Abstract

We propose to achieve nonreciprocal quantum control of photons in a quadratic optomechanical (QOM) system based on directional nonlinear interactions. We show that by optically pumping the QOM system in one side, the effective QOM coupling can be enhanced significantly in that side, but not for the other side. This, contrary to the intuitive picture, allows the emergence of a nonreciprocal photon blockade in such optomechanical devices with weak single-photon QOM coupling. Our proposal opens up the prospect of exploring and utilizing quantum nonreciprocal optomechanics, with applications ranging from single-photon nonreciprocal devices to on-chip chiral quantum engineering.

© 2020 Chinese Laser Press

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2018 (11)

Y.-L. Liu, C. Wang, J. Zhang, and Y.-X. Liu, “Cavity optomechanics: manipulating photons and phonons towards the single-photon strong coupling,” Chin. Phys. B 27, 024204 (2018).
[Crossref]

G. Li, X. Xiao, Y. Li, and X. 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]

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]

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

X.-W. Xu, H.-Q. Shi, A.-X. Chen, and Y.-X. Liu, “Cross-correlation between photons and phonons in quadratically coupled optomechanical systems,” Phys. Rev. A 98, 013821 (2018).
[Crossref]

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

H. Jing, H. Lü, S. K. Özdemir, T. Carmon, and F. Nori, “Nanoparticle sensing with a spinning resonator,” Optica 5, 1424–1430 (2018).
[Crossref]

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

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

Y.-L. Zhang, C.-L. Zou, C.-S. Yang, H. Jing, C.-H. Dong, G.-C. Guo, and X.-B. Zou, “Phase-controlled phonon laser,” New J. Phys. 20, 093005 (2018).
[Crossref]

J. Zhang, B. Peng, S. Ozdemir, K. Pichler, D. Krimer, G. Zhao, F. Nori, Y.-X. Liu, S. Rotter, and L. Yang, “A phonon laser operating at an exceptional point,” Nat. Photonics 12, 479–484 (2018).
[Crossref]

2017 (12)

H. Zhang, X. Zhao, Y. Wang, Q. Huang, and J. Xia, “Femtogram scale high frequency nano-optomechanical resonators in water,” Opt. Express 25, 821–830 (2017).
[Crossref]

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

H. Lü, Y. Jiang, Y.-Z. Wang, and H. Jing, “Optomechanically induced transparency in a spinning resonator,” Photon. Res. 5, 367–371 (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]

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]

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).
[Crossref]

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]

H. Xie, C.-G. Liao, X. Shang, M.-Y. Ye, and X.-M. Lin, “Phonon blockade in a quadratically coupled optomechanical system,” Phys. Rev. A 96, 013861 (2017).
[Crossref]

A. Metelmann and A. A. Clerk, “Nonreciprocal quantum interactions and devices via autonomous feedforward,” Phys. Rev. A 95, 013837 (2017).
[Crossref]

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

H. Qiu, J. Dong, L. Liu, and X. Zhang, “Energy-efficient on-chip optical diode based on the optomechanical effect,” Opt. Express 25, 8975–8985 (2017).
[Crossref]

2016 (8)

K. Fang, M. Matheny, X. Luan, and O. Painter, “Optical transduction and routing of microwave phonons in cavity-optomechanical circuits,” Nat. Photonics 10, 489–496 (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]

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, 657–661 (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]

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

G. Brawley, M. Vanner, P. Larsen, S. Schmid, A. Boisen, and W. Bowen, “Non-linear optomechanical measurement of mechanical motion,” Nat. Commun. 7, 10988 (2016).
[Crossref]

M. J. Burek, J. D. Cohen, S. M. Meenehan, N. El-Sawah, C. Chia, T. Ruelle, S. Meesala, J. Rochman, H. A. Atikian, M. Markham, D. J. Twitchen, M. D. Lukin, O. Painter, and M. Lončar, “Diamond optomechanical crystals,” Optica 3, 1404–1411 (2016).
[Crossref]

S. Hua, J. Wen, X. Jiang, Q. Hua, L. Jiang, and M. Xiao, “Demonstration of a chip-based optical isolator with parametric amplification,” Nat. Commun. 7, 13657 (2016).
[Crossref]

2015 (12)

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).
[Crossref]

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 520, 522–525 (2015).
[Crossref]

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

X.-Y. Lü, Y. Wu, J. R. Johansson, H. Jing, J. Zhang, and F. Nori, “Squeezed optomechanics with phase-matched amplification and dissipation,” Phys. Rev. Lett. 114, 093602 (2015).
[Crossref]

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

Y. Shi, Z. Yu, and S. Fan, “Limitations of nonlinear optical isolators due to dynamic reciprocity,” Nat. Photonics 9, 388–392 (2015).
[Crossref]

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

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

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

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

2014 (5)

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

M. Metcalfe, “Applications of cavity optomechanics,” Appl. Phys. Rev. 1, 031105 (2014).
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C. Doolin, B. D. Hauer, P. H. Kim, A. J. R. MacDonald, H. Ramp, and J. P. Davis, “Nonlinear optomechanics in the stationary regime,” Phys. Rev. A 89, 053838 (2014).
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H. Jing, S. K. Özdemir, X.-Y. Lü, J. Zhang, L. Yang, and F. Nori, “PT-symmetric phonon laser,” Phys. Rev. Lett. 113, 053604 (2014).
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H. Wang, Z. Wang, J. Zhang, S. K. Özdemir, L. Yang, and Y.-X. Liu, “Phonon amplification in two coupled cavities containing one mechanical resonator,” Phys. Rev. A 90, 053814 (2014).
[Crossref]

2013 (2)

H. Wu, G. Heinrich, and F. Marquardt, “The effect of Landau-Zener dynamics on phonon lasing,” New J. Phys. 15, 123022 (2013).
[Crossref]

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popovic, A. Melloni, J. Joannopoulos, M. Vanwolleghem, C. Doerr, and H. Renner, “What is–and what is not–an optical isolator,” Nat. Photonics 7, 579–582 (2013).
[Crossref]

2012 (4)

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

M. Hafezi and P. Rabl, “Optomechanically induced non-reciprocity in microring resonators,” Opt. Express 20, 7672–7684 (2012).
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H.-K. Li, Y.-C. Liu, X. Yi, C.-L. Zou, X.-X. Ren, and Y.-F. Xiao, “Proposal for a near-field optomechanical system with enhanced linear and quadratic coupling,” Phys. Rev. A 85, 053832 (2012).
[Crossref]

A. Xuereb, C. Genes, and A. Dantan, “Strong coupling and long-range collective interactions in optomechanical arrays,” Phys. Rev. Lett. 109, 223601 (2012).
[Crossref]

2011 (2)

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 472, 69–73 (2011).
[Crossref]

H. Jing, D. S. Goldbaum, L. Buchmann, and P. Meystre, “Quantum optomechanics of a Bose-Einstein antiferromagnet,” Phys. Rev. Lett. 106, 223601 (2011).
[Crossref]

2010 (4)

G. Heinrich, J. G. E. Harris, and F. Marquardt, “Photon shuttle: Landau-Zener-Stückelberg dynamics in an optomechanical system,” Phys. Rev. A 81, 011801 (2010).
[Crossref]

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

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

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).
[Crossref]

2009 (3)

S. Manipatruni, J. T. Robinson, and M. Lipson, “Optical nonreciprocity in optomechanical structures,” Phys. Rev. Lett. 102, 213903 (2009).
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F. Marquardt and S. M. Girvin, “Optomechanics,” Physics 2, 40 (2009).
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G. Anetsberger, O. Arcizet, Q. P. Unterreithmeier, R. Rivière, A. Schliesser, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, “Near-field cavity optomechanics with nanomechanical oscillators,” Nat. Phys. 5, 909–914 (2009).
[Crossref]

2008 (2)

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science 321, 1172–1176 (2008).
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J. Thompson, B. Zwickl, A. Jayich, F. Marquardt, S. Girvin, and J. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008).
[Crossref]

2005 (1)

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
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2000 (1)

G. B. Malykin, “The Sagnac effect: correct and incorrect explanations,” Phys. Usp. 43, 1229–1252 (2000).
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1997 (1)

S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50, 36–42 (1997).
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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, 3761–3774 (1985).
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Agarwal, G. S.

G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81, 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 472, 69–73 (2011).
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Alù, A.

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

G. Anetsberger, O. Arcizet, Q. P. Unterreithmeier, R. Rivière, A. Schliesser, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, “Near-field cavity optomechanics with nanomechanical oscillators,” Nat. Phys. 5, 909–914 (2009).
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Arcizet, O.

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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G. Anetsberger, O. Arcizet, Q. P. Unterreithmeier, R. Rivière, A. Schliesser, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, “Near-field cavity optomechanics with nanomechanical oscillators,” Nat. Phys. 5, 909–914 (2009).
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Aspelmeyer, M.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
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M. Aspelmeyer, P. Meystre, and K. Schwab, “Quantum optomechanics,” Phys. Today 65, 29–35 (2012).
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Atikian, H. A.

Aumentado, J.

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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Baets, R.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popovic, A. Melloni, J. Joannopoulos, M. Vanwolleghem, C. Doerr, and H. Renner, “What is–and what is not–an optical isolator,” Nat. Photonics 7, 579–582 (2013).
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Bahl, G.

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

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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Bernier, N. R.

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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Boisen, A.

G. Brawley, M. Vanner, P. Larsen, S. Schmid, A. Boisen, and W. Bowen, “Non-linear optomechanical measurement of mechanical motion,” Nat. Commun. 7, 10988 (2016).
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G. Brawley, M. Vanner, P. Larsen, S. Schmid, A. Boisen, and W. Bowen, “Non-linear optomechanical measurement of mechanical motion,” Nat. Commun. 7, 10988 (2016).
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G. Brawley, M. Vanner, P. Larsen, S. Schmid, A. Boisen, and W. Bowen, “Non-linear optomechanical measurement of mechanical motion,” Nat. Commun. 7, 10988 (2016).
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Buchmann, L.

H. Jing, D. S. Goldbaum, L. Buchmann, and P. Meystre, “Quantum optomechanics of a Bose-Einstein antiferromagnet,” Phys. Rev. Lett. 106, 223601 (2011).
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S. Maayani, R. Dahan, Y. Kligerman, E. Moses, A. Hassan, H. Jing, F. Nori, D. Christodoulides, and T. Carmon, “Flying couplers above spinning resonators generate irreversible refraction,” Nature 558, 569–572 (2018).
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H. Jing, H. Lü, S. K. Özdemir, T. Carmon, and F. Nori, “Nanoparticle sensing with a spinning resonator,” Optica 5, 1424–1430 (2018).
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Y. Jiang, S. Maayani, T. Carmon, F. Nori, and H. Jing, “Nonreciprocal phonon laser,” Phys. Rev. Appl. 10, 064037 (2018).
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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 472, 69–73 (2011).
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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 472, 69–73 (2011).
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X.-W. Xu, H.-Q. Shi, A.-X. Chen, and Y.-X. Liu, “Cross-correlation between photons and phonons in quadratically coupled optomechanical systems,” Phys. Rev. A 98, 013821 (2018).
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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. Photonics 10, 657–661 (2016).
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Chia, C.

Cho, S.

J.-M. Pirkkalainen, S. Cho, F. Massel, J. Tuorila, T. Heikkila, P. Hakonen, and M. Sillanpaa, “Cavity optomechanics mediated by a quantum two-level system,” Nat. Commun. 6, 6981 (2015).
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Christodoulides, D.

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

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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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 quantum interactions and devices via autonomous feedforward,” Phys. Rev. A 95, 013837 (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).
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M. J. Burek, J. D. Cohen, S. M. Meenehan, N. El-Sawah, C. Chia, T. Ruelle, S. Meesala, J. Rochman, H. A. Atikian, M. Markham, D. J. Twitchen, M. D. Lukin, O. Painter, and M. Lončar, “Diamond optomechanical crystals,” Optica 3, 1404–1411 (2016).
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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 520, 522–525 (2015).
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Collett, M. J.

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, 3761–3774 (1985).
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Dahan, R.

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

A. Xuereb, C. Genes, and A. Dantan, “Strong coupling and long-range collective interactions in optomechanical arrays,” Phys. Rev. Lett. 109, 223601 (2012).
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Davis, J. P.

C. Doolin, B. D. Hauer, P. H. Kim, A. J. R. MacDonald, H. Ramp, and J. P. Davis, “Nonlinear optomechanics in the stationary regime,” Phys. Rev. A 89, 053838 (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, 1520–1523 (2010).
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Devoret, M. H.

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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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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Doerr, C.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popovic, A. Melloni, J. Joannopoulos, M. Vanwolleghem, C. Doerr, and H. Renner, “What is–and what is not–an optical isolator,” Nat. Photonics 7, 579–582 (2013).
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Dong, C.-H.

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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Y.-L. Zhang, C.-L. Zou, C.-S. Yang, H. Jing, C.-H. Dong, G.-C. Guo, and X.-B. Zou, “Phase-controlled phonon laser,” New J. Phys. 20, 093005 (2018).
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Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10, 657–661 (2016).
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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).
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Dong, J.

Doolin, C.

C. Doolin, B. D. Hauer, P. H. Kim, A. J. R. MacDonald, H. Ramp, and J. P. Davis, “Nonlinear optomechanics in the stationary regime,” Phys. Rev. A 89, 053838 (2014).
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Eich, M.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popovic, A. Melloni, J. Joannopoulos, M. Vanwolleghem, C. Doerr, and H. Renner, “What is–and what is not–an optical isolator,” Nat. Photonics 7, 579–582 (2013).
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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 472, 69–73 (2011).
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El-Sawah, N.

Fan, S.

Y. Shi, Z. Yu, and S. Fan, “Limitations of nonlinear optical isolators due to dynamic reciprocity,” Nat. Photonics 9, 388–392 (2015).
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D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popovic, A. Melloni, J. Joannopoulos, M. Vanwolleghem, C. Doerr, and H. Renner, “What is–and what is not–an optical isolator,” Nat. Photonics 7, 579–582 (2013).
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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, M. Matheny, X. Luan, and O. Painter, “Optical transduction and routing of microwave phonons in cavity-optomechanical circuits,” Nat. Photonics 10, 489–496 (2016).
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Feofanov, A. K.

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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Fink, J. M.

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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Fleischhauer, M.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
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Freude, W.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popovic, A. Melloni, J. Joannopoulos, M. Vanwolleghem, C. Doerr, and H. Renner, “What is–and what is not–an optical isolator,” Nat. Photonics 7, 579–582 (2013).
[Crossref]

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).
[Crossref]

Fu, W.

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]

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, 3761–3774 (1985).
[Crossref]

Gavartin, E.

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

Genes, C.

A. Xuereb, C. Genes, and A. Dantan, “Strong coupling and long-range collective interactions in optomechanical arrays,” Phys. Rev. Lett. 109, 223601 (2012).
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R. Schilling, H. Schütz, A. H. Ghadimi, V. Sudhir, D. J. Wilson, and T. J. Kippenberg, “Near-field integration of a SiN nanobeam and a SiO2 microcavity for Heisenberg-limited displacement sensing,” Phys. Rev. Appl. 5, 054019 (2016).
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Girvin, S.

J. Thompson, B. Zwickl, A. Jayich, F. Marquardt, S. Girvin, and J. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008).
[Crossref]

Girvin, S. M.

F. Marquardt and S. M. Girvin, “Optomechanics,” Physics 2, 40 (2009).
[Crossref]

Goldbaum, D. S.

H. Jing, D. S. Goldbaum, L. Buchmann, and P. Meystre, “Quantum optomechanics of a Bose-Einstein antiferromagnet,” Phys. Rev. Lett. 106, 223601 (2011).
[Crossref]

Gröblacher, S.

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 520, 522–525 (2015).
[Crossref]

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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).
[Crossref]

Guo, G.-C.

Y.-L. Zhang, C.-L. Zou, C.-S. Yang, H. Jing, C.-H. Dong, G.-C. Guo, and X.-B. Zou, “Phase-controlled phonon laser,” New J. Phys. 20, 093005 (2018).
[Crossref]

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

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

Hafezi, M.

Hakonen, P.

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

Han, K.

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

Harris, J.

J. Thompson, B. Zwickl, A. Jayich, F. Marquardt, S. Girvin, and J. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008).
[Crossref]

Harris, J. G. E.

G. Heinrich, J. G. E. Harris, and F. Marquardt, “Photon shuttle: Landau-Zener-Stückelberg dynamics in an optomechanical system,” Phys. Rev. A 81, 011801 (2010).
[Crossref]

Harris, S. E.

S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50, 36–42 (1997).
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Figures (4)

Fig. 1.
Fig. 1. (a), (b) Schematic diagram for generating QOM coupling, where a mechanical nanostring oscillator is placed between two whispering gallery mode (WGM) resonators. (c), (d) Dispersion of the optical modes as a function of the displacement.
Fig. 2.
Fig. 2. (a) The transmission coefficients T21 (solid black curve) and T12 (dashed red curve) as a function of the detuning Δ/G. (b) The isolation as a function of the detuning Δ/G. (c) The equal-time second-order correlation function log10[gij(2)(0)] (ij=12,21) as a function of the detuning Δ/G. (d) The second-order correlation function log10[g21(2)(τ)] as a function of the normalized time delay γcτ/(2π) at detuning Δ=2G. The other parameters are Δm=Δ/2, G=3γc, ε=γc/10, γm=γc/100, and nth=0.
Fig. 3.
Fig. 3. Schematic energy spectrum of the linearized QOM coupling between optical mode aL and mechanical resonator b, where |00|0,0, |10|0,1, |2±1(|1,0±|0,2)/2, |3±1(|1,1±|0,3)/2, |40(3|2,0+|0,4)/2, |4±1(|2,0±2|1,2+3|0,4)/(22), and |n,m represents the Fock state with n photons in aL and m phonons in b.
Fig. 4.
Fig. 4. (a) Transmission coefficient T21. (b) The equal-time second-order correlation function log10[g21(2)(0)] versus the detuning Δ/G with different mean thermal phonon number (nth=0,0.1,1). (c) The isolation T21/T12. (d) The equal-time second-order correlation function log10[g21(2)(0)] versus the mean thermal phonon number nth with different detuning (Δ=0,2G,6G). The other parameters are the same as in Fig. 2.

Equations (30)

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Hom=j=1,2λ=cw,ccw[ω0+(1)jg0q]aj,λaj,λ+J(a1,ccwa2,cw+a1,cwa2,ccw+H.c.)+12ωm(q2+p2),
Hom(ω++gq2)(aL,+aL,++aR,+aR,+)+(ωgq2)(aL,aL,+aR,aR,)+12ωm(q2+p2)
Hom+driv=(Δa+gq2)(aLaL+aRaR)+12ωm(q2+p2)+ΩaL+ΩaL,
Heff=Δa(aLaL+aRaR)+ωmbb+GaLb2+G*aLb2,
Heff=ΔaLaL+ΔaRaR+Δmbb+GaLb2+G*aLb2,
T21a2,outa2,outa1,ina1,in=γc24ε2aLaL,
T12a1,outa1,outa2,ina2,in=γc24ε2aRaR,
g21(2)(τ)a2,out(t)a2,out(t+τ)a2,out(t+τ)a2,out(t)a2,out(t)a2,out(t)2=aL(t)aL(t+τ)aL(t+τ)aL(t)aL(t)aL(t)2,
g12(2)(τ)a1,out(t)a1,out(t+τ)a1,out(t+τ)a1,out(t)a1,out(t)a1,out(t)2=aR(t)aR(t+τ)aR(t+τ)aR(t)aR(t)aR(t)2,
ρt=i[Htot,ρ]+γcL[aL]ρ+γcL[aR]ρ+γm(nth+1)L[b]ρ+γmnthL[b]ρ,
Hom=(a1,ccwa2,cw)(ω0g0qJJω0+g0q)(a1,ccwa2,cw)+(a1,cwa2,ccw)(ω0g0qJJω0+g0q)(a1,cwa2,ccw)+12ωm(q2+p2),
Hom=(aL,+aL,)(ω+(q)00ω(q))(aL,+aL,)+(aR,+aR,)(ω+(q)00ω(q))(aR,+aR,)+12ωm(q2+p2),
aL,±=1D±[Ja1,ccw+(g0q±J2+(g0q)2)a2,cw]
aR,±=1D±[Ja1,cw+(g0q±J2+(g0q)2)a2,ccw],
D±2=J2+(g0q±J2+(g0q)2)2,
ω±(q)=ω0±J2+(g0q)2.
ω±(q)ω±±g022Jq2,
αL=i2Ωγc+i2Δa
αR=qs=ps=0.
Heff=ΔaaLaL+ΔaaRaR+ωmbb+g2(|αL|2+aLaL+aRaR)(b+b)2+g2(αLaL+αL*aL)(b+b)2.
|00|0,0
|10|0,1,
|2±112(|1,0±|0,2),
Δ2,±=±2G.
|3±112(|1,1±|0,3),
Δ3,±=±6G.
|4012(3|2,0+|0,4),
|4±1122(|2,0±2|1,2+3|0,4),
Δ4,0=0,
Δ4,±=±4G.

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