Abstract

We present a dual-gate optical transistor based on a multimode optomechanical system, composed of three indirectly coupled cavities and an intermediate mechanical resonator pumped by a frequency-matched field. In this system, two cavities driven on the red mechanical sidebands are regarded as input/ouput gates/poles and the third one on the blue sideband as a basic/control gate/pole, while the resonator as the other basic/control gate/pole. As a nonreciprocal scheme, the significant unidirectional amplification can be resulted by controlling the two control gates/poles. In particular, the nonreciprocal direction of the optical amplification/rectification can be controlled by adjusting the phase differences between two red-sideband driving fields (the pumping and probe fields). Meanwhile, the narrow window that can be analyzed by the effective mechanical damping rate, arises from the extra blue-sideband cavity. Moreover, the tunable slow/fast light effect can be observed, i.e, the group velocity of the unidirectional transmission can be controlled, and thus the switching scheme of slow/fast light effect can also utilized to realize both slow and fast lights through opposite propagation directions, respectively. Such an amplification transistor scheme of controllable amplitude, direction and velocity may imply exciting opportunities for potential applications in photon networks and quantum information processing.

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

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References

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2019 (5)

B. Abdo, N. T. Bronn, O. Jinka, S. Olivadese, A. D. Córcoles, V. P. Adiga, M. Brink, R. E. Lake, X. Wu, D. P. Pappas, and J. M. Chow, “Active protection of a superconducting qubit with an interferometric josephson isolator,” Nat. Commun. 10(1), 3154 (2019).
[Crossref]

I. M. Mirza, W. Ge, and H. Jing, “Optical nonreciprocity and slow light in coupled spinning optomechanical resonators,” Opt. Express 27(18), 25515–25530 (2019).
[Crossref]

C. Jiang, L. N. Song, and Y. Li, “Directional phase-sensitive amplifier between microwave and optical photons,” Phys. Rev. A 99(2), 023823 (2019).
[Crossref]

H. Xu, L. Jiang, A. Clerk, and J. Harris, “Nonreciprocal control and cooling of phonon modes in an optomechanical system,” Nature 568(7750), 65–69 (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]

2018 (14)

L. Du, Y.-M. Liu, B. Jiang, and Y. Zhang, “All-optical photon switching, router and amplifier using a passive-active optomechanical system,” EPL 122(2), 24001 (2018).
[Crossref]

B. He, L. Yang, X. Jiang, and M. Xiao, “Transmission nonreciprocity in a mutually coupled circulating structure,” Phys. Rev. Lett. 120(20), 203904 (2018).
[Crossref]

D. Malz, L. D. Tóth, N. R. Bernier, A. K. Feofanov, T. J. Kippenberg, and A. Nunnenkamp, “Quantum-limited directional amplifiers with optomechanics,” Phys. Rev. Lett. 120(2), 023601 (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]

Y. Jiang, S. Maayani, T. Carmon, F. Nori, and H. Jing, “Nonreciprocal phonon laser,” Phys. Rev. Appl. 10(6), 064037 (2018).
[Crossref]

F. Ruesink, J. P. Mathew, M.-A. Miri, A. Alù, and E. Verhagen, “Optical circulation in a multimode optomechanical resonator,” Nat. Commun. 9(1), 1798 (2018).
[Crossref]

I. Moaddel Haghighi, N. Malossi, R. Natali, G. Di Giuseppe, and D. Vitali, “Sensitivity-bandwidth limit in a multimode optoelectromechanical transducer,” Phys. Rev. Appl. 9(3), 034031 (2018).
[Crossref]

X. Z. Zhang, L. Tian, and Y. Li, “Optomechanical transistor with mechanical gain,” Phys. Rev. A 97(4), 043818 (2018).
[Crossref]

Y. Tokura and N. Nagaosa, “Nonreciprocal responses from non-centrosymmetric quantum materials,” Nat. Commun. 9(1), 3740 (2018).
[Crossref]

I. Liberal and N. Engheta, “Multiqubit subradiant states in n-port waveguide devices: $\epsilon$ϵ-and-μ-near-zero hubs and nonreciprocal circulators,” Phys. Rev. A 97(2), 022309 (2018).
[Crossref]

Z. Shen, Y.-L. Zhang, Y. Chen, F.-W. Sun, X.-B. Zou, G.-C. Guo, C.-L. Zou, and C.-H. Dong, “Reconfigurable optomechanical circulator and directional amplifier,” Nat. Commun. 9(1), 1797 (2018).
[Crossref]

A. R. Hamann, C. Müller, M. Jerger, M. Zanner, J. Combes, M. Pletyukhov, M. Weides, T. M. Stace, and A. Fedorov, “Nonreciprocity realized with quantum nonlinearity,” Phys. Rev. Lett. 121(12), 123601 (2018).
[Crossref]

A. Roulet and C. Bruder, “Quantum synchronization and entanglement generation,” Phys. Rev. Lett. 121(6), 063601 (2018).
[Crossref]

Y. Tokura and N. Nagaosa, “Nonreciprocal responses from non-centrosymmetric quantum materials,” Nat. Commun. 9(1), 3740 (2018).
[Crossref]

2017 (7)

H. Lü, Y. Jiang, Y.-Z. Wang, and H. Jing, “Optomechanically induced transparency in a spinning resonator,” Photonics Res. 5(4), 367–371 (2017).
[Crossref]

L. Du, C.-H. Fan, H.-X. Zhang, and J.-H. Wu, “Synchronization enhancement of indirectly coupled oscillators via periodic modulation in an optomechanical system,” Sci. Rep. 7(1), 15834 (2017).
[Crossref]

S. Huang and G. S. Agarwal, “Robust force sensing for a free particle in a dissipative optomechanical system with a parametric amplifier,” Phys. Rev. A 95(2), 023844 (2017).
[Crossref]

Z. Wu, R.-H. Luo, J.-Q. Zhang, Y.-H. Wang, W. Yang, and M. Feng, “Force-induced transparency and conversion between slow and fast light in optomechanics,” Phys. Rev. A 96(3), 033832 (2017).
[Crossref]

K. Fang, J. Luo, A. Metelmann, M. H. Matheny, F. Marquardt, A. A. Clerk, and O. Painter, “Generalized non-reciprocity in an optomechanical circuit via synthetic magnetism and reservoir engineering,” Nat. Phys. 13(5), 465–471 (2017).
[Crossref]

N. R. Bernier, L. D. Toth, A. Koottandavida, M. A. Ioannou, D. Malz, A. Nunnenkamp, A. Feofanov, and T. Kippenberg, “Nonreciprocal reconfigurable microwave optomechanical circuit,” Nat. Commun. 8(1), 604 (2017).
[Crossref]

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

2016 (5)

F. Ruesink, M.-A. Miri, A. Alu, and E. Verhagen, “Nonreciprocity and magnetic-free isolation based on optomechanical interactions,” Nat. Commun. 7(1), 13662 (2016).
[Crossref]

C. Benedetti, F. Galve, A. Mandarino, M. G. Paris, and R. Zambrini, “Minimal model for spontaneous quantum synchronization,” Phys. Rev. A 94(5), 052118 (2016).
[Crossref]

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10(10), 657–661 (2016).
[Crossref]

S. Mahmoodian, P. Lodahl, and A. S. Sørensen, “Quantum networks with chiral-light–matter interaction in waveguides,” Phys. Rev. Lett. 117(24), 240501 (2016).
[Crossref]

X. Guo, C.-L. Zou, H. Jung, and H. X. Tang, “On-chip strong coupling and efficient frequency conversion between telecom and visible optical modes,” Phys. Rev. Lett. 117(12), 123902 (2016).
[Crossref]

2015 (7)

S. Yoo and Q.-H. Park, “Chiral light-matter interaction in optical resonators,” Phys. Rev. Lett. 114(20), 203003 (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]

K. M. Sliwa, M. Hatridge, A. Narla, S. Shankar, L. Frunzio, R. J. Schoelkopf, and M. H. Devoret, “Reconfigurable josephson circulator/directional amplifier,” Phys. Rev. X 5(4), 041020 (2015).
[Crossref]

A. Metelmann and A. A. Clerk, “Nonreciprocal photon transmission and amplification via reservoir engineering,” Phys. Rev. X 5(2), 021025 (2015).
[Crossref]

M. J. Akram, M. M. Khan, and F. Saif, “Tunable fast and slow light in a hybrid optomechanical system,” Phys. Rev. A 92(2), 023846 (2015).
[Crossref]

M. Schmidt, S. Kessler, V. Peano, O. Painter, and F. Marquardt, “Optomechanical creation of magnetic fields for photons on a lattice,” Optica 2(7), 635–641 (2015).
[Crossref]

B. P. Hou, L. F. Wei, and S. J. Wang, “Optomechanically induced transparency and absorption in hybridized optomechanical systems,” Phys. Rev. A 92(3), 033829 (2015).
[Crossref]

2014 (9)

X. Xu and J. M. Taylor, “Squeezing in a coupled two-mode optomechanical system for force sensing below the standard quantum limit,” Phys. Rev. A 90(4), 043848 (2014).
[Crossref]

R. Fleury, D. L. Sounas, C. F. Sieck, M. R. Haberman, and A. Alù, “Sound isolation and giant linear nonreciprocity in a compact acoustic circulator,” Science 343(6170), 516–519 (2014).
[Crossref]

N. A. Estep, D. L. Sounas, J. Soric, and A. Alù, “Magnetic-free non-reciprocity and isolation based on parametrically modulated coupled-resonator loops,” Nat. Phys. 10(12), 923–927 (2014).
[Crossref]

L. D. Tzuang, K. Fang, P. Nussenzveig, S. Fan, and M. Lipson, “Non-reciprocal phase shift induced by an effective magnetic flux for light,” Nat. Photonics 8(9), 701–705 (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]

P. Stadler, W. Belzig, and G. Rastelli, “Ground-state cooling of a carbon nanomechanical resonator by spin-polarized current,” Phys. Rev. Lett. 113(4), 047201 (2014).
[Crossref]

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

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2013 (3)

S. Horsley, J.-H. Wu, M. Artoni, and G. La Rocca, “Optical nonreciprocity of cold atom bragg mirrors in motion,” Phys. Rev. Lett. 110(22), 223602 (2013).
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D.-W. Wang, H.-T. Zhou, M.-J. Guo, J.-X. Zhang, J. Evers, and S.-Y. Zhu, “Optical diode made from a moving photonic crystal,” Phys. Rev. Lett. 110(9), 093901 (2013).
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S. Shahidani, M. H. Naderi, and M. Soltanolkotabi, “Control and manipulation of electromagnetically induced transparency in a nonlinear optomechanical system with two movable mirrors,” Phys. Rev. A 88(5), 053813 (2013).
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2011 (2)

B. Chen, C. Jiang, and K.-D. Zhu, “Slow light in a cavity optomechanical system with a bose-einstein condensate,” Phys. Rev. A 83(5), 055803 (2011).
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J. Chan, T. 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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2010 (4)

G. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81(4), 041803 (2010).
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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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Y. Hadad and B. Z. Steinberg, “Magnetized spiral chains of plasmonic ellipsoids for one-way optical waveguides,” Phys. Rev. Lett. 105(23), 233904 (2010).
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2009 (3)

Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photonics 3(2), 91–94 (2009).
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Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461(7265), 772–775 (2009).
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S. Manipatruni, J. T. Robinson, and M. Lipson, “Optical nonreciprocity in optomechanical structures,” Phys. Rev. Lett. 102(21), 213903 (2009).
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2004 (1)

R. J. Potton, “Reciprocity in optics,” Rep. Prog. Phys. 67(5), 717–754 (2004).
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2003 (1)

S. Spillane, T. Kippenberg, O. Painter, and K. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91(4), 043902 (2003).
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2000 (1)

M. Cai, O. Painter, and K. J. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-gallery mode system,” Phys. Rev. Lett. 85(1), 74–77 (2000).
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1987 (1)

E. X. DeJesus and C. Kaufman, “Routh-hurwitz criterion in the examination of eigenvalues of a system of nonlinear ordinary differential equations,” Phys. Rev. A 35(12), 5288–5290 (1987).
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1985 (1)

C. Gardiner and M. Collett, “Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation,” Phys. Rev. A 31(6), 3761–3774 (1985).
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B. Abdo, N. T. Bronn, O. Jinka, S. Olivadese, A. D. Córcoles, V. P. Adiga, M. Brink, R. E. Lake, X. Wu, D. P. Pappas, and J. M. Chow, “Active protection of a superconducting qubit with an interferometric josephson isolator,” Nat. Commun. 10(1), 3154 (2019).
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G. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81(4), 041803 (2010).
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S. Huang and G. S. Agarwal, “Robust force sensing for a free particle in a dissipative optomechanical system with a parametric amplifier,” Phys. Rev. A 95(2), 023844 (2017).
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M. J. Akram, M. M. Khan, and F. Saif, “Tunable fast and slow light in a hybrid optomechanical system,” Phys. Rev. A 92(2), 023846 (2015).
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J. Chan, T. 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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F. Ruesink, M.-A. Miri, A. Alu, and E. Verhagen, “Nonreciprocity and magnetic-free isolation based on optomechanical interactions,” Nat. Commun. 7(1), 13662 (2016).
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F. Ruesink, J. P. Mathew, M.-A. Miri, A. Alù, and E. Verhagen, “Optical circulation in a multimode optomechanical resonator,” Nat. Commun. 9(1), 1798 (2018).
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R. Fleury, D. L. Sounas, C. F. Sieck, M. R. Haberman, and A. Alù, “Sound isolation and giant linear nonreciprocity in a compact acoustic circulator,” Science 343(6170), 516–519 (2014).
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N. A. Estep, D. L. Sounas, J. Soric, and A. Alù, “Magnetic-free non-reciprocity and isolation based on parametrically modulated coupled-resonator loops,” Nat. Phys. 10(12), 923–927 (2014).
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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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Artoni, M.

S. Horsley, J.-H. Wu, M. Artoni, and G. La Rocca, “Optical nonreciprocity of cold atom bragg mirrors in motion,” Phys. Rev. Lett. 110(22), 223602 (2013).
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Aspelmeyer, M.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86(4), 1391–1452 (2014).
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J. Chan, T. 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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P. Stadler, W. Belzig, and G. Rastelli, “Ground-state cooling of a carbon nanomechanical resonator by spin-polarized current,” Phys. Rev. Lett. 113(4), 047201 (2014).
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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).
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C. Benedetti, F. Galve, A. Mandarino, M. G. Paris, and R. Zambrini, “Minimal model for spontaneous quantum synchronization,” Phys. Rev. A 94(5), 052118 (2016).
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D. Malz, L. D. Tóth, N. R. Bernier, A. K. Feofanov, T. J. Kippenberg, and A. Nunnenkamp, “Quantum-limited directional amplifiers with optomechanics,” Phys. Rev. Lett. 120(2), 023601 (2018).
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B. Abdo, N. T. Bronn, O. Jinka, S. Olivadese, A. D. Córcoles, V. P. Adiga, M. Brink, R. E. Lake, X. Wu, D. P. Pappas, and J. M. Chow, “Active protection of a superconducting qubit with an interferometric josephson isolator,” Nat. Commun. 10(1), 3154 (2019).
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Bronn, N. T.

B. Abdo, N. T. Bronn, O. Jinka, S. Olivadese, A. D. Córcoles, V. P. Adiga, M. Brink, R. E. Lake, X. Wu, D. P. Pappas, and J. M. Chow, “Active protection of a superconducting qubit with an interferometric josephson isolator,” Nat. Commun. 10(1), 3154 (2019).
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A. Roulet and C. Bruder, “Quantum synchronization and entanglement generation,” Phys. Rev. Lett. 121(6), 063601 (2018).
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M. Cai, O. Painter, and K. J. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-gallery mode system,” Phys. Rev. Lett. 85(1), 74–77 (2000).
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Y. Jiang, S. Maayani, T. Carmon, F. Nori, and H. Jing, “Nonreciprocal phonon laser,” Phys. Rev. Appl. 10(6), 064037 (2018).
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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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Carson, J.

Chan, J.

J. Chan, T. 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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Chang, L.

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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Chen, B.

B. Chen, C. Jiang, and K.-D. Zhu, “Slow light in a cavity optomechanical system with a bose-einstein condensate,” Phys. Rev. A 83(5), 055803 (2011).
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Chen, Y.

Z. Shen, Y.-L. Zhang, Y. Chen, F.-W. Sun, X.-B. Zou, G.-C. Guo, C.-L. Zou, and C.-H. Dong, “Reconfigurable optomechanical circulator and directional amplifier,” Nat. Commun. 9(1), 1797 (2018).
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Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10(10), 657–661 (2016).
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Chen, Y.-T.

L. Du, Y.-T. Chen, J.-H. Wu, and Y. Li, “Nonreciprocal quantum interference and coherent photon routing in a three-port optomechanical system,” arXiv:1909.07753 (2019).

Chong, Y.

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461(7265), 772–775 (2009).
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Chow, J. M.

B. Abdo, N. T. Bronn, O. Jinka, S. Olivadese, A. D. Córcoles, V. P. Adiga, M. Brink, R. E. Lake, X. Wu, D. P. Pappas, and J. M. Chow, “Active protection of a superconducting qubit with an interferometric josephson isolator,” Nat. Commun. 10(1), 3154 (2019).
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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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Clerk, A.

H. Xu, L. Jiang, A. Clerk, and J. Harris, “Nonreciprocal control and cooling of phonon modes in an optomechanical system,” Nature 568(7750), 65–69 (2019).
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Clerk, A. A.

K. Fang, J. Luo, A. Metelmann, M. H. Matheny, F. Marquardt, A. A. Clerk, and O. Painter, “Generalized non-reciprocity in an optomechanical circuit via synthetic magnetism and reservoir engineering,” Nat. Phys. 13(5), 465–471 (2017).
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A. Metelmann and A. A. Clerk, “Nonreciprocal photon transmission and amplification via reservoir engineering,” Phys. Rev. X 5(2), 021025 (2015).
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C. Gardiner and M. Collett, “Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation,” Phys. Rev. A 31(6), 3761–3774 (1985).
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A. R. Hamann, C. Müller, M. Jerger, M. Zanner, J. Combes, M. Pletyukhov, M. Weides, T. M. Stace, and A. Fedorov, “Nonreciprocity realized with quantum nonlinearity,” Phys. Rev. Lett. 121(12), 123601 (2018).
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Córcoles, A. D.

B. Abdo, N. T. Bronn, O. Jinka, S. Olivadese, A. D. Córcoles, V. P. Adiga, M. Brink, R. E. Lake, X. Wu, D. P. Pappas, and J. M. Chow, “Active protection of a superconducting qubit with an interferometric josephson isolator,” Nat. Commun. 10(1), 3154 (2019).
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Cui, C.-L.

Dahan, R.

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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DeJesus, E. X.

E. X. DeJesus and C. Kaufman, “Routh-hurwitz criterion in the examination of eigenvalues of a system of nonlinear ordinary differential equations,” Phys. Rev. A 35(12), 5288–5290 (1987).
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Deléglise, S.

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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K. M. Sliwa, M. Hatridge, A. Narla, S. Shankar, L. Frunzio, R. J. Schoelkopf, and M. H. Devoret, “Reconfigurable josephson circulator/directional amplifier,” Phys. Rev. X 5(4), 041020 (2015).
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I. Moaddel Haghighi, N. Malossi, R. Natali, G. Di Giuseppe, and D. Vitali, “Sensitivity-bandwidth limit in a multimode optoelectromechanical transducer,” Phys. Rev. Appl. 9(3), 034031 (2018).
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Z. Shen, Y.-L. Zhang, Y. Chen, F.-W. Sun, X.-B. Zou, G.-C. Guo, C.-L. Zou, and C.-H. Dong, “Reconfigurable optomechanical circulator and directional amplifier,” Nat. Commun. 9(1), 1797 (2018).
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Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10(10), 657–661 (2016).
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L. Du, Y.-M. Liu, B. Jiang, and Y. Zhang, “All-optical photon switching, router and amplifier using a passive-active optomechanical system,” EPL 122(2), 24001 (2018).
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L. Du, C.-H. Fan, H.-X. Zhang, and J.-H. Wu, “Synchronization enhancement of indirectly coupled oscillators via periodic modulation in an optomechanical system,” Sci. Rep. 7(1), 15834 (2017).
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L. Du, Y.-T. Chen, J.-H. Wu, and Y. Li, “Nonreciprocal quantum interference and coherent photon routing in a three-port optomechanical system,” arXiv:1909.07753 (2019).

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I. Liberal and N. Engheta, “Multiqubit subradiant states in n-port waveguide devices: $\epsilon$ϵ-and-μ-near-zero hubs and nonreciprocal circulators,” Phys. Rev. A 97(2), 022309 (2018).
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N. A. Estep, D. L. Sounas, J. Soric, and A. Alù, “Magnetic-free non-reciprocity and isolation based on parametrically modulated coupled-resonator loops,” Nat. Phys. 10(12), 923–927 (2014).
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Evers, J.

D.-W. Wang, H.-T. Zhou, M.-J. Guo, J.-X. Zhang, J. Evers, and S.-Y. Zhu, “Optical diode made from a moving photonic crystal,” Phys. Rev. Lett. 110(9), 093901 (2013).
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Fan, C.-H.

L. Du, C.-H. Fan, H.-X. Zhang, and J.-H. Wu, “Synchronization enhancement of indirectly coupled oscillators via periodic modulation in an optomechanical system,” Sci. Rep. 7(1), 15834 (2017).
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Fan, S.

L. D. Tzuang, K. Fang, P. Nussenzveig, S. Fan, and M. Lipson, “Non-reciprocal phase shift induced by an effective magnetic flux for light,” Nat. Photonics 8(9), 701–705 (2014).
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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).
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Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photonics 3(2), 91–94 (2009).
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Fang, K.

K. Fang, J. Luo, A. Metelmann, M. H. Matheny, F. Marquardt, A. A. Clerk, and O. Painter, “Generalized non-reciprocity in an optomechanical circuit via synthetic magnetism and reservoir engineering,” Nat. Phys. 13(5), 465–471 (2017).
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L. D. Tzuang, K. Fang, P. Nussenzveig, S. Fan, and M. Lipson, “Non-reciprocal phase shift induced by an effective magnetic flux for light,” Nat. Photonics 8(9), 701–705 (2014).
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A. R. Hamann, C. Müller, M. Jerger, M. Zanner, J. Combes, M. Pletyukhov, M. Weides, T. M. Stace, and A. Fedorov, “Nonreciprocity realized with quantum nonlinearity,” Phys. Rev. Lett. 121(12), 123601 (2018).
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Z. Wu, R.-H. Luo, J.-Q. Zhang, Y.-H. Wang, W. Yang, and M. Feng, “Force-induced transparency and conversion between slow and fast light in optomechanics,” Phys. Rev. A 96(3), 033832 (2017).
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N. R. Bernier, L. D. Toth, A. Koottandavida, M. A. Ioannou, D. Malz, A. Nunnenkamp, A. Feofanov, and T. Kippenberg, “Nonreciprocal reconfigurable microwave optomechanical circuit,” Nat. Commun. 8(1), 604 (2017).
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Feofanov, A. K.

D. Malz, L. D. Tóth, N. R. Bernier, A. K. Feofanov, T. J. Kippenberg, and A. Nunnenkamp, “Quantum-limited directional amplifiers with optomechanics,” Phys. Rev. Lett. 120(2), 023601 (2018).
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R. Fleury, D. L. Sounas, C. F. Sieck, M. R. Haberman, and A. Alù, “Sound isolation and giant linear nonreciprocity in a compact acoustic circulator,” Science 343(6170), 516–519 (2014).
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Frunzio, L.

K. M. Sliwa, M. Hatridge, A. Narla, S. Shankar, L. Frunzio, R. J. Schoelkopf, and M. H. Devoret, “Reconfigurable josephson circulator/directional amplifier,” Phys. Rev. X 5(4), 041020 (2015).
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Fu, C.-B.

Galve, F.

C. Benedetti, F. Galve, A. Mandarino, M. G. Paris, and R. Zambrini, “Minimal model for spontaneous quantum synchronization,” Phys. Rev. A 94(5), 052118 (2016).
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Gardiner, C.

C. Gardiner and M. Collett, “Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation,” Phys. Rev. A 31(6), 3761–3774 (1985).
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Gavartin, E.

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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Ge, W.

Gianfreda, M.

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).
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Gröblacher, S.

J. Chan, T. 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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Gu, K.-H.

Guo, G.-C.

Z. Shen, Y.-L. Zhang, Y. Chen, F.-W. Sun, X.-B. Zou, G.-C. Guo, C.-L. Zou, and C.-H. Dong, “Reconfigurable optomechanical circulator and directional amplifier,” Nat. Commun. 9(1), 1797 (2018).
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Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10(10), 657–661 (2016).
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Guo, M.-J.

D.-W. Wang, H.-T. Zhou, M.-J. Guo, J.-X. Zhang, J. Evers, and S.-Y. Zhu, “Optical diode made from a moving photonic crystal,” Phys. Rev. Lett. 110(9), 093901 (2013).
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X. Guo, C.-L. Zou, H. Jung, and H. X. Tang, “On-chip strong coupling and efficient frequency conversion between telecom and visible optical modes,” Phys. Rev. Lett. 117(12), 123902 (2016).
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R. Fleury, D. L. Sounas, C. F. Sieck, M. R. Haberman, and A. Alù, “Sound isolation and giant linear nonreciprocity in a compact acoustic circulator,” Science 343(6170), 516–519 (2014).
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Hadad, Y.

Y. Hadad and B. Z. Steinberg, “Magnetized spiral chains of plasmonic ellipsoids for one-way optical waveguides,” Phys. Rev. Lett. 105(23), 233904 (2010).
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Hamann, A. R.

A. R. Hamann, C. Müller, M. Jerger, M. Zanner, J. Combes, M. Pletyukhov, M. Weides, T. M. Stace, and A. Fedorov, “Nonreciprocity realized with quantum nonlinearity,” Phys. Rev. Lett. 121(12), 123601 (2018).
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Harris, J.

H. Xu, L. Jiang, A. Clerk, and J. Harris, “Nonreciprocal control and cooling of phonon modes in an optomechanical system,” Nature 568(7750), 65–69 (2019).
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H. Xu, D. Mason, L. Jiang, and J. G. E. Harris, “Topological dynamics in an optomechanical system with highly non-degenerate modes,” arXiv:1703.07374 (2017).

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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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Hatridge, M.

K. M. Sliwa, M. Hatridge, A. Narla, S. Shankar, L. Frunzio, R. J. Schoelkopf, and M. H. Devoret, “Reconfigurable josephson circulator/directional amplifier,” Phys. Rev. X 5(4), 041020 (2015).
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B. He, L. Yang, X. Jiang, and M. Xiao, “Transmission nonreciprocity in a mutually coupled circulating structure,” Phys. Rev. Lett. 120(20), 203904 (2018).
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Hill, J. T.

J. Chan, T. 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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Figures (6)

Fig. 1.
Fig. 1. (a) Schematic diagram of an optomechanical system composed of three indirectly coupled cavities and a mechanical resonator. (b) Schematic illustration of this system composed of three Fabry-Pérot-type resonators with a common movable mirror driven by a mechanical field. (c) Transition relation diagram of internal states in this multi-mode system where $n_{ak}$ and $n_{b}$ are the excitation numbers of the three cavity modes and the mechanical mode, respectively.
Fig. 2.
Fig. 2. (a) Stability diagram of the system on the plane of coupling strengths $G_{1}$ and $G_{3}$ for $\phi =0$ (The diagram is the same as those with $\phi$ of any value.). (b) Stability diagram of the system on the plane of phase difference $\phi$ and the coupling strength $G_{1}$ with $n=2$ . The other parameter is $\gamma _{m}=\kappa /100$ . The yellow regions are stable and the green regions are unstable.
Fig. 3.
Fig. 3. Logarithms of transmissivities (a) $T_{12}$ and (b) $T_{21}$ versus detuning $\delta$ and coupling strength $G_{3}$ for cavity $a_{3}$ on blue sideband with $y=2$ . Logarithms of the transmissivities $T_{12}$ and $T_{21}$ with $G_{3}=2.8\kappa$ and $y=2$ (c) for $a_{3}$ on blue sideband; (d) for $a_{3}$ on red sideband. The transmissivity $T_{12}$ versus $\delta$ with (e) $G_{3}=0$ and (f) $G_3=2.8\kappa$ for cavity $a_{3}$ on blue sideband. Other parameters are $\phi =\pi$ , $\beta =\pi /2$ , $\gamma _{m}=\kappa /100$ and $G=2\kappa$ .
Fig. 4.
Fig. 4. (a) Logarithms of the transmissivities $T_{21}$ and $T_{12}$ and (b) isolation ratio $I$ versus relative ratio $y$ with $\delta =0$ . Other parameters are the same as those in Fig. 3.
Fig. 5.
Fig. 5. Logarithms of transmissivities $T_{21}$ and $T_{12}$ verse $\delta$ for $\phi =\pi /2$ and (a) $\beta =0$ ; (b) $\beta =\pi /2$ . Logarithms of transmissivities $T_{21}$ and $T_{12}$ verse $\beta$ for $\delta =0$ and (c) $\phi =\pi /2$ ; (d) $\phi =\pi$ . Logarithms of transmissivities $T_{21}$ and $T_{12}$ verse $\phi$ for $\delta =0$ and (e) $\beta =3\pi /2$ ; (f) $\beta =\pi$ . (g) Isolation ratio $I$ verse $\beta$ for $\phi =\pi /2$ and $\delta =0$ . (h) Transmissivities $T_{21}$ and $T_{12}$ verse $\phi$ for $\beta =\pi /2$ . Other parameters are the same as those in Fig. 3.
Fig. 6.
Fig. 6. (a) Group delay $\tau _{12}$ versus coupling strength $G_{3}$ for $\phi =\pi /2$ , $\beta =\pi /2$ and $\delta =0$ . (b) Group delays $\tau _{12}$ and $\tau _{21}$ versus detuning $\delta$ for $\phi =\pi$ and $\beta =0$ . $\tau _{12(21)}=0$ is the transition point of slow and fast lights. Other parameters are the same as those in Fig. 3.

Equations (19)

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H = H 0 + H I + H d .
H 0 = k = 1 3 ω k a k a k + ω m b b ;
H I = g 1 a 1 a 1 ( b + b ) + g 2 a 2 a 2 ( b + b ) + g 3 a 3 a 3 ( b + b ) ;
H d = k = 1 3 ε c k ( i a k e i ω c k t e i θ k + H . c . ) + ε p ( i a 1 e i ω p t + H . c . ) + ε b ( i b e i ω b t + H . c . ) ,
a 1 ˙ = ( κ + i Δ 1 ) a 1 + i g 1 a 1 ( b + b ) + ε c 1 e i θ 1 + ε p e i σ t + 2 κ a 1 i n , a 2 ˙ = ( κ + i Δ 2 ) a 2 i g 2 a 2 ( b + b ) + ε c 2 e i θ 2 + 2 κ a 2 i n , a 3 ˙ = ( κ + i Δ 3 ) a 3 i g 3 a 3 ( b + b ) + ε c 3 e i θ 3 + 2 κ a 3 i n , b ˙ = ( γ m + i ω m ) b + i ( g 1 a 1 a 1 g 2 a 2 a 2 g 3 a 3 a 3 ) + ε b e i ω b t + 2 γ m b i n ,
a 1 = ε c 1 e i θ 1 κ + i Δ 1 , a 2 = ε c 2 e i θ 2 κ + i Δ 2 , a 3 = ε c 3 e i θ 3 κ + i Δ 3 , b = i ( g 1 | α 1 | 2 g 2 | α 2 | 2 g 3 | α 3 | 2 ) γ m + i ω m ,
δ a 1 ˙ = ( κ + i Δ 1 ) δ a 1 + i G 1 ( δ b + δ b ) + ε p e i σ t + 2 κ a 1 i n , δ a 2 ˙ = ( κ + i Δ 2 ) δ a 2 i G 2 ( δ b + δ b ) + 2 κ a 2 i n , δ a 3 ˙ = ( κ + i Δ 3 ) δ a 3 i G 3 ( δ b + δ b ) + 2 κ a 3 i n , δ b ˙ = ( γ m + i ω m ) δ b + i ( G 1 δ a 1 + G 1 δ a 1 G 2 δ a 2 G 2 δ a 2 G 3 δ a 3 G 3 δ a 3 ) + ε b e i ω b t + 2 γ m b i n ,
δ a 1 ˙ = κ δ a 1 + i G 1 δ b + ε p e i δ t + 2 κ a 1 i n , δ a 2 ˙ = κ δ a 2 i G 2 δ b + 2 κ a 2 i n , δ a 3 ˙ = κ δ a 3 + i G 3 δ b + 2 κ a 3 i n , δ b ˙ = γ m δ b + i G 1 δ a 1 i G 2 δ a 2 i G 3 δ a 3 + ε b e i δ t + 2 γ m b i n ,
δ a 1 ˙ = ( κ i δ ) δ a 1 + i G 1 δ b + ε p , δ a 2 ˙ = ( κ i δ ) δ a 2 i G 2 δ b , δ a 3 ˙ = ( κ i δ ) δ a 3 + i G 3 δ b , δ b ˙ = ( γ m i δ ) δ b + i G 1 δ a 1 i G 2 δ a 2 i G 3 δ a 3 + ε b .
δ a 1 = | G | 2 ε p + i G ε b ( κ i δ ) N + ε p κ i δ , δ a 2 = G 2 e i ϕ ε p i G e i ϕ ε b ( κ i δ ) N
t 21 = 2 κ [ G 2 e i ϕ i G y e i ( ϕ + β ) ( κ i δ ) ] N
t 12 = 2 κ [ G 2 e i ϕ i G y e i β ( κ i δ ) ] N .
τ ˙ = M τ + ν + μ ,
M = ( κ 0 0 i G 1 0 κ 0 i G 2 0 0 κ i G 3 i G 1 i G 2 i G 3 γ m   ) .
( γ m i δ + | G 1 | 2 + | G 2 | 2 | G 3 | 2 κ i δ ) δ b = ( i G 1 κ i δ + y e i β ) ε p ,
Γ b = γ m + κ ( 2 | G | 2 | G 3 | 2 ) κ 2 + δ 2 ,
Γ r = γ m + κ ( 2 | G | 2 + | G 3 | 2 ) κ 2 + δ 2 .
I = T 12 T 21 T 12 + T 21
τ 12 ( 21 ) = d Θ 12 ( 21 ) d ω p ,

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