1. Introduction

In cavity optomechanics, the interaction of optical fields with mechanical oscillators provides an excellent research platform for the study of fundamental physics and applications [1–3]. The radiation pressure or gradient forces [4] of the conventional optomechanical interactions lead to a dispersive shift of the cavity frequency. It has been employed for squeezing of optical mode [5,6], entanglement between optical and mechanical mode [7,8], cooling the mechanical oscillators to the quantum ground state [9,10], and optomechanical normal mode splitting [11,12]. Moreover, many interesting applications have been demonstrated due to the nonlinear optomechanical interaction [1], such as photon blockade [13–17], Kerr nonlinearity [18,19] and optomechanical induced transparency [20–23].

In general, optomechanical interactions mainly manifest in two distinct forms of coupling: dispersive coupling [1,24] and dissipative coupling [25,26]. Dispersive coupling typically occurs in conventional optomechanical systems, in which displacement of the mechanical oscillator results in a shift in the optical cavity resonant frequency. In contrast, the dissipative coupling characterizes the dependence of the cavity decay rate on the displacement of the mechanical oscillator. The dissipative coupled linearized optomechanical system has recently attracted considerable attention, e.g., cooling the mechanical oscillators in the unresolved sideband regime [26–28], electromagnetically induced transparency [12,29] and the squeezing of the output light [30–32]. Dissipative coupling has been experimentally achieved with different realizations, such as a waveguide coupled to a microdisk resonator [25], a monolayer graphene membrane in an optical resonator [33], a tapered fiber coupled to a whispering-gallery mode [34–36], and a photonic crystal cavity [37]. Among these, an attractive scheme in the optical domain with a Michelson-Sagnac interferometer (MSI) to realize dispersive and dissipative coupling was first proposed in [27], and later experimentally investigated in [10] that can cool the mechanical oscillator from room temperature (293 K) to 126 mK.

In the dissipative coupled systems, the classical nonlinear effects have been well studied. However, to our knowledge, very few studies have discussed the properties the property of quantum nonlinear effects, which can be generally characterized by the photon anti-bunching effect. Recently, the second-order correlation of the output light [31] and the nonreciprocal photon blockade [38] were studied in dissipative coupled systems. It is noteworthy that, the nonlinear effect discussed in Ref. [31] is studied under the linearization approximation, while in Ref. [38], a single-mode approximation of the environment is made. These two treatments either lack an effective description of the quantum nonlinearity, or face difficulties in experimental realizations. Therefore, an experimentally realizable scheme with a general theoretical approach, including the description of quantum nonlinearity, is urgently needed.

In this paper, we demonstrate that strong quantum nonlinearity can be realized in the dissipative coupling regime in the MSI optomechanical system. The photon anti-bunching effect of the composite system, consisting of a dissipative coupled compound cavity and an auxiliary cavity to achieve unconventional photon blockade, is studied by utilizing the full-quantum approach. Our results show that the quantum nonlinear effect is highly dependent on the dispersive coupling strength. More importantly, the interplay between dispersive and dissipative couplings may produce strong anti-bunching (blockade) even in the weak coupling regime. We also identify the optimal parameter condition for producing a strong blockade effect, i.e., $\Delta _c \Delta _e=J^2$. Moreover, the dissipative coupling plays a key role in protecting the quantum nature of the system. If the gain of the dissipative coupling strength is equal to the loss caused by the conventional dissipation, the dissipative coupling will balance the effect of system loss. Then the system can be approximately described as a closed system (without considering quantum fluctuations of the environment [39]), and the quantum nonlinear effect is highlighted.

The paper is organized as follows. In Sec. II, we introduce the system Hamiltonian and derive the effective Hamiltonian. In Sec. III, both numerical and analytical methods are used to study the second-order correlation function. In Sec. IV, we discuss the feasibility of the experiment. Finally, we conclude and discuss the application prospects of our scheme in Sec. V.

2. Model and Hamiltonian

A compound MSI optomechanical model [40], capable of both dispersive and dissipative coupling, is depicted in Fig. 1. Our model contains a standard optical cavity (auxiliary cavity) and an MSI optomechanical system (compound cavity) that can realize both dispersive and dissipative coupling. In the MSI optomechanical system (surrounded by blue dotted line in Fig. 1), $\text {M}_1$, $\text {M}_2$ and $\text {M}_3$ are perfect total reflection mirrors, while the movable membrane $\text {M}$ is a semitransparent membrane. As is shown in the diagram, the beam splitter (BS) in the optical path is used to control the input and output of the MSI-cavity. The reflectivity $R$ and transmissivity $T$ of the BS can determine the magnitude of the dispersive and dissipative coupling [41]. In extreme situations, a purely dissipative coupling can even be achieved [27] (see Sec. IV for a specific discussion). The effective MSI-cavity length can be obtained by halving the length of the Sagnac mode $M_1\to BS\to M_2\to M_3 \to BS$ (or its chiral pair $M_1\to BS\to M_3\to M\to M_2\to BS$) [27]. The displacement of the membrane $M$ can change the cavity length as well as the the interference optical path of the MSI. Thus the membrane will affect the cavity frequency and the input-output relationship, resulting in dispersive and dissipative coupling. Besides, the compound cavity is coupled to the auxiliary cavity with the coupling strength $J$, and the auxiliary cavity is an optical empty cavity. The total Hamiltonian of the model is read [26] ($\hbar =1$)

The first three terms in Eq. (1) represent the free energy of the compound cavity, auxiliary cavity and the mechanical oscillator, respectively. In our model, an auxiliary cavity is introduced to construct multipath quantum coherent to achieve unconventional blockade with lower nonlinearity coefficient requirements [42]. $\hat {a}_c(\hat {a}_e)$ is the bosonic operator of the compound cavity (auxiliary cavity) with frequency $\omega _c(\omega _e)$. $\hat {Q}=\hat {x}/x_{zpf}$ and $\hat {P}=\hat {p}/(m\omega _m x_{zpf})$ are the dimensionless displacement and momentum operators, respectively. $\hat {x}=x_{zpf}(\hat {b}+\hat {b}^{\dagger})$ is the displacement of mechanical oscillator, and $\hat {b}$ is the annihilation operator of the mechanical oscillator with frequency $\omega _m$. $x_{zpf}=(2m\omega _m)^{-\frac {1}{2}}$ denotes the size of the zero-point fluctuations and $m$ is the mass of the mechanical oscillator. The fourth term represents the total cavity-mechanical coupling. The parameters $A$ and $B$, which can be derived from the frequency and dissipative shifts, denote the weights of the dispersive and dissipative couplings, respectively. The dispersive coupling is described by the optical frequency shift per zero-point fluctuation $g_\omega =\frac {\partial \omega _c(Q)}{\partial x}x_{zpf}$ and the dissipative coupling by the dissipation shift per zero-point fluctuation $g_\kappa =\frac {\partial \kappa _c(Q)}{\partial x}x_{zpf}$ (the specific expressions for $\omega _c(Q)$ and $\kappa _c(Q)$ are in the description of Eq. (2)). The fifth term represents the cavity-bath interaction with $\hat {H}_{int}=i\sqrt {\frac {\kappa _c}{2\pi \rho }}\sum _{q}(\hat {a}_c^{\dagger}\hat {b}_{q}-\hat {b}_{q}^{\dagger}\hat {a}_c)$ [43,44], where $\hat {b}_q$ is bosonic annihilation operator of the optical bath coupled to the compound cavity. $\hat {H}_{diss}$ represents the commonly bath of the auxiliary-cavity and oscillator as well as the system-bath interactions [45]. The second last term represents the tunneling coupling of compound cavity and auxiliary cavity with the strength $J$. The last term denotes the laser driving, which can be introduced by the BS as a semi-classical input to the MSI in our model, where $\omega _d$ denotes the frequency of driving laser and $\epsilon _j$ denotes the laser driving strength. In fact, $\epsilon _c=2\sqrt {\frac {P(\kappa _c+g_\kappa \hat {Q})}{\hbar \omega _d}}$ [46,40] and $\epsilon _e=2\sqrt {\frac {P\kappa _e}{\hbar \omega _d}}$ with the input power $P$ of the laser. Under the weak driving approximation [47,48], we can get $\epsilon _c \approx 2\sqrt {\frac {P\kappa _c}{\hbar \omega _d}}$.

 

Fig. 1. The sketch of Michelson-Sagnac interferometer [27] with a movable membrane M between the mirror $M_2$ and $M_3$ as the effective MSI mirror(the area surrounded by black dotted line) combined with $M_1$ into a Fabry-P$\mathrm {\acute {e}}$rot cavity which coupled to an auxiliary cavity with the tunneling coupling strength $J$. Q represent the displacement of the membrane M.

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With the displacement-modified input-output relation in the Markovian regime [27,45], the Hamiltonian can be reduced by equating the environmental operators $\sqrt {\frac {\kappa _c}{2\pi \rho }}\sum _{q}\hat {b}_q$ in the system-environment interaction to $\sqrt {\kappa _c}\hat {a}_{c,in}-\frac {\kappa _c}{2}\hat {a}_{c}\left (1+\frac {g_\kappa }{2\kappa _c}\hat {Q}\right )$ [12]. After eliminating the bath of MSI-cavity and in the rotating frame, the reduced Hamiltonian is then given by [27,29]

Bringing Eq. (4) back to Eq. (3c), the mechanical mode can be reduced and thus obtained the dynamical equations for the long-time evolution of the optical modes,

The effective noise operator reads

The corresponding quantum correlation can be obtained by using $\delta$-correlation functions of optical and mechanical noise,

This effective Hamiltonian describes the square-coupled nonlinear form of the optical modes in the dissipative coupled system. According to the last term in Eq. (8), the dispersive coupling strength $g_\omega \neq 0$ is a prerequisite for the nonlinear effect of the system. Moreover, this dispersive coupling affects both the amplitude and the phase of the light (corresponding to the imaginary and real parts of the nonlinear term), whereas the dissipative coupling affects only the amplitude. It is worth noting that this dissipative nonlinear effect is the opposite of the loss effect of the cavity field to the environment.

Again, it is visible from the comparison of the sign of $\frac {\kappa _c}{2} \hat {a}_c$ and $\frac {g_\kappa g_\omega \hat {N}_c}{2\omega _m}\hat {a}_{c}$ in Eqs. (5a). This provides us with the possibility that if we choose the suitable strength of the dissipative nonlinearity, it is able to resist the loss from the environment and thus protect the quantum property of the system. This effect of resisting noise is very similar to the processing of non-Hermitian Hamiltonian [51] and is verified in Fig. 6 of our discussion later.

The effective noise as a function of the original parameters are illustrated in Fig. 2. Obviously, as shown in Fig. 2(a), the lower the average photon number or the phonon temperature, the more helpful it is to obtain a lower environmental excitation number. For example, $n_{eff}\left (n_{th}=100,\bar {N}=1\right )\approx 1.04\times 10^{-3}$, the effective environmental excitation number is almost 5 orders of magnitude smaller than the average photon number. The effective environmental temperature is 3 orders of magnitude smaller than the real temperature (e.g. the temperature is about $5$mK with $\omega _m=10^{6}$Hz and the corresponding effective environmental temperature is $7\times 10^{-6}$K). As shown in Fig. 2(b), to realize a lower effective environmental noise, the weak coupling condition is necessary. As a result, our system has much stronger temperature robustness. Compared with the general experimental phonon temperatures [1], the effective environmental temperature is close to zero as long as the driving field is weak enough to ensure a low level of average intracavity photon number.

 

Fig. 2. (a) Effective thermal phonon number $n_{eff}$ as a function of the photon number $\bar {N}$ and thermal phonon number $n_{th}$. The coupling strength are $g_\omega /\omega _m=2\times 10^{-4}$ and $g_\kappa /\omega _m=5\times 10^{-4}$. (b) $n_{eff}$ as a function of the dispersive coupling strength $g_\omega$ and dissipative coupling strength $g_\kappa$. The thermal phonon number $n_{th}=100$ and the photon number $\bar {N}=1$. The other parameters are $\kappa /\omega _m=5\times 10^{-3}$, $\gamma /\omega _m=10^{-5}$.

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To confirm the validity of our approximation, we numerically compare the original Hamiltonian with our effective Hamiltonian in the calculation of the steady-state second-order correlation function in Fig. 3. The corresponding master equations for the numerical simulations are,

 

Fig. 3. The first three figures represent (a) the mean photon number $\langle \hat {a}_c^{\dagger}\hat {a}_c\rangle$, (b) the mean value of the nonlinear term $\langle \hat {a}_c^{\dagger}\hat {a}_c^{\dagger}\hat {a}_c\hat {a}_c\rangle$, (c) the second-order correlation function $g_c^{(2)}(0)$ as the function of the detuning $\Delta _c$ compared the numerical solution of $H_{eff}$ [Eq. (8)] with $H_{R}$ [Eq. (2)], respectively. (d) The ratio of $g^{(2)}_{eff}(0)$ to $g^{(2)}_{R}(0)$ as the function of the detuning $\Delta _c$ and $\Delta _e$, and the six curves originate from taking six identical intervals for $\Delta _e$, from left to right are $\Delta _e/\omega _m=$ −0.5, −0.5+1/6, −0.5+1/3, −0.5+1/2, −0.5+2/3, −0.5+5/6. (e) is the Y-Z view of (d) and (f) is the X-Z view of (d). The other parameters are $\Delta _e/\omega _m=0.2, \kappa _c/\omega _m=5\times 10^{-3}, \kappa _e/\omega _m=5\times 10^{-3}, g_\omega /\omega _m=2\times 10^{-4}, g_\kappa /\omega _m=5\times 10^{-4}, \gamma /\omega _m=1\times 10^{-5}, n_{th}=100, J/\omega _m=0.2, \epsilon _c/\omega _m=5\times 10^{-3}, \epsilon _e/\omega _m=5\times 10^{-3}.$

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As shown in Fig. 3, (a), (b), (c) provides a good visualisation of the well-fitting results of our effective Hamiltonian and the original Hamiltonian. To give a more specific representation of the viability of our effective method, we define the second-order correlation function ratio $g_{eff}^{(2)}(0)/g_{R}^{(2)}(0)$ under the effective Hamiltonian and the original Hamiltonian and plot the figure (d), (e) and (f). When this ratio is 1, it means that our method is perfect. By traversing the parameters, the average error of our method is $\pm 0.08{\%}$ and the maximum error range is $\pm 7{\%}$.

3. Second order correlation function for optical mode

The second-order correlation function is a very effective formula to evaluate quantum nonlinearity, and is easily detected in experiments [52]. To investigate the properties of the optical quantum nonlinear effects in dissipative coupled systems, both numerical and analytical methods are used to solve the second-order correlation function for specific analysis. For simplicity, the state truncation method [48] is used to obtain the analytical solution.

In the weak pumping conditions $C_0\gg C_{j,ce}\gg C_{jj}, \{j=c,e\}$ and after utilising the truncation method, the general optical quantum state is expressed as,

To correctly account for the nonlinear effects character of the system, we introduce the quantum master equation [47] for the system density matrix by using Eq. (9a). Then, we can derive the corresponding numerical solutions of Eq. (9a) to analyse the effect of the parameters on the second-order correlation function. The comparison of the analytical and numerical solutions is shown in Fig. 4. These two results are well matched in the figures. Surprisingly, there is a sharp decrease in both the analytical and numerical solutions at $\Delta /\omega _m = -J/\omega _m = -0.2$, which corresponds to $D_J = 0$ (as we have analysed in our analytical solution), i.e., $g^{(2)}(0) = 0$. At this singularity, the assumption of $\kappa = 0$ in the analytical solution causes the stronger anti-bunching effect from the analytical solution than from the numerical ones, corresponding to the black-dashed line being much deeper than the red-solid line. Moreover, $g^{(2)}_c(0)$ from both the numerical and analytical solutions can reach their minimum in Fig. 4(a), which can be explained by the previous analytical solution i.e., $g^{(2)}_c(0)=0$ with $K=0$. When $\Delta _j/\omega _m>0.2$, we can see that $g^{(2)}_c(0)$ from both the numerical and analytical solutions converge to 1, which means that the light is in the coherent state for the large detuning case. The zero-time second-order correlation of photons in the compound cavity and the auxiliary cavity as a function of the two cavities’ detuning are shown in Fig. 5, respectively. As shown in Fig. 5, the near-photon blockade region (the region surrounded by the white dashed line) matches well with the analytical solution, corresponding to the hyperbolic region and the linear region. The simplified expressions between $\Delta _c$ and $\Delta _e$ can be obtained according to Eqs. (12): $\Delta _c=J^2/\Delta _e$ and $\Delta _c=-\Delta _e$ for the hyperbolic region and for the linear region, respectively. Except for above special conditions, a rapid enhancement of $g_c^{(2)}(0)$ will cause the appearance of classical effects for the cavity. Especially when $-\Delta _e = J$, Fig. 5 shows a strong bunching effect. This special condition can be understood from the effective Hamiltonian in Eq. (8), when $-\Delta _e=J$, the detuning energy of the auxiliary cavity is exactly the same as the energy of the BS interaction. Therefore, the energy of the auxiliary cavity can be converted into the compound cavity through a resonance-like effect. Due to the condition $J\gg \{g_\kappa,g_\omega \}$, the converted energy can eliminate the blockade effect due to nonlinear energy shift ($\propto n^2$) and excite the photons in the compound cavity, thus to exhibit a bunching effect.

 

Fig. 4. The comparison of the numerical and analytical solutions in the compound cavity (a) and auxiliary cavity (b) with $\Delta _c=\Delta _e=\Delta$. The vertical coordinate is zero-time second-order correlation $\log _{10}g^{(2)}(0)$. The other parameters are the same as in Fig. 3.

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Fig. 5. Zero-time second-order correlation $g^2(0)$ as a function of the detuning $\Delta _c/\omega _m$ and $\Delta _e/\omega _m$ in compound [(a),(b),(c),(d)] and auxiliary [(e),(f),(g),(h)] cavity. Where $g_\omega /\omega _m=2\times 10^{-4}$ remains unchanged and $g_\kappa /\omega _m$ is 0, $2\times 10^{-4}$, $4\times 10^{-4}$ and $6\times 10^{-4}$, respectively. The white dotted line represents the value of $log_{10}[g^2(0)] = -1.5$. The other parameters are the same as in Fig. 3.

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Comparing the plots in Fig. 5, it shows that $g_\kappa$ has an optimising effect on $g_c^{(2)}(0)$. From the four plots of the compound cavity, although the increase of $g_\kappa$ can not result a significant decrease of $g_c^{(2)}(0)$, while can increase the range of parameters in the near-photon blockade region. It can be seen from the expression $g^{(2)}_c(0) \propto \frac {1}{|(D_J-\Delta _e^2)G+2\omega _mKD_J|^2}$. Obviously, $g^{(2)}_c(0)$ is decreasing with the increase of $|G|$ ($|G|$ increases with $g_\kappa$). Note that this increasing trend is relatively saturated when $g_\kappa /\omega _m$ exceeds 4$\times 10^{-4}$. A similar conclusion can be obtained in the auxiliary cavity, as shown in Fig. 5, where the region of the minimum value of the second-order correlation function (the blue region in the figure) exhibits a clear hyperbolic function characteristic. Differently, the near-photon blockade region is gradually becoming smaller as $g_\kappa$ increases, which is widely divergent to compound cavity, due to the different expression $g^{(2)}_e(0)\propto \left |\frac {D_J f_B G+X}{ (J+\Delta _c)^2(D_J-\Delta _e^2)G+Y}\right |^2$ ($X$ and $Y$ are constant), and $|D_J f_B|> |(J+\Delta _c)^2(D_J-\Delta _e^2)|$, thus $g^{(2)}_e(0)$ is increasing with the increase of $|G|$. Therefore, $g_\kappa$ has a negative damping of the photon blockade effect in the auxiliary cavity.

Besides the effect of detuning on the system, we still need to explore the contribution of dissipative and dispersive coupling to the second-order correlation function. We select the detuning parameters of the near-blockade region in Fig. 5 with $\Delta _c=\Delta _e=-J$ to investigate this contribution and illustrate it in Fig. 6.

 

Fig. 6. Zero-time second-order correlation $g^{(2)}(0)$ as a function of the dispersive coupling strength $g_\omega$ and dissipative coupling strength $g_\kappa$ in compound-cavity (a) and auxiliary cavity (b). The blank dotted line represents the value of $\log _{10}[g^{(2)}(0)]= -3$, the white dotted line represents the value of $\log _{10}[g^{(2)}(0)] = 0$, and the red point represent the minimum of the $g^{(2)}(0)$ (It’s size and coordinates is are also shown in the figure). The detuning $\Delta _c=\Delta _e=-J$. The other parameters are the same as in Fig. 3.

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As shown in Fig. 6, the second-order correlation function $g^{(2)}_c(0)$ is the complex nonlinear relationship with the parameters $g_\omega$ and $g_\kappa$ for both the compound- and auxiliary-cavities. Obviously, a smaller $g_\omega$ requires a lager $g_\kappa$ to obtain photon blockade effect for both the compound- and auxiliary-cavities, as is marked in I-regions of Fig. 6(a) and Fig. 6(b). The special regions (Region-II in Fig. 6) with $\log _{10}[g^{(2)}(0)] \le -3$ are also marked to observe the optimal $g_c^{(2)}(0)$. Compared with the case of non-dissipative coupling (corresponding to the regions with $g_\kappa =0$ in the figure), an improved photon blockade is obtained with the help of dissipative coupling. Note that, with the parameters in the figure, for the non-dissipative coupling case, $g^{(2)}(0)$ cannot reach the value in Region-II even if $g_\omega$ increases close to the unstable region. This conclusion is consistent with the trend in Fig. 4, i.e., the minimum value of the numerical solution in (b) is smaller than that in (a) at $\Delta =-J$.

The impact of dissipation on the system is shown in Fig. 7. The second-order correlation function $g_c^{(2)}(0)$ will reaches the minimum values as $\kappa$ increases. In addition, the minimal values decrease as the dissipative coupling strength $g_\kappa$ increases, which is described by the red- dotted line in the graph. This phenomenon can be explained by Eq. (5a). When the dissipative coupling term $g_\kappa g_\omega /2\omega _m \hat {a}^{\dagger} _c \hat {a}_c \hat {a}_c$ is equal to the optical dissipative term $\kappa _c/2 \hat {a}_c$, the system can be treated as a dissipation-free system. Then the nonlinear effect of the system can be strengthened, thus the minimum values of the second-order correlation function $g_c^{(2)}(0)$ will increase. This process is similar to the inverse dissipation in non-Hermitian Hamiltonian and is discussed in the analysis of Eq. (8). The minimum second-order correlation function is derived as $\kappa _{min} \propto \frac {g_\kappa g_\omega }{2\omega _m}$. Obviously, $\kappa _{min}$ is increased with the rising of $g_\kappa g_\omega$.

 

Fig. 7. Zero-time second-order correlation $g^{(2)}(0)$ as a function of the dissipation $\kappa /\omega _m$ in two cavities. Where $g_\omega /\omega _m=2\times 10^{-4}$ remains unchanged and $g_\kappa$ is 0, $2\times 10^{-4}$, $4\times 10^{-4}$ and $6\times 10^{-4}$, respectively. The other parameters are the same as in Fig. 6.

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According to the previous analysis, $J$ is a very important parameter which can be tuned in waveguide-coupled and close-coupled optical systems [53,54]. As shown in Fig. 8, the effect of $J$ and the detuning on the second-order correlation function $g^{(2)}(0)$ is analysed. We can see that the photon blockade effect (black-dashed line) only occurs when $J = -\Delta$. While the second-order correlation function $g^{(2)}(0)$ increases rapidly when $J\neq -\Delta$. Moreover, compound cavity shows a anti-bunching effect while the auxiliary cavity shows a bunching effect (white-dashed line) for $\Delta =0$. This difference can be understood by the analytical solution. In Eq. (12), we have derived $g_c^{(2)}(0)=0$ and $g_e^{(2)}(0)=4$ at $\Delta = 0$.

 

Fig. 8. Zero-time second-order correlation $g^{(2)}(0)$ as a function of the cavity coupling strength J and detuning $\Delta$ in compound cavity (a) and auxiliary cavity (b). The black dashed line represents $log_{10}[g^2(0)] = -1$ and the white dashed line represents $log_{10}[g^2(0)] = 0.602$. The other parameters are the same as in Fig. 3.

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4. Experimental feasibility discussion

For the realization of our scheme, the required MSI has been experimentally implemented using a macroscopic silicon nitride membrane with dimensions of $(1.5 \text {mm})^2$ in a cavity-enhanced Michelson-type interferometer [10]. This setup, combined with an additional mirror (with a reflectivity of $0.9997$ according to Ref. [10]), forms an optical cavity with an effective length equal to half the photon in/out. The displacement of the mirror (labeled as M in Fig. 1) changes the linewidth of the cavity, leading to the so-called dissipative coupling cavity linewidth [10,26].

The key parameters in our discussion, $g_\omega$ and $g_{\kappa }$, can be modulated by the transmissivity-reflectivity of the BS ($T$ and $R$) and M ($\tau$ and $\rho$). The relationships are detailed in Ref. [27], where $g_\omega =\sqrt {2}g_0 \alpha$ and $g_{\kappa }\approx -i g_0 |\tau |\beta$. Here, $\alpha =2[(|R|^2-|T|^2)+\tau e^{-i v}\cos (\chi )]$, $\beta =2[2RT+\rho e^{-i v}\cos (\chi )]e^{i(\sigma +\chi +v)}$, $g_0=\omega _c x_{zpf}/(\sqrt {2}L)$, $\chi$ is the argument of transmission coefficient, $v$ is the phase-delay of $M_1-BS$ and $\sigma$ is the arm-length sum phase for the loop in the interferometer. With the experimentally reported parameters [55] ($x_{zpf}=1\text {fm}$, driving wavelength $\lambda _c=1064 \text {nm}$ and $L=7.5\text {cm}$), setting $|R|^2/|T|^2=0.643/0.357$ can achieve the coupling $g_{\kappa }: g_\omega =5: 2$ as used in our figure.

The eigenfrequency of the mechanical oscillator we used is on the order of $10^6$Hz, which is realized in experiments for many optomechanical systems [1,58,59]. Recent experimental reports indicate that the frequency of a silicon nitride membrane in MSI-based dissipative coupled systems is $\omega _m=2\pi \times 136$kHz [10]. In our numerical simulations the relative parameters actually work. The eigenfrequency of the mechanical oscillator is only used to calculate the temperature. According to the detailed discussion in Fig. 7, we can achieve $g^{(2)}_j(0)<10^{-1}$ with $\kappa /\omega _m<10^{-2}$. This requirement under resolved sideband conditions has been realized in some optomechanical systems [60–62]. In Fig. 6, we show that it is possible to realize a strong anti-bunching effect in the weak coupling regime with $g_{\omega (\kappa )}/\omega _m$ in the order of $10^{-4}-10^{-3}$. For single-photon coupling rates, experiments are still in the weak coupling regime, i.e. $g/\omega _m\in \{10^{-9}-10^{-1} \}$ [10,37,53–58,62–63]. Among the systems that can achieve dissipative coupling require special mechanical oscillators, and current experimental realisations include membrane and crystal systems. Some of the dissipative coupled experimental system parameters are reviewed in Table 1. Membrane-based systems can achieve arbitrary ratios of $g_{\kappa }: g_\omega$ by adjusting the transmission/reflection coefficient, but the experimental coupling strength are too weak to achieve strong anti-bunching effect. Correspondingly, the coupling strength of the crystal-based systems is sufficient to achieve anti-bunching effect, but the ratio $g_{\kappa }: g_\omega <1$. We labelled the experimental parameters reported by Meyer et al., 2016 [53], Sawadsky et al., 2015 [10] and Primo et al., 2023 [56] in Fig. 6 as ‘$\circ$’, ‘$\vartriangle$’ and ‘$\square$’, respectively. These experimental parameters allow weak anti-bunching effect. We find that a crystal-like adjustable coupling strength is given in two photonic crystal membranes based optomechanical platform [64] may meet our requirements. Additionally, the temperature requirement in our system is compatible with related experiments [58,65]. The effective environmental temperature is 7 to 8 orders of magnitude smaller than the real temperature (see Fig. 2). The introduction of an auxiliary cavity can be coupled via low-loss optical fiber [66,67] or waveguide [68]. For the detection of output light, we can use a non-polarizing beamsplitter and two single-photon detectors to record photon counts and photon auto-correlations, as reported in experiments to observe unconventional photon blockade [69].

Table 1. A brief review of experimentally achieved coupling strengths in dissipative coupled optomechanical systems.

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5. Conclusion

In conclusion, a full-quantum method is applied to study the dissipative coupled system without using linearized approximation. A strong anti-bunching effect is realized in the nonlinear weak coupling regime. The optimal region of second-order correlation function is a hyperbolic form with the relation $J^2 = \Delta _c \Delta _e$. Moreover, the system exhibits the photon blockade effect with the appropriate parameters for $g_\omega$ and $g_\kappa$. This is different from the nonlinear effects in pure dispersion-coupled systems [13,70,71], where the quantum nonlinearity is proportional to the coupling strength $g_{\omega }$ (i.e., the case $g_\kappa =0$ in Fig. 6). Dissipative-coupling-assisted optomechanical system can achieve strong quantum nonlinearity ($g^{(2)}(0)\rightarrow 0$) under weak coupling conditions (see Region-II in Fig. 6). Especially, the dissipative coupling in our system can resist the destruction of the quantum effects by optical dissipation (see Fig. 7).

Going forward, we confirm the validity of the dissipative coupling theory to study the quantum nonlinear effects in optomechanical systems. Our quantum nonlinear effects require neither strong single-photon coupling coefficients as in conventional blockade [13,72,73] nor stringent parametrics condition as in unconventional blockade [42,74–76]. This dissipative coupling-assisted blockade effect is very favorable for performing single-photon transmission control [47] and implementing optical quantum diodes [77,78]. In quantum information processing, MSI-based optomechanical systems can also serve as a core to realize optical and mechanical-optical controlled gates [79–81]. Moreover, this strong nonlinearity is a necessary resource in quantum synchronization [82–85] and the study of quantum chaos effects [86]. Furthermore, MSI can serve as a kindly platform for exploring classical and quantum transitions in the nonlinear regime [87,88].