1. Introduction

Photons are the ideal information carriers for long-distance quantum communication and play special role in information processing [1–4] . The strong nonlinear interaction between photons is very important for single-photon information processing as well as for the observation of strict quantum effects [5–7]. Unfortunately, the interaction between photons is very weak, so it is still a challenge to enhance the interaction between photons at the level of few photons. Natural Kerr media have extremely weak nonlinearity [1], therefore, it is of great significance to enhance the nonlinearity and find the equivalent Kerr nonlinear system. A lot of efforts had focused on improving the nonlinearity of photons such as using multi-level atom structure [8–11], employing two coupled quantum modes [12,13], meanwhile photon antibunching under nonlinear medium have been studied [14–16].

Cavity optomechanics due to the nonlinear interaction induced by radiation pressure has aroused wide interest over the past decades [17–19], including precision control and measurement [20,21], testing of the quantum-classical boundary [22,23] and quantum information processing [24,25]. Different from other systems, the characteristics of the optomechanical system itself are of great significance for the study of photon nonlinearity. Many studies are devoted to enhancing single photon nonlinearity. Both Gong et al. [26] and Rabl [27] have shown that the single-photon optomechanical inducing nonlinearity is very weak due to the weak single-photon optomechanical coupling. Samuel et al. have exhibited that an optomechanical system can be equivalent to a Kerr medium [28]. Zhou et al. have proved that the nonlinearity can be enhanced by atomic coherence [29], and a lot of schemes are also proposed that the coherent dipole coupling, photon-photon and photon-phonon cross-Kerr nonlinearities not only can be enhanced but also can be controlled [30–32]. One of the potential application of nonlinearity is to generate single-photon source. The photon blockade effect has been widely studied in theory and experiment in various systems, such as multimode optomechanical system [33–35], quadratically coupled optomechanical system [36] and so on. In addition, multi-photon blockade [37–39] and nonreciprocal photon blockade [40,41] have been extensively studied recently. In analogy to photon blockade, phonon blockade has also been studied [42,43]. All of the above work shows that the nonlinearity of photons is of special meaning.

For the purpose of enhancing nonlinearity of photons, here we put forward a scheme in hybrid cavity optomechanical system. Inspired by enhancing spin-phonon interaction in a hybrid spin-mechanical setups [44], we joint the spin-mechanical system with optomechanical interaction. Employing a time-dependent two-phonon pump to drive cantilever [45–49] , we can modulate the spring constant of the mechanical oscillator. We derive an effective Hamiltonian and demonstrate that the self-Kerr nonlinearity and cross-Kerr nonlinearity of the two cavity modes can be both enhanced by adjusting the classical drive amplitude as well as increasing the spin-phonon coupling strength. Since all implements are under weak coupling mechanism, the current scheme may be easier to be realized. We also show the optical nonlinearity can be used to achieve an important optical application, photon blockade, which provides a possible way to realize the single-photon source experimentally.

This paper is organized as follows. In Sec. 2., the hybrid optomechanical system is introduced and the effective Hamiltonian is derive. The validity of the effective Hamiltonian is proved. Then, the discussion of the influence of some parameters on enhancing nonlinearity is given. In Sec. 3., the second order correlation function is calculated both analytically and numerically to confirm the photon blockade. The cross correlation and multi-photon blockade are also discussed. Finally, a brief summary of our work is given in Sec. 4.

2. Model and effective interaction

We consider a hybrid optomechanical system illustrated in Fig. 1, where a spin qubit is coupled to the mechanical motion of a cantilever via a sharp magnet tip attached to its end [50]. In the meantime, the mechanical resonator interacts with two toroidal microcavitys, this kind of optomechanical coupling has been used in experiments by evanescently coupling high-Q nanomechanical oscillators to the tightly confined optical field of an ultrahigh-finesse toroidal silica microresonator [51], tunneling of photons between the two cavities is allowed. In addition, to create a mechanical parametric amplifier effect, a periodic pump is used to demodulate the spring constant of the cantilever, which can be achieved by placing a electrode to the cantilever and applying a time-varying voltage. The spring constant can be modified by adjust the gradient of the electrostatic force from the electrode [52]. The system Hamiltonian can be written as $H_{\mathrm {tot}}=H_{0}+H_{\mathrm {int}}+H_{\mathrm {dr-m}} (\bar {h} =1)$ with

 

Fig. 1. Schematic of a hybrid cavity-spin-mechanics system.

Download Full Size | PDF

In the rotating frame defined by $V_{1}=\mathrm {exp}[-i\omega _{p}tb^{\dagger }b-i\omega _{e}|e\rangle \langle e|]$, the transformed Hamiltonian $H_{1}$ reads

The detailed deduction and the expressions of the parameter are given in Appendix A. $\eta _{1}$ and $\eta _{2}$ are the strengths of effective self-Kerr nonlinearity and the cross-Kerr nonlinearity between the two cavity modes. If the thermal environment of the mechanical mode can not be ignored, one can rewrite (6) with high mean value of $\langle b^{\dagger }b\rangle$, and the cross-nonlinear terms between mechanical mode and optical mode will be resulted, which can be obtained from (1) in Appendix A. In order to observe the mechanism of nonlinear enhancement more clearly, we simplify the parameters $\eta _1$ and $\eta _2$ under the conditions $\Delta _{m}\ll \delta \ll \omega _{p}$. The nonlinear coefficients

As we have pointed out $\{|\omega _{p}-\delta |,\omega _{p}+\delta \}\gg \frac {g_{+}e^{r}\Lambda }{\Delta _{m}}$, then $\mu =\frac {g_{+}e^{r}\Lambda }{ \Delta _{m}}(\frac {1}{\omega _{p}-\delta }-\frac {1}{\omega _{p}+\delta })|$ $\ll 1$, but $\eta _{1}=\frac {\mu g_{+}e^{r}\Lambda }{\Delta _{m}}$ still can be amplified due to $e^{r}$ and $\Lambda$ increasing with the increasing of $r$. Similarly, the cross-Kerr nonlinearity $\eta _{2}$ also can be enhanced. That is to say, if $r=0$ ($\Omega _{p}=0$), the nonlinear coefficients will lose the exponential enhancement behavior. We can see that the nonlinearity is resulted from the jointing effect of optomechanical interaction and spin-mechanical interaction and is enhanced by the parametric amplification of mechanical oscillator, which is obviously different from single-photon optomechanical induced nonlinearity ($\sim \frac {g^{2}}{\omega _{m}}$), where the nonlinear coefficient is very weak due to the weak single-photon optomechanical coupling [27].

 

Fig. 2. Evolutions $\langle a_{1}^{\dagger }a_{1}\rangle$ (a) and the fidelity (b) with total Hamiltonian in (1) and effective Hamiltonian in (6) respectively. The inset of Fig. 2(b) shows the time evolution of $\langle b^{\dagger }b\rangle$ and $\langle \sigma _{+}\sigma _- \rangle$. Initially, the mode $a_1$ and $b$ are both in the vacuum state, mode $a_2$ is in Fock state $|1\rangle$, and two-level atom is in its ground state. The parameters are $J=1$, $r=1$, $\lambda =2J$, $\omega =10J,$ $\Omega _{p}=0.05J$, $\omega _{m}=20.05J$, $\omega _{e}=21J$, $\omega _{p}=20J$, $g_{1}=0.1224J$ and $g_{2}=0.0101J$.

Download Full Size | PDF

In order to verify the validity of the effective Hamiltonian, the time evolution of $\langle a_{1}^{\dagger }a_{1}\rangle$ is plotted with $H_{tot}$ and $H_{eff}$, respectively, shown in Fig. 2(a). We find that the result corresponding to $H_{eff}$ agrees well with that of full Hamiltonian. In addition, the time evolution of fidelity $F=\{\mathrm {Tr} [(\sqrt {\rho _{\mathrm {eff}}}\rho _{\mathrm {tot}}\sqrt {\rho _{\mathrm {eff}}})^{1/2}]\}^2$ [54] between $H_{tot}$ and $H_{eff}$ is plotted in Fig. 2(b). It can be seen that the fidelity keep high value in a long time. In addition, in the subgraph, we plot the time evolution of $\langle b^{\dagger }b\rangle$ and $\langle \sigma _{+}\sigma _- \rangle$, which proves that as long as appropriate parameters are selected and the above approximate conditions are met, the oscillator will be in the virtual excitation, and the population of the qubit in the excited state is also close to 0, then we reasonably believe that $\langle b^{\dagger }b\rangle =0$ and $\langle \sigma _{+}\sigma _- \rangle =0$ are satisfied. It also illustrates that effective Hamilton is trustable.

In Fig. 3, we plot the self-Kerr nonlinearity strength $\eta _{1}$, the cross-Kerr nonlinearity strength $\eta _{2}$ and the effective frequencies of the two cavity modes $\xi _{1,2}$ versus the squeezing parameter $r$. We show that $\eta _{1}$ and $\eta _{2}$ enables an exponential enhancement with increasing squeezing parameter $r$ by adjusting the parametric driving $\Omega _{p}$ and the detuning $\delta _{m}$. However, as we can see from the figure, when the squeezing parameter is relatively small, the enhancement of nonlinear strength is very weak. Next, in Fig. 4, we plot $\eta _{1,2}$ and $\xi _{1,2}$ versus the spin-phonon coupling strength $\lambda$ under squeezing parameter $r=1$. We demonstrate that by introducing the spin-mechanical interaction, the self-Kerr and cross-Kerr nonlinearity strength $\eta _{1}$ and $\eta _{2}$ can also be amplified further by increasing the coupling strength $\lambda$ even if the squeezing parameter is relatively small. In addition, one can note that $\eta _{1,2}$ is amplified greatly, but the values of $\xi _{1,2}$ is small. This is because $\eta _{1,2}$ proportion to $g_{+}$ and $\xi _{1,2}$ only contains $g_{-}$. If $g_{1}=g_{2}$, the effective frequency $\xi _{1,2}=0$, but the nonlinearity $\eta _{1,2}$ is still amplified. Therefore, the small and adjustable effective frequency and the enhancement of nonlinearity have its advantage, because it is strong nonlinearity playing the key role in quantum gate rather than the high frequency [55]. Usually, the damping rate $\{\kappa _{1},\kappa _{2}\}<J$ , thus, by jointing the optomechanical spin-mechanical interactions, self-Kerr nonlinearity strength $\eta _{1}$, the cross-Kerr nonlinearity strength $\eta _{2}$ step into strong coupling regime, i.e., $\{\eta _{1},\eta _{2}\}>\{\xi _{1},\xi _{2},\kappa _{1},\kappa _{2}\}$

 

Fig. 3. The self-Kerr nonlinearity strength $\eta _{1}$, the cross-Kerr nonlinearity strength $\eta _{2}$ and the frequencies of the two cavity modes $\xi _{1,2}$ versus the squeezing parameter $r$. The parameters are taken as $J=1$, $\lambda =2J$, other parameters are the same as above.

Download Full Size | PDF

 

Fig. 4. The self-Kerr nonlinearity strength $\eta _{1}$, the cross-Kerr nonlinearity strength $\eta _{2}$ and the frequencies of the two cavity modes $\xi _{1,2}$ versus the spin-phonon coupling $\lambda$. The parameters are taken as $J=1$, $r=1$, other parameters are the same as Fig. 3.

Download Full Size | PDF

3. Photon blockade with the enhancement nonlinearity

Using the effective Hamiltonian Eq. (6), we now discuss photon statistical properties including photon blockade and the cross correlation between the two cavities. Introducing the driving of two cavities $H_{\mathrm {dri-c}}=\sum _{i=1,2}\Omega _{i}(a_{i}e^{i\omega _{li}t}+H.c.)$. In the frame rotating at the driving laser frequency $\omega _{l}$, the Hamiltonian can be written as

In the weak pumping regime, only the lower energy levels of the cavity modes are occupied. The general state of the system in few-photon subspace can be written as

The values of the coefficients can be determined by solving the Schrödinger equation. The steady-state solution of the probability amplitudes is given in Appexdix B. Due to the coupling between the two cavities, the eigen-mode of the two cavities is modulated as two super modes $a_+$ and $a_-$ with frequencies $\omega +J$ and $\omega -J$. To characterize the photon statistical properties of the two supermodes, we employ the equal-time second-order correlation function

This quantity characterizes self correlation and mutual correlation respectively, which is also interpreted as the joint probability of detecting two photons at the same time. If $g_{a_i}^{(2)}(0)<1$ corresponds to sub-Poisson statistics of the cavity mode, which is a nonclassical effect often referred to as photon anti-bunching effect. $g_{a_{i}}^{(2)}(0)>1$ corresponds to photon bunching effect. The limit $g_{a_{i}}^{(2)}(0)\rightarrow 0$ indicates complete photon blockade, in which two photons never occupy the cavity mode at the same time. Then, the steady-state correlation functions of the two cavity modes can be analytically obtained as

Now, considering the amplified nonlinearity parameters $\{\eta _{1},\eta _{2}\}\gg \kappa$, we can easy see that $\{g_{a_{+}}^{(2)}(0),g_{a_{-}}^{(2)}(0)\}\approx \frac {\kappa ^{2}}{4\eta _{1}^{2}}\rightarrow 0$, when the detuning $\Delta _{1,2}=-\eta _{1}$. Under this condition, $g_{a_{+}a_{-}}^{(2)}(0)\approx \frac {\kappa ^{2}}{\eta _{2}^{2}}$. Therefore, the larger values of $\eta _{1}$ and $\eta _{2}$, the best of the photon blockade. Otherwise, the weak nonlinearity ($\{\eta _{1},\eta _{2}\}<\kappa$) can not result in photon blockade. On the other hand, we can understand it from the eigenvalues of Hamiltonian. When $\Delta _{1,2}=-\eta _{1}$, $E_{10}=E_{01}=E_{00}=0$, which means that $|01\rangle$ and $|10\rangle$ are easy populated; meanwhile $E_{11}=\eta _{2}$, $|11\rangle$ is not easy populated for large value of $\eta _{2}$. Therefore, the amplified nonlinearity parameters $\eta _{1}$ and $\eta _{2}$ are the necessary condition to generate ideal photon blockade.

We now numerically simulate the photon statistical properties. Due to the atom and the mechanical mode keeping in the virtual excitation, the dissipation of qubit under zero temperature environment has no effect for the ground state atom. Since we only care about the property of the optical modes, for the Hamiltonian (6), the thermal environment of mechanical mode has no effect on the optical fields. So, we reasonably ignore the dissipation of qubit and the mechanical mode and write the master equation as

In Fig. 5(a) and (b), $g_{a_{i}}^{(2)}(0)$ and $\langle a_{i}^{\dagger }a_{i}\rangle$ are plotted versus the effective self-Kerr nonlinear strength $\eta _{1}$ for $a_{\pm }$ modes, where the numerical results with the quantum master equation (blue-solid curves) coincide with the analytical simulation of $g_{a_{i}}^{(2)}(0)$ (yellow-triangle). As shown in the Fig. 5(a) and (b), the minimum value of $g_{a_{i}}^{(2)}(0)$ reaches $10^{-4}$ for both the two supermodes under the single photon resonance condition $\Delta _{1,2}=-\eta _{1}$. At this point, $\langle a_{i}^{\dagger }a_{i}\rangle$ achieves its maximum values between 0.1 and 1, and thus confirms the optimal photon blockade. In Fig. 5(c) and (d), we employ master equation to numerical simulate $g_{a_{i}}^{(2)}(0)$ as a function of $\eta _{1}$ and the effective $\Delta _{1}$ and $\Delta _{2}$ for supermodes $a_{\pm }$. Noticeably, when $\mathrm {log} _{10}[g_{a_{i}}^{(2)}(0)]$ achieve its minimum values, there are always of the relation $\Delta _{1,2}\approx -\eta _{1}$, which means that the numerical simulation is always consistent with the analytical result.

 

Fig. 5. $g_{a_{i}}^{(2)}(0)$ and $\langle a_{i}^{\dagger }a_{i}\rangle$ as functions of $\eta _{1}$ for $i=+$ (a), $-$ (b) where the yellow-triangle lines are the analytical solutions, the solid-blue lines and the red-dashed lines are numerical results with (12). Contour plot $\mathrm {log}_{10}[g_{a_{i}}^{(2)}(0)]$ as a function of $\eta _{1}$ and $\Delta _{1}$ (c) and $\Delta _{2}$ (d). The parameters are $\Delta _{1}=-5.9654J$, $\Delta _{2}=-5.9644J$, $\omega _{l1}=12J$, $\omega _{l2}=10J$, $\Omega _{+}=\Omega _{-}=0.1J,$ $\kappa =0.3J.$

Download Full Size | PDF

As we have analyzed that when $\Delta _{1}+\Delta _{2}=-2\eta _{1}$, the cross correlation function $g_{a_{+}a\_}^{(2)}(0)$ also should be depressed, which is numerically shown in Fig. 6(a). Thus, when {$g_{a_{+}}^{(2)}(0),g_{a\_}^{(2)}(0), g_{a_{+}a\_}^{(2)}(0)$}$\sim 0$, the probability amplitude {$C_{11},C_{02},C_{20}\}$ are very small. The state is approximately $|\psi \rangle \simeq C_{00}|{0,0}\rangle +C_{10}|{1,0} \rangle +C_{01}|{0,1}\rangle$. Obviously, it is entangled state. In order to measure the entanglement, we employ logarithmic negativity [56], $E_{N}=\mathrm {log_{2}}\parallel \rho _{AB}^{T_{A}}\parallel _{1}$,where the symbol $\parallel \cdot \parallel _{1}$ donates the trace norm, and $\rho _{AB}^{T_{A}}$ is the partial transpose of the reduced density matrix $\rho _{AB}$ of the two supermodes. The logarithmic negativity $E_{N}$ is non-negative for entangled states. Figure 6(b) exhibits $E_{N}$ obtaining its maximum value when $\Delta _{1}+\Delta _{2}=-2\eta _{1}$, $\Omega _{+}=\Omega _{-}$. See the expression ( 5), under this condition, $C_{10}=C_{01}$, therefore, the entanglement achieve its maximum value.

 

Fig. 6. (a) The equal-time cross correlation function $g_{a_{+}a_{-}}^{(2)}(0)$ and (b) the logarithmic negativity $E_{N}$ versus the effective self-Kerr nonlinear strength $\eta _{1}$. The parameters are the same as Fig. 5.

Download Full Size | PDF

We also discuss the two-photon blockade (2PB) affected by nonlinearity. By defining $g_{a_{i}}^{(3)}(0)=\frac {\langle a_{i}^{\dagger }a_{i}^{\dagger }a_{i}^{\dagger }a_{i}a_{i}a_{i}\rangle }{\langle a_{i}^{\dagger }a_{i}\rangle ^{3} },(i=+,-)$, we see that 2PB can be judged by the standards $g_{a_{i}}^{(3)}(0)<1$ meanwhile $g_{a_{i}}^{(2)}(0)\geq 1$, which means allowing the absorption of two photons but blocking the subsequent photons. In Fig. 7(a) and (b), the second-order and third-order correlation function $g_{a_{i}}^{(2)}(0)$ and $g_{a_{i}}^{(3)}(0)$ is shown as a function of $\eta _{1}$ for both two super modes respectively. When $\eta _{1}$ is within the regions 3.574$\sim 3.975$ for $a_+$ (3.573$\sim 3.974$ for $a_-$), $g_{a_{i}}^{(3)}(0)<1$ but $g_{a_{i}}^{(2)}(0)\geq 1$, it belongs to the 2PB regime. Increasing $\eta _{1}$ further, it is single-photon blockade (1PB).

 

Fig. 7. $g_{a_{i}}^{n}(0)$ versus $\eta _{1}$ where $i=+$ (a), $-$ (b), and the blue (red) shadow region indicates 1PB (2PB). The parameters are the same as above.

Download Full Size | PDF

4. Discussions and conclusions

As to the feasible in experiment, the atom-phonon system can be realized in [50], in which a Si cantilever with fundamental frequency of $\omega _{m}\sim 2\pi \times 7$MHz is coupled to a magnetic tip by producing magnetic gradients at a appropriate distances away from the tip with coupling strength $\lambda \approx 2\pi \times 115$ kHz, which is a little bit smaller than our requirement. For superconducting charge qubits the electrostatic coupling can be substantially stronger as $\lambda \approx 2\pi \times 5$MHz, and the mechanical frequency is $\omega _{m} \approx 2\pi \times 58$MHz. Then $\lambda /\omega _m\sim 0. 08$ which fits better with the parameters of Fig. 2 [57]. As for the periodic pump, the pump frequency and the amplitude is adjustable by periodic modulation of the spring constant of the mechanical element [52]. So that the rotating-wave approximation conditions in current scheme can be satisfy by selecting the proper detuning $\delta _{m}$ and $\delta$. In addition, one need couple the the above two atom-phonon structures with two cavities so as to satisfy our system. The optomechanical coupling strengths in current scheme are about $g\sim 10^{-4}\omega _{m}-10^{-3}\omega _{m}$, which greatly relaxes the requirement of the coupling [58] and can be satisfied. We also concern a spin-optomechanical-crystal hybrid system that contains all the interaction forms in our article where an array of SiV centers are integrated to a 1D optomechanical crystal [59,60]. In the diamond optomechanical crystal [61], the mechanical and optical frequencies are about $2\pi \times 46$ GHz (or few GHz) and $2\pi \times 200$THz. With a high quality factor $Q\sim 10^{7}$, the optical decay rate is $\kappa \sim 2\pi \times 20$MHz. By targeted ion implantation, SiV centers can be accurately implanted into the diamond crystal. The strain-induced spin-phonon coupling strength can be calculated as $\lambda \sim 2\pi \times 30$MHz, and the optomechanical coupling $g\sim 2\pi \times 100$MHz. Then $g/\omega _m \sim 0.002$ is larger than that in Fig. 2, but $\lambda /\omega _m$ is far below the parameters in Fig. 2. And it remains to be studied how to introducing two-phonon drive to this system. We expect that the current scheme will be feasible in the near future.

In conclusions, we have proposed a scheme by employing optomechanical and spin-mechanical interactions to enhance the nonlinearity of the hybrid optomechanical system. By applying a tunable and nonlinear pump laser to the mechanical resonator, corresponding to a mechanical parameter amplification, the spin-mechanical coupling can be directly enhanced. Through a series of deductions, the effective Hamiltonian is given and the validity of the approximation is proved by the comparing the results of full interaction with that of effective Hamiltonian. Importantly, we show that the self-Kerr and cross-Kerr nonlinearity strengths can be enhanced by increasing the squeezing strength $r$ and the spin-mechanical coupling strength. As an application of optical nonlinearity, we investigate photon blockade phenomenon. Our results demonstrate that the single and two photon blockade can occur in different nonlinear regime.