Abstract

We theoretically propose a scheme for realizing a quantum-limited directional amplifier in a triple-cavity optomechanical system, where one microwave cavity and two optical cavities are, respectively, coupled to a common mechanical resonator. Moreover, the two optical cavities are coupled directly to facilitate the directional amplification between microwave and optical photons. We find that directional amplification between the three cavity modes is achieved with two gain process and one conversion process, and the direction of amplification can be modulated by controlling the phase difference between the field-enhanced optomechanical coupling strengths. Furthermore, with increasing the optomechanical cooperativity, both gain and bandwidth of the directional amplifier can be enhanced, and the noise added to the amplifier can be suppressed to approach the standard quantum limit on the phase-preserving linear amplifier.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2018 (3)

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, 1797 (2018).
[Crossref] [PubMed]

D. B. Sohn, S. Kim, and Gaurav Bahl, “Time-reversal symmetry breaking with acoustic pumping of nanophotonic circuits,” Nat. Photon. 12, 91–97 (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, 023601 (2018).
[Crossref] [PubMed]

2017 (11)

G. A. Peterson, F. Lecocq, K. Cicak, R. W. Simmonds, J. Aumentado, and J. D. Teufel, “Demonstration of efficient nonreciprocity in a microwave optomechanical circuit,” Phys. Rev. X 7, 031001 (2017).

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

S. Barzanjeh, M. Wulf, M. Peruzzo, M. Kalaee, P. B. Dieterle, O. Painter, and J. M. Fink, “Mechanical on-chip microwave circulator,” Nat. Commun. 8, 953 (2017).
[Crossref] [PubMed]

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

H. Lü, S. K. Özdemir, L.-M. Kuang, F. Nori, and H. Jing, “Exceptional points in random-defect phonon lasers,” Phys. Rev. Applied 8, 044020 (2017).
[Crossref]

M.-A. Miri, F. Ruesink, E. Verhagen, and A. Alù, “Optical nonreciprocity based on optomechanical coupling,” Phys. Rev. Appl. 7, 064014 (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, 465–471 (2017).
[Crossref]

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

F. Lecocq, L. Ranzani, G. A. Peterson, K. Cicak, R. W. Simmonds, J. D. Teufel, and J. Aumentado, “Nonreciprocal microwave signal processing with a field-programmable Josephson amplifier,” Phys. Rev. Appl. 7, 024028 (2017).
[Crossref]

A. C. Mahoney, J. I. Colless, S. J. Pauka, J. M. Hornibrook, J. D. Watson, G. C. Gardner, M. J. Manfra, A. C. Doherty, and D. J. Reilly, “On-Chip microwave quantum Hall circulator,” Phys. Rev. X 7, 011007 (2017).

L. D. Tóth, N. R. Bernier, A. Nunnenkamp, A. K. Feofanov, and T. J. Kippenberg, “A dissipative quantum reservoir for microwave light using a mechanical oscillator,” Nat. Phys. 13, 787–793 (2017).
[Crossref]

2016 (6)

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, 083034 (2016).
[Crossref]

C. F. Ockeloen-Korppi, E. Damskägg, J.-M. Pirkkalainen, T. T. Heikkilä, F. Massel, and M. A. Sillanpää, “Low-noise amplification and frequency conversion with a multiport microwave optomechanical device,” Phys. Rev. X 6, 041024 (2016).

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, 123902 (2016).
[Crossref] [PubMed]

F. Ruesink, M.-A. Miri, A. Alù, and E. Verhagen, “Nonreciprocity and magnetic-free isolation based on optomechanical interactions,” Nat. Commun. 7, 13662 (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. Photon. 10, 657 (2016).
[Crossref]

X.-W. Xu, Y. Li, A.-X. Chen, and Y.-x. Liu, “Nonreciprocal conversion between microwave and optical photons in electro-optomechanical systems,” Phys. Rev. A 93, 023827 (2016).
[Crossref]

2015 (11)

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

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

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

J. Kim, M. C. Kuzyk, K. Han, H. Wang, and G. Bahl, “Non-reciprocal Brillouin scattering induced transparency,” Nat. Phys. 11, 275 (2015).
[Crossref]

C. H. Dong, Z. Shen, C. L. Zou, Y. L. Zhang, W. Fu, and G. C. Guo, “Brillouin-scattering-induced transparency and non-reciprocal light storage,” Nat. Commun. 6, 6193 (2015).
[Crossref] [PubMed]

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, 041020 (2015).

H. Xiong, L. G. Si, X. Y. Lv, 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, 1–13 (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, 9663 (2015).
[Crossref] [PubMed]

J. D. Cohen, S. M. Meenehan, G. S. MacCabe, S. Gröblacher, A. H. Safavi-Naeini, F. Marsili, M. D. Shaw, and O. Painter, “Phonon counting and intensity interferometry of a nanomechanical resonator,” Nature (London) 520, 522–525 (2015).
[Crossref]

L. Tian, “Optoelectromechanical transducer: Reversible conversion between microwave and optical photons,” Ann. Phys. (Berlin) 527, 1 (2015).
[Crossref]

C. Dong, V. Fiore, M. C. Kuzyk, L. Tian, and H. Wang, “Optical wavelength conversion via optomechanical coupling in a silica resonator,” Ann. Phys. (Berlin) 527, 100 (2015).
[Crossref]

2014 (8)

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

R. W. Andrews, R. W. Peterson, T. P. Purdy, K. Cicak, R. W. Simmonds, C. A. Regal, and K. W. Lehnert, “Bidirectional and efficient conversion between microwave and optical light,” Nat. Phys. 10, 321–326 (2014).
[Crossref]

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391 (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, 516–519 (2014).
[Crossref] [PubMed]

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, 923–927 (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. Photon. 8, 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, 394–398 (2014).
[Crossref]

A. Metelmann and A. A. Clerk, “Quantum-limited amplification via reservoir engineering,” Phys. Rev. Lett. 112, 133904 (2014).
[Crossref] [PubMed]

2013 (5)

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

T. A. Palomaki, J. D. Teufel, R. W. Simmonds, and K. W. Lehnert, “Entangling mechanical motion with microwave fields,” Science 342, 710–713 (2013).
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B. Abdo, K. Sliwa, L. Frunzio, and M. Devoret, “Directional amplification with a Josephson circuit,” Phys. Rev. X 3, 031001 (2013).

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, 093901 (2013).
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X. Zhou, F. Hocke, A. Schliesser, A. Marx, H. Huebl, R. Gross, and T. J. Kippenberg, “Slowing, advancing and switching of microwave signals using circuit nanoelectromechanics,” Nat. Phys. 9, 179–184 (2013).
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2012 (6)

J. T. Hill, A. H. Safavi-Naeini, J. Chan, and O. Painter, “Coherent optical wavelength conversion via cavity optomechanics,” Nat. Commun. 3, 1196 (2012).
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H. Lira, Z. Yu, S. Fan, and M. Lipson, “Electrically driven nonreciprocity induced by interband photonic transition on a dilicon vhip,” Phys. Rev. Lett. 109, 033901 (2012).
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L. Deák and T. Fülöp, “Reciprocity in quantum, electromagnetic and other wave scattering,” Ann. Phys. 327, 1050–1077 (2012).
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M. Hafezi and P. Rabl, “Optomechanically induced non-reciprocity in microring resonators,” Opt. Express 20, 7672–7684 (2012).
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Y. Sato, Y. Tanaka, J. Upham, Y. Takahashi, T. Asano, and S. Noda, “Strong coupling between distant photonic nanocavities and its dynamic control,” Nat. Photon. 6, 56–61 (2012).
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K. Fang, Z. Yu, and S. Fan, “Realizing effective magnetic field for photons by controlling the phase of dynamic modulation,” Nat. Photon. 6, 782–787 (2012).
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2011 (5)

L. Bi, J. Hu, P. Jiang, D. H. Kim, G. F. Dionne, L. C. Kimerling, and C. A. Ross, “On-chip optical isolation in monolithically integrated non-reciprocal optical resonators,” Nat. Photon. 5, 758–762 (2011).
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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 (London) 478, 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 (London) 475, 359–363 (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 (London) 472, 69–73 (2011).
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F. Massel, T. T. Heikkilä, J.-M. Pirkkalainen, S. U. Cho, H. Saloniemi, P. J. Hakonen, and M. A. Sillanpää, “Microwave amplification with nanomechanical resonators,” Nature (London) 480, 351–354 (2011).
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2010 (4)

I. S. Grudinin, H. Lee, O. Painter, and K. J. Vahala, “Phonon laser action in a tunable two-level system,” Phys. Rev. Lett. 104, 083901 (2010).
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G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81, 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, 1520–1523 (2010).
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A. A. Clerk, M. H. Devoret, S. M. Girvin, Florian Marquardt, and R. J. Schoelkopf, “Introduction to quantum noise, measurement, and amplification,” Rev. Mod. Phys. 82, 1155 (2010).
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2009 (2)

Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photon. 3, 91–94 (2009).
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F. Marquardt and S. M. Girvin, “Optomechanics,” Physics 2, 40 (2009).
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2008 (1)

C. Genes, A. Mari, P. Tombesi, and D. Vitali, “Robust entanglement of a micromechanical resonator with output optical fields,” Phys. Rev. A 78, 032316 (2008).
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2004 (1)

R. J. Potton, “Reciprocity in optics,” Rep. Prog. Phys. 67, 717 (2004).
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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, 5288 (1987).
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1985 (1)

C. W. Gardiner and M. J. Collett, “Input and output in damped quantum system: quantum stochastic differential equations and the master equation,” Phys. Rev. A 31, 3761 (1985).
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1982 (1)

C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D 26, 1817 (1982).
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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 (London) 475, 359–363 (2011).
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M.-A. Miri, F. Ruesink, E. Verhagen, and A. Alù, “Optical nonreciprocity based on optomechanical coupling,” Phys. Rev. Appl. 7, 064014 (2017).
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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, 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, 923–927 (2014).
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Andrews, R. W.

R. W. Andrews, R. W. Peterson, T. P. Purdy, K. Cicak, R. W. Simmonds, C. A. Regal, and K. W. Lehnert, “Bidirectional and efficient conversion between microwave and optical light,” Nat. Phys. 10, 321–326 (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, 1520–1523 (2010).
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Asano, T.

Y. Sato, Y. Tanaka, J. Upham, Y. Takahashi, T. Asano, and S. Noda, “Strong coupling between distant photonic nanocavities and its dynamic control,” Nat. Photon. 6, 56–61 (2012).
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F. Lecocq, L. Ranzani, G. A. Peterson, K. Cicak, R. W. Simmonds, J. D. Teufel, and J. Aumentado, “Nonreciprocal microwave signal processing with a field-programmable Josephson amplifier,” Phys. Rev. Appl. 7, 024028 (2017).
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G. A. Peterson, F. Lecocq, K. Cicak, R. W. Simmonds, J. Aumentado, and J. D. Teufel, “Demonstration of efficient nonreciprocity in a microwave optomechanical circuit,” Phys. Rev. X 7, 031001 (2017).

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J. Bochmann, A. Vainsencher, D. D. Awschalom, and A. N. Cleland, “Nanomechanical coupling between microwave and optical photons,” Nat. Phys. 9, 712–716 (2013).
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D. B. Sohn, S. Kim, and Gaurav Bahl, “Time-reversal symmetry breaking with acoustic pumping of nanophotonic circuits,” Nat. Photon. 12, 91–97 (2018).
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S. Barzanjeh, M. Wulf, M. Peruzzo, M. Kalaee, P. B. Dieterle, O. Painter, and J. M. Fink, “Mechanical on-chip microwave circulator,” Nat. Commun. 8, 953 (2017).
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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, 394–398 (2014).
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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, 023601 (2018).
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N. R. Bernier, L. D. Tóth, A. Koottandavida, M. A. Ioannou, D. Malz, A. Nunnenkamp, A. K. Feofanov, and T. J. Kippenberg, “Nonreciprocal reconfigurable microwave optomechanical circuit,” Nat. Commun. 8, 604 (2017).
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L. D. Tóth, N. R. Bernier, A. Nunnenkamp, A. K. Feofanov, and T. J. Kippenberg, “A dissipative quantum reservoir for microwave light using a mechanical oscillator,” Nat. Phys. 13, 787–793 (2017).
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L. Bi, J. Hu, P. Jiang, D. H. Kim, G. F. Dionne, L. C. Kimerling, and C. A. Ross, “On-chip optical isolation in monolithically integrated non-reciprocal optical resonators,” Nat. Photon. 5, 758–762 (2011).
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J. Bochmann, A. Vainsencher, D. D. Awschalom, and A. N. Cleland, “Nanomechanical coupling between microwave and optical photons,” Nat. Phys. 9, 712–716 (2013).
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C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D 26, 1817 (1982).
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J. T. Hill, A. H. Safavi-Naeini, J. Chan, and O. Painter, “Coherent optical wavelength conversion via cavity optomechanics,” Nat. Commun. 3, 1196 (2012).
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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 (London) 478, 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 (London) 472, 69–73 (2011).
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Chang, D. E.

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 (London) 472, 69–73 (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. Photon. 8, 524–529 (2014).
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X.-W. Xu, Y. Li, A.-X. Chen, and Y.-x. Liu, “Nonreciprocal conversion between microwave and optical photons in electro-optomechanical systems,” Phys. Rev. A 93, 023827 (2016).
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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, 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. Photon. 10, 657 (2016).
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Cho, S. U.

F. Massel, T. T. Heikkilä, J.-M. Pirkkalainen, S. U. Cho, H. Saloniemi, P. J. Hakonen, and M. A. Sillanpää, “Microwave amplification with nanomechanical resonators,” Nature (London) 480, 351–354 (2011).
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Cicak, K.

F. Lecocq, L. Ranzani, G. A. Peterson, K. Cicak, R. W. Simmonds, J. D. Teufel, and J. Aumentado, “Nonreciprocal microwave signal processing with a field-programmable Josephson amplifier,” Phys. Rev. Appl. 7, 024028 (2017).
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G. A. Peterson, F. Lecocq, K. Cicak, R. W. Simmonds, J. Aumentado, and J. D. Teufel, “Demonstration of efficient nonreciprocity in a microwave optomechanical circuit,” Phys. Rev. X 7, 031001 (2017).

R. W. Andrews, R. W. Peterson, T. P. Purdy, K. Cicak, R. W. Simmonds, C. A. Regal, and K. W. Lehnert, “Bidirectional and efficient conversion between microwave and optical light,” Nat. Phys. 10, 321–326 (2014).
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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 (London) 475, 359–363 (2011).
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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, 712–716 (2013).
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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, 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, 021025 (2015).

A. Metelmann and A. A. Clerk, “Quantum-limited amplification via reservoir engineering,” Phys. Rev. Lett. 112, 133904 (2014).
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A. A. Clerk, M. H. Devoret, S. M. Girvin, Florian Marquardt, and R. J. Schoelkopf, “Introduction to quantum noise, measurement, and amplification,” Rev. Mod. Phys. 82, 1155 (2010).
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J. D. Cohen, S. M. Meenehan, G. S. MacCabe, S. Gröblacher, A. H. Safavi-Naeini, F. Marsili, M. D. Shaw, and O. Painter, “Phonon counting and intensity interferometry of a nanomechanical resonator,” Nature (London) 520, 522–525 (2015).
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A. C. Mahoney, J. I. Colless, S. J. Pauka, J. M. Hornibrook, J. D. Watson, G. C. Gardner, M. J. Manfra, A. C. Doherty, and D. J. Reilly, “On-Chip microwave quantum Hall circulator,” Phys. Rev. X 7, 011007 (2017).

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C. W. Gardiner and M. J. Collett, “Input and output in damped quantum system: quantum stochastic differential equations and the master equation,” Phys. Rev. A 31, 3761 (1985).
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C. F. Ockeloen-Korppi, E. Damskägg, J.-M. Pirkkalainen, T. T. Heikkilä, F. Massel, and M. A. Sillanpää, “Low-noise amplification and frequency conversion with a multiport microwave optomechanical device,” Phys. Rev. X 6, 041024 (2016).

Deák, L.

L. Deák and T. Fülöp, “Reciprocity in quantum, electromagnetic and other wave scattering,” Ann. Phys. 327, 1050–1077 (2012).
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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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Deléglise, S.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
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Devoret, M.

B. Abdo, K. Sliwa, L. Frunzio, and M. Devoret, “Directional amplification with a Josephson circuit,” Phys. Rev. X 3, 031001 (2013).

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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, 041020 (2015).

A. A. Clerk, M. H. Devoret, S. M. Girvin, Florian Marquardt, and R. J. Schoelkopf, “Introduction to quantum noise, measurement, and amplification,” Rev. Mod. Phys. 82, 1155 (2010).
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S. Barzanjeh, M. Wulf, M. Peruzzo, M. Kalaee, P. B. Dieterle, O. Painter, and J. M. Fink, “Mechanical on-chip microwave circulator,” Nat. Commun. 8, 953 (2017).
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L. Bi, J. Hu, P. Jiang, D. H. Kim, G. F. Dionne, L. C. Kimerling, and C. A. Ross, “On-chip optical isolation in monolithically integrated non-reciprocal optical resonators,” Nat. Photon. 5, 758–762 (2011).
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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. Photon. 10, 657 (2016).
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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 (London) 472, 69–73 (2011).
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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, 923–927 (2014).
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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, 093901 (2013).
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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, 394–398 (2014).
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H. Lira, Z. Yu, S. Fan, and M. Lipson, “Electrically driven nonreciprocity induced by interband photonic transition on a dilicon vhip,” Phys. Rev. Lett. 109, 033901 (2012).
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K. Fang, Z. Yu, and S. Fan, “Realizing effective magnetic field for photons by controlling the phase of dynamic modulation,” Nat. Photon. 6, 782–787 (2012).
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Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photon. 3, 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, 465–471 (2017).
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K. Fang, Z. Yu, and S. Fan, “Realizing effective magnetic field for photons by controlling the phase of dynamic modulation,” Nat. Photon. 6, 782–787 (2012).
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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, 023601 (2018).
[Crossref] [PubMed]

N. R. Bernier, L. D. Tóth, A. Koottandavida, M. A. Ioannou, D. Malz, A. Nunnenkamp, A. K. Feofanov, and T. J. Kippenberg, “Nonreciprocal reconfigurable microwave optomechanical circuit,” Nat. Commun. 8, 604 (2017).
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L. D. Tóth, N. R. Bernier, A. Nunnenkamp, A. K. Feofanov, and T. J. Kippenberg, “A dissipative quantum reservoir for microwave light using a mechanical oscillator,” Nat. Phys. 13, 787–793 (2017).
[Crossref]

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S. Barzanjeh, M. Wulf, M. Peruzzo, M. Kalaee, P. B. Dieterle, O. Painter, and J. M. Fink, “Mechanical on-chip microwave circulator,” Nat. Commun. 8, 953 (2017).
[Crossref] [PubMed]

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C. Dong, V. Fiore, M. C. Kuzyk, L. Tian, and H. Wang, “Optical wavelength conversion via optomechanical coupling in a silica resonator,” Ann. Phys. (Berlin) 527, 100 (2015).
[Crossref]

Fleury, R.

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, 516–519 (2014).
[Crossref] [PubMed]

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, 041020 (2015).

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[Crossref]

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Sci. China Phys. Mech. Astron. (1)

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[Crossref]

Sci. Rep. (1)

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Figures (5)

Fig. 1
Fig. 1 Schematic diagram of the triple-cavity optomechanical system. One microwave cavity a1 and two optical cavities a2 and a3 are respectively coupled to the common mechanical resonator with the field-enhanced optomechanical coupling strength Gk(k = 1, 2, 3), where ϕ is the phase difference between Gk. Moreover, the two optical cavities are coupled directly via hopping interaction J to facilitate the nonreciprocal transmission. Cavity a1 is driven on its red sideband by a pump field at the frequency ωd,1, and cavities a2 and a3 are driven on their respective blue sidebands by the pump fields at the frequencies ωd,2 and ωd,3. The solid arrows represent the transmission elements between different modes.
Fig. 2
Fig. 2 Stability diagram with respect to C1 and C2. Other parameters are κ1/2π = 2 MHz, κ2/2π = 3 MHz, κ3/2π = 3 MHz, γm = κ2/100, ω = 0, ϕ = −π/2, G3 = G2κ3/(2J), and J κ 2 κ 3 / 2.
Fig. 3
Fig. 3 Transmission probabilities |T12|2 and |T21|2 as functions of the probe detuning ω for ϕ = −π/2, 0, π/2, and π, respectively. Other parameters are κ1/2π = 2 MHz, κ2/2π = 3 MHz, κ3/2π = 3 MHz, γm = κ2/100, C1 = 10, C 2 = C 1 0.1 C 1, G3 = G2κ3/(2J), and J κ 2 κ 3 / 2.
Fig. 4
Fig. 4 Directional amplifier. (a)–(c) Transmission probabilities |Tij|2 (i, j = 1, 2, 3) as functions of the probe detuning ω/2π. (d) Graphical representation of the amplification process, where S, I, and V represent/the “Signal”, “Idler”, and “Vacuum” ports, respectively. G and C correspond to the gain and conversion process. Other parameters used are ϕ = −π/2, κ1/2π = 2 MHz, κ2/2π = 3 MHz, κ3/2π = 3 MHz, γm = κ2/100, C1 = 3, C 2 = C 1 0.1 C 1, G3 = G2κ3/(2J), and J κ 2 κ 3 / 2.
Fig. 5
Fig. 5 (a) Transmission probability |T13|2 and (b) added noise N 1 of the cavity a1 as a function of the probe detuning ω with C1 = {1, 5, 10, 50} and C 2 = C 1 0.1 C 1. Other parameters used are the same as those in Fig. 4 except nm = 50.

Equations (31)

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H = k = 1 3 ω k a k a k + ω m b b + k = 1 3 g k a k a k ( b + b ) + J ( a 2 a 3 + a 3 a 2 ) + k = 1 3 ε k ( a k e i ω d , k t + a k e i ω d , k t ) ,
H rot = k = 1 3 Δ k a k a k + ω m b b + k = 1 3 g k a k a k ( b + b ) + J ( a 2 a 3 + a 3 a 2 ) + k = 1 3 ε k ( a k + a k ) ,
a ˙ 1 = ( κ 1 / 2 i Δ 1 ) a 1 + i g 1 a 1 ( b + b ) + i ε 1 + κ 1 a 1 , in ,
a ˙ 2 = ( κ 2 / 2 + i Δ 2 ) a 2 i g 2 a 2 ( b + b ) i J a 3 i ε 2 + κ 2 a 2 , in ,
a ˙ 3 = ( κ 3 / 2 + i Δ 3 ) a 3 i g 3 a 3 ( b + b ) i J a 2 i ε 3 + κ 3 a 3 , in ,
b ˙ = ( γ m / 2 i ω m ) b + i k g k a k a k + γ m b in .
a k , in ( t ) a k , in ( t ) = δ ( t t ) , a k , in ( t ) a k , in ( t ) = 0 , b in ( t ) b in ( t ) = ( n m + 1 ) δ ( t t ) , b in ( t ) b in ( t ) = n m δ ( t t ) ,
α 1 * = i ε 1 κ 1 / 2 i Δ 1 , α 2 = i J α 3 + i ε 2 κ 2 / 2 + i Δ 2 , α 3 = i J α 2 + i ε 3 κ 3 / 2 + i Δ 3 , β * = i k g k | α k | 2 γ m / 2 i ω m ,
δ a ˙ 1 = ( κ 1 / 2 i Δ 1 ) δ a 1 + i g α 1 * ( δ b + δ b ) + κ 1 a 1 , in ,
δ a ˙ 2 = ( κ 2 / 2 + i Δ 2 ) δ a 2 i g 2 α 2 ( δ b + δ b ) i J δ a 3 + κ 2 a 2 , in ,
δ a ˙ 3 = ( κ 3 / 2 + i Δ 3 ) δ a 3 i g 3 α 3 ( δ b + δ b ) i J δ a 2 + κ 3 a 3 , in ,
δ b ˙ = ( γ m / 2 i ω m ) δ b + i k g k ( α k * δ a k + α k δ a k ) + γ m b in .
δ a ˙ 1 = κ 1 2 δ a 1 + i G 1 e i ϕ 1 δ b + κ 1 a 1 , in ,
δ a ˙ 2 = κ 2 2 δ a 2 i G 2 e i ϕ 2 δ b i J δ a 3 + κ 2 a 2 , in ,
δ a ˙ 3 = κ 3 2 δ a 3 i G 3 e i ϕ 3 δ b i J δ a 2 + κ 3 a 3 , in ,
δ b ˙ = γ m 2 δ b + i ( G 1 e i ϕ 1 δ a 1 + G 2 e i ϕ 2 δ a 2 + G 3 e i ϕ 3 δ a 3 ) + γ m b i n ,
H eff = G 1 δ a 1 δ b e i ϕ 1 + G 2 δ a 2 δ b e i ϕ 2 + G 3 δ a 3 δ b e i ϕ 3 + J δ a 2 δ a 3 + H . c ..
μ ˙ = M μ + K μ i n ,
M = ( κ 1 / 2 0 0 i G 1 0 κ 2 / 2 i J i G 2 0 i J κ 3 / 2 i G 3 e i ϕ i G 1 i G 2 i G 3 e i ϕ γ m / 2 ) .
o ( ω ) = + o ( t ) e i ω t d t ,
o ( ω ) = + o ( t ) e i ω t d t ,
μ ( ω ) = ( M + i ω I ) 1 K μ in ( ω ) ,
μ out ( ω ) = T ( ω ) μ in ( ω ) ,
T ( ω ) = I + K ( M + i ω I ) 1 K .
T 12 ( ω ) = κ 1 κ 2 A ( ω ) G 1 Γ 3 ( G 2 + i J e i ϕ G 3 / Γ 3 ) ,
T 21 ( ω ) = κ 1 κ 2 A ( ω ) G 1 Γ 3 ( G 2 + i J e i ϕ G 3 / Γ 3 ) ,
T 21 ( 0 ) = 8 κ 1 κ 2 G 1 G 2 4 κ 2 G 1 2 4 κ 1 G 2 2 + κ 1 κ 2 γ m = 2 C 1 C 2 C 1 C 2 + 1 ,
T ( 0 ) = ( C 1 + C 2 1 C 1 C 2 + 1 0 2 i C 1 C 2 C 1 C 2 + 1 2 i C 1 C 1 C 2 + 1 2 C 1 C 2 C 1 C 2 + 1 0 i C 1 + C 2 + 1 C 1 C 2 + 1 2 i C 2 C 1 C 2 + 1 0 i 0 0 2 i C 1 C 1 C 2 + 1 0 2 C 2 C 1 C 2 + 1 C 1 C 2 1 C 1 C 2 + 1 ) ,
S 1 , out ( ω ) = 1 2 d Ω 2 π a 1 , out ( ω ) a 1 , out ( Ω ) + a 1 , out ( Ω ) a 1 , out ( ω ) = 1 2 | T 11 ( ω ) | 2 + 1 2 | T 12 ( ω ) | 2 + 1 2 | T 13 ( ω ) | 2 + ( n m + 1 2 ) | T 14 ( ω ) | 2 ,
G = | T 31 ( 0 ) | 2 = 4 C 1 C 2 ( C 1 C 2 + 1 ) 2 .
N 1 ( 0 ) = 1 2 ( C 1 + C 2 1 ) 2 4 C 1 C 2 + ( n m + 1 2 ) 1 C 2 ,

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