Abstract

We study the directional amplification of an optical probe field in a three-mode optomechanical system, where the mechanical resonator interacts with two linearly-coupled optical cavities and the cavities are driven by strong optical pump fields. The optical probe field is injected into one of the cavity modes, and at the same time, the mechanical resonator is subject to a mechanical drive with the driving frequency equal to the frequency difference between the optical probe and pump fields. We show that the transmission of the probe field can be amplified in one direction and de-amplified in the opposite direction. This directional amplification or de-amplification results from the constructive or destruction interference between different transmission paths in this three-mode optomechanical system.

© 2017 Optical Society of America

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2017 (5)

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

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. Applied 7(2), 024028 (2017).
[Crossref]

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

Y. L. Zhang, C. H. Dong, C. L. Zou, X. B. Zou, Y. D. Wang, and G. C. Guo, “Optomechanical devices based on traveling-wave microresonators,” Phys. Rev. A 95(4), 043815 (2017).
[Crossref]

M. A. Miri, F. Ruesink, E. Verhagen, and A. Alù, “Fundamentals of optical non-reciprocity based on optomechanical coupling,” Phys. Rev. Applied 7(6), 064014 (2017).
[Crossref]

2016 (5)

H. Fu, Z. Gong, T. Mao, C. Sun, S. Yi, Y. Li, and G. Cao, “Classical analog of Stuckelberg interferometry in a two-coupled-cantilever based optomechanical system,” Phys. Rev. A 94(4), 043855 (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(2), 023827 (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. Photon. 10(10), 657–661 (2016).
[Crossref]

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]

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] [PubMed]

2015 (9)

I. Söllner, S. Mahmoodian, S. L. Hansen, L. Midolo, A. Javadi, G. Kiršanskė, T. Pregnolato, H. El-Ella, E. H. Lee, J. D. Song, Søren Stobbe, and P. Lodahl, “Deterministic photon–emitter coupling in chiral photonic circuits,” Nat. Nanotechnol. 10(9), 775–778 (2015).
[Crossref] [PubMed]

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

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

Sh. Barzanjeh, S. Guha, C. Weedbrook, D. Vitali, J. H. Shapiro, and S. Pirandola, “Microwave quantum illumination,” Phys. Rev. Lett. 114(8), 080503 (2015).
[Crossref] [PubMed]

L. Tian, “Optoelectromechanical transducer: Reversible conversion between microwave and optical photons,” Ann. Phys. (Berlin) 527(1–2), 1–14 (2015).
[Crossref]

J. Kim, M. C. Kuzyk, K. Han, H. Wang, and G. Bahl, “Non-reciprocal Brillouin scattering induced transparency,” Nat. Phys. 11(3), 275–280 (2015).
[Crossref]

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

X. W. Xu and Y. Li, “Optical nonreciprocity and optomechanical circulator in three-mode optomechanical systems,” Phys. Rev. A 91(5), 053854 (2015).
[Crossref]

L. Ranzani and J. Aumentado, “Graph-based analysis of nonreciprocity in coupled-mode systems,” New J. Physics,  17(2), 023024 (2015).
[Crossref]

2014 (5)

A. Metelmann and A. A. Clerk, “Quantum-limited amplification via reservoir engineering,” Phys. Rev. Lett. 112(13), 133904 (2014).
[Crossref] [PubMed]

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]

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86(4), 1391 (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. 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(7), 524–529 (2014).
[Crossref]

2013 (9)

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).
[Crossref] [PubMed]

S. A. R. Horsley, J.-H. Wu, M. Artoni, and G. C. La Rocca, “Optical nonreciprocity of cold atom Bragg mirrors in motion,” Phys. Rev. Lett. 110(22), 223602 (2013).
[Crossref] [PubMed]

K. Qu and G. S. Agarwal, “Phonon-mediated electromagnetically induced absorption in hybrid optoelectromechanical systems,” Phys. Rev. A 87(3), 031802 (2013).
[Crossref]

P. Meystre, “A short walk through quantum optomechanics,” Ann. Phys. (Berlin) 525(3), 215–233 (2013).
[Crossref]

A. Arvanitaki and A. A. Geraci, “Detecting high-frequency gravitational waves with optically levitated sensors,” Phys. Rev. Lett. 110(7), 071105 (2013).
[Crossref] [PubMed]

M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88(1), 013804 (2013).
[Crossref]

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(3), 179–184 (2013).
[Crossref]

H. Okamoto, A. Gourgout, C. Y. Chang, K. Onomitsu, I. Mahboob, E. Y. Chang, and H. Yamaguchi, “Coherent phonon manipulation in coupled mechanical resonators,” Nat. Phys. 9(8), 480–484 (2013).
[Crossref]

T. Faust, J. Rieger, M. J. Seitner, J. P. Kotthaus, and E. M. Weig, “Coherent control of a classical nanomechanical two-level system,” Nat. Phys. 9(8), 485–488 (2013).
[Crossref]

2012 (5)

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

A. G. Krause, M. Winger, T. D. Blasius, Q. Lin, and O. Painter, “A high-resolution microchip optomechanical accelerometer,” Nat. Photon. 6(11), 768–772 (2012).
[Crossref]

M. Aspelmeyer, P. Meystre, and K. C. Schwab, “Quantum optomechanics,” Phys. Today 65(7), 29–35 (2012).
[Crossref]

S. Forstner, S. Prams, J. Knittel, E. D. van Ooijen, J. D. Swaim, G. I. Harris, A. Szorkovszky, W. P. Bowen, and H. Rubinsztein-Dunlop, “Cavity optomechanical magnetometer,” Phys. Rev. Lett. 108(12), 120801 (2012).
[Crossref] [PubMed]

F. Hocke, X. Zhou, A. Schliesser, T. J. Kippenberg, H. Huebl, and R. Gross, “Electromechanically induced absorption in a circuit nano-electromechanical system,” New J. Phys. 14(12), 123037 (2012).
[Crossref]

2011 (5)

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(7377), 351–354 (2011).
[Crossref]

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(12), 758–762 (2011).
[Crossref]

D. E. Chang, A. H. Safavi-Naeini, M. Hafezi, and O. Painter, “Slowing and stopping light using an optomechanical crystal array,” New J. Phys. 13(2), 023003 (2011).
[Crossref]

J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature (London),  471(7337), 204–208 (2011).
[Crossref]

A. H. Safavi-Naeini, T. P. M. 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(7341), 69–73 (2011).
[Crossref]

2010 (2)

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

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

2009 (3)

J. D. Teufel, T. Donner, M. A. Castellanos-Beltran, J. W. Harlow, and K. W. Lehnert, “Nanomechanical motion measured with an imprecision below that at the standard quantum limit,” Nat. Nanotechnol. 4(12), 820–823 (2009).
[Crossref] [PubMed]

Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photon. 3(2), 91–94 (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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2008 (2)

C. A. Regal, J. D. Teufel, and K. W. Lehnert, “Measuring nanomechanical motion with a microwave cavity interferometer,” Nat. Phys. 4(7), 555–560 (2008).
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T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science 321(5893), 1172–1176 (2008).
[Crossref] [PubMed]

2007 (1)

D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98(3), 030405 (2007).
[Crossref] [PubMed]

2004 (1)

D. Rugar, R. Budakian, H. J. Mamin, and B. W. Chui, “Single spin detection by magnetic resonance force microscopy,” Nature (London) 430(6997), 329–332 (2004).
[Crossref]

2003 (2)

S. Mancini, D. Vitali, and P. Tombesi, “Scheme for teleportation of quantum states onto a mechanical resonator,” Phys. Rev. Lett. 90(13), 137901 (2003).
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S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91(4), 043902 (2003).
[Crossref] [PubMed]

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(1), 74 (2000).
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1991 (1)

D. Rugar and P. Grütter, “Mechanical parametric amplification and thermomechanical noise squeezing,” Phys. Rev. Lett. 67(6), 699 (1991).
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Agarwal, G. S.

K. Qu and G. S. Agarwal, “Phonon-mediated electromagnetically induced absorption in hybrid optoelectromechanical systems,” Phys. Rev. A 87(3), 031802 (2013).
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G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81(4), 041803 (2010).
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Alegre, T. P. M.

A. H. Safavi-Naeini, T. P. M. 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(7341), 69–73 (2011).
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Allman, M. S.

J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature (London),  471(7337), 204–208 (2011).
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Alù, A.

M. A. Miri, F. Ruesink, E. Verhagen, and A. Alù, “Fundamentals of optical non-reciprocity based on optomechanical coupling,” Phys. Rev. Applied 7(6), 064014 (2017).
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F. Ruesink, M. A. Miri, A. Alù, and E. Verhagen, “Nonreciprocity and magnetic-free isolation based on optomechanical interactions,” Nat. Commun. 7, 13662 (2016).
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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).
[Crossref]

Arcizet, O.

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

A. Arvanitaki and A. A. Geraci, “Detecting high-frequency gravitational waves with optically levitated sensors,” Phys. Rev. Lett. 110(7), 071105 (2013).
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Aspelmeyer, M.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86(4), 1391 (2014).
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M. Aspelmeyer, P. Meystre, and K. C. Schwab, “Quantum optomechanics,” Phys. Today 65(7), 29–35 (2012).
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D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98(3), 030405 (2007).
[Crossref] [PubMed]

Aumentado, J.

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. Applied 7(2), 024028 (2017).
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L. Ranzani and J. Aumentado, “Graph-based analysis of nonreciprocity in coupled-mode systems,” New J. Physics,  17(2), 023024 (2015).
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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,” arXiv: 1703.05269.

Bahl, G.

J. Kim, M. C. Kuzyk, K. Han, H. Wang, and G. Bahl, “Non-reciprocal Brillouin scattering induced transparency,” Nat. Phys. 11(3), 275–280 (2015).
[Crossref]

Barzanjeh, Sh.

Sh. Barzanjeh, S. Guha, C. Weedbrook, D. Vitali, J. H. Shapiro, and S. Pirandola, “Microwave quantum illumination,” Phys. Rev. Lett. 114(8), 080503 (2015).
[Crossref] [PubMed]

Bawaj, M.

M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88(1), 013804 (2013).
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Bernier, N. R.

D. Malz, L. D. Toth, N. R. Bernier, A. K. Feofanov, T. J. Kippenberg, and A. Nunnenkamp, “Quantum-limited directional amplifiers with optomechanics,” arXiv: 1705.00436.

N. R. Bernier, L. D. Tóth, A. Koottandavida, M. Ioannou, D. Malz, A. Nunnenkamp, A. K. Feofanov, and T. J. Kippenberg, “Nonreciprocal reconfigurable microwave optomechanical circuit,” arXiv: 1612.08223 (2016).

Bi, L.

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(12), 758–762 (2011).
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Biancofiore, C.

M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88(1), 013804 (2013).
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Blasius, T. D.

A. G. Krause, M. Winger, T. D. Blasius, Q. Lin, and O. Painter, “A high-resolution microchip optomechanical accelerometer,” Nat. Photon. 6(11), 768–772 (2012).
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Böhm, H. R.

D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98(3), 030405 (2007).
[Crossref] [PubMed]

Bowen, W. P.

S. Forstner, S. Prams, J. Knittel, E. D. van Ooijen, J. D. Swaim, G. I. Harris, A. Szorkovszky, W. P. Bowen, and H. Rubinsztein-Dunlop, “Cavity optomechanical magnetometer,” Phys. Rev. Lett. 108(12), 120801 (2012).
[Crossref] [PubMed]

Budakian, R.

D. Rugar, R. Budakian, H. J. Mamin, and B. W. Chui, “Single spin detection by magnetic resonance force microscopy,” Nature (London) 430(6997), 329–332 (2004).
[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(1), 74 (2000).
[Crossref] [PubMed]

Cao, G.

H. Fu, Z. Gong, T. Mao, C. Sun, S. Yi, Y. Li, and G. Cao, “Classical analog of Stuckelberg interferometry in a two-coupled-cantilever based optomechanical system,” Phys. Rev. A 94(4), 043855 (2016).
[Crossref]

Castellanos-Beltran, M. A.

J. D. Teufel, T. Donner, M. A. Castellanos-Beltran, J. W. Harlow, and K. W. Lehnert, “Nanomechanical motion measured with an imprecision below that at the standard quantum limit,” Nat. Nanotechnol. 4(12), 820–823 (2009).
[Crossref] [PubMed]

Chan, J.

A. H. Safavi-Naeini, T. P. M. 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(7341), 69–73 (2011).
[Crossref]

Chang, C. Y.

H. Okamoto, A. Gourgout, C. Y. Chang, K. Onomitsu, I. Mahboob, E. Y. Chang, and H. Yamaguchi, “Coherent phonon manipulation in coupled mechanical resonators,” Nat. Phys. 9(8), 480–484 (2013).
[Crossref]

Chang, D. E.

A. H. Safavi-Naeini, T. P. M. 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(7341), 69–73 (2011).
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D. E. Chang, A. H. Safavi-Naeini, M. Hafezi, and O. Painter, “Slowing and stopping light using an optomechanical crystal array,” New J. Phys. 13(2), 023003 (2011).
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Chang, E. Y.

H. Okamoto, A. Gourgout, C. Y. Chang, K. Onomitsu, I. Mahboob, E. Y. Chang, and H. Yamaguchi, “Coherent phonon manipulation in coupled mechanical resonators,” Nat. Phys. 9(8), 480–484 (2013).
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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(7), 524–529 (2014).
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Chen, A. X.

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(2), 023827 (2016).
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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. Photon. 10(10), 657–661 (2016).
[Crossref]

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(7377), 351–354 (2011).
[Crossref]

Chui, B. W.

D. Rugar, R. Budakian, H. J. Mamin, and B. W. Chui, “Single spin detection by magnetic resonance force microscopy,” Nature (London) 430(6997), 329–332 (2004).
[Crossref]

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. Applied 7(2), 024028 (2017).
[Crossref]

J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature (London),  471(7337), 204–208 (2011).
[Crossref]

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,” arXiv: 1703.05269.

Clerk, A. A.

K. Fang, J. Luo, A. Metelmann, M. H. Matheny, F. Marquardt, A. A. Clerk, and O. Painter, “Generalized nonreciprocity in an optomechanical circuit via synthetic magnetism and reservoir engineering,” Nat. Phys. 13, 465 (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).

A. Metelmann and A. A. Clerk, “Quantum-limited amplification via reservoir engineering,” Phys. Rev. Lett. 112(13), 133904 (2014).
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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).
[Crossref] [PubMed]

Dionne, G. F.

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(12), 758–762 (2011).
[Crossref]

Dong, C. H.

Y. L. Zhang, C. H. Dong, C. L. Zou, X. B. Zou, Y. D. Wang, and G. C. Guo, “Optomechanical devices based on traveling-wave microresonators,” Phys. Rev. A 95(4), 043815 (2017).
[Crossref]

Dong, C.-H.

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(10), 657–661 (2016).
[Crossref]

Donner, T.

J. D. Teufel, T. Donner, M. A. Castellanos-Beltran, J. W. Harlow, and K. W. Lehnert, “Nanomechanical motion measured with an imprecision below that at the standard quantum limit,” Nat. Nanotechnol. 4(12), 820–823 (2009).
[Crossref] [PubMed]

Eichenfield, M.

A. H. Safavi-Naeini, T. P. M. 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(7341), 69–73 (2011).
[Crossref]

El-Ella, H.

I. Söllner, S. Mahmoodian, S. L. Hansen, L. Midolo, A. Javadi, G. Kiršanskė, T. Pregnolato, H. El-Ella, E. H. Lee, J. D. Song, Søren Stobbe, and P. Lodahl, “Deterministic photon–emitter coupling in chiral photonic circuits,” Nat. Nanotechnol. 10(9), 775–778 (2015).
[Crossref] [PubMed]

Estep, N. A.

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]

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).
[Crossref] [PubMed]

Fan, S.

Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photon. 3(2), 91–94 (2009).
[Crossref]

Fang, K.

K. Fang, J. Luo, A. Metelmann, M. H. Matheny, F. Marquardt, A. A. Clerk, and O. Painter, “Generalized nonreciprocity in an optomechanical circuit via synthetic magnetism and reservoir engineering,” Nat. Phys. 13, 465 (2017).
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Faust, T.

T. Faust, J. Rieger, M. J. Seitner, J. P. Kotthaus, and E. M. Weig, “Coherent control of a classical nanomechanical two-level system,” Nat. Phys. 9(8), 485–488 (2013).
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Feofanov, A. K.

N. R. Bernier, L. D. Tóth, A. Koottandavida, M. Ioannou, D. Malz, A. Nunnenkamp, A. K. Feofanov, and T. J. Kippenberg, “Nonreciprocal reconfigurable microwave optomechanical circuit,” arXiv: 1612.08223 (2016).

D. Malz, L. D. Toth, N. R. Bernier, A. K. Feofanov, T. J. Kippenberg, and A. Nunnenkamp, “Quantum-limited directional amplifiers with optomechanics,” arXiv: 1705.00436.

Ferreira, A.

D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98(3), 030405 (2007).
[Crossref] [PubMed]

Forstner, S.

S. Forstner, S. Prams, J. Knittel, E. D. van Ooijen, J. D. Swaim, G. I. Harris, A. Szorkovszky, W. P. Bowen, and H. Rubinsztein-Dunlop, “Cavity optomechanical magnetometer,” Phys. Rev. Lett. 108(12), 120801 (2012).
[Crossref] [PubMed]

Fu, H.

H. Fu, Z. Gong, T. Mao, C. Sun, S. Yi, Y. Li, and G. Cao, “Classical analog of Stuckelberg interferometry in a two-coupled-cantilever based optomechanical system,” Phys. Rev. A 94(4), 043855 (2016).
[Crossref]

Galassi, M.

M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88(1), 013804 (2013).
[Crossref]

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).
[Crossref] [PubMed]

Geraci, A. A.

A. Arvanitaki and A. A. Geraci, “Detecting high-frequency gravitational waves with optically levitated sensors,” Phys. Rev. Lett. 110(7), 071105 (2013).
[Crossref] [PubMed]

Gigan, S.

D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98(3), 030405 (2007).
[Crossref] [PubMed]

Giuseppe, G. Di

M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88(1), 013804 (2013).
[Crossref]

Gong, Z.

H. Fu, Z. Gong, T. Mao, C. Sun, S. Yi, Y. Li, and G. Cao, “Classical analog of Stuckelberg interferometry in a two-coupled-cantilever based optomechanical system,” Phys. Rev. A 94(4), 043855 (2016).
[Crossref]

Gourgout, A.

H. Okamoto, A. Gourgout, C. Y. Chang, K. Onomitsu, I. Mahboob, E. Y. Chang, and H. Yamaguchi, “Coherent phonon manipulation in coupled mechanical resonators,” Nat. Phys. 9(8), 480–484 (2013).
[Crossref]

Gross, R.

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(3), 179–184 (2013).
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F. Hocke, X. Zhou, A. Schliesser, T. J. Kippenberg, H. Huebl, and R. Gross, “Electromechanically induced absorption in a circuit nano-electromechanical system,” New J. Phys. 14(12), 123037 (2012).
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Grütter, P.

D. Rugar and P. Grütter, “Mechanical parametric amplification and thermomechanical noise squeezing,” Phys. Rev. Lett. 67(6), 699 (1991).
[Crossref] [PubMed]

Guerreiro, A.

D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98(3), 030405 (2007).
[Crossref] [PubMed]

Guha, S.

Sh. Barzanjeh, S. Guha, C. Weedbrook, D. Vitali, J. H. Shapiro, and S. Pirandola, “Microwave quantum illumination,” Phys. Rev. Lett. 114(8), 080503 (2015).
[Crossref] [PubMed]

Guo, G. C.

Y. L. Zhang, C. H. Dong, C. L. Zou, X. B. Zou, Y. D. Wang, and G. C. Guo, “Optomechanical devices based on traveling-wave microresonators,” Phys. Rev. A 95(4), 043815 (2017).
[Crossref]

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. Photon. 10(10), 657–661 (2016).
[Crossref]

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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Guo, X.

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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Hafezi, M.

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

D. E. Chang, A. H. Safavi-Naeini, M. Hafezi, and O. Painter, “Slowing and stopping light using an optomechanical crystal array,” New J. Phys. 13(2), 023003 (2011).
[Crossref]

Hakonen, P. J.

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(7377), 351–354 (2011).
[Crossref]

Han, K.

J. Kim, M. C. Kuzyk, K. Han, H. Wang, and G. Bahl, “Non-reciprocal Brillouin scattering induced transparency,” Nat. Phys. 11(3), 275–280 (2015).
[Crossref]

Hansen, S. L.

I. Söllner, S. Mahmoodian, S. L. Hansen, L. Midolo, A. Javadi, G. Kiršanskė, T. Pregnolato, H. El-Ella, E. H. Lee, J. D. Song, Søren Stobbe, and P. Lodahl, “Deterministic photon–emitter coupling in chiral photonic circuits,” Nat. Nanotechnol. 10(9), 775–778 (2015).
[Crossref] [PubMed]

Harlow, J. W.

J. D. Teufel, T. Donner, M. A. Castellanos-Beltran, J. W. Harlow, and K. W. Lehnert, “Nanomechanical motion measured with an imprecision below that at the standard quantum limit,” Nat. Nanotechnol. 4(12), 820–823 (2009).
[Crossref] [PubMed]

Harris, G. I.

S. Forstner, S. Prams, J. Knittel, E. D. van Ooijen, J. D. Swaim, G. I. Harris, A. Szorkovszky, W. P. Bowen, and H. Rubinsztein-Dunlop, “Cavity optomechanical magnetometer,” Phys. Rev. Lett. 108(12), 120801 (2012).
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Figures (4)

Fig. 1
Fig. 1 Schematic of a three-mode optomechanical system driven by two pump fields with the same frequency ωd. A probe field with frequency ωp is applied to one of the two cavities, that is, incident in cavity 1 from the left side (the thin solid arrow) or incident in cavity 2 from the right side (the thin dashed arrow). The mechanical resonator is subject to a mechanical drive with the driving frequency ωb. The cavities and the mechanical resonator are coupled via radiation-pressure forces, and the cavities are directly coupled to each other.
Fig. 2
Fig. 2 The transmission probabilities T21 and T12 versus Δm = ωm (ωp − ωd) for different values of θ and φ: (a) θ = 0, φ = π/2; (b) θ = π/2, φ = 0; (c) θ−= π/2, φ = π/2. Other parameters are y = 20, η1,2 = 1, γ1 = 1.1γm, γ2 = 1.5γm, G = |G1,2| = J = γm, and Δ 1 , 2 = Δ m.
Fig. 3
Fig. 3 Plot of the probability of transmission T21 and T12 as functions of θ and φ, respectively. (a) φ = π/2. (b) θ = π/2. Other parameters are y = 20, η1,2 = 1, G = | G 1 , 2 | = J = γ m, Δ m = Δ 1 , 2 = 0, γ1 = 1.1γm, and γ2 = 1.5γm. One can see that at certain optimal values of θ and φ, e.g., θ = π/2, φ = π/2, T12 → 0 and T21 ≫ 1.
Fig. 4
Fig. 4 The transmission probabilities T21 and T12 versus y. Other parameters are θ = π/2, φ = π/2, G = |G1,2| = J = γm, Δ m = Δ 1 , 2 = 0, γ1 = 1.1γm, and γ2 = 1.5γm.

Equations (27)

Equations on this page are rendered with MathJax. Learn more.

H = H 0 + H I + H d .
H 0 = ω 1 a 1 a 1 + ω 2 a 2 a 2 + ω m b b ,
H I = J ( a 1 a 2 + a 1 a 2 ) + i g i a i a i ( b + b )
H d = i ( i ε i a i e i ω d t e i θ i + h . c . ) + ( i ε p a 1 e i ω p t + i ε b b e i ω b t + h . c . ) ,
a ˙ 1 = { γ 1 i [ Δ 1 + g 1 ( b + b ) ] } a 1 i J a 2 + ε 1 e i θ 1 + ε p e i ( ω d ω p ) t + ξ 1 ,
a ˙ 2 = { γ 2 i [ Δ 2 + g 2 ( b + b ) ] } a 2 i J a 1 + ε 2 e i θ 2 + ξ 2 ,
b ˙ = ( γ m i ω m ) b i ( g 1 a 1 a 1 + g 2 a 2 a 2 ) + ε b e i ω b t + ξ m .
a 1 = ( γ 2 + i Δ 2 ) ε 1 e i θ 1 i J ε 2 e i θ 2 ( γ 1 + i Δ 1 ) ( γ 2 + i Δ 2 ) + J 2 ,
a 2 = ( γ 1 + i Δ 1 ) ε 2 e i θ 2 i J ε 1 e i θ 1 ( γ 1 + i Δ 1 ) ( γ 2 + i Δ 2 ) + J 2 ,
b = i ( g 1 | a 1 | 2 + g 2 | a 2 | 2 ) γ m + i ω m ,
δ a ˙ 1 = ( γ 1 i Δ 1 ) δ a 1 i G 1 ( δ b + δ b ) i J δ a 2 + ε p e i ( ω d ω p ) t + ξ 1 ,
δ a ˙ 2 = ( γ 2 i Δ 2 ) δ a 2 i G 2 ( δ b + δ b ) i J δ a 1 + ξ 2 ,
δ b ˙ = ( γ m i ω m ) δ b i ( G 1 δ a 1 + G 1 * δ a 1 ) i ( G 2 δ a 2 + G 2 * δ a 2 ) + ε b e i ω b t + ξ m ,
δ a ˙ 1 = Γ 1 δ a 1 i G 1 δ b i J δ a 2 + ε p + ξ 1 ,
δ a ˙ 2 = Γ 2 δ a 2 i G 2 δ b i J δ a 1 + ξ 2 ,
δ b ˙ = Γ m δ b i G 1 * δ a 1 i G 2 * δ a 2 + ε b + ξ m ,
δ a 1 = i G 2 ε b ( i J Γ m + G 1 G 2 * ) + ( Γ 2 Γ m + | G 2 | 2 ) ( ε p Γ m i G 1 ε b ) ( Γ 1 Γ m + | G 1 | 2 ) ( Γ 2 Γ m + | G 2 | 2 ) ( i J Γ m + G 1 G 2 * ) ( i J Γ m + G 1 * G 2 ) ,
δ a 2 = ( i J Γ m + G 1 * G 2 ) ( ε p Γ m i G 1 ε b ) i G 2 ε b ( Γ 1 Γ m + | G 1 | 2 ) ( Γ 1 Γ m + | G 1 | 2 ) ( Γ 2 Γ m + | G 2 | 2 ) ( i J Γ m + G 1 G 2 * ) ( i J Γ m + G 1 * G 2 ) ,
δ b = J Γ m ( ε b J ε p G 2 * ) + Γ 2 Γ m ( ε b Γ 1 i ε p G 1 * ) ( Γ 1 Γ m + | G 1 | 2 ) ( Γ 2 Γ m + | G 2 | 2 ) ( i J Γ m + G 1 G 2 * ) ( i J Γ m + G 1 * G 2 ) .
δ a i o u t + δ a i i n = 2 γ i e δ a i ,
t 21 δ a 2 o u t / δ a 1 i n .
t 21 = 2 γ 1 e γ 2 e [ ( i J Γ m + G 1 * G 2 ) ( Γ m i G 1 y e i φ ) + i G 2 y e i φ ( Γ 1 Γ m + | G 1 | 2 ) ( Γ 1 Γ m + | G 1 | 2 ) ( Γ 2 Γ m + | G 2 | 2 ) ( i J Γ m + G 1 G 2 * ) ( i J Γ m + G 1 * G 2 ) ] ,
t 12 = 2 γ 1 e γ 2 e [ ( i J Γ m + G 2 * G 1 ) ( Γ m i G 2 y e i φ ) + i G 1 y e i φ ( Γ 2 Γ m + | G 2 | 2 ) ( Γ 2 Γ m + | G 2 | 2 ) ( Γ 1 Γ m + | G 1 | 2 ) ( i J Γ m + G 2 G 1 * ) ( i J Γ m + G 2 * G 1 ) ] .
t 21 = 2 γ 1 e γ 2 e [ ( i J Γ m + G 2 e i θ ) ( Γ m i G y e i φ ) + i G ( Γ 1 Γ m + G 2 ) y e i ( θ + φ ) ( Γ 1 Γ m + G 2 ) ( Γ 2 Γ m + G 2 ) ( i J Γ m + G 2 e i θ ) ( i J Γ m + G 2 e i θ ) ] ,
t 12 = 2 γ 1 e γ 2 e [ ( i J Γ m + G 2 e i θ ) ( Γ m i G y e i ( θ + φ ) ) + i G ( Γ 2 Γ m + G 2 ) y e i φ ( Γ 1 Γ m + G 2 ) ( Γ 2 Γ m + G 2 ) ( i J Γ m + G 2 e i θ ) ( i J Γ m + G 2 e i θ ) ] .
T 21 = 4 γ 1 γ 2 ( 2 γ m ( y + 1 ) y ( γ 1 + γ m ) ( γ 1 + γ m ) ( γ 2 + γ m ) ) 2 ,
T 12 = 4 y 2 γ 1 γ 2 ( γ 1 + γ m ) 2 .

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