## Abstract

A significantly low value of the single-photon coupling constant is a major challenge in the creation of a single-photon source via photon blockade. Here, we propose a photon blockade scheme composed of a weakly second-order nonlinear medium with an optical parametric amplification in a low-frequency cavity. Unlike the traditional weakly coupled system, the effective coupling strength in the proposed scheme can be significantly higher than the decay rate of the cavity mode. This can be achieved by adjusting the squeezing parameter even if the original coupling strength is weak. The thermal noise of the squeezed cavity mode can be suppressed by a squeezed vacuum field. Using a probability amplitude method, we obtain the optimal condition of photon blockade in the steady-state. By solving the master equation numerically in the steady-state, a strong photon antibunching effect that is consistent with the optimal conditions can be obtained in the cavity with low frequency. Besides, the photon blockade phenomenon and cross-correlation of two cavities can be significantly enhanced under a specific squeezing parameter. Our results may be useful for future studies on the characteristics of photon statistics.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

The production of a single-photon source is essential in the application of photons in quantum information and quantum communication. Perfect single-photon sources emit photons sequentially. In recent years, the antibunching effect (ABE) has drawn much attention as one of the new phenomena in photon statistics for creating a single-photon source. Imamoḡlu et al. proposed using a high-finesse cavity containing a low-density four-level atomic medium [1]. They found that the transmitted photons demonstrated a clear ABE. ABE is a kind of quantum effect that plays a vital theoretical role in revealing the quantum nature of light. Additionally, it has attractive potential applications in the foundation tests of quantum theory [2–4], quantum in formation processing [5], high-precision measurements [6], and optomechanical storage [7].

One of the mechanisms for creating antibunching photons is the photon blockade (PB), which is also known as the conventional PB. It states that the excitation of a first photon blocks the transport of a second photon in the cavity [1]. It is essential to note that the PB refers to nonlinear quantum scissors based on optical-state truncation [8–10]. Based on this mechanism, the PB is theoretically predicted in many different systems, such as cavity quantum electrodynamics [11–14] and quantum optomechanical systems [15–18]. Furthermore, the PB first observed in the optical cavity coupled to a single trapped atom [19]. Many experimental groups have observed PB effects in different systems, such as circuit cavity quantum electrodynamics [20–22], quantum dots in photonic crystal systems [23], optical nano-cavities [24], and photonic crystal cavities [25]. The potential applications of the PB include interferometers [26], nonlinear scissors [27], and semiconductor microcavities [28]. For the system mentioned, to enter the quantum nonlinear region, the single-photon coupling strength should be comparable to the decay rate of the cavity mode [29]. Unfortunately, the coupling constant is significantly smaller than the decay rate of the cavity. Therefore, it is necessary to increase the coupling strength in the experiment.

Recently, Liew and Savona [30] presented a new mechanism known as the unconventional photon blockade (UPB). They found that a photonic molecule consisting of two linearly coupled nonlinear cavity modes can give rise to strong photon antibunching, even with nonlinearities that are much smaller than the decay rates of the cavity modes. This physical mechanism can be regarded as a destructive quantum interference between different excitation pathways [31,32]. Based on this mechanism, different systems have been proposed to realize the UPB, such as coupled optomechanical systems [33], bimodal optical cavities with quantum dots [34,35], symmetric and antisymmetric modes in weakly nonlinear photonic molecules [36], optimized squeezing states [37], a double quantum well embedded in a micropillar optical cavity [38], and coupled single-mode cavities with third-order nonlinearities [37]. However, there is one limitation in the original UPB scheme. A weak nonlinearity induces large coupling, resulting in the rapid oscillation of the second-order correlation function [36,37,39–43]. This obstacle can be overcome by mutually driving the modes and mixing the output [43]. In our scheme, we overcome this limitation using phase matching.

In this study, we propose a scheme to significantly enhance coupling strength by placing an optical parametric amplification (OPA) in a second-order nonlinear system. In the system, two lasers with the same amplitude drive two cavities. Moreover, even if the original nonlinear coupling is considerably weak, we can also obtain a significant PB phenomenon. We derive the optimal conditions analytically for PB in steady-state. Next, by solving the master equation numerically in the steady-state, PB phenomenon can be obtained in the cavity with low frequency, and that is consistent with the optimal conditions. Unlike the traditional weakly coupled system, the effective coupling strength in the proposed scheme can be significantly higher than the decay rate of the cavity mode. Finally, in the appropriate squeezing parameter conditions, the second-order correlations and cross-correlation of two cavities can be reduced significantly. Our results demonstrate that by introducing OPA and adjusting the squeezed parameters, the initial weak coupling coefficient can be amplified effectively. These results are helpful in future studies on the characteristics of photon statistics.

The rest of this paper is organized as follows: In Sec. 2, we describe the physical model and obtain the effective Hamiltonian of the system. In Sec. 3, we present the second-order correlation functions and the master equation of the system. In Sec. 4, we solved the second-order correlation functions numerically with master equation. We also give the optimal parameter condition to achieve the optimal photon blockade effect, and discuss the influence of system parameters on the photon blockade effect. Experimental feasibility and conclusions are given in Sec. 5.

## 2. Model and Hamiltonian

We consider a system that includes double coupled cavities with frequencies $\omega _{a}$ and $\omega _{c}$, as shown in Fig. 1(a). The two cavities coupled via $\chi ^{(2)}$ nonlinearity mediate the conversion of the photon in cavity mode $a$ into two photons in cavity mode $c$. To make the work effective, it is necessary to drive the cavities using two continuous-wave input lasers with the driving frequency $2\omega _{l}$ and $\omega _{l}$, respectively, and the same driving amplitude $\Omega =\sqrt {\kappa P_{in}/\hbar \omega _{l}}$, $P_{in}$ and $\kappa =\omega /Q$ being the input power and optical mode decay rate respectively. $Q$ is the quality factor of the optical cavity. Meanwhile, an OPA is placed in the cavity mode $c$ and is pumped using a driving field with the driving frequency $2\omega _{l}$, amplitude $\lambda /2$, and phase $\phi$. The total system Hamiltonian can be written as ($\hbar =1$)

Moreover, the thermal noise of the cavity $c$ related to the parametric squeezing effect is enlarged. By introducing the broadband squeezed vacuum field $c_0$, the thermal noise of the cavity can be suppressed (where $\omega _{0}$ is the central frequency, $r_0$ is the squeezing parameter, and $\phi _0$ is the reference phase). The cavity $c$ was driven by injecting the squeezed field $c_0$ as an input field. Experimentally, a squeezed bandwidth of up to gigahertz has been realized via OPA [48,49]. Because the typical line width of optical cavities is of the order of megahertz, the squeezed input field is well approximated as having an infinite bandwidth [50]. Thus, it can be regarded as a squeezed reservoir. Next, under the ideal parameter conditions $r=r_0$ and $\phi -\phi _0=\pm n\pi$ $(n=1,3,5,\ldots )$, the thermal noise of the squeezed cavity mode $\tilde {c}$ can be suppressed entirely. Qualitatively, this result can be comprehended using phase-matching [51]. In the following text, we can regard the squeezed mode as equivalently coupled to a vacuum reservoir.

## 3. Second-order correlations and master equation

To quantify the photons statistics in the system, we consider the second-order correlation functions defined by

## 4. Optimal condition for UPB: analytical results

In this section, we analytically derive the optimal condition for UPB using the equal-time correlations in a truncated Fock state basis [39,40]. The Fock state basis of the system is denoted by $\vert m~n \rangle$, with the number $m$ representing the photon number in the cavity $a$ and $n$ representing the photon number in cavity $c$. To study the UPB in cavity $c$, we can notice that there are two paths for generating two photons in the cavity $c$, as shown in Fig. 1(b), i.e., $\vert {00}\rangle \rightarrow \vert {01}\rangle \rightarrow \vert {02}\rangle$ and $\vert {00}\rangle \rightarrow \vert {10}\rangle \rightarrow \vert {02}\rangle$, which is supposed to cause the UPB in the cavity $c$. We expand the wave function of the whole system in the few-photon subspace as

Next, to acquire the optimal conditions for a strong photon blockade effect, we numerically simulate the second-order correlation function of the equal-time by employing the master equation demonstrated in Eq. (8). The equal-time second-order correlation function of the cavity $c$ is displayed in Fig. 3(a), where the white dashed line represents the derived optimal condition in Eq. (14). The optimal PB effect obtained using numerical simulation is always consistent with the optimal analytical condition. It is noteworthy that when $\Omega$ and $J$ approach zero, the system is isolated. Therefore, it does not satisfy the optimal conditions. Figure 3(b) shows the equal-time second-order correlation function versus the $\Delta ^{'}_{c}$ with different $\Delta _{a}$. As shown in the diagram, the second-order correlation function changes along with the change in $\Delta ^{'}_{c}$ ; thus satisfying the optimum condition for driving the amplitude $\Omega /\kappa =0.01$, the coupling strength $J/\kappa =0.04$, and the squeezing parameter $r=3$. In the detuning value $\Delta _{a}=0$ of the cavity $a$, the lowest point of the equal-time second-order correlation function corresponds to the detuning $\Delta ^{'}_{c}=0$ of the cavity $c$. Similarly, in the detuning value $\Delta _{a}/\kappa =\pm 0.5$ of the cavity $a$, the lowest point of the equal-time second-order correlation function corresponds to the detuning $\Delta ^{'}_{c}/\kappa =\pm 0.5$ of the cavity $c$. The detuning change satisfies the first equation as shown in Eq. (14).

In Fig. 4, the delay-time second-order correlation function $g_{cc}^{(2)}(\tau )$ is plotted as a function of the time delay $\tau$. When other parameters are fixed, the $g_{cc}^{(2)}(\tau )$ first increases and then decreases with the increase in $\tau$, then repeat this trend and gradually level off as shown in Fig. 4(a). Similar to the reports given by Lemonde et al. and Liew et al. [30,31], $g_{cc}^{(2)}(\tau )$ shows an oscillation behavior as the delay time, as shown in Fig. 4(b). The magnitude of the oscillation decreases as $\tau$ increases, and approaches approximately unite when $\tau >5$. This is the approximate lifetime of the photons in the cavities. This oscillation behavior originates from the Rabi oscillation between the photon states. We can also use the same method to calculate $g_{aa}^{(2)}(\tau )$ and $g_{ac}^{(2)}(\tau )$. More notably, we can see that the final $g_{cc}^{(2)}(\tau )$ is going to approach 1. This mean that the distribution of photon number of coherent states is random (Poisson statistics distribution). In other words, as the time delay increases, the photons with time interval will eventually tend to coherent. Figure 5(a) shows a logarithmic plot (of base 10) of the equal-time second-order correlation function, which is plotted as the function of the coupling strength $J$ under the condition of $r=0$. The black dotted lines, the green solid lines, and the red dotted dash lines represent $g_{aa}^{(2)}(0)$, $g_{cc}^{(2)}(0)$, and $g_{ac}^{(2)}(0)$, respectively. When other parameters are fixed, the $g_{cc}^{(2)}(0)$ first decreases and then increase with the increase in $J$. It can be concluded that the lowest point is nearly $J=0.03$. The other two lines are all around $g^{(2)}(0)=1$, that is, there is only a blockade effect in the cavity $c$ without the squeezing parameter. Figure 5(b) displays the second-order correlation function of the steady-state under $r=3$. The black dotted lines, the green solid lines, and the red dotted dash lines represent $g_{aa}^{(2)}(0)$, $g_{cc}^{(2)}(0)$, and $g_{ac}^{(2)}(0)$, respectively. When other parameters are fixed, the $g_{aa}^{(2)}(0)$ first decreases and then decreases with the increase in $J$. The $g_{cc}^{(2)}(0)$ increases with the increase in $J$, and the $g_{ac}^{(2)}(0)$ first decreases and then increases gradually with the increase in $J$. For $g^{(2)}(0)$, the lowest point is between $10^{-2}$ and $10^{-1}$ at $r=0$, as shown in Fig. 5(a), and the lowest point is between $10^{-5}$ and $10^{-3}$ at $r=3$, as shown in Fig. 5(b). Contrary to the above, we can not only see that there are photon blockade effects in both cavities and a cross-correlation of two cavities under the squeezed parameter, but also an enhancement of the photon antibunching effect. Even if the initial coupling is in the weak situation, we may enhance the coupling strength through adjusting the squeezed parameter to improve the blockade effect. This proves that the increased coupling strength is helpful in studying the photon blockade effect.

## 5. Conclusions

Now, we compare the state-of-art experimental parameters with our proposed parameters. $\chi ^{(2)}$ nonlinear materials, such as GaAs [52], AlGaAs [53], GaN, and AlN [54], can be used in the present proposal. $\chi ^{(2)}$ nonlinear materials as the cavity mirror between two cavities can be achieved by coating. The values of the nonlinear coupling coefficient $J$ depend on the material choice. Assuming the frequency of the cavity mode as $\omega _{a}=2\pi \times 163.195$THz, The double optical cavities quality factor are defined as $Q_a=1.6\times 10^{-4}$, $Q_c=3.2\times 10^{-4}$. Therefore, $\kappa _1=\kappa '$ is experimentally feasible.

We investigate a scheme to significantly enhance coupling strength by placing an optical parametric amplification in a second-order nonlinear system. The efficient Hamiltonian was obtained using the squeezing transformation. The thermal noise of the squeezed cavity mode can be suppressed totally with the help of a squeezed vacuum field. We obtain the optimal condition of photon blockade using the probability amplitude method. Next, by numerically solving the master equation, a strong photon antibunching effect that is consistent with the optimal conditions can be obtained in the cavity with low frequency. We discover that even if the original nonlinear coupling is considerably weak, we can also obtain a significant PB phenomenon. Besides, under an appropriate squeezing parameter condition, the second-order correlations and cross-correlation of two cavities can be ameliorated significantly. These results are helpful in future experimental studies on the characteristics of photon statistics.

## Funding

Outstanding Young Talent Fund Project of Jilin Province (20180520223JH); Science and Technology project of Jilin Provincial Education Department of China during the 13th Five-Year Plan Period (JJKH20200510KJ).

## Disclosures

The authors declare that there are no conflicts of interest related to this article.

## References

**1. **A. Imamoḡlu, H. Schmidt, G. Woods, and M. Deutsch, “Strongly interacting photons in a nonlinear cavity,” Phys. Rev. A **79**(8), 1467–1470 (1997). [CrossRef]

**2. **O. Romero-Isart, A. C. Pflanzer, F. Blaser, R. Kaltenbaek, M. Aspelmeyer, and J. I. Cirac, “Large quantum superpositions and interference of massive nanometer-sized objects,” Phys. Rev. Lett. **107**(2), 020405 (2011). [CrossRef]

**3. **Z. Q. Yin, T. Li, and L. M. Duan, “Large quantum superpositions of a levitated nanodiamond through spin-optomechanical coupling,” Phys. Rev. A **88**(3), 033614 (2013). [CrossRef]

**4. **C. H. Bai, D. Y. Wang, S. Zhang, S. Liu, and H. F. Wang, “Modulation Based Atom Mirror Entanglement and Mechanical Squeezing in an Unresolved Sideband Optomechanical System,” Ann. Phys. (Berlin, Ger.) **531**(7), 1800271 (2019). [CrossRef]

**5. **K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical quantum information processing with photons and phonons,” Phys. Rev. Lett. **109**(1), 013603 (2012). [CrossRef]

**6. **T. P. Purdy, R. W. Peterson, and C. A. Regal, “Observation of radiation pressure shot noise on a macroscopic object,” Science **339**(6121), 801–804 (2013). [CrossRef]

**7. **V. Fiore, Y. Yang, M. C. Kuzyk, R. Barbour, L. Tian, and H. Wang, “Storing optical information as a mechanical excitation in a silica optomechanical resonator,” Phys. Rev. Lett. **107**(13), 133601 (2011). [CrossRef]

**8. **A. Miranowicz, M. Paprzycka, Y. X. Liu, J. Bajer, and F. Nori, “Two photon and three photon blockades in driven nonlinear systems,” Phys. Rev. A **87**(2), 023809 (2013). [CrossRef]

**9. **W. Leoński and R. Tanaś, “Possibility of producing the one-photon state in a kicked cavity with a nonlinear Kerr medium,” Phys. Rev. A **49**(1), R20–R23 (1994). [CrossRef]

**10. **G. H. Hovsepyan, A. R. Shahinyan, and G. Y. Kryuchkyan, “Multiphoton blockades in pulsed regimes beyond stationary limits,” Phys. Rev. A **90**(1), 013839 (2014). [CrossRef]

**11. **L. Tian and H. J. Carmichael, “Quantum trajectory simulations of two-state behavior in an optical cavity containing one atom,” Phys. Rev. A **46**(11), R6801–R6804 (1992). [CrossRef]

**12. **M. J. Werner and A. Imamoḡlu, “Photon-photon interactions in cavity electromagnetically induced transparency,” Phys. Rev. A **61**(1), 011801 (1999). [CrossRef]

**13. **R. J. Brecha, P. R. Rice, and M. Xiao, “Two level atoms in a driven optical cavity: Quantum dynamics of forward photon scattering for weak incident fields,” Phys. Rev. A **59**(3), 2392–2417 (1999). [CrossRef]

**14. **S. Rosenblum, S. Parkins, and B. Dayan, “Photon routing in cavity QED: Beyond the fundamental limit of photon blockade,” Phys. Rev. A **84**(3), 033854 (2011). [CrossRef]

**15. **P. Rabl, “Photon blockade effect in optomechanical systems,” Phys. Rev. Lett. **107**(6), 063601 (2011). [CrossRef]

**16. **A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett. **107**(6), 063602 (2011). [CrossRef]

**17. **J. Q. Liao and F. Nori, “Photon blockade in quadratically coupled optomechanical systems,” Phys. Rev. A **88**(2), 023853 (2013). [CrossRef]

**18. **H. Wang, X. Gu, Y. X. Liu, A. Miranowicz, and F. Nori, “Tunable photon blockade in a hybrid system consisting of an optomechanical device coupled to a two-level system,” Phys. Rev. A **92**(3), 033806 (2015). [CrossRef]

**19. **K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, “Photon blockade in an optical cavity with one trapped atom,” Nature (London) **436**(7047), 87–90 (2005). [CrossRef]

**20. **C. Lang, D. Bozyigit, C. Eichler, L. Steffen, J. M. Fink, A. A. Abdumalikov Jr, M. Baur, S. Filipp, M. P. da Silva, A. Blais, and A. Wallraff, “Observation of resonant photon blockade at microwave frequencies using correlation function measurements,” Phys. Rev. Lett. **106**(24), 243601 (2011). [CrossRef]

**21. **A. J. Hoffman, S. J. Srinivasan, S. Schmidt, L. Spietz, J. Aumentado, H. E. Türeci, and A. A. Houck, “Dispersive photon blockade in a superconducting circuit,” Phys. Rev. Lett. **107**(5), 053602 (2011). [CrossRef]

**22. **Y. X. Liu, X. W. Xu, A. Miranowicz, and F. Nori, “From blockade to transparency: Controllable photon transmission through a circuit-QED system,” Phys. Rev. A **89**(4), 043818 (2014). [CrossRef]

**23. **A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vučković, “Coherent generation of non-classical light on a chip via photon-induced tunnelling and blockade,” Nat. Phys. **4**(11), 859–863 (2008). [CrossRef]

**24. **A. Faraon, A. Majumdar, and J. Vučković, “Generation of nonclassical states of light via photon blockade in optical nanocavities,” Phys. Rev. A **81**(3), 033838 (2010). [CrossRef]

**25. **K. Müller, A. Rundquist, K. A. Fischer, T. Sarmiento, K. G. Lagoudakis, Y. A. Kelaita, C. S. Muñoz, E. D. Valle, F. P. Laussy, and J. Vučković, “Coherent generation of nonclassical light on chip via detuned photon blockade,” Phys. Rev. Lett. **114**(23), 233601 (2015). [CrossRef]

**26. **D. E. Chang, A. S. Sorensen, E. A. Demler, and M. D. Lukin, “A single-photon transistor using nanoscale surface plasmons,” Nat. Phys. **3**(11), 807–812 (2007). [CrossRef]

**27. **A. Miranowicz, M. Paprzycka, Y. X. Liu, J. Bajer, and F. Nori, “Two-photon and three-photon blockades in driven nonlinear systems,” Phys. Rev. A **87**(2), 023809 (2013). [CrossRef]

**28. **O. Kyriienko and T. C. H. Liew, “Triggered single-photon emitters based on stimulated parametric scattering in weakly nonlinear systems,” Phys. Rev. A **90**(6), 063805 (2014). [CrossRef]

**29. **M. A. Lemonde, N. Didier, and A. A. Clerk, “Enhanced nonlinear interactions in quantum optomechanics via mechanical amplification,” Nat. Commun. **7**(1), 11338 (2016). [CrossRef]

**30. **T. C. H. Liew and V. Savona, “Single photons from coupled quantum modes,” Phys. Rev. Lett. **104**(18), 183601 (2010). [CrossRef]

**31. **M. Bamba, A. Imamoglu, I. Carusotto, and C. Ciuti, “Origin of strong photon antibunching in weakly nonlinear photonic molecules,” Phys. Rev. A **83**(2), 021802 (2011). [CrossRef]

**32. **I. Carusotto and C. Ciuti, “Quantum fluids of light,” Rev. Mod. Phys. **85**(1), 299–366 (2013). [CrossRef]

**33. **X. W. Xu and Y. J. Li, “Antibunching photons in a cavity coupled to an optomechanical system,” J. Phys. B: At., Mol. Opt. Phys. **46**(3), 035502 (2013). [CrossRef]

**34. **A. Majumdar, M. Bajcsy, A. Rundquist, and J. Vučković, “Loss-enabled sub-Poissonian light generation in a bimodal nanocavity,” Phys. Rev. Lett. **108**(18), 183601 (2012). [CrossRef]

**35. **W. Zhang, Z. Y. Yu, Y. M. Liu, and Y. W. Peng, “Optimal photon antibunching in a quantum dot bimodal cavity system,” Phys. Rev. A **89**(4), 043832 (2014). [CrossRef]

**36. **X. W. Xu and Y. Li, “Strong photon antibunching of symmetric and antisymmetric modes in weakly nonlinear photonic molecules,” Phys. Rev. A **90**(3), 033809 (2014). [CrossRef]

**37. **H. Z. Shen, Y. H. Zhou, and X. X. Yi, “Tunable photon blockade in coupled semiconductor cavities,” Phys. Rev. A **91**(6), 063808 (2015). [CrossRef]

**38. **O. Kyriienko, I. A. Shelykh, and T. C. H. Liew, “Tunable single-photon emission from dipolaritons,” Phys. Rev. A **90**(3), 033807 (2014). [CrossRef]

**39. **T. C. H. Liew and V. Savona, “Single photons from coupled quantum modes,” Phys. Rev. Lett. **104**(18), 183601 (2010). [CrossRef]

**40. **M. Bamba, A. Imamoḡlu, I. Carusotto, and C. Ciuti, “Origin of strong photon antibunching in weakly nonlinear photonic molecules,” Phys. Rev. A **83**(2), 021802 (2011). [CrossRef]

**41. **H. Flayac and V. Savona, “Input-output theory of the unconventional photon blockade,” Phys. Rev. A **88**(3), 033836 (2013). [CrossRef]

**42. **H. Flayac and V. Savona, “Single photons from dissipation in coupled cavities,” Phys. Rev. A **94**(1), 013815 (2016). [CrossRef]

**43. **H. Flayac and V. Savona, “Unconventional photon blockade,” Phys. Rev. A **96**(5), 053810 (2017). [CrossRef]

**44. **D. Gerace and V. Savona, “Unconventional photon blockade in doubly resonant microcavities with second-order nonlinearity,” Phys. Rev. A **89**(3), 031803 (2014). [CrossRef]

**45. **A. Majumdar and D. Gerace, “Single-photon blockade in doubly resonant nanocavities with second-order nonlinearity,” Phys. Rev. B **87**(23), 235319 (2013). [CrossRef]

**46. **Y. Y. Jiang, A. D. Ludlow, N. D. Lemke, R. W. Fox, J. A. Sherman, L.-S. Ma, and C. W. Oates, “Making optical atomic clocks more stable with 10^{−16}- level laser stabilization,” Nat. Photonics **5**(3), 158–161 (2011). [CrossRef]

**47. **T. M. Fortier, M. S. Kirchner, F. Quinlan, J. Taylor, J. C. Bergquist, T. Rosenband, N. Lemke, A. Ludlow, Y. Jiang, C. W. Oates, and S. A. Diddams, “Generation of ultrastable microwaves via optical frequency division,” Nat. Photonics **5**(7), 425–429 (2011). [CrossRef]

**48. **S. Ast, M. Mehmet, and R. Schnabel, “High-bandwidth squeezed light at 1550 nm from a compact monolithic PPKTP cavity,” Opt. Express **21**(11), 13572 (2013). [CrossRef]

**49. **T. Serikawa, J. Yoshikawa, K. Makino, and A. Furusawa, “Creation and measurement of broadband squeezed vacuum from a ring optical parametric oscillator,” Opt. Express **24**(25), 28383 (2016). [CrossRef]

**50. **K. W. Murch, S. J. Weber, K. M. Beck, E. Ginossar, and I. Siddiqi, “Reduction of the radiative decay of atomic coherence in squeezed vacuum,” Nature **499**(7456), 62–65 (2013). [CrossRef]

**51. **X. Y. Lü, Y. Wu, J. R. Johansson, H. Jing, J. Zhang, and F. Nori, “Squeezed optomechanics with phase-matched amplification and dissipation,” Phys. Rev. Lett. **114**(9), 093602 (2015). [CrossRef]

**52. **S. Bergfeld and W. Daum, “Second-Harmonic Generation in GaAs: Experiment versus Theoretical Predictions of $\chi ^{(2)}_{xyz}$,” Phys. Rev. Lett. **90**(3), 036801 (2003). [CrossRef]

**53. **Z. Yang, P. Chak, A. D. Bristow, H. M. van Driel, R. Iyer, J. S. Aitchison, A. L. Smirl, and J. E. Sipe, “Enhanced second-harmonic generation in AlGaAs microring resonators,” Opt. Lett. **32**(7), 826 (2007). [CrossRef]

**54. **J. Chen, Z. H. Levine, and J. W. Wilkins, “Calculated second-harmonic susceptibilities of BN, AlN, and GaN,” Appl. Phys. Lett. **66**(9), 1129–1131 (1995). [CrossRef]