mathrm{p}\mathrm{t}}\approx 0.096\kappa. By comparing the numerical results g_c^{(2)}(0) under two sets of optimal parameters, we find that there are no significant differences between them. From Fig.4, it is evident that the approximate results corresponding to the effective Hamiltonian H_{\mathrm{e}\mathrm{f}\mathrm{f}} are in excellent agreement with the exact results using the full Hamiltonian H. We also note that when \kappa t=15, the correlation function g_c^{(2)}(0) approaches a steady value {10}^{-4}. For the decay rate of the cavity mode \kappa =0.1 MHz, the corresponding relaxation time is t\approx 150 µs.

In view of rapid progress of optomechanical technology, we find that our system is also possibly realized in the multimode optomechanical system [33, 34]. To further obtain the pure optical system, we use the mechanical frequency {\omega }_m= {10}^2\kappa for the realization of the BO approximation. Currently, several optomechanical systems, such as photonic and phononic crystals [35] or microtoroid [36], have the high-frequency mechanical modes (on the order of ~ MHz−GHz), so that the condition of the BO approximation is satisfied. For observation of the UPB effect, we need to satisfy the optimal condition, namely the strong or weak nonlinearity U (i.e., optomechanical coupling g). However, except for cold-atomic experiments [37, 38], the strong nonlinear coupling in the optomechanical experiments is not so easy to realize. To achieve strong coupling, it has been suggested theoretically to use the coalescence effect [8], the Josephson effect [9–11], the squeezing effect [12–16], and so on. These methods could significantly enhance coupling strength by several orders of magnitude. Thus, our used parameters could be implemented in the currently available optomechanical systems.

Fig.5 plots the second-order correlation function g_c^{(2)}(0) as a function of \mathrm{\Delta }/\kappa for different values U. Here, we observe the difference between the numerical and the analytical results for g_c^{(2)}(0). This is because the Hilbert space of the optical system can be truncated to the finite dimension. Specifically, in the approximate analytical calculation, we can truncate the photon state according to the low-excitation number n_{a_1}+n_{a_2}+n_c\le 2. However, in the numerical simulation, we have taken into account the higher state for n_{a_1}+n_{a_2}+n_c\ge 3. In Fig.5(a) and (b), for the photon tunneling J=2\kappa, the optimal nonlinearities are U_{\mathrm{o}\mathrm{p}\mathrm{t}}\approx 13.637\kappa and U_{\mathrm{o}\mathrm{p}\mathrm{t}}\approx 0.096\kappa that correspond to the optimal detunings of {\mathrm{\Delta }}_{\mathrm{o}\mathrm{p}\mathrm{t}}\approx 4.934\kappa and {\mathrm{\Delta }}_{\mathrm{o}\mathrm{p}\mathrm{t}}\approx-0.233\kappa, respectively. These optimal values are calculated analytically according to Eq. (13). As expected, g_c^{(2)}(0) reaches the minimal value {10}^{-4} at the optimal detuning {\mathrm{\Delta }}_{\mathrm{o}\mathrm{p}\mathrm{t}}, and shows strong photon blockade in the strong- or weak-coupling regime. On the other hand, when we consider the case of vanishing nonlinearity (U=0), we find g_c^{(2)}(0) \to 1, i.e., there is no photon blockade effect. Therefore, the nonlinearity is necessary for generating photon blockade.

To more clearly visualize photon blockade, we show the second- and third-order correlation functions g_c^{(2)}(0) and g_c^{(3)}(0) as functions of \mathrm{\Delta }/\kappa. From Fig.6(a) and (b), whether in the strong-coupling regime (U>\kappa) or in the weak-coupling regime (), we can observe and g_c^{(3)}(0) >1 around the optimum value {\mathrm{\Delta }}_{\mathrm{o}\mathrm{p}\mathrm{t}}, which indicates that two-photon emission is suppressed but three-photon emission is enhanced. It should be noted that this phenomenon corresponds to the UPB effect, coming from the destructive quantum interference.

Finally, we study the influence of pure dephasing on the UPB effect [22, 39]. To illustrate how the pure dephasing affects UPB, we introduce another Lindblad term defined by \frac{{\gamma }_p}{2} L[c^{†} c]\rho into the master equation (14), where {\gamma }_p is the pure dephasing rate of the cavity mode c. In Fig.7(a) and (b), we plot the second-order correlation function g_c^{(2)}(0) as a function of \mathrm{\Delta }/\kappa for different pure-dephasing rates {\gamma }_p=0, {\gamma }_p=0.01 \kappa, {\gamma }_p =0.1\kappa. It can be seen that with the increase of the pure-dephasing rate, g_c^{(2)}(0) increases gradually at the optimal detuning ({\mathrm{\Delta }}_{\mathrm{o}\mathrm{p}\mathrm{t}}\approx 4.934\kappa or {\mathrm{\Delta }}_{\mathrm{o}\mathrm{p}\mathrm{t}}\approx -0.233\kappa). This means that the strong pure dephasing can attenuate the UPB effect, leading to the weak photon antibunching.

4 Conclusion

In summary, we have analyzed the unconventional photon blockade effect (UPB) in the four-mode optomechanical system. In this system, the self- and cross-Kerr nonlinearities result from the BO adiabatic separation between the optical and mechanical modes, which is the important factor to produce the UPB effect. By minimizing the result of the second-order correlation function, we derive the two sets of the optimal parameter conditions required for the UPB. It is found that under these conditions, the UPB can be observed in the strong or weak nonlinearities. This is due to the destructive quantum interference between the three different paths for two-photon excitation in the mode c. Additionally, we also find that the pure dephasing has an undesirable effect on the UPB.

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