Abstract

In this work, we propose a scheme in three-mode optical systems to simulate a strongly coupled optomechanical system. The nonreciprocity observed in such a three-mode optomechanical circulatory system (OMCS) is explored. To be specific, we first derive a quantum Langevin equation (QLE) for the strongly coupled OMCS by suitably choosing the laser field, then we give a condition for the frequency of the laser and the mechanical decay rate, beyond which the optomechanical system has a unidirectional transmission regardless of how strong the optomechanical coupling is. The optomechanically induced transparency is also studied. The present results can be extended to a more general two-dimensional optomechanical system and a planar quantum network, and the prediction is possible to be observed in an optomechanical crystal or integrated quantum superconducting circuit. This scheme paves a way for the construction of various quantum devices that are necessary for quantum information processing.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2019 (2)

L. Tang, J. S. Tang, W. D. Zhang, G. W. Lu, H. Zhang, Y. Zhang, K. Y. Xia, and M. Xiao, “On-chip chiral single photon interface: Isolation and unidirectional emission,” Phys. Rev. A 99, 043833 (2019).
[Crossref]

B. J. Li, R. Huang, X. W. Xu, A. Miranowicz, and H. Jing, “Nonreciprocal unconventional photon blockade in a spinning optomechanical system,” Photon. Res. 7, 630 (2019).
[Crossref]

2018 (13)

L. D. Bino, J. M. Silver, M. T. M. Woodley, S. L. Stebbings, X. Zhao, and P. Del’Haye, Microresonator isolators and circulators based on the intrinsic nonreciprocity of the Kerr effect, Optica 5, 279 (2018).
[Crossref]

H. Z. Shen, S. Xu, Y. H. Zhou, G. C. Wang, and X. X. Yi, “Unconventional photon blockade from bimodal driving and dissipations in coupled semiconductor microcavities,” J. Phys. B 51, 035503 (2018).
[Crossref]

H. Z. Shen, C. Shang, Y. H. Zhou, and X. X. Yi, “Unconventional single-photon blockade in non-Markovian systems,” Phys. Rev. A 98, 023856 (2018).
[Crossref]

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

X. W Xu, L. N. Song, Qiang Zheng, Z. H. Wang, and Yong Li, “Optomechanically induced nonreciprocity in a three-mode optomechanical system,” Phys. Rev. A 98, 063845 (2018).
[Crossref]

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

R. Huang, A. Miranowicz, J. Q. Liao, F. Nori, and H. Jing, “Nonreciprocal Photon Blockade,” Phys. Rev. Lett. 121, 153601 (2018).
[Crossref] [PubMed]

S. Zippilli, N. Kralj, M. Rossi, G. D. Giuseppe, and D. Vitali, “Cavity optomechanics with feedback-controlled in-loop light,” Phys. Rev. A 98, 023828 (2018).
[Crossref]

Y. H. Zhou, H. Z. Shen, X. Y. Zhang, and X. X. Yi,“ Zero eigenvalues of a photon blockade induced by a non-Hermitian Hamiltonian with a gain cavity,” Phys. Rev. A 97, 043819 (2018).
[Crossref]

I. Cohen and K. Mølmer, “Deterministic quantum network for distributed entanglement and quantum computation,” Phys. Rev. A 98, 030302 (R) (2018).
[Crossref]

Z. C. Shi, D. Ran, L. T. Shen, Y. Xia, and X. X. Yi, “Quantum state engineering by periodical two-step modulation in an atomic system,” Optics Express 26(26), 34789 (2018).
[Crossref]

G. L. Li, X. Xiao, Y. Li, and X. G. Wang, “Tunable optical nonreciprocity and a phonon-photon router in an optomechanical system with coupled mechanical and optical modes,” Phys. Rev. A 97, 023801 (2018).
[Crossref]

B. He, L. Yang, X. S. Jiang, and M. Xiao, “Transmission Nonreciprocity in a Mutually Coupled Circulating Structure,” Phys. Rev. Lett. 120, 203904 (2018).
[Crossref] [PubMed]

2017 (4)

Q. T. Cao, H. Wang, C. H. Dong, H. Jing, R. S. Liu, X. Chen, L. Ge, Q. Gong, and Y. F. Xiao, “Experimental Demonstration of Spontaneous Chirality in a Nonlinear Microresonator,” Phys. Rev. Lett. 118, 033901 (2017)
[Crossref] [PubMed]

X. W. Xu, A. X. Chen, Y. Li, and Y. X Liu, “Single-photon nonreciprocal transport in one-dimensional coupled-resonator waveguides,” Phys. Rev. A 95, 063808 (2017).
[Crossref]

L. Tian and Z. Li, “Nonreciprocal quantum-state conversion between microwave and optical photons,” Phys. Rev. A 96, 013808 (2017).
[Crossref]

Y. H. Zhou, S. S. Zhang, H. Z. Shen, and X. X. Yi, “Second-order nonlinearity induced transparency,” Opt. Lett. 42(7), 1289 (2017).
[Crossref] [PubMed]

2016 (8)

Y. H. Zhou, H. Z. Shen, X. Q. Shao, and X. X. Yi, “Strong photon antibunching with weak second-order nonlinearity under dissipation and coherent driving,” Opt. Express 24, 17332 (2016).
[Crossref]

W. Xiong, D. Y. Jin, Y. Y. Qiu, C. H. Lam, and J. Q. You, “Cross-Kerr effect on an optomechanical system,” Phys. Rev. A 93, 023844 (2016).
[Crossref]

Y. Jiang, S. Maayani, T. Carmon, F. Nori, and H. Jing, “Nonreciprocal Photon Laser,” Phys. Rev. Appl. 6, 040001 (2016).

M. Scheucher, A. Hilico, E. Will, J. Volz, and A. Rauschenbeutel, “Quantum optical circulator controlled by a single chirally coupled atom,” Science 354, 1577 (2016).
[Crossref] [PubMed]

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, 657 (2016).
[Crossref]

M. Scheucher, A. Hilico, E. Will, J. Volz, and A. Rauschenbeutel, “Quantum optical circulator controlled by a single chirally coupled atom,” Science 354, 1577–1578 (2016).
[Crossref] [PubMed]

I. A. Walmsley and J. Nunn, “Editorial: Building Quantum Networks,” Phys. Rev. Appl. 6, 040001 (2016).
[Crossref]

Q. C. Wu, Y. H. Chen, B. H. Huang, J. Song, Y. Xia, and S. B. Zheng, “Improving the stimulated Raman adiabatic passage via dissipative quantum dynamics,” Optics Express 20(7), 22847 (2016).
[Crossref]

2015 (11)

Y. H. Chen, Y. Xia, Q. Q. Chen, and J. Song, “Fast and noise-resistant implementation of quantum phase gates and creation of quantum entangled states,” Phys. Rev. A 91, 012325 (2015).
[Crossref]

Z. Wang, L. Shi, Y. Liu, X. Xu, and X. Zhang, “Optical Nonreciprocity in Asymmetric Optomechanical Couplers,” Sci. Rep. 5, 8657 (2015).
[Crossref] [PubMed]

Z. Q. Yin, W. L. Yang, L. Sun, and L. M. Duan, “Quantum network of superconducting qubits through an optomechanical interface,” Phys. Rev. A 91, 012333 (2015).
[Crossref]

H. Flayac, D. Gerace, and V. Savona, “An all-silicon single photon source by unconventional photon blockade,” Sci. Rep. 5, 11223 (2015).
[Crossref]

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

T. K. Paraïso, M. Kalaee, L. Zang, H. Pfeifer, F. Marquardt, and O. Painter, “Position-Squared Coupling in a Tunable Photonic Crystal Optomechanical Cavity,” Phys. Rev. X 5, 041024 (2015).

J. M. Pirkkalainen, S. U. Cho, F. Massel, J. Tuorila, T. T. Heikkilä, P. J. Hakonen, and M. A. Sillanpää, “Cavity optomechanics mediated by a quantum two-level system,” Nat. Commun. 6, 6981 (2015).
[Crossref] [PubMed]

R. Khan, F. Massel, and T. T. Heikkilä, “Cross-Kerr nonlinearity in optomechanical systems,” Phys. Rev. A 91, 043822 (2015).
[Crossref]

H. Z. Shen, Y. H. Zhou, H. D. Liu, G. C. Wang, and X. X. Yi, “Exact optimal control of photon blockade with weakly nonlinear coupled cavities,” Opt. Express 23, 32835 (2015).
[Crossref] [PubMed]

X. W. Xu and Y. Li, “Optical nonreciprocity and optomechanical circulator in three-mode optomechanical systems,” Phys. Rev. A 91, 053854 (2015).
[Crossref]

Y. H. Zhou, H. Z. Shen, and X. X. Yi, “Unconventional photon blockade with second-order nonlinearity,” Phys. Rev. A 92, 023838 (2015).
[Crossref]

2014 (4)

H. Z. Shen, Y. H. Zhou, and X. X. Yi, “Quantum optical diode with semiconductor microcavities,” Phys. Rev. A 90, 023849 (2014).
[Crossref]

J. Gough, “Feedback network models for quantum transport,” Phys. Rev. E 90, 062109 (2014).
[Crossref]

Y. H. Chen, Y. Xia, Q. Q. Chen, and J. Song, “Effcient shortcuts to adiabatic passage for fast population transfer in multiparticle systems,” Phys. Rev. A 89, 033856 (2014).
[Crossref]

T. T. Heikkilä, F. Massel, J. Tuorila, R. Khan, and M. A. Sillanpää, “Enhancing Optomechanical Coupling via the Josephson Effect,” Phys. Rev. Lett. 112, 203603 (2014).
[Crossref]

2013 (7)

J. Q. Liao and C. K. Law, “Correlated two-photon scattering in cavity optomechanics,” Phys. Rev. A 87, 043809 (2013).
[Crossref]

T. Hong, H. Yang, H. X. Miao, and Y. Chen, “Open quantum dynamics of single-photon optomechanical devices,” Phys. Rev. A 88, 023812 (2013).
[Crossref]

M. Ludwig and F. Marquardt, “Quantum Many-Body Dynamics in Optomechanical Arrays,” Phys. Rev. Lett. 111, 073603 (2013).
[Crossref] [PubMed]

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, 053813 (2013).
[Crossref]

M. A. Lemonde, N. Didier, and A. A. Clerk, “Nonlinear Interaction Effects in a Strongly Driven Optomechanical Cavity,” Phys. Rev. Lett. 111, 053602 (2013).
[Crossref] [PubMed]

S. Aldana, C. Bruder, and A. Nunnenkamp, “Equivalence between an optomechanical system and a Kerr medium,” Phys. Rev. A 88, 043826 (2013).
[Crossref]

X. W. Xu, Y. J. Li, and Y. X. Liu, “Photon-induced tunneling in optomechanical systems,” Phys. Rev. A 87, 025803 (2013).
[Crossref]

2012 (4)

G. S. Agarwal and S. Huang, “Optomechanical systems as single-photon routers,” Phys. Rev. A 85, 021801 (R) (2012).
[Crossref]

M. Hafezi and P. Rabl, “Optomechanically induced nonreciprocity in microring resonators,” Opt. Express 20(7), 7672–7684 (2012).
[Crossref] [PubMed]

J. Q. Liao, H. K. Cheung, and C. K. Law, “Spectrum of single-photon emission and scattering in cavity optomechanics,” Phys. Rev. A 85, 025803 (2012).
[Crossref]

B. He, “Quantum optomechanics beyond linearization,” Phys. Rev. A 85, 063820 (2012).
[Crossref]

2011 (3)

M. Bamba, A. Imamođlu, I. Carusotto, and C. Ciuti, “Origin of strong photon antibunching in weakly nonlinear photonic molecules,” Phys. Rev. A 83, 021802(R) (2011).
[Crossref]

A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Single-Photon Optomechanics,” Phys. Rev. Lett. 107, 063602 (2011).
[Crossref] [PubMed]

P. Rabl, “Photon Blockade Effect in Optomechanical Systems,” Phys. Rev. Lett. 107, 063601 (2011).
[Crossref] [PubMed]

2010 (1)

S. Weis, R. Rivière, S. Deléglise, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010).
[Crossref] [PubMed]

2009 (2)

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462(7269), 78–82 (2009).
[Crossref] [PubMed]

S. Manipatruni, J. T. Robinson, and M. Lipson, “Optical Nonreciprocity in Optomechanical Structures,” Phys. Rev. Lett. 102, 213903 (2009).
[Crossref] [PubMed]

2008 (6)

T. J. Kippenberg and K. J. Vahala, “Cavity Optomechanics: Back-Action at the Mesoscale,” Science 321, 1172–1176 (2008).
[Crossref] [PubMed]

H. J. Kimble, “The quantum internet,” Nature 453(7198), 1023–1030 (2008).
[Crossref] [PubMed]

K. W. Murch, K. L. Moore, S. Gupta, and D. M. Stamper-Kurn, “Observation of quantum-measurement backaction with an ultracold atomic gas,” Nat. Phys. 4(7), 561–564 (2008).
[Crossref]

F. Brennecke, S. Ritter, T. Donner, and T. Esslinger, “Cavity optomechanics with a Bose-Einstein condensate,” Science 322(5899), 235–238 (2008).
[Crossref] [PubMed]

J. K. Asbóth, H. Ritsch, and P. Domokos, “Optomechanical coupling in a one-dimensional optical lattice,” Phys. Rev. A 77, 063424 (2008).
[Crossref]

T. R. Zaman, X. Y. Guo, and R. J. Ram, “Semiconductor Waveguide Isolators,” J. Lightwave Technol.,” 26(2), 291–301 (2008).
[Crossref]

2006 (1)

K. Koshini, “Semiclassical evaluation of two-photon cross-Kerr effect,” Phys. Rev. A 74, 053818 (2006).
[Crossref]

2000 (1)

M. Cai, O. J. 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, 74 (2000).
[Crossref] [PubMed]

1997 (1)

A. Imamođlu, H. Schmidt, G. Woods, and M. Deutsch, “Strongly Interacting Photons in a Nonlinear Cavity,” Phys. Rev. Lett. 79, 1467 (1997).
[Crossref]

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, 5288 (1987).
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1985 (1)

C. W. Gardiner and M. J. Collett, “Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation,” Phys. Rev. A 31, 3761 (1985).
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G. S. Agarwal and S. Huang, “Optomechanical systems as single-photon routers,” Phys. Rev. A 85, 021801 (R) (2012).
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S. Aldana, C. Bruder, and A. Nunnenkamp, “Equivalence between an optomechanical system and a Kerr medium,” Phys. Rev. A 88, 043826 (2013).
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J. K. Asbóth, H. Ritsch, and P. Domokos, “Optomechanical coupling in a one-dimensional optical lattice,” Phys. Rev. A 77, 063424 (2008).
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M. Bamba, A. Imamođlu, I. Carusotto, and C. Ciuti, “Origin of strong photon antibunching in weakly nonlinear photonic molecules,” Phys. Rev. A 83, 021802(R) (2011).
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A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Single-Photon Optomechanics,” Phys. Rev. Lett. 107, 063602 (2011).
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W. P. Bowen and G. J. Milburn, Quantum Optomechanics (CRC, USA, 2015).
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F. Brennecke, S. Ritter, T. Donner, and T. Esslinger, “Cavity optomechanics with a Bose-Einstein condensate,” Science 322(5899), 235–238 (2008).
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Bruder, C.

S. Aldana, C. Bruder, and A. Nunnenkamp, “Equivalence between an optomechanical system and a Kerr medium,” Phys. Rev. A 88, 043826 (2013).
[Crossref]

Cai, M.

M. Cai, O. J. 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, 74 (2000).
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Camacho, R. M.

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462(7269), 78–82 (2009).
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Cao, Q. T.

Q. T. Cao, H. Wang, C. H. Dong, H. Jing, R. S. Liu, X. Chen, L. Ge, Q. Gong, and Y. F. Xiao, “Experimental Demonstration of Spontaneous Chirality in a Nonlinear Microresonator,” Phys. Rev. Lett. 118, 033901 (2017)
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Carmon, T.

Y. Jiang, S. Maayani, T. Carmon, F. Nori, and H. Jing, “Nonreciprocal Photon Laser,” Phys. Rev. Appl. 6, 040001 (2016).

Carusotto, I.

M. Bamba, A. Imamođlu, I. Carusotto, and C. Ciuti, “Origin of strong photon antibunching in weakly nonlinear photonic molecules,” Phys. Rev. A 83, 021802(R) (2011).
[Crossref]

Chan, J.

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462(7269), 78–82 (2009).
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X. W. Xu, A. X. Chen, Y. Li, and Y. X Liu, “Single-photon nonreciprocal transport in one-dimensional coupled-resonator waveguides,” Phys. Rev. A 95, 063808 (2017).
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J. S. Zhang and A. X. Chen, “Enhancing quadratic optomechanical coupling via nonlinear medium and lasers,” arXiv: 1810.13052 (2018).

X. W. Xu, Y. J. Zhao, H. Wang, H. Jing, and A. X. Chen, “Nonreciprocal photon blockade via quadratic optomechanical coupling,” arXiv:1809.07596 (2018).

Chen, Q. Q.

Y. H. Chen, Y. Xia, Q. Q. Chen, and J. Song, “Fast and noise-resistant implementation of quantum phase gates and creation of quantum entangled states,” Phys. Rev. A 91, 012325 (2015).
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Y. H. Chen, Y. Xia, Q. Q. Chen, and J. Song, “Effcient shortcuts to adiabatic passage for fast population transfer in multiparticle systems,” Phys. Rev. A 89, 033856 (2014).
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Chen, X.

Q. T. Cao, H. Wang, C. H. Dong, H. Jing, R. S. Liu, X. Chen, L. Ge, Q. Gong, and Y. F. Xiao, “Experimental Demonstration of Spontaneous Chirality in a Nonlinear Microresonator,” Phys. Rev. Lett. 118, 033901 (2017)
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Chen, Y.

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, 657 (2016).
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T. Hong, H. Yang, H. X. Miao, and Y. Chen, “Open quantum dynamics of single-photon optomechanical devices,” Phys. Rev. A 88, 023812 (2013).
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Chen, Y. H.

Q. C. Wu, Y. H. Chen, B. H. Huang, J. Song, Y. Xia, and S. B. Zheng, “Improving the stimulated Raman adiabatic passage via dissipative quantum dynamics,” Optics Express 20(7), 22847 (2016).
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Y. H. Chen, Y. Xia, Q. Q. Chen, and J. Song, “Fast and noise-resistant implementation of quantum phase gates and creation of quantum entangled states,” Phys. Rev. A 91, 012325 (2015).
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Y. H. Chen, Y. Xia, Q. Q. Chen, and J. Song, “Effcient shortcuts to adiabatic passage for fast population transfer in multiparticle systems,” Phys. Rev. A 89, 033856 (2014).
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J. Q. Liao, H. K. Cheung, and C. K. Law, “Spectrum of single-photon emission and scattering in cavity optomechanics,” Phys. Rev. A 85, 025803 (2012).
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Cho, S. U.

J. M. Pirkkalainen, S. U. Cho, F. Massel, J. Tuorila, T. T. Heikkilä, P. J. Hakonen, and M. A. Sillanpää, “Cavity optomechanics mediated by a quantum two-level system,” Nat. Commun. 6, 6981 (2015).
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Ciuti, C.

M. Bamba, A. Imamođlu, I. Carusotto, and C. Ciuti, “Origin of strong photon antibunching in weakly nonlinear photonic molecules,” Phys. Rev. A 83, 021802(R) (2011).
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Clerk, A. A.

M. A. Lemonde, N. Didier, and A. A. Clerk, “Nonlinear Interaction Effects in a Strongly Driven Optomechanical Cavity,” Phys. Rev. Lett. 111, 053602 (2013).
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I. Cohen and K. Mølmer, “Deterministic quantum network for distributed entanglement and quantum computation,” Phys. Rev. A 98, 030302 (R) (2018).
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C. W. Gardiner and M. J. Collett, “Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation,” Phys. Rev. A 31, 3761 (1985).
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S. Maayani, R. Dahan, Y. Kligerman, E. Moses, A. U. Hassan, H. Jing, and F. Nori, “Flying couplers above spinning resonators generate irreversible refraction,” Nature (London) 558, 569 (2018).
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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, 5288 (1987).
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Del’Haye, P.

Deléglise, S.

S. Weis, R. Rivière, S. Deléglise, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010).
[Crossref] [PubMed]

Deutsch, M.

A. Imamođlu, H. Schmidt, G. Woods, and M. Deutsch, “Strongly Interacting Photons in a Nonlinear Cavity,” Phys. Rev. Lett. 79, 1467 (1997).
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Didier, N.

M. A. Lemonde, N. Didier, and A. A. Clerk, “Nonlinear Interaction Effects in a Strongly Driven Optomechanical Cavity,” Phys. Rev. Lett. 111, 053602 (2013).
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Domokos, P.

J. K. Asbóth, H. Ritsch, and P. Domokos, “Optomechanical coupling in a one-dimensional optical lattice,” Phys. Rev. A 77, 063424 (2008).
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Dong, C. H.

Q. T. Cao, H. Wang, C. H. Dong, H. Jing, R. S. Liu, X. Chen, L. Ge, Q. Gong, and Y. F. Xiao, “Experimental Demonstration of Spontaneous Chirality in a Nonlinear Microresonator,” Phys. Rev. Lett. 118, 033901 (2017)
[Crossref] [PubMed]

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, 657 (2016).
[Crossref]

Donner, T.

F. Brennecke, S. Ritter, T. Donner, and T. Esslinger, “Cavity optomechanics with a Bose-Einstein condensate,” Science 322(5899), 235–238 (2008).
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Duan, L. M.

Z. Q. Yin, W. L. Yang, L. Sun, and L. M. Duan, “Quantum network of superconducting qubits through an optomechanical interface,” Phys. Rev. A 91, 012333 (2015).
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Eichenfield, M.

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462(7269), 78–82 (2009).
[Crossref] [PubMed]

Esslinger, T.

F. Brennecke, S. Ritter, T. Donner, and T. Esslinger, “Cavity optomechanics with a Bose-Einstein condensate,” Science 322(5899), 235–238 (2008).
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H. Flayac, D. Gerace, and V. Savona, “An all-silicon single photon source by unconventional photon blockade,” Sci. Rep. 5, 11223 (2015).
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Gardiner, C. W.

C. W. Gardiner and M. J. Collett, “Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation,” Phys. Rev. A 31, 3761 (1985).
[Crossref]

Ge, L.

Q. T. Cao, H. Wang, C. H. Dong, H. Jing, R. S. Liu, X. Chen, L. Ge, Q. Gong, and Y. F. Xiao, “Experimental Demonstration of Spontaneous Chirality in a Nonlinear Microresonator,” Phys. Rev. Lett. 118, 033901 (2017)
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Gerace, D.

H. Flayac, D. Gerace, and V. Savona, “An all-silicon single photon source by unconventional photon blockade,” Sci. Rep. 5, 11223 (2015).
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Girvin, S. M.

A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Single-Photon Optomechanics,” Phys. Rev. Lett. 107, 063602 (2011).
[Crossref] [PubMed]

Giuseppe, G. D.

S. Zippilli, N. Kralj, M. Rossi, G. D. Giuseppe, and D. Vitali, “Cavity optomechanics with feedback-controlled in-loop light,” Phys. Rev. A 98, 023828 (2018).
[Crossref]

Gong, Q.

Q. T. Cao, H. Wang, C. H. Dong, H. Jing, R. S. Liu, X. Chen, L. Ge, Q. Gong, and Y. F. Xiao, “Experimental Demonstration of Spontaneous Chirality in a Nonlinear Microresonator,” Phys. Rev. Lett. 118, 033901 (2017)
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J. Gough, “Feedback network models for quantum transport,” Phys. Rev. E 90, 062109 (2014).
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Guo, G. C.

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, 657 (2016).
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Guo, X. Y.

Gupta, S.

K. W. Murch, K. L. Moore, S. Gupta, and D. M. Stamper-Kurn, “Observation of quantum-measurement backaction with an ultracold atomic gas,” Nat. Phys. 4(7), 561–564 (2008).
[Crossref]

Hafezi, M.

Hakonen, P. J.

J. M. Pirkkalainen, S. U. Cho, F. Massel, J. Tuorila, T. T. Heikkilä, P. J. Hakonen, and M. A. Sillanpää, “Cavity optomechanics mediated by a quantum two-level system,” Nat. Commun. 6, 6981 (2015).
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Hassan, A. U.

S. Maayani, R. Dahan, Y. Kligerman, E. Moses, A. U. Hassan, H. Jing, and F. Nori, “Flying couplers above spinning resonators generate irreversible refraction,” Nature (London) 558, 569 (2018).
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He, B.

B. He, L. Yang, X. S. Jiang, and M. Xiao, “Transmission Nonreciprocity in a Mutually Coupled Circulating Structure,” Phys. Rev. Lett. 120, 203904 (2018).
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B. He, “Quantum optomechanics beyond linearization,” Phys. Rev. A 85, 063820 (2012).
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Heikkilä, T. T.

J. M. Pirkkalainen, S. U. Cho, F. Massel, J. Tuorila, T. T. Heikkilä, P. J. Hakonen, and M. A. Sillanpää, “Cavity optomechanics mediated by a quantum two-level system,” Nat. Commun. 6, 6981 (2015).
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R. Khan, F. Massel, and T. T. Heikkilä, “Cross-Kerr nonlinearity in optomechanical systems,” Phys. Rev. A 91, 043822 (2015).
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T. T. Heikkilä, F. Massel, J. Tuorila, R. Khan, and M. A. Sillanpää, “Enhancing Optomechanical Coupling via the Josephson Effect,” Phys. Rev. Lett. 112, 203603 (2014).
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Hilico, A.

M. Scheucher, A. Hilico, E. Will, J. Volz, and A. Rauschenbeutel, “Quantum optical circulator controlled by a single chirally coupled atom,” Science 354, 1577–1578 (2016).
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M. Scheucher, A. Hilico, E. Will, J. Volz, and A. Rauschenbeutel, “Quantum optical circulator controlled by a single chirally coupled atom,” Science 354, 1577 (2016).
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Hong, T.

T. Hong, H. Yang, H. X. Miao, and Y. Chen, “Open quantum dynamics of single-photon optomechanical devices,” Phys. Rev. A 88, 023812 (2013).
[Crossref]

Huang, B. H.

Q. C. Wu, Y. H. Chen, B. H. Huang, J. Song, Y. Xia, and S. B. Zheng, “Improving the stimulated Raman adiabatic passage via dissipative quantum dynamics,” Optics Express 20(7), 22847 (2016).
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Huang, J. F.

J. Q. Liao, J. F. Huang, L. Tian, L. M. Kuang, and C. P. Sun, “Generalized Ultrastrong Optomechanics,” arXiv: 1802.09254 (2018).

Huang, R.

Huang, S.

G. S. Agarwal and S. Huang, “Optomechanical systems as single-photon routers,” Phys. Rev. A 85, 021801 (R) (2012).
[Crossref]

Imamodlu, A.

M. Bamba, A. Imamođlu, I. Carusotto, and C. Ciuti, “Origin of strong photon antibunching in weakly nonlinear photonic molecules,” Phys. Rev. A 83, 021802(R) (2011).
[Crossref]

A. Imamođlu, H. Schmidt, G. Woods, and M. Deutsch, “Strongly Interacting Photons in a Nonlinear Cavity,” Phys. Rev. Lett. 79, 1467 (1997).
[Crossref]

Jiang, X. S.

B. He, L. Yang, X. S. Jiang, and M. Xiao, “Transmission Nonreciprocity in a Mutually Coupled Circulating Structure,” Phys. Rev. Lett. 120, 203904 (2018).
[Crossref] [PubMed]

Jiang, Y.

Y. Jiang, S. Maayani, T. Carmon, F. Nori, and H. Jing, “Nonreciprocal Photon Laser,” Phys. Rev. Appl. 6, 040001 (2016).

Jin, D. Y.

W. Xiong, D. Y. Jin, Y. Y. Qiu, C. H. Lam, and J. Q. You, “Cross-Kerr effect on an optomechanical system,” Phys. Rev. A 93, 023844 (2016).
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Jing, H.

B. J. Li, R. Huang, X. W. Xu, A. Miranowicz, and H. Jing, “Nonreciprocal unconventional photon blockade in a spinning optomechanical system,” Photon. Res. 7, 630 (2019).
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S. Maayani, R. Dahan, Y. Kligerman, E. Moses, A. U. Hassan, H. Jing, and F. Nori, “Flying couplers above spinning resonators generate irreversible refraction,” Nature (London) 558, 569 (2018).
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R. Huang, A. Miranowicz, J. Q. Liao, F. Nori, and H. Jing, “Nonreciprocal Photon Blockade,” Phys. Rev. Lett. 121, 153601 (2018).
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Q. T. Cao, H. Wang, C. H. Dong, H. Jing, R. S. Liu, X. Chen, L. Ge, Q. Gong, and Y. F. Xiao, “Experimental Demonstration of Spontaneous Chirality in a Nonlinear Microresonator,” Phys. Rev. Lett. 118, 033901 (2017)
[Crossref] [PubMed]

Y. Jiang, S. Maayani, T. Carmon, F. Nori, and H. Jing, “Nonreciprocal Photon Laser,” Phys. Rev. Appl. 6, 040001 (2016).

X. W. Xu, Y. J. Zhao, H. Wang, H. Jing, and A. X. Chen, “Nonreciprocal photon blockade via quadratic optomechanical coupling,” arXiv:1809.07596 (2018).

Kalaee, M.

T. K. Paraïso, M. Kalaee, L. Zang, H. Pfeifer, F. Marquardt, and O. Painter, “Position-Squared Coupling in a Tunable Photonic Crystal Optomechanical Cavity,” Phys. Rev. X 5, 041024 (2015).

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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, 5288 (1987).
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Khan, R.

R. Khan, F. Massel, and T. T. Heikkilä, “Cross-Kerr nonlinearity in optomechanical systems,” Phys. Rev. A 91, 043822 (2015).
[Crossref]

T. T. Heikkilä, F. Massel, J. Tuorila, R. Khan, and M. A. Sillanpää, “Enhancing Optomechanical Coupling via the Josephson Effect,” Phys. Rev. Lett. 112, 203603 (2014).
[Crossref]

Kimble, H. J.

H. J. Kimble, “The quantum internet,” Nature 453(7198), 1023–1030 (2008).
[Crossref] [PubMed]

Kippenberg, T. J.

S. Weis, R. Rivière, S. Deléglise, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010).
[Crossref] [PubMed]

T. J. Kippenberg and K. J. Vahala, “Cavity Optomechanics: Back-Action at the Mesoscale,” Science 321, 1172–1176 (2008).
[Crossref] [PubMed]

Kligerman, Y.

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

Koshini, K.

K. Koshini, “Semiclassical evaluation of two-photon cross-Kerr effect,” Phys. Rev. A 74, 053818 (2006).
[Crossref]

Kralj, N.

S. Zippilli, N. Kralj, M. Rossi, G. D. Giuseppe, and D. Vitali, “Cavity optomechanics with feedback-controlled in-loop light,” Phys. Rev. A 98, 023828 (2018).
[Crossref]

Kuang, L. M.

J. Q. Liao, J. F. Huang, L. Tian, L. M. Kuang, and C. P. Sun, “Generalized Ultrastrong Optomechanics,” arXiv: 1802.09254 (2018).

Lam, C. H.

W. Xiong, D. Y. Jin, Y. Y. Qiu, C. H. Lam, and J. Q. You, “Cross-Kerr effect on an optomechanical system,” Phys. Rev. A 93, 023844 (2016).
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J. Q. Liao and C. K. Law, “Correlated two-photon scattering in cavity optomechanics,” Phys. Rev. A 87, 043809 (2013).
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J. Q. Liao, H. K. Cheung, and C. K. Law, “Spectrum of single-photon emission and scattering in cavity optomechanics,” Phys. Rev. A 85, 025803 (2012).
[Crossref]

Lemonde, M. A.

M. A. Lemonde, N. Didier, and A. A. Clerk, “Nonlinear Interaction Effects in a Strongly Driven Optomechanical Cavity,” Phys. Rev. Lett. 111, 053602 (2013).
[Crossref] [PubMed]

Li, B. J.

Li, G. L.

G. L. Li, X. Xiao, Y. Li, and X. G. Wang, “Tunable optical nonreciprocity and a phonon-photon router in an optomechanical system with coupled mechanical and optical modes,” Phys. Rev. A 97, 023801 (2018).
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Li, Y.

G. L. Li, X. Xiao, Y. Li, and X. G. Wang, “Tunable optical nonreciprocity and a phonon-photon router in an optomechanical system with coupled mechanical and optical modes,” Phys. Rev. A 97, 023801 (2018).
[Crossref]

X. W. Xu, A. X. Chen, Y. Li, and Y. X Liu, “Single-photon nonreciprocal transport in one-dimensional coupled-resonator waveguides,” Phys. Rev. A 95, 063808 (2017).
[Crossref]

X. W. Xu and Y. Li, “Optical nonreciprocity and optomechanical circulator in three-mode optomechanical systems,” Phys. Rev. A 91, 053854 (2015).
[Crossref]

Li, Y. J.

X. W. Xu, Y. J. Li, and Y. X. Liu, “Photon-induced tunneling in optomechanical systems,” Phys. Rev. A 87, 025803 (2013).
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Li, Yong

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Figures (7)

Fig. 1
Fig. 1 The schematic illustration of the model. A cyclic three-mode optical system driven by three pump fields of amplitudes εa1, εa2, and εa3 with frequency ωd. Two of the optical modes are linearly coupled with each other, while they are coupled to another described by cross-Kerr (CK) nonlinear terms.
Fig. 2
Fig. 2 Schematic diagram of typical experimental configurations of the all-optical CK nonlinear coupling system. (a) The mirrors on both sides of the cavity are fixed, and the mirrors in the middle can move. The assisting qubits are used to induce the CK effect between the two cavity modes. (b) An equivalent quantum circuit with superconducting charge qubits. (c) Two optomechanical crystals are coupled to each other.
Fig. 3
Fig. 3 Scattering probabilities T a 2 a 3 ( ω ) and T a 3 a 2 ( ω ) as functions of the frequency of the incoming signal ω (in units of γa3) for different phase different φ from 0 to 1 (in units of π). The other parameters are γaj = γa3, Δ a j σ s = Δ a 3 = 10 γ a 3, J = Ga2 = Ga1e = 0.5γa3. The right-hand panel is a contour plot for the left-hand panel, and the purple portion of the figure is replaced by white.
Fig. 4
Fig. 4 Scattering probabilities T a 2 a 3 ( ω ) and T a 3 a 2 ( ω ) as a function of the frequency of the incoming signal ω (units of γa3) for different optomechanical coupling constant Ga2(units of γa3). The other parameters chosen are γaj = γa3, Δ a j σ s = Δ a 3 = 10 γ a 3, J = 0.5γa3, Ga1 = −iGa2, Ga2 ∈ [0, 2γa3]. The right-hand panel is a contour plot for the left-hand panel, and the dark blue part of the left figure corresponds to the white in the right figure.
Fig. 5
Fig. 5 Numerical calculations for the transmission rates. Fig. (a)–(d) shows the transmission probability of optical and optomechanical system as a function of the frequency of the incoming signal ω(units of γa3) for different optomechanical coupling constant Ga1. (a) shows the optical transmission given by calculations with RWA. (b) is for the transmission considering the anti-rotational terms. (c) shows the optomechanical transmission with RWA. (d) shows the optomechanical transmission with anti-rotational wave terms. The other parameters chosen are set to meet the optimal parameter conditions.
Fig. 6
Fig. 6 Numerical calculation Re[εT ] as a function of Ω (units of γa3) for different optomechanical coupling constant: Ga1 = 0.1γa3 (blue solid line with stars), Ga1 = 0.2γa3 (red solid line with circles), Ga1 = 0.3γa3 (orange solid line with squares), Ga1 = 0.4γa3 (solid brown line), Ga1 = 0.5γa3 (green dotted line).
Fig. 7
Fig. 7 (a) A loop system where an equivalent mechanical mode is coupled to each cavity in an optical cavity-array. (b) The three-mode optomechanical system consists of two mechanical modes and one optical mode. (c) The general form of a planar quantum network, which is composed of arbitrary optical mode and a mechanical mode.

Equations (53)

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H ^ T = H ^ 0 + H ^ I + H ^ d ,
H ^ 0 = i = 1 3 ω a i a ^ i a ^ i + k i = 1 3 ω i k Γ ^ i k Γ ^ i k , H ^ d = i i = 1 3 ( ε a i a ^ i e i ( φ a i ω d t ) H . c ) , H ^ I = j = 1 2 χ a j a ^ j a ^ j a ^ 3 a ^ 3 + J ( a ^ 1 a ^ 2 + a ^ 2 a ^ 1 ) + k i = 1 3 g i k ( a ^ i Γ ^ i k + Γ ^ i k a ^ i ) ,
H ^ = H ^ 0 + H ^ I + H ^ d ( B ^ 3 + B ^ 3 ) + F ,
H ^ 0 = j = 1 2 Δ a j a ^ j a ^ j + Δ a 3 a ^ 3 a ^ 3 + k i = 1 3 ω i k Γ ^ i k Γ ^ i k , H ^ I = H ^ I j = 1 2 χ a j | α 3 | ( a ^ 3 e i θ + a ^ 3 e i θ ) a ^ j a ^ j , H ^ d = i i = 1 3 ( ε a i a ^ i e i φ a i H . c ) Δ a 3 ( α 3 a ^ 3 + α 3 * a ^ 3 ) ,
ε a 3 e i φ a 3 + ( i Δ a 3 γ a 3 / 2 ) α 3 s = 0 .
H ^ = H ^ 0 s + H ^ I s + H ^ d s ( B ^ 3 s + B ^ 3 s ) + F s ,
d d t a ^ 3 = k a ^ 3 + i j = 1 2 g a j s e i θ a ^ j a ^ j + γ a 3 a ^ 3 in + ( i Δ a 3 γ a 3 2 ) α 3 s + ε a 3 e i φ a 3 .
θ = arctan ( γ a 3 2 sin φ a 3 + Δ a 3 cos φ a 3 γ a 3 2 cos φ a 3 Δ a 3 sin φ a 3 ) , | α 3 s | = ε a 3 | k | .
d d t a ^ 1 = + Γ ^ 1 a ^ 1 i J a ^ 2 + ε a 1 e i φ a 1 + γ a 1 a ^ 1 in , d d t a ^ 2 = + Γ ^ 2 a ^ 2 i J a ^ 1 + ε a 2 e i φ a 2 + γ a 2 a ^ 2 in , d d t a ^ 3 = k a ^ 3 + i j = 1 2 g a j s e i θ a ^ j a ^ j + γ a 3 a ^ 3 in ,
η 1 = k 2 ε a 1 e i φ a 1 i J ε a 2 e i φ a 2 k 1 k 2 + J 2 , η 2 = k 1 ε a 2 e i φ a 2 i J ε a 1 e i φ a 1 k 1 k 2 + J 2 , η 3 = i j = 1 2 g a j s | η j | 2 e i θ k ,
d d t δ a ^ 1 = k 1 δ a ^ 1 + i G a 1 σ ^ i J δ a ^ 2 + γ a 1 δ a ^ 1 in , d d t δ a ^ 2 = k 2 δ a ^ 2 + i G a 2 σ ^ i J δ a ^ 1 + γ a 2 δ a ^ 2 in , d d t δ a ^ 3 = k δ a ^ 3 + i j = 1 2 τ ^ j e i θ + γ a 3 δ a ^ 3 in ,
d d t υ ^ = μ υ ^ + L υ ^ in ,
μ = [ k 1 i J i G a 1 e i θ i J k 2 i G a 2 e i θ i G a 1 * e i θ i G a 2 * e i θ k ] ,
τ ( ω ) = [ τ ( t ) ] = + τ ( t ) e i ω t d t ,
υ ( ω ) = ( μ i ω I ) 1 L υ in ( ω ) ,
υ o u t ( ω ) = Γ ( ω ) υ in ( ω ) ,
Γ ( ω ) = L T [ μ i ω I ] 1 L I .
1 | A | [ γ a 1 2 A 11 * | A | γ a 1 γ a 2 A 12 * γ a 1 γ a 3 A 13 * γ a 2 γ a 1 A 21 * γ a 2 2 A 22 * | A | γ a 2 γ a 3 A 23 * γ a 3 γ a 1 A 31 * γ a 3 γ a 2 A 32 * γ a 3 2 A 33 * | A | ] ,
A j j * = ( k j i ω ) ( k i ω ) + | G a j | 2 e 2 i θ , A 33 * = ( k 1 i ω ) ( k 2 i ω ) + J 2 , A 12 * = J ( i k + ω ) G a 1 * G a 2 e 2 i θ , A 21 * = J ( i k + ω ) G a 1 G a 2 * e 2 i θ , A 13 * = { G a 1 * ( i k 2 + ω ) + J G a 2 * } e i θ , A 31 * = { G a 1 ( i k 2 + ω ) + J G a 2 } e i θ , A 23 * = { G a 2 * ( i k 1 + ω ) J G a 1 * } e i θ , A 32 * = { G a 2 ( i k 1 + ω ) + J G a 1 } e i θ .
S o u t ( ω ) = d ω υ o u t ( ω ) υ o u t ( ω ) .
S out ( ω ) = T ( ω ) S in ( ω ) + S Q ( ω ) + S T ( ω ) ,
S out ( ω ) = T ( ω ) S in ( ω ) ,
A 23 * A 23 * = J 2 | G a 1 | 2 2 J Re [ G a 2 * G a 1 ( i k 1 + ω ) ] + | G a 2 | 2 ( | k 1 | 2 + ω 2 2 ω Im [ k 1 ] ) .
A 32 * A 32 * = J 2 | G a 1 | 2 2 J Re [ G a 1 * G a 2 ( i k 1 + ω ) ] + | G a 2 | 2 ( | k 1 | 2 + ω 2 2 ω Im [ k 1 ] ) .
Γ + = [ 0 0 i i 0 0 0 1 0 ] or Γ = [ 0 i 0 0 0 1 i 0 0 ] ,
T a 2 a 1 = γ a 1 γ a 2 | β | 2 { | G a 1 | 2 | G a 2 | 2 + | J k | 2 2 J σ } ,
T a 3 a 2 = γ a 3 γ a 2 | G a 1 | 2 | k | 2 + | J G a 2 | 2 2 J σ | G a 1 | 2 | G a 2 | 2 + | J k | 2 2 J σ ,
β = k 2 ( k 1 k + J 2 ) + j = 1 2 k j | G a j + 1 | 2 2 i J Re ( G a 1 G a 2 * ) ,
H ^ a = H ^ a 0 + H ^ a I + H ^ a d ,
H ^ a 0 = τ = 1 N Δ a τ a ^ τ a ^ τ + Δ a m a ^ m a ^ m , H ^ a I = τ = 1 N J ( a ^ τ a ^ τ + 1 + a ^ τ + 1 a ^ τ ) τ = 1 N g a m s ^ m a ^ τ a ^ τ , H ^ a d = τ = 1 N i ε a τ a ^ τ e i φ a τ + κ m a ^ m + H . c ,
0 = ε a m e i φ a m + ( i Δ a m γ a m 2 ) α m s ,
H ^ 2 = H ^ 2 0 + H ^ 2 I + H ^ 2 d ,
H ^ 2 0 = j = 1 2 Δ a j a ^ j a ^ j + Δ a 3 s s a ^ 3 a ^ 3 , H ^ 2 I = J ( a ^ 1 a ^ 2 + a ^ 2 a ^ 1 ) j = 1 2 g a j s s ^ j a ^ 3 a ^ 3 , H ^ 2 d = i = 1 3 i ε a i a ^ i e i φ a i j = 1 2 o j a ^ j + H . c ,
0 = + ε a 1 e i φ a 1 + ( i Δ a 1 γ a 1 2 ) α 1 s s , 0 = + ε a 2 e i φ a 2 + ( i Δ a 2 γ a 2 2 ) α 2 s s + i J α 1 s s .
H ^ s = i = 1 3 ω a i a ^ i a ^ i + j = 1 2 χ a j a ^ j a ^ j a ^ 3 a ^ 3 + J ( a ^ 1 a ^ 2 + a ^ 2 a ^ 1 ) + i i = 1 3 ( ε a i a ^ i e i ( φ a i ω d ) H . c . ) .
H ^ s = i = 1 3 Δ a i a ^ i a ^ i + j = 1 2 χ a j a ^ j a ^ j a ^ 3 a ^ 3 + J ( a ^ 1 a ^ 2 + a ^ 2 a ^ 1 ) + i i = 1 3 ( ε a i a ^ i e i φ a i H . c . ) ,
ρ ˙ = i [ ρ , H ^ s ] + i = 1 3 γ a i { ( n ¯ a i + 1 ) ^ [ a ^ 3 ] + n ¯ a i ^ [ a ^ i ] } ρ .
ρ = 𝒟 ^ a 3 ( α 3 ) ρ 𝒟 ^ a 3 ( α 3 ) ,
ρ ˙ = t 𝒟 ^ a 3 ( α 3 ) ρ 𝒟 ^ a 3 ( α 3 ) + 𝒟 ^ a 3 ( α 3 ) ρ t 𝒟 ^ a 3 ( α 3 ) + 𝒟 ^ a 3 ( α 3 ) ρ ˙ 𝒟 ^ a 3 ( α 3 ) .
t 𝒟 ^ a 3 ( α 3 ) = 1 2 ( α 3 α ˙ 3 * α ˙ 3 α 3 * ) 𝒟 ^ a 3 ( α 3 ) 𝒟 ^ a 3 ( α 3 ) ( a ^ 3 α ˙ 3 * a ^ 3 α ˙ 3 ) ,
ρ ˙ = [ 𝒟 ^ a 3 ( α 3 ) ρ ˙ 𝒟 ^ a 3 ( α 3 ) , ( a ^ 3 α ˙ 3 a ^ 3 α ˙ 3 * ) ] + 𝒟 ^ a 3 ( α 3 ) ρ ˙ 𝒟 ^ a 3 ( α 3 ) .
ρ ˙ = + i [ ρ , H ^ s ] + i = 1 3 γ a i { ( n ¯ a i + 1 ) ^ [ a ^ i ] + n ¯ a i ^ [ a ^ i ] } ρ + { α ˙ 3 + ( i Δ a 3 γ a 3 2 ) α 3 + ε a 3 e i φ a 3 } [ a ^ 3 , ρ ] { α ˙ 3 * + ( i Δ a 3 γ a 3 2 ) α 3 * + ε a 3 e i φ a 3 } [ a ^ 3 , ρ ] ,
H ^ s = + j = 1 2 Δ a j a ^ j a ^ j + Δ a 3 a ^ 3 a ^ 3 + j = 1 2 χ a j a ^ j a ^ j a ^ 3 a ^ 3 + J ( a ^ 1 a ^ 2 + a ^ 2 a ^ 1 ) j = 1 2 χ a j a ^ j a ^ j a ^ 3 a ^ 3 j = 1 2 g a j s ( a ^ 3 e i θ + a ^ 3 e i θ ) a ^ j a ^ j .
α ˙ 3 s + ( i Δ a 3 γ a 3 2 ) α 3 s + ε a 3 e i φ a 3 = 0 .
d d t a ^ 3 = i Δ a 3 a ^ 3 i k g 3 k Γ ^ 3 k + i Δ a 3 α 3 s + i j = 1 2 g a j s e i θ a ^ j a ^ j + ε a 3 e i ϕ a 3 ,
d d t Γ ^ 3 k = i ω 3 k Γ ^ 3 k i g 3 k a ^ 3 i g 3 k α 3 s .
Γ ^ 3 k = i g 3 k 0 t { a ^ 3 ( τ ) + α 3 s } e i ω 3 k ( t τ ) d τ + Γ ^ 3 k ( 0 ) e i ω 3 k t .
d d t a ^ 3 = + f ^ a 3 k g 3 k 2 0 t { a ^ 3 ( τ ) + α 3 s } e i ω 3 k ( t τ ) d τ + i Δ a 3 ( α 3 s a ^ 3 ) + i j = 1 2 g a j s e i θ a ^ j a ^ j + ε a 3 e i ϕ a 3 ,
f ^ a 3 ( t ) = i k g 3 k Γ ^ 3 k ( 0 ) e i ω 3 k t ,
d d t a ^ 3 = + f ^ a 3 ( t ) 0 t g 3 2 ( ω ) π δ ( t τ ) { a ^ 3 ( τ ) + α 3 s } d τ + i Δ a 3 ( α 3 s a ^ 3 ) + i j = 1 2 g a j s e i θ a ^ j a ^ j + ε a 3 e i ϕ a 3 ,
f ^ a 3 ( t ) = i 0 g 3 ( ω ) Γ ^ 30 ( ω ) e i ω t d ω .
d d t a ^ 3 = ( i Δ a 3 γ a 3 2 ) a ^ 3 + i j = 1 2 g a j s e i ω a ^ j a ^ j + ε a 3 e i ϕ a 3 + ( i Δ a 3 γ a 3 2 ) α 3 s + γ a 3 a ^ 3 in ,
a ^ 3 in = 1 2 π 0 i g 3 ( ω ) Γ ^ 30 ( ω ) e i ω t d ω .

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