Abstract

We study the nonreciprocal transmission and the fast-slow light effects in a cavity optomechanical system, in which the cavity supports a clockwise and a counter-clockwise circulating optical mode; both the modes are driven simultaneously by a strong pump field and a weak signal field. We find that the system reveals a nonreciprocal transmission of the signal fields when the intrinsic photon loss of the cavity is equal to the external coupling loss of the cavity. However, when the intrinsic photon loss is much less than the external coupling loss, the nonreciprocity about the transmission properties almost disappears, the nonreciprocity is shown in the group delay properties of the signal fields, and the system exhibits a nonreciprocal fast-slow light propagation phenomenon.

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

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References

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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
  31. 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(7341), 69–73 (2011).
    [Crossref]
  32. 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]
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    [Crossref] [PubMed]
  34. It should be point out that Pc cannot be zero, since the linearization demands a strong the pump fields. For the parameters here, we take Pc = 100 nW, and we have C0 ≈ 28.
  35. J. M. Dobrindt, I. Wilson-Rae, and T. J. Kippenberg, “Parametric normal-mode splitting in cavity optomechanics,” Phys. Rev. Lett. 101(26), 263602 (2008).
    [Crossref] [PubMed]

2019 (3)

Q. Bin, X.-Y. Lü, T.-S. Yin, Y. Li, and Y. Wu, “Collective radiance effects in the ultrastrong-coupling regime,” Phys. Rev. A 99(3), 033809 (2019).
[Crossref]

L.-L. Zheng, T.-S. Yin, Q. Bin, X.-Y. Lü, and Y. Wu, “Single-photon-induced phonon blockade in a hybrid spin-optomechanical system,” Phys. Rev. A 99(1), 013804 (2019).
[Crossref]

L. M. de Lépinay, E. Damskägg, C. F. Ockeloen-Korppi, and M. A. Sillanpää, “Realization of directional amplification in a microwave optomechanical device,” Phys. Rev. Appl. 11(3), 034027 (2019).
[Crossref]

2018 (2)

R. Huang, A. Miranowicz, J.-Q. Liao, F. Nori, and H. Jing, “Nonreciprocal photon blockade,” Phys. Rev. Lett. 121(15), 153601 (2018).
[Crossref] [PubMed]

D. Malz, L. D. Tóth, N. R. Bernier, A. K. Feofanov, T. J. Kippenberg, and A. Nunnenkamp, “Quantum-limited directional amplifiers with optomechanics,” Phys. Rev. Lett. 120(2), 023601 (2018).
[Crossref] [PubMed]

2017 (4)

L. Tian and Z. Li, “Nonreciprocal quantum-state conversion between microwave and optical photons,” Phys. Rev. A 96(1), 013808 (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(1), 953 (2017).
[Crossref] [PubMed]

Q. Yang, B. P. Hou, and D. G. Lai, “Local modulation of double optomechanically induced transparency and amplification,” Opt. Express 25(9), 9697–9711 (2017).
[Crossref] [PubMed]

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(3), 031001 (2017).

2016 (3)

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. Alu, 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 (2)

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

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

2014 (1)

J.-H. Wu, M. Artoni, and G. C. La Rocca, “Non-Hermitian degeneracies and unidirectional reflectionless atomic lattices,” Phys. Rev. Lett. 113(12), 123004 (2014).
[Crossref] [PubMed]

2013 (2)

K. Xia, M. Alamri, and M. S. Zubairy, “Ultrabroadband nonreciprocal transverse energy flow of light in linear passive photonic circuits,” Opt. Express 21(22), 25619–25631 (2013).
[Crossref] [PubMed]

T. A. Palomaki, J. D. Teufel, R. W. Simmonds, and K. W. Lehnert, “Entangling mechanical motion with microwave fields,” Science 342(6159), 710–713 (2013).
[Crossref] [PubMed]

2012 (1)

2011 (4)

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(7341), 69–73 (2011).
[Crossref]

M. S. Kang, A. Butsch, and P. S. J. Russell, “Reconfigurable light-driven opto-acoustic isolators in photonic crystal fibre,” Nat. Photon. 5(9), 549–553 (2011).
[Crossref]

C. Wang, C.-Z. Zhou, and Z.-Y. Li, “On-chip optical diode based on silicon photonic crystal heterojunctions,” Opt. Express 19(27), 26948–26955 (2011).
[Crossref]

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333(6043), 729–733 (2011).
[Crossref] [PubMed]

2010 (3)

C. E. Eüter, K. G. Makris, R. EI-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010).
[Crossref]

H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear PT-symmetric optical structures,” Phys. Rev. A 82(4), 043803 (2010).
[Crossref]

Q. Wang, F. Xu, Z.-Y. Yu, X.-S. Qian, X.-K. Hu, Y.-Q. Lu, and H.-T. Wang, “A bidirectional tunable optical diode based on periodically poled LiNbO3,” Opt. Express 18(7), 7340–7346 (2010).
[Crossref] [PubMed]

2009 (1)

S. Manipatruni, J. T. Robinson, and M. Lipson, “Optical nonreciprocity in optomechanical structures,” Phys. Rev. Lett. 102(21), 213903 (2009).
[Crossref] [PubMed]

2008 (1)

J. M. Dobrindt, I. Wilson-Rae, and T. J. Kippenberg, “Parametric normal-mode splitting in cavity optomechanics,” Phys. Rev. Lett. 101(26), 263602 (2008).
[Crossref] [PubMed]

2007 (2)

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]

T. R. Zaman, X. Guo, and R. J. Ram, “Faraday rotation in an InP waveguide,” Appl. Phys. Lett. 90(2), 023514 (2007).
[Crossref]

2005 (1)

2004 (1)

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(6), 3761 (1985).
[Crossref]

Alamri, M.

Alu, A.

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

Artoni, M.

J.-H. Wu, M. Artoni, and G. C. La Rocca, “Non-Hermitian degeneracies and unidirectional reflectionless atomic lattices,” Phys. Rev. Lett. 113(12), 123004 (2014).
[Crossref] [PubMed]

Aspelmeyer, M.

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.

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(3), 031001 (2017).

Ayache, M.

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333(6043), 729–733 (2011).
[Crossref] [PubMed]

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(1), 953 (2017).
[Crossref] [PubMed]

Bernier, N. R.

D. Malz, L. D. Tóth, N. R. Bernier, A. K. Feofanov, T. J. Kippenberg, and A. Nunnenkamp, “Quantum-limited directional amplifiers with optomechanics,” Phys. Rev. Lett. 120(2), 023601 (2018).
[Crossref] [PubMed]

Bin, Q.

L.-L. Zheng, T.-S. Yin, Q. Bin, X.-Y. Lü, and Y. Wu, “Single-photon-induced phonon blockade in a hybrid spin-optomechanical system,” Phys. Rev. A 99(1), 013804 (2019).
[Crossref]

Q. Bin, X.-Y. Lü, T.-S. Yin, Y. Li, and Y. Wu, “Collective radiance effects in the ultrastrong-coupling regime,” Phys. Rev. A 99(3), 033809 (2019).
[Crossref]

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]

Butsch, A.

M. S. Kang, A. Butsch, and P. S. J. Russell, “Reconfigurable light-driven opto-acoustic isolators in photonic crystal fibre,” Nat. Photon. 5(9), 549–553 (2011).
[Crossref]

Chan, J.

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(7341), 69–73 (2011).
[Crossref]

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(7341), 69–73 (2011).
[Crossref]

Chen, A.-X.

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

Chen, 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]

Chen, Y.-F.

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333(6043), 729–733 (2011).
[Crossref] [PubMed]

Christodoulides, D. N.

C. E. Eüter, K. G. Makris, R. EI-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010).
[Crossref]

H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear PT-symmetric optical structures,” Phys. Rev. A 82(4), 043803 (2010).
[Crossref]

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(3), 031001 (2017).

Clerk, A. A.

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

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(6), 3761 (1985).
[Crossref]

Damskägg, E.

L. M. de Lépinay, E. Damskägg, C. F. Ockeloen-Korppi, and M. A. Sillanpää, “Realization of directional amplification in a microwave optomechanical device,” Phys. Rev. Appl. 11(3), 034027 (2019).
[Crossref]

de Lépinay, L. M.

L. M. de Lépinay, E. Damskägg, C. F. Ockeloen-Korppi, and M. A. Sillanpää, “Realization of directional amplification in a microwave optomechanical device,” Phys. Rev. Appl. 11(3), 034027 (2019).
[Crossref]

Dieterle, P. B.

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(1), 953 (2017).
[Crossref] [PubMed]

Dobrindt, J. M.

J. M. Dobrindt, I. Wilson-Rae, and T. J. Kippenberg, “Parametric normal-mode splitting in cavity optomechanics,” Phys. Rev. Lett. 101(26), 263602 (2008).
[Crossref] [PubMed]

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]

Dötsch, H.

Eichenfield, M.

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(7341), 69–73 (2011).
[Crossref]

EI-Ganainy, R.

C. E. Eüter, K. G. Makris, R. EI-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010).
[Crossref]

El-Ganainy, R.

H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear PT-symmetric optical structures,” Phys. Rev. A 82(4), 043803 (2010).
[Crossref]

Espinola, R. L.

Eüter, C. E.

C. E. Eüter, K. G. Makris, R. EI-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010).
[Crossref]

Fainman, Y.

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333(6043), 729–733 (2011).
[Crossref] [PubMed]

Feng, L.

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333(6043), 729–733 (2011).
[Crossref] [PubMed]

Feofanov, A. K.

D. Malz, L. D. Tóth, N. R. Bernier, A. K. Feofanov, T. J. Kippenberg, and A. Nunnenkamp, “Quantum-limited directional amplifiers with optomechanics,” Phys. Rev. Lett. 120(2), 023601 (2018).
[Crossref] [PubMed]

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]

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(1), 953 (2017).
[Crossref] [PubMed]

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(6), 3761 (1985).
[Crossref]

Ge, W.

I. M. Mirza, W. Ge, and H. Jing, “On the optical nonreciprocity and slow light propagation in coupled spinning optomechanical resonators,” arXiv:1810.03709 (2018).

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]

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]

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, 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).
[Crossref] [PubMed]

T. R. Zaman, X. Guo, and R. J. Ram, “Faraday rotation in an InP waveguide,” Appl. Phys. Lett. 90(2), 023514 (2007).
[Crossref]

Hafezi, M.

Hill, J. T.

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(7341), 69–73 (2011).
[Crossref]

Hou, B. P.

Hu, X.-K.

Huang, J.

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333(6043), 729–733 (2011).
[Crossref] [PubMed]

Huang, R.

R. Huang, A. Miranowicz, J.-Q. Liao, F. Nori, and H. Jing, “Nonreciprocal photon blockade,” Phys. Rev. Lett. 121(15), 153601 (2018).
[Crossref] [PubMed]

Izuhara, T.

Jing, H.

R. Huang, A. Miranowicz, J.-Q. Liao, F. Nori, and H. Jing, “Nonreciprocal photon blockade,” Phys. Rev. Lett. 121(15), 153601 (2018).
[Crossref] [PubMed]

I. M. Mirza, W. Ge, and H. Jing, “On the optical nonreciprocity and slow light propagation in coupled spinning optomechanical resonators,” arXiv:1810.03709 (2018).

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

Jung, H.

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]

Kalaee, 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(1), 953 (2017).
[Crossref] [PubMed]

Kang, M. S.

M. S. Kang, A. Butsch, and P. S. J. Russell, “Reconfigurable light-driven opto-acoustic isolators in photonic crystal fibre,” Nat. Photon. 5(9), 549–553 (2011).
[Crossref]

Kip, D.

C. E. Eüter, K. G. Makris, R. EI-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010).
[Crossref]

Kippenberg, T. J.

D. Malz, L. D. Tóth, N. R. Bernier, A. K. Feofanov, T. J. Kippenberg, and A. Nunnenkamp, “Quantum-limited directional amplifiers with optomechanics,” Phys. Rev. Lett. 120(2), 023601 (2018).
[Crossref] [PubMed]

J. M. Dobrindt, I. Wilson-Rae, and T. J. Kippenberg, “Parametric normal-mode splitting in cavity optomechanics,” Phys. Rev. Lett. 101(26), 263602 (2008).
[Crossref] [PubMed]

Kottos, T.

H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear PT-symmetric optical structures,” Phys. Rev. A 82(4), 043803 (2010).
[Crossref]

La Rocca, G. C.

J.-H. Wu, M. Artoni, and G. C. La Rocca, “Non-Hermitian degeneracies and unidirectional reflectionless atomic lattices,” Phys. Rev. Lett. 113(12), 123004 (2014).
[Crossref] [PubMed]

Lai, D. G.

Lecocq, F.

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(3), 031001 (2017).

Lehnert, K. W.

T. A. Palomaki, J. D. Teufel, R. W. Simmonds, and K. W. Lehnert, “Entangling mechanical motion with microwave fields,” Science 342(6159), 710–713 (2013).
[Crossref] [PubMed]

Levy, M.

Li, Y.

Q. Bin, X.-Y. Lü, T.-S. Yin, Y. Li, and Y. Wu, “Collective radiance effects in the ultrastrong-coupling regime,” Phys. Rev. A 99(3), 033809 (2019).
[Crossref]

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

Li, Z.

L. Tian and Z. Li, “Nonreciprocal quantum-state conversion between microwave and optical photons,” Phys. Rev. A 96(1), 013808 (2017).
[Crossref]

Li, Z.-Y.

Liao, J.-Q.

R. Huang, A. Miranowicz, J.-Q. Liao, F. Nori, and H. Jing, “Nonreciprocal photon blockade,” Phys. Rev. Lett. 121(15), 153601 (2018).
[Crossref] [PubMed]

Lin, Q.

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(7341), 69–73 (2011).
[Crossref]

Lipson, M.

S. Manipatruni, J. T. Robinson, and M. Lipson, “Optical nonreciprocity in optomechanical structures,” Phys. Rev. Lett. 102(21), 213903 (2009).
[Crossref] [PubMed]

Lu, M.-H.

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333(6043), 729–733 (2011).
[Crossref] [PubMed]

Lu, Y.-Q.

Lü, X.-Y.

Q. Bin, X.-Y. Lü, T.-S. Yin, Y. Li, and Y. Wu, “Collective radiance effects in the ultrastrong-coupling regime,” Phys. Rev. A 99(3), 033809 (2019).
[Crossref]

L.-L. Zheng, T.-S. Yin, Q. Bin, X.-Y. Lü, and Y. Wu, “Single-photon-induced phonon blockade in a hybrid spin-optomechanical system,” Phys. Rev. A 99(1), 013804 (2019).
[Crossref]

Makris, K. G.

C. E. Eüter, K. G. Makris, R. EI-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010).
[Crossref]

Malz, D.

D. Malz, L. D. Tóth, N. R. Bernier, A. K. Feofanov, T. J. Kippenberg, and A. Nunnenkamp, “Quantum-limited directional amplifiers with optomechanics,” Phys. Rev. Lett. 120(2), 023601 (2018).
[Crossref] [PubMed]

Manipatruni, S.

S. Manipatruni, J. T. Robinson, and M. Lipson, “Optical nonreciprocity in optomechanical structures,” Phys. Rev. Lett. 102(21), 213903 (2009).
[Crossref] [PubMed]

Mayer Alegre, T. P.

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(7341), 69–73 (2011).
[Crossref]

Metelmann, A.

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

Miranowicz, A.

R. Huang, A. Miranowicz, J.-Q. Liao, F. Nori, and H. Jing, “Nonreciprocal photon blockade,” Phys. Rev. Lett. 121(15), 153601 (2018).
[Crossref] [PubMed]

Miri, M. A.

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

Mirza, I. M.

I. M. Mirza, W. Ge, and H. Jing, “On the optical nonreciprocity and slow light propagation in coupled spinning optomechanical resonators,” arXiv:1810.03709 (2018).

Nori, F.

R. Huang, A. Miranowicz, J.-Q. Liao, F. Nori, and H. Jing, “Nonreciprocal photon blockade,” Phys. Rev. Lett. 121(15), 153601 (2018).
[Crossref] [PubMed]

Nunnenkamp, A.

D. Malz, L. D. Tóth, N. R. Bernier, A. K. Feofanov, T. J. Kippenberg, and A. Nunnenkamp, “Quantum-limited directional amplifiers with optomechanics,” Phys. Rev. Lett. 120(2), 023601 (2018).
[Crossref] [PubMed]

Ockeloen-Korppi, C. F.

L. M. de Lépinay, E. Damskägg, C. F. Ockeloen-Korppi, and M. A. Sillanpää, “Realization of directional amplification in a microwave optomechanical device,” Phys. Rev. Appl. 11(3), 034027 (2019).
[Crossref]

Osgood, R. M.

Painter, O.

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(1), 953 (2017).
[Crossref] [PubMed]

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(7341), 69–73 (2011).
[Crossref]

Palomaki, T. A.

T. A. Palomaki, J. D. Teufel, R. W. Simmonds, and K. W. Lehnert, “Entangling mechanical motion with microwave fields,” Science 342(6159), 710–713 (2013).
[Crossref] [PubMed]

Peruzzo, 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(1), 953 (2017).
[Crossref] [PubMed]

Peterson, G. A.

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(3), 031001 (2017).

Qian, X.-S.

Rabl, P.

Ram, R. J.

T. R. Zaman, X. Guo, and R. J. Ram, “Faraday rotation in an InP waveguide,” Appl. Phys. Lett. 90(2), 023514 (2007).
[Crossref]

Ramezani, H.

H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear PT-symmetric optical structures,” Phys. Rev. A 82(4), 043803 (2010).
[Crossref]

Robinson, J. T.

S. Manipatruni, J. T. Robinson, and M. Lipson, “Optical nonreciprocity in optomechanical structures,” Phys. Rev. Lett. 102(21), 213903 (2009).
[Crossref] [PubMed]

Ruesink, F.

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

Russell, P. S. J.

M. S. Kang, A. Butsch, and P. S. J. Russell, “Reconfigurable light-driven opto-acoustic isolators in photonic crystal fibre,” Nat. Photon. 5(9), 549–553 (2011).
[Crossref]

Safavi-Naeini, A. H.

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(7341), 69–73 (2011).
[Crossref]

Scherer, A.

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333(6043), 729–733 (2011).
[Crossref] [PubMed]

Segev, M.

C. E. Eüter, K. G. Makris, R. EI-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010).
[Crossref]

Shen, Z.

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]

Sillanpää, M. A.

L. M. de Lépinay, E. Damskägg, C. F. Ockeloen-Korppi, and M. A. Sillanpää, “Realization of directional amplification in a microwave optomechanical device,” Phys. Rev. Appl. 11(3), 034027 (2019).
[Crossref]

Simmonds, R. W.

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(3), 031001 (2017).

T. A. Palomaki, J. D. Teufel, R. W. Simmonds, and K. W. Lehnert, “Entangling mechanical motion with microwave fields,” Science 342(6159), 710–713 (2013).
[Crossref] [PubMed]

Sun, F.-W.

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]

Tang, H. 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).
[Crossref] [PubMed]

Teufel, J. D.

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(3), 031001 (2017).

T. A. Palomaki, J. D. Teufel, R. W. Simmonds, and K. W. Lehnert, “Entangling mechanical motion with microwave fields,” Science 342(6159), 710–713 (2013).
[Crossref] [PubMed]

Tian, L.

L. Tian and Z. Li, “Nonreciprocal quantum-state conversion between microwave and optical photons,” Phys. Rev. A 96(1), 013808 (2017).
[Crossref]

Tombesi, P.

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]

Tóth, L. D.

D. Malz, L. D. Tóth, N. R. Bernier, A. K. Feofanov, T. J. Kippenberg, and A. Nunnenkamp, “Quantum-limited directional amplifiers with optomechanics,” Phys. Rev. Lett. 120(2), 023601 (2018).
[Crossref] [PubMed]

Tsai, M.-C.

Vedral, V.

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]

Verhagen, E.

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

Vitali, D.

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]

Wang, C.

Wang, H.

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

Wang, H.-T.

Wang, Q.

Wilson-Rae, I.

J. M. Dobrindt, I. Wilson-Rae, and T. J. Kippenberg, “Parametric normal-mode splitting in cavity optomechanics,” Phys. Rev. Lett. 101(26), 263602 (2008).
[Crossref] [PubMed]

Winger, M.

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(7341), 69–73 (2011).
[Crossref]

Wu, J.-H.

J.-H. Wu, M. Artoni, and G. C. La Rocca, “Non-Hermitian degeneracies and unidirectional reflectionless atomic lattices,” Phys. Rev. Lett. 113(12), 123004 (2014).
[Crossref] [PubMed]

Wu, Y.

Q. Bin, X.-Y. Lü, T.-S. Yin, Y. Li, and Y. Wu, “Collective radiance effects in the ultrastrong-coupling regime,” Phys. Rev. A 99(3), 033809 (2019).
[Crossref]

L.-L. Zheng, T.-S. Yin, Q. Bin, X.-Y. Lü, and Y. Wu, “Single-photon-induced phonon blockade in a hybrid spin-optomechanical system,” Phys. Rev. A 99(1), 013804 (2019).
[Crossref]

Wulf, 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(1), 953 (2017).
[Crossref] [PubMed]

Xia, K.

Xiao, Y.-F.

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]

Xu, F.

Xu, X.-W.

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

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

Xu, Y.-L.

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333(6043), 729–733 (2011).
[Crossref] [PubMed]

Yang, Q.

Yin, T.-S.

Q. Bin, X.-Y. Lü, T.-S. Yin, Y. Li, and Y. Wu, “Collective radiance effects in the ultrastrong-coupling regime,” Phys. Rev. A 99(3), 033809 (2019).
[Crossref]

L.-L. Zheng, T.-S. Yin, Q. Bin, X.-Y. Lü, and Y. Wu, “Single-photon-induced phonon blockade in a hybrid spin-optomechanical system,” Phys. Rev. A 99(1), 013804 (2019).
[Crossref]

Yu, Z.-Y.

Zaman, T. R.

T. R. Zaman, X. Guo, and R. J. Ram, “Faraday rotation in an InP waveguide,” Appl. Phys. Lett. 90(2), 023514 (2007).
[Crossref]

Zeilinger, 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]

Zhang, Y.-L.

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]

Zhao, Y.-J.

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

Zheng, L.-L.

L.-L. Zheng, T.-S. Yin, Q. Bin, X.-Y. Lü, and Y. Wu, “Single-photon-induced phonon blockade in a hybrid spin-optomechanical system,” Phys. Rev. A 99(1), 013804 (2019).
[Crossref]

Zhou, C.-Z.

Zou, C.-L.

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]

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]

Zou, X.-B.

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]

Zubairy, M. S.

Appl. Phys. Lett. (1)

T. R. Zaman, X. Guo, and R. J. Ram, “Faraday rotation in an InP waveguide,” Appl. Phys. Lett. 90(2), 023514 (2007).
[Crossref]

J. Opt. Soc. Am. B (1)

Nat. Commun. (2)

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(1), 953 (2017).
[Crossref] [PubMed]

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

Nat. Photon. (2)

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]

M. S. Kang, A. Butsch, and P. S. J. Russell, “Reconfigurable light-driven opto-acoustic isolators in photonic crystal fibre,” Nat. Photon. 5(9), 549–553 (2011).
[Crossref]

Nat. Phys. (1)

C. E. Eüter, K. G. Makris, R. EI-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010).
[Crossref]

Nature(London) (1)

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(7341), 69–73 (2011).
[Crossref]

Opt. Express (5)

Opt. Lett. (1)

Phys. Rev. A (6)

H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear PT-symmetric optical structures,” Phys. Rev. A 82(4), 043803 (2010).
[Crossref]

Q. Bin, X.-Y. Lü, T.-S. Yin, Y. Li, and Y. Wu, “Collective radiance effects in the ultrastrong-coupling regime,” Phys. Rev. A 99(3), 033809 (2019).
[Crossref]

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. Tian and Z. Li, “Nonreciprocal quantum-state conversion between microwave and optical photons,” Phys. Rev. A 96(1), 013808 (2017).
[Crossref]

L.-L. Zheng, T.-S. Yin, Q. Bin, X.-Y. Lü, and Y. Wu, “Single-photon-induced phonon blockade in a hybrid spin-optomechanical system,” Phys. Rev. A 99(1), 013804 (2019).
[Crossref]

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[Crossref]

Phys. Rev. Appl. (1)

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[Crossref]

Phys. Rev. Lett. (7)

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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).
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[Crossref] [PubMed]

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Phys. Rev. X (2)

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

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Science (2)

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[Crossref] [PubMed]

Other (3)

It should be point out that Pc cannot be zero, since the linearization demands a strong the pump fields. For the parameters here, we take Pc = 100 nW, and we have C0 ≈ 28.

I. M. Mirza, W. Ge, and H. Jing, “On the optical nonreciprocity and slow light propagation in coupled spinning optomechanical resonators,” arXiv:1810.03709 (2018).

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

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Figures (5)

Fig. 1
Fig. 1 (a) Schematic diagram of our proposed model. An optomechanical microtoroid cavity supports a clockwise circulating mode (â) and a counter-clockwise circulating mode (ĉ), both the two cavity modes couple with the mechanical mode () via the radiation pressure. The pump fields (εap, εcp) and signal fields (εas, εcs) couple with the cavity modes by an optical fiber. (b) The nonreciprocal transmission: the right-moving signal field is completely transmitted (Ta = 1), while the left-moving signal field is blocking-up (Tc = 0). (c) The nonreciprocal fast-slow light: both the right-moving field and the left-moving signal field are transmitted (Ta = Tc = 1). However, the group delay of the right-moving signal field is negative (τa < 0), that corresponds to the fast light. The group delay of the left-moving signal field is positive (τc > 0), that corresponds to the slow light.
Fig. 2
Fig. 2 The transmission spectra Ta (red solid lines) and Tc (blue dashed lines) as a function of δas/ωm and δcs/ωm, respectively. The system works near the red sideband (Δ = ωm). The parameters are: (a) Pa = Pc, (b) Pa = 102Pc, (c) Pa = 104Pc, (d) Pa = 105Pc. The other parameters are stated in the text.
Fig. 3
Fig. 3 The transmission spectra Ta (red solid lines) and Tc (blue dashed lines) as a function of δas/ωm and δcs/ωm, respectively. The system works near the blue sideband (Δ = −ωm). The parameters are: (a) Pa = Pc, (b) Pa = 6Pc, (c) Pa = 8.5Pc, (d) Pa = 9.5Pc, (e) Pa = 5 × 102Pc, (f) Pa = 104Pc. The other parameters are stated in the text.
Fig. 4
Fig. 4 The transmission spectra Ta (red solid lines) and Tc (blue dashed lines) as a function of δas/ωm and δcs/ωm, respectively, for different intrinsic photon loss rate κin. The system works near the blue sideband (Δ = −ωm). The parameters are: (a) κin = κ, (b) κin = 10−1κ„ (c) κin = 10−2κ, (d) κin = 10−3κ. The other parameters are stated in the text.
Fig. 5
Fig. 5 The group delay τa (red solid lines) and τc (blue dashed lines) as a function of δas/ωm and δcs/ωm, respectively. The system works near the blue sideband (Δ = −ωm). The parameters are: (a) Pa = 5 × 104Pc, (b) Pa = 1 × 105Pc, (c) Pa = 2 × 105Pc, (d) Pa = 5 × 105Pc. The other parameters are stated in the text.

Equations (10)

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H total = H om + H aps + H cps + H ac ,
H = Δ a ^ a ^ + Δ c ^ c ^ + ω m b ^ b ^ + g ( a ^ a ^ + c ^ c ^ ) ( b ^ + b ^ ) + i ε ap ( a ^ H . c . ) + i ε as ( a ^ e i δ as t H . c . ) + i ε cp ( c ^ H . c . ) + i ε cs ( c ^ e i δ cs t H . c . ) + J ( a ^ c ^ + c ^ a ^ ) ,
d a ^ d t = ( i Δ + κ t ) a ^ i g a ^ ( b ^ + b ^ ) i J c ^ + ε ap + ε as e i δ as t + 2 κ a ^ in , d c ^ d t = ( i Δ + κ t ) c ^ i g c ^ ( b ^ + b ^ ) i J a ^ + ε cp + ε c s e i δ cs t + 2 κ c ^ in , d b ^ d t = ( i ω m + γ ) b ^ i g ( a ^ a ^ + c ^ c ^ ) + 2 γ b ^ in ,
d A d t = ( i Δ + κ t ) A i g A ( B + B * ) i J C + ε ap + ε as e i δ as t , d C d t = ( i Δ + κ t ) C i g C ( B + B * ) i J A + ε cp + ε cs e i δ cs t , d B d t = ( i ω m + γ ) B i g ( | A | 2 + | C | 2 ) .
X = X 0 + X a + e i δ as t + X a e i δ as t + X c + e i δ c s t + X c e i δ c s t ,
T a = | t a ( δ as ) | 2 = | 1 2 κ η ( δ as ) | 2 , T c = | t c ( δ cs ) | 2 = | 1 2 κ ξ ( δ cs ) | 2 .
τ a = d ϕ a ( ω as ) d ω as , τ c = d ϕ c ( ω cs ) d ω cs .
0 = ( i Δ + κ t ) A 0 i g A 0 ( B 0 + B 0 * ) i J C 0 + ε ap , 0 = ( i Δ + κ t ) C 0 i g C 0 ( B 0 + B 0 * ) i J A 0 + ε cp , 0 = ( i ω m + γ ) B 0 i g ( | A 0 | 2 + | C 0 | 2 ) .
Φ a B a + = i g ( A 0 * A a + + A a + * A 0 + C 0 * C a + + C a + * C 0 ) , Ω a A a + = i g A 0 ( B a + + B a + * ) i J C a + + ε as , Ω a C a + = i g C 0 ( B a + + B a + * ) i J A a + ,
Φ c B c + = i g ( A 0 * A c + + A c + * A 0 + C 0 * C c + + C c + * C 0 ) , Ω c C c + = i g C 0 ( B c + + B c + * ) i J A c + + ε cs , Ω c A c + = i g A 0 ( B c + + B c + * ) i J C c + ,