Abstract

We investigate the enhanced four-wave mixing (FWM) process in a parity-time ($\mathcal{PT}$)-symmetric optomechanical system, where an active cavity is coupled to a passive cavity supporting a mechanical mode. The passive cavity is optically driven by a strong control field and a weak probe field, and the mechanical mode is excited by a weak coherent driving field. By tuning the coupling strength between the two cavities with balanced gain and loss, we find that the FWM intensity can be significantly enhanced near the exceptional points (EPs) at low control power, which is about 12 orders of magnitude higher than that of the single-cavity case. Due to the interference effect induced by the optical and mechanical driving field, it is shown that the FWM intensity can be further enhanced or suppressed by tuning the amplitude and phase of the mechanical driving field. Moreover, the dependence of the FWM intensity on the frequency and power of the control field is also discussed. Our work provides a route to enhance the four-wave mixing process in a flexible way.

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

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2020 (1)

X. W. Xu, Y. J. Zhao, H. Wang, H. Jing, and A. X. Chen, “Quantum nonreciprocality in quadratic optomechanics,” Photonics Res. 8(2), 143–150 (2020).
[Crossref]

2019 (6)

L. Y. He, “Parity-time-symmetry-enhanced sideband generation in an optomechanical system,” Phys. Rev. A 99(3), 033843 (2019).
[Crossref]

P. Renault, H. Yamaguchi, and I. Mahboob, “Virtual exceptional points in an electromechanical system,” Phys. Rev. Appl. 11(2), 024007 (2019).
[Crossref]

T. X. Lu, Y. F. Jiao, H. L. Zhang, F. Saif, and H. Jing, “Selective and switchable optical amplification with mechanical driven oscillators,” Phys. Rev. A 100(1), 013813 (2019).
[Crossref]

C. Zhai, R. Huang, H. Jing, and L.-M. Kuang, “Mechanical switch of photon blockade and photon-induced tunneling,” Opt. Express 27(20), 27649–27662 (2019).
[Crossref]

X. F. Wang and B. Chen, “Four-wave mixing response in a hybrid atom-optomechanical system,” J. Opt. Soc. Am. B 36(2), 162–167 (2019).
[Crossref]

X. Y. Zhang, Q. T. Cao, Z. Wang, Y. X. Liu, C. W. Qiu, L. Yang, Q. H. Gong, and Y. F. Xiao, “Symmetry-breaking-induced nonlinear optics at a microcavity surface,” Nat. Photonics 13(1), 21–24 (2019).
[Crossref]

2018 (3)

Z. Y. Li, X. You, Y. M. Li, Y. C. Liu, and K. C. Peng, “Multimode four-wave mixing in an unresolved sideband optomechanical system,” Phys. Rev. A 97(3), 033806 (2018).
[Crossref]

S. Maayani, R. Dahan, Y. Kligerman, E. Moses, A. U. Hassan, H. Jing, F. Nori, D. N. Christodoulides, and T. Carmon, “Flying couplers above spinning resonators generate irreversible refraction,” Nature 558(7711), 569–572 (2018).
[Crossref]

D. B. Sohn, S. Kim, and G. Bahl, “Time-reversal symmetry breaking with acoustic pumping of nanophotonic circuits,” Nat. Photonics 12(2), 91–97 (2018).
[Crossref]

2017 (9)

Y. Li, Y. Y. Huang, X. Z. Zhang, and L. Tian, “Optical directional amplification in a three-mode optomechanical system,” Opt. Express 25(16), 18907–18916 (2017).
[Crossref]

L. G. Si, H. Xiong, M. S. Zubairy, and Y. Wu, “Optomechanically induced opacity and amplification in a quadratically coupled optomechanical system,” Phys. Rev. A 95(3), 033803 (2017).
[Crossref]

C. Bekker, R. Kalra, C. Baker, and W. P. Bowen, “Injection locking of an electro-optomechanical device,” Optica 4(10), 1196–1204 (2017).
[Crossref]

W. Chen, Ş. K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature 548(7666), 192–196 (2017).
[Crossref]

H. Hodaei, A. U. Hassan, S. Wittek, H. G.-Gracia, R. E.-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548(7666), 187–191 (2017).
[Crossref]

X. Y. Zhang, Y. Q. Guo, P. Pei, and X. X. Yi, “Optomechanically induced absorption in parity-time-symmetric optomechanical systems,” Phys. Rev. A 95(6), 063825 (2017).
[Crossref]

H. Jing, Ş. K. Özdemir, H. Lü, and F. Nori, “High-order exceptional points in optomechanics,” Sci. Rep. 7(1), 3386 (2017).
[Crossref]

Y. L. Liu and Y.-x. Liu, “Energy-localization-enhanced ground-state cooling of a mechanical resonator from room temperature in optomechanics using a gain cavity,” Phys. Rev. A 96(2), 023812 (2017).
[Crossref]

S. Hong, R. Riedinger, I. Marinkovic, A. Wallucks, S. G. Hofer, R. A. Norte, M. Aspelmeyer, and S. Gröblacher, “Hanbury Brown and Twiss interferometry of single phonons from an optomechanical resonator,” Science 358(6360), 203–206 (2017).
[Crossref]

2016 (7)

Y. Jiao, H. Lü, J. Qian, Y. Li, and H. Jing, “Nonlinear optomechanics with gain and loss: amplifying higher-order sideband and group delay,” New J. Phys. 18(8), 083034 (2016).
[Crossref]

H. Xu, D. Mason, L. Y. Jiang, and J. G. E. Harris, “Topological energy transfer in an optomechanical system with exceptional points,” Nature 537(7618), 80–83 (2016).
[Crossref]

W. Li, Y. Jiang, C. Li, and H. Song, “Parity-time-symmetry enhanced optomechanically-induced-transparency,” Sci. Rep. 6(1), 31095 (2016).
[Crossref]

Z. P. Liu, J. Zhang, S. K. Özdemir, B. Peng, H. Jing, X.-Y. Lü, C.-W. Li, L. Yang, F. Nori, and Y.-x. Liu, “Metrology with PT-symmetric cavities: enhanced sensitivity near the PT-phase transition,” Phys. Rev. Lett. 117(11), 110802 (2016).
[Crossref]

B. He, L. Yang, and M. Xiao, “Dynamical phonon laser in coupled active-passive microresonators,” Phys. Rev. A 94(3), 031802 (2016).
[Crossref]

C. Jiang, Y. S. Cui, X. T. Bian, F. Zuo, H. L. Yu, and G. B. Chen, “Phase-dependent multiple optomechanically induced absorption in multimode optomechanical systems with mechanical driving,” Phys. Rev. A 94(2), 023837 (2016).
[Crossref]

R. Schilling, H. Schütz, A. H. Ghadimi, V. Sudhir, D. J. Wilson, and T. J. Kippenberg, “Near-field integration of a SiN nanobeam and a SiO2 microcavity for Heisenberg-limited displacement sensing,” Phys. Rev. Appl. 5(5), 054019 (2016).
[Crossref]

2015 (9)

W. Z. Jia, L. F. Wei, Y. Li, and Y.-x. Liu, “Phase-dependent optical response properties in an optomechanical system by coherently driving the mechanical resonator,” Phys. Rev. A 91(4), 043843 (2015).
[Crossref]

J. Y. Ma, C. You, L. G. Si, H. Xiong, J. H. Li, X. X. Yang, and Y. Wu, “Optomechanically induced transparency in the presence of an external time-harmonic-driving force,” Sci. Rep. 5(1), 11278 (2015).
[Crossref]

X.-Y. Lü, H. Jing, J.-Y. Ma, and Y. Wu, “PT-symmetry-breaking chaos in optomechanics,” Phys. Rev. Lett. 114(25), 253601 (2015).
[Crossref]

H. Jing, Ş. K. Özdemir, Z. Geng, J. Zhang, X.-Y. Lü, B. Peng, L. Yang, and F. Nori, “Optomechanically-induced transparency in parity-time-symmetric microresonators,” Sci. Rep. 5(1), 9663 (2015).
[Crossref]

L. Fan, K. Y. Fong, M. Poot, and H. X. Tang, “Cascaded optical transparency in multimode-cavity optomechanical systems,” Nat. Commun. 6(1), 5850 (2015).
[Crossref]

J. Zhang, B. Peng, Ş. K. Özdemir, Y.-x. Liu, H. Jing, X.-y. Lü, Y.-l. Liu, L. Yang, and F. Nori, “Giant nonlinearity via breaking parity-time symmetry: A route to low-threshold phonon diodes,” Phys. Rev. B 92(11), 115407 (2015).
[Crossref]

T. Wasak, P. Szańkowski, V. V. Konotop, and M. Trippenbach, “Four-wave mixing in a parity-time ($\mathcal{PT}$PT)-symmetric coupler,” Opt. Lett. 40(22), 5291–5294 (2015).
[Crossref]

H. Xiong, L. G. Si, X. Y. Lü, X. X. Yang, and Y. Wu, “Review of cavity optomechanics in the weak-coupling regime: from linearization to intrinsic nonlinear interactions,” Sci. China: Phys., Mech. Astron. 58(5), 1–13 (2015).
[Crossref]

X. W. Xu and Y. Li, “Controllable optical output fields from an optomechanical system with mechanical driving,” Phys. Rev. A 92(2), 023855 (2015).
[Crossref]

2014 (4)

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86(4), 1391–1452 (2014).
[Crossref]

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8(7), 524–529 (2014).
[Crossref]

B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

H. Jing, S. K. Özdemir, X.-Y. Lü, J. Zhang, L. Yang, and F. Nori, “PT-symmetric phonon laser,” Phys. Rev. Lett. 113(5), 053604 (2014).
[Crossref]

2013 (4)

A. H. Safavi-Naeini, S. Gröblacher, J. H. Hill, J. Chan, M. Aspelmeyer, and O. Painter, “Squeezed light from a silicon micromechanical resonator,” Nature 500(7461), 185–189 (2013).
[Crossref]

T. P. Purdy, P.-L. Yu, R. W. Peterson, N. S. Kampel, and C. A. Regal, “Strong optomechanical squeezing of light,” Phys. Rev. X 3(3), 031012 (2013).
[Crossref]

C. Jiang, Y. S. Cui, and H. X. Liu, “Controllable four-wave mixing based on mechanical vibration in two-mode optomechanical systems,” Europhys. Lett. 104(3), 34004 (2013).
[Crossref]

J. Bochmann, A. Vainsencher, D. D. Awschalom, and A. N. Cleland, “Nanomechanical coupling between microwave and optical photons,” Nat. Phys. 9(11), 712–716 (2013).
[Crossref]

2012 (1)

E. Verhagen, S. Deléglise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” Nature 482(7383), 63–67 (2012).
[Crossref]

2011 (3)

A. H. Safavi-Naeini, T. P. Mayer Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472(7341), 69–73 (2011).
[Crossref]

J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478(7367), 89–92 (2011).
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J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475(7356), 359–363 (2011).
[Crossref]

2010 (4)

G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81(4), 041803 (2010).
[Crossref]

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010).
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S. M. Huang and G. S. Agarwal, “Normal-mode splitting and antibunching in Stokes and anti-Stokes processes in cavity optomechanics: Radiation-pressure-induced four-wave-mixing cavity optomechanics,” Phys. Rev. A 81(3), 033830 (2010).
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A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, “Introduction to quantum noise, measurement, and amplification,” Rev. Mod. Phys. 82(2), 1155–1208 (2010).
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2009 (3)

G. Anetsberger, O. Arcizet, Q. P. Unterreithmeier, R. Rivière, A. Schliesser, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, “Near-field cavity optomechanics with nanomechanical oscillators,” Nat. Phys. 5(12), 909–914 (2009).
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F. Marquardt and S. M. Girvin, “Optomechanics,” Physics 2, 40 (2009).
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A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
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2004 (1)

2003 (1)

Y. Wu, J. Saldana, and Y. F. Zhu, “Large enhancement of four-wave mixing by suppression of photon absorption from electromagnetically induced transparency,” Phys. Rev. A 67(1), 013811 (2003).
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1998 (1)

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having $\mathcal{PT}$PT-Symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998).
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1996 (1)

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A. H. Safavi-Naeini, S. Gröblacher, J. H. Hill, J. Chan, M. Aspelmeyer, and O. Painter, “Squeezed light from a silicon micromechanical resonator,” Nature 500(7461), 185–189 (2013).
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J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478(7367), 89–92 (2011).
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C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having $\mathcal{PT}$PT-Symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998).
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C. Jiang, Y. S. Cui, X. T. Bian, F. Zuo, H. L. Yu, and G. B. Chen, “Phase-dependent multiple optomechanically induced absorption in multimode optomechanical systems with mechanical driving,” Phys. Rev. A 94(2), 023837 (2016).
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C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having $\mathcal{PT}$PT-Symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998).
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X. Y. Zhang, Q. T. Cao, Z. Wang, Y. X. Liu, C. W. Qiu, L. Yang, Q. H. Gong, and Y. F. Xiao, “Symmetry-breaking-induced nonlinear optics at a microcavity surface,” Nat. Photonics 13(1), 21–24 (2019).
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A. H. Safavi-Naeini, S. Gröblacher, J. H. Hill, J. Chan, M. Aspelmeyer, and O. Painter, “Squeezed light from a silicon micromechanical resonator,” Nature 500(7461), 185–189 (2013).
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A. H. Safavi-Naeini, T. P. Mayer Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472(7341), 69–73 (2011).
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L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8(7), 524–529 (2014).
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H. Hodaei, A. U. Hassan, S. Wittek, H. G.-Gracia, R. E.-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548(7666), 187–191 (2017).
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A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
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Cleland, A. N.

J. Bochmann, A. Vainsencher, D. D. Awschalom, and A. N. Cleland, “Nanomechanical coupling between microwave and optical photons,” Nat. Phys. 9(11), 712–716 (2013).
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A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, “Introduction to quantum noise, measurement, and amplification,” Rev. Mod. Phys. 82(2), 1155–1208 (2010).
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C. Jiang, Y. S. Cui, X. T. Bian, F. Zuo, H. L. Yu, and G. B. Chen, “Phase-dependent multiple optomechanically induced absorption in multimode optomechanical systems with mechanical driving,” Phys. Rev. A 94(2), 023837 (2016).
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C. Jiang, Y. S. Cui, and H. X. Liu, “Controllable four-wave mixing based on mechanical vibration in two-mode optomechanical systems,” Europhys. Lett. 104(3), 34004 (2013).
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S. Maayani, R. Dahan, Y. Kligerman, E. Moses, A. U. Hassan, H. Jing, F. Nori, D. N. Christodoulides, and T. Carmon, “Flying couplers above spinning resonators generate irreversible refraction,” Nature 558(7711), 569–572 (2018).
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S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010).
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A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, “Introduction to quantum noise, measurement, and amplification,” Rev. Mod. Phys. 82(2), 1155–1208 (2010).
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A. H. Safavi-Naeini, T. P. Mayer Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472(7341), 69–73 (2011).
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S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010).
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A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, “Introduction to quantum noise, measurement, and amplification,” Rev. Mod. Phys. 82(2), 1155–1208 (2010).
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S. Hong, R. Riedinger, I. Marinkovic, A. Wallucks, S. G. Hofer, R. A. Norte, M. Aspelmeyer, and S. Gröblacher, “Hanbury Brown and Twiss interferometry of single phonons from an optomechanical resonator,” Science 358(6360), 203–206 (2017).
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A. H. Safavi-Naeini, S. Gröblacher, J. H. Hill, J. Chan, M. Aspelmeyer, and O. Painter, “Squeezed light from a silicon micromechanical resonator,” Nature 500(7461), 185–189 (2013).
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J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478(7367), 89–92 (2011).
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A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
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S. Maayani, R. Dahan, Y. Kligerman, E. Moses, A. U. Hassan, H. Jing, F. Nori, D. N. Christodoulides, and T. Carmon, “Flying couplers above spinning resonators generate irreversible refraction,” Nature 558(7711), 569–572 (2018).
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H. Hodaei, A. U. Hassan, S. Wittek, H. G.-Gracia, R. E.-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548(7666), 187–191 (2017).
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A. H. Safavi-Naeini, S. Gröblacher, J. H. Hill, J. Chan, M. Aspelmeyer, and O. Painter, “Squeezed light from a silicon micromechanical resonator,” Nature 500(7461), 185–189 (2013).
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J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478(7367), 89–92 (2011).
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A. H. Safavi-Naeini, T. P. Mayer Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472(7341), 69–73 (2011).
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H. Hodaei, A. U. Hassan, S. Wittek, H. G.-Gracia, R. E.-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548(7666), 187–191 (2017).
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Huang, S.

G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81(4), 041803 (2010).
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S. M. Huang and G. S. Agarwal, “Normal-mode splitting and antibunching in Stokes and anti-Stokes processes in cavity optomechanics: Radiation-pressure-induced four-wave-mixing cavity optomechanics,” Phys. Rev. A 81(3), 033830 (2010).
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D. Bothner, S. Yanai, A. Iniguez-Rabago, M. Yuan, Y. M. Blanter, and G. A. Steele, “Cavity electromechanics with parametric mechanical driving,” arXiv:1908.08496v1.

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C. Jiang, Y. S. Cui, X. T. Bian, F. Zuo, H. L. Yu, and G. B. Chen, “Phase-dependent multiple optomechanically induced absorption in multimode optomechanical systems with mechanical driving,” Phys. Rev. A 94(2), 023837 (2016).
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C. Jiang, Y. S. Cui, and H. X. Liu, “Controllable four-wave mixing based on mechanical vibration in two-mode optomechanical systems,” Europhys. Lett. 104(3), 34004 (2013).
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L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8(7), 524–529 (2014).
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Jiang, L. Y.

H. Xu, D. Mason, L. Y. Jiang, and J. G. E. Harris, “Topological energy transfer in an optomechanical system with exceptional points,” Nature 537(7618), 80–83 (2016).
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Figures (7)
Fig. 1.
Fig. 1. Schematic illustration of the $\mathcal {PT}$-symmetric optomechanical system. The passive cavity with loss rate $\kappa _1$ interacts with a mechanical mode with resonance frequency $\omega _m$ and damping rate $\gamma _m$ via radiation pressure. The active cavity with tunable gain rate $\kappa _2$ is coupled to the passive cavity, and the coupling strength $J$ can be adjusted by the distance between them. In addition, the passive cavity is driven by a strong control field at frequency $\omega _c$ and a weak probe field at frequency $\omega _p$, and the mechanical resonator is excited by a weak coherent mechanical driving field at frequency $\Omega =\omega _p-\omega _c$.

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Fig. 2.
Fig. 2. FWM intensitiy $\log _{10}|\mathrm {FWM_p}|^2$ as a function of (a) $(\Omega -\omega _m)/\gamma _m$ for different values of coupling strength $J$ and (b) $J/(\kappa _1+\kappa _2)$ with $\Omega =\omega _m$. Other parameters are $\Delta _1^{'}=\Delta _2=0$, $\omega _1=2\pi c/\lambda$ with $\lambda =1550$ nm, $R=15$ $\mu$m, $g_1=\omega _1/R$, $\kappa _1/2\pi =6$ MHz, $\kappa _2=0.99\kappa _1$, $m=6.2$ ng, $\omega _m/2\pi =78$ MHz, $\gamma _m/2\pi =12$ kHz, $P_c=50~$nW, $\eta _c=0.4$, $\varepsilon _p=\varepsilon _c/1000$, and $\varepsilon _m=0.$

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Fig. 3.
Fig. 3. Contour plots of FWM intensity $|\mathrm {FWM_p}|^2$ at $\Omega =\omega _m$ as functions of the amplitude $\varepsilon _m$ and phase difference $\phi /\pi$ with (a) $\Delta _1^{'}=\Delta _2=0$ and (b) $\Delta _1^{'}=\Delta _2=0.5\omega _m$. The other parameters are the same as those in Fig. 2 except $J=0.251(\kappa _1+\kappa _2)$.

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Fig. 4.
Fig. 4. Plots of $|\mathrm {FWM}_1|^2$, $|\mathrm {FWM}_2|^2$, and $|\mathrm {FWM_p}|^2$ at $\Omega =\omega _m$ as functions of the mechanical driving amplitude $\varepsilon _m$ with $\phi =\pi /2$ in (a) and (b) and $\phi =3\pi /2$ in (c) and (d). Here we choose $\Delta _1^{'}=\Delta _2=0$ and $J=0.251(\kappa _1+\kappa _2)$. The other parameters are the same as those in Fig. 2.

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Fig. 5.
Fig. 5. FWM intensity $\log _{10}|\mathrm {FWM_p}|^2$ at $\Omega =\omega _m$ as a function of gain-loss ratio $\kappa _2/\kappa _1$ for different values of $\varepsilon _m$ and $\phi$. The other parameters are the same as those in Fig. 4.

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Fig. 6.
Fig. 6. FWM intensity $\log _{10}|\mathrm {FWM_p}|^2$ at $\Omega =\omega _m$ as a function of $\Delta _{1}^{'}/\omega _m$ for different values of $\varepsilon _m$ and $\phi$. Here we fix $\Delta _2=\Delta _1^{'}$, and the other parameters are the same as those in Fig. 4.

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Fig. 7.
Fig. 7. FWM intensity $\log _{10}|\mathrm {FWM_p}|^2$ as a function of $(\Omega -\omega _m)/\gamma _m$ for different control power $P_c$. The inset shows the FWM intensity $\log _{10}|\mathrm {FWM_p}|^2$ at $\Omega =\omega _m$ versus the control power $P_c$. The other parameters are the same as those in Fig. 4 except $\Delta _1^{'}=\Delta _2=\omega _m$, $\varepsilon _m=4$ pN and $\phi =\pi /2$.

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Equations (20)

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H = H 0 + H I + H d r , H 0 = ℏ Δ 1 a 1 † a 1 + ℏ Δ 2 a 2 † a 2 + p 2 2 m + 1 2 m ω m 2 x 2 , H I = − ℏ J ( a 1 † a 2 + a 1 a 2 † ) − ℏ g 1 a 1 † a 1 x , H d r = i ℏ η c κ 1 [ ( ε c + ε p e − i Ω t − i ϕ p c ) a 1 † − H . c . ] − x ε m c o s ( Ω t + ϕ m ) .
a 1 ˙ = − ( κ 1 2 + i Δ 1 − g 1 x ) a 1 + i J a 2 + η c κ 1 ( ε c + ε p e − i Ω t − i ϕ p ) ,
a 2 ˙ = ( κ 2 2 − i Δ 2 ) a 2 + i J a 1 ,
x ˙ = p m ,
p ˙ = − m ω m 2 x − γ m p + ℏ g 1 a 1 † a 1 + ε m cos ⁡ ( Ω t + ϕ m ) ,
a 1 s = η c κ 1 ε c ( κ 2 / 2 − i Δ 2 ) ( κ 1 / 2 + i Δ 1 ′ ) ( κ 2 / 2 − i Δ 2 ) − J 2 , a 2 s = − i J η c κ 1 ε c ( κ 1 / 2 + i Δ 1 ′ ) ( κ 2 / 2 − i Δ 2 ) − J 2 , x s = ℏ g 1 | a 1 s | 2 m ω m 2 , p s = 0 ,
δ a 1 ˙ = − ( κ 1 / 2 + i Δ 1 ′ ) δ a 1 + i g 1 a 1 s δ x + i J δ a 2 + η c κ 1 ε p e − i Ω t − i ϕ p c ,
δ a 2 ˙ = ( κ 2 / 2 − i Δ 2 ) δ a 2 + i J δ a 1 ,
δ x ˙ = δ p m ,
δ p ˙ = − m ω m 2 δ x − γ m δ p + ℏ g 1 ( a 1 s ∗ δ a 1 + a 1 s δ a 1 † ) + ε m c o s ( Ω t + ϕ m ) .
a 1 + = χ 1 ( Ω ) 1 + i f ∗ ( − Ω ) − i f ( Ω ) { [ 1 + i f ∗ ( − Ω ) ] η c κ 1 ε p e − i ϕ p + i g 1 a 1 s χ m ( Ω ) ε m 2 e − i ϕ m } ,
a 1 − = χ 1 ( − Ω ) 1 − i f ( − Ω ) + i f ∗ ( Ω ) [ i f ∗ ( Ω ) a 1 s a 1 s ∗ η c κ 1 ε p e i ϕ p + i g 1 a 1 s χ m ∗ ( Ω ) ε m 2 e i ϕ m ] .
χ m ( Ω ) = 1 m ( ω m 2 − Ω 2 − i γ m Ω ) , χ 2 ( Ω ) = 1 κ 2 / 2 − i Δ 2 + i Ω , χ 1 ( Ω ) = 1 κ 1 / 2 + i ( Δ 1 ′ − Ω ) − J 2 χ 2 ( Ω ) , f ( Ω ) = ℏ g 1 2 | a 1 s | 2 χ m ( Ω ) κ 1 / 2 + i ( Δ 1 ′ − Ω ) − J 2 χ 2 ( Ω ) .
a 1 , o u t ( t ) = ( ε c − η c κ 1 a 1 s ) e − i ω c t ⏟ C o n t r o l   f i e l d + ( ε p e − i ϕ p c − η c κ 1 a 1 + ) e − i ω p t ⏟ P r o b e   f i e l d − η c κ 1 a 1 − e − i ( 2 ω c − ω p ) t ⏟ F W M   f i e l d .
t p = ε p e − i ϕ p c − η c κ 1 a 1 + ε p e − i ϕ p c = t 1 + t 2 ,
t 1 = 1 − 1 + i f ∗ ( − Ω ) 1 + i f ∗ ( − Ω ) − i f ( Ω ) χ 1 ( Ω ) η c κ 1 ,
t 2 = − i g 1 a 1 s χ 1 ( Ω ) χ m ( Ω ) ε m / ( 2 ε p ) 1 + i f ∗ ( − Ω ) − i f ( Ω ) η c κ 1 e − i ϕ .
F W M p = η c κ 1 a 1 − ε p e i ϕ p = F W M 1 + F W M 2 ,
F W M 1 = i f ∗ ( Ω ) χ 1 ( − Ω ) a 1 s / a 1 s ∗ 1 − i f ( − Ω ) + i f ∗ ( Ω ) η c κ 1 ,
F W M 2 = i g 1 a 1 s χ 1 ( − Ω ) χ m ∗ ε m / ( 2 ε p ) 1 − i f ( − Ω ) + i f ∗ ( Ω ) η c κ 1 e i ϕ .

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