Abstract

We study the optical response of a double optomechanical cavity system assisted by two Rydberg atoms. The target atom is only coupled with one side cavity by a single cavity mode, and gate one is outside the cavities. It has been realized that a long-range manipulation of optical properties of a hybrid system, by controlling the Rydberg atom decoupled with the optomechanical cavity. Switching on the coupling between atoms and cavity mode, the original spatial inversion symmetry of the double cavity structure has been broken. Combining the controllable optical non-reciprocity with the coherent perfect absorption/transmission/synthesis effect (CPA/CPT/CPS reported by [ X.-B. YanOpt. Express 22, 4886 (2014)], we put forward the theoretical schemes of an all-optical transistor which contains functions such as a controllable diode, rectifier, and amplifier by controlling a single gate photon.

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

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References

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2018 (1)

L. Du, Y. Zhang, C.-H. Fan, Y.-M. Liu, F. Gao, and J.-H. Wu, “Enhanced nonlinear characteristics with the assistance of a PT-symmetric trimer system,” Sci. Rep. 8, 2933 (2018).
[Crossref]

2017 (1)

J. Restrepo, I. Favero, and C. Ciuti, “Fully coupled hybrid cavity optomechanics: Quantum interferences and correlations,” Phys. Rev. A 95, 023832 (2017).
[Crossref]

2016 (1)

2015 (3)

I. M. Mirza, “Real-time emission spectrum of a hybrid atom-optomechanical cavity,” J. Opt. Soc. Am. B: Opt. Phys. 32, 1604–1614 (2015).
[Crossref]

D. Yan, Z.-H. Wang, C.-N. Ren, H. Gao, Y. Li, and J.-H. Wu, “Duality and bistability in an optomechanical cavity coupled to a Rydberg superatom,” Phys. Rev. A 91, 023813 (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]

2014 (14)

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

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. H. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

J. H. Wu, M. Artoni, and G. C. La Rocca, “Non-Hermitian Degeneracies and Unidirectional Reflectionless Atomic Lattices,” Phys. Rev. Lett. 113, 123004 (2014).
[Crossref] [PubMed]

D. Tiarks, S. Baur, K. Schneider, S. Durr, and G. Rempe, “Single-Photon Transistor Using a Forster Resonance,” Phys. Rev. Lett. 113, 053602 (2014).
[Crossref]

H. Gorniaczyk, C. Tresp, J. Schmidt, H. Fedder, and S. Hofferberth, “Single-Photon Transistor Mediated by Interstate Rydberg Interactions,” Phys. Rev. Lett. 113, 053601 (2014).
[Crossref] [PubMed]

E. Li, B. J. Eggleton, K. Fang, and S. Fan, “Photonic Aharonov-Bohm effect in photon-phonon interactions,” Nat. Commun. 5, 3225 (2014).
[PubMed]

M. Castellanos Munoz, A. Y. Petrov, L. OFaolain, J. Li, T. F. Krauss, and M. Eich, “Optically Induced Indirect Photonic Transitions in a Slow Light Photonic Crystal Waveguide,” Phys. Rev. Lett. 112, 053904 (2014).
[Crossref] [PubMed]

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

E. J. Lenferink, G. Wei, and N. P. Stern, “Coherent optical non-reciprocity in axisymmetric resonators,” Phys. Rev. Lett. 22, 16099 (2014).

I. M. Mirza and S. J. van Enk, “Single-photon time-dependent spectra in quantum optomechanics,” Phys. Rev. A 90, 043831 (2014).
[Crossref]

H. Wang, Z. X. Wang, J. Zhang, S. K. Ozdemir, L. Yang, and Y. X. Liu, “Phonon amplification in two coupled cavities containing one mechanical resonator,” Phys. Rev. A 90, 053814 (2014).
[Crossref]

X. B. Yan, C. L. Cui, K. H. Gu, X. D. Tian, C. B. Fu, and J. H. Wu, “Coherent perfect absorption, transmission, and synthesis in a double-cavity optomechanical system,” Opt. Express 22, 4886–4895 (2014).
[Crossref] [PubMed]

G. S. Agarwal and Sumei Huang, “Nanomechanical inverse electromagnetically induced transparency and confinement of light in normal modes,” New J. Phys. 16, 033023 (2014).
[Crossref]

A. Carmele, B. Vogell, K. Stannigel, and P. Zoller, “Opto-nanomechanics strongly coupled to a Rydberg superatom: coherent versus incoherent dynamics,” New J. Phys. 16, 063042 (2014).
[Crossref]

2013 (5)

C. B. Fu, X. B. Yan, K. H. Gu, C. L. Cui, J. H. Wu, and T. D. Fu, “Steady-state solutions of a hybrid system involving atom-light and optomechanical interactions: Beyond the weak-cavity-field approximation,” Phys. Rev. A 87, 053841 (2013).
[Crossref]

D. W. Wang, H. T. Zhou, M. J. Guo, J. X. Zhang, J. Evers, and S. Y. Zhu, “Optical Diode Made from a Moving Photonic Crystal,” Phys. Rev. Lett. 110, 093901 (2013).
[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, 223602 (2013).
[Crossref] [PubMed]

B. Anand, R. Podila, K. Lingam, S. R. Krishnan, S. S. S. Sai, R. Philip, and A. M. Rao, “Optical Diode Action from Axially Asymmetric Nonlinearity in an All-Carbon Solid-State Device,” Nano Lett. 13, 5771–5776 (2013).
[Crossref] [PubMed]

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

2012 (6)

Y. D. Wang and A. A. Clerk, “Using Interference for High Fidelity Quantum State Transfer in Optomechanics,” Phys. Rev. Lett. 108, 153603 (2012).
[Crossref] [PubMed]

L. Tian, “Adiabatic State Conversion and Pulse Transmission in Optomechanical Systems,” Phys. Rev. Lett. 108, 153604 (2012).
[Crossref] [PubMed]

M. Ludwig, A. H. Safavi-Naeini, O. Painter, and F. Marquardt, “Enhanced Quantum Nonlinearities in a Two-Mode Optomechanical System,” Phys. Rev. Lett. 109, 063601 (2012).
[Crossref] [PubMed]

K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical Quantum Information Processing with Photons and Phonons,” Phys. Rev. Lett. 109, 013603 (2012).
[Crossref] [PubMed]

L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. Qi, “An All-Silicon Passive Optical Diode,” Science 335, 447–450 (2012).
[Crossref]

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

2011 (7)

Y. Chang, T. Shi, Y.-X. Liu, C.-P. Sun, and F. Nori, “Multistability of electromagnetically induced transparency in atom-assisted optomechanical cavities,” Phys. Rev. A 83, 063826 (2011).
[Crossref]

D. Petrosyan, J. Otterbach, and M. Fleischhauer, “Electromagnetically Induced Transparency with Rydberg Atoms,” Phys. Rev. Lett. 107, 213601 (2011).
[Crossref] [PubMed]

H. Weimer, M. Muller, H. P. Buchler, and I. Lesanovsky, “Digital quantum simulation with Rydberg atoms,” Quantum Inf. Process. 10, 885–906 (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. Photonics 5, 758–762 (2011).
[Crossref]

I. V. Shadrivov, V. A. Fedotov, D. A. Powell, Y. S. Kivshar, and N. I. Zheludev, “Electromagnetic wave analogue of an electronic diode,” New J. Phys. 13, 033025 (2011).
[Crossref]

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

Y. Shen, M. Bradford, and J. T. Shen, “Single-Photon Diode by Exploiting the Photon Polarization in a Waveguide,” Phys. Rev. Lett. 107, 173902 (2011).
[Crossref] [PubMed]

2010 (11)

A. E. Miroshnichenko, E. Brasselet, and Y. S. Kivshar, “Reversible optical nonreciprocity in periodic structures with liquid crystals,” Appl. Phys. Lett. 96, 063302 (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] [PubMed]

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, 7340–7356 (2010).
[Crossref] [PubMed]

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

A. B. Khanikaev, S. H. Mousavi, G. Shvets, and Y. S. Kivshar, “One-Way Extraordinary Optical Transmission and Nonreciprocal Spoof Plasmons,” Phys. Rev. Lett. 105, 126804 (2010).
[Crossref] [PubMed]

L. Isenhower, E. Urban, X. L. Zhang, A. T. Gill, T. Henage, T. A. Johnson, T. G. Walker, and M. Saffman, “Demonstration of a Neutral Atom Controlled-NOT Quantum Gate,” Phys. Rev. Lett. 104, 010503 (2010).
[Crossref] [PubMed]

H. Weimer, M. Muller, I. Lesanovsky, P. Zoller, and H. P. Buchler, “A Rydberg quantum simulator,” Nat. Phys. 6, 382–388 (2010).
[Crossref]

M. Saffman, T. G. Walker, and K. Molmer, “Quantum information with Rydberg atoms,” Rev. Mod. Phys. 82, 2313 (2010).
[Crossref]

J. D. Pritchard, D. Maxwell, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Cooperative Atom-Light Interaction in a Blockaded Rydberg Ensemble,” Phys. Rev. Lett. 105, 193603 (2010).
[Crossref]

C. Guerlin, E. Brion, T. Esslinger, and K. Molmer, “Cavity quantum electrodynamics with a Rydberg-blocked atomic ensemble,” Phys. Rev. A 82, 053832 (2010).
[Crossref]

Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent Perfect Absorbers: Time-Reversed Lasers,” Phys. Rev. Lett. 105, 053901 (2010).
[Crossref] [PubMed]

2009 (7)

L. H. Pedersen and K. Molmer, “Few qubit atom-light interfaces with collective encoding,” Phys. Rev. A 79, 012320 (2009).
[Crossref]

E. Urban, T. A. Johnson, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker, and M. Saffman, “Observation of Rydberg blockade between two atoms,” Nat. Phys. 5, 110–114 (2009).
[Crossref]

A. Gaetan, A. Y. Miroshnychenko, T. Wilk, A. Chotia, M. Viteau, D. Comparat, P. Pillet, A. Browaeys, and P. Grangier, “Observation of collective excitation of two individual atoms in the Rydberg blockade regime,” Nat. Phys. 5, 115–118 (2009).
[Crossref]

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljacic, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature (London) 461, 772–776 (2009).
[Crossref]

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

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

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

2008 (4)

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

F. D. M. Haldane and S. Raghu, “Possible Realization of Directional Optical Waveguides in Photonic Crystals with Broken Time-Reversal Symmetry,” Phys. Rev. Lett. 100, 013904 (2008).
[Crossref] [PubMed]

H. Ian, Z. R. Gong, Y. X. Liu, C. P. Sun, and F. Nori, “Cavity optomechanical coupling assisted by an atomic gas,” Phys. Rev. A 78, 013824 (2008).
[Crossref]

C. Genes, D. Vitali, and P. Tombesi, “Emergence of atom-light-mirror entanglement inside an optical cavity,” Phys. Rev. A 77, 050307(R) (2008).
[Crossref]

2004 (3)

D. Tong, S. M. Farooqi, J. Stanojevic, S. Krishnan, Y. P. Zhang, R. Cote, E. E. Eyler, and P. L. Gould, “Local Blockade of Rydberg Excitation in an Ultracold Gas,” Phys. Rev. Lett. 93, 063001 (2004).
[Crossref] [PubMed]

K. Singer, M. Reetz-Lamour, T. Amthor, L. G. Marcassa, and M. Weidemuller, “Suppression of Excitation and Spectral Broadening Induced by Interactions in a Cold Gas of Rydberg Atoms,” Phys. Rev. Lett. 93, 163001 (2004).
[Crossref] [PubMed]

R. J. Potton, “Reciprocity in optics,” Rep. Prog. Phys. 67, 717–754 (2004).
[Crossref]

2002 (1)

M. Saffman and T. G. Walker, “Creating single-atom and single-photon sources from entangled atomic ensembles,” Phys. Rev. A 66, 065403 (2002).
[Crossref]

2001 (1)

K. Gallo, G. Assanto, K. R. Parameswaran, and M. M. Fejer, “All-optical diode in a periodically poled lithium niobate waveguide,” Appl. Phys. Lett. 79, 314–316 (2001).
[Crossref]

2000 (1)

D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Cote, and M. D. Lukin, “Fast Quantum Gates for Neutral Atoms,” Phys. Rev. Lett. 85, 2208 (2000).
[Crossref] [PubMed]

Adams, C. S.

J. D. Pritchard, D. Maxwell, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Cooperative Atom-Light Interaction in a Blockaded Rydberg Ensemble,” Phys. Rev. Lett. 105, 193603 (2010).
[Crossref]

Agarwal, G. S.

G. S. Agarwal and Sumei Huang, “Nanomechanical inverse electromagnetically induced transparency and confinement of light in normal modes,” New J. Phys. 16, 033023 (2014).
[Crossref]

Alamri, M.

Amthor, T.

K. Singer, M. Reetz-Lamour, T. Amthor, L. G. Marcassa, and M. Weidemuller, “Suppression of Excitation and Spectral Broadening Induced by Interactions in a Cold Gas of Rydberg Atoms,” Phys. Rev. Lett. 93, 163001 (2004).
[Crossref] [PubMed]

Anand, B.

B. Anand, R. Podila, K. Lingam, S. R. Krishnan, S. S. S. Sai, R. Philip, and A. M. Rao, “Optical Diode Action from Axially Asymmetric Nonlinearity in an All-Carbon Solid-State Device,” Nano Lett. 13, 5771–5776 (2013).
[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, 123004 (2014).
[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, 223602 (2013).
[Crossref] [PubMed]

Aspelmeyer, M.

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

Assanto, G.

K. Gallo, G. Assanto, K. R. Parameswaran, and M. M. Fejer, “All-optical diode in a periodically poled lithium niobate waveguide,” Appl. Phys. Lett. 79, 314–316 (2001).
[Crossref]

Baur, S.

D. Tiarks, S. Baur, K. Schneider, S. Durr, and G. Rempe, “Single-Photon Transistor Using a Forster Resonance,” Phys. Rev. Lett. 113, 053602 (2014).
[Crossref]

Bender, C. M.

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. H. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

Bennett, S. D.

K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical Quantum Information Processing with Photons and Phonons,” Phys. Rev. Lett. 109, 013603 (2012).
[Crossref] [PubMed]

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. Photonics 5, 758–762 (2011).
[Crossref]

Bradford, M.

Y. Shen, M. Bradford, and J. T. Shen, “Single-Photon Diode by Exploiting the Photon Polarization in a Waveguide,” Phys. Rev. Lett. 107, 173902 (2011).
[Crossref] [PubMed]

Brasselet, E.

A. E. Miroshnichenko, E. Brasselet, and Y. S. Kivshar, “Reversible optical nonreciprocity in periodic structures with liquid crystals,” Appl. Phys. Lett. 96, 063302 (2010).
[Crossref]

Brion, E.

C. Guerlin, E. Brion, T. Esslinger, and K. Molmer, “Cavity quantum electrodynamics with a Rydberg-blocked atomic ensemble,” Phys. Rev. A 82, 053832 (2010).
[Crossref]

Browaeys, A.

A. Gaetan, A. Y. Miroshnychenko, T. Wilk, A. Chotia, M. Viteau, D. Comparat, P. Pillet, A. Browaeys, and P. Grangier, “Observation of collective excitation of two individual atoms in the Rydberg blockade regime,” Nat. Phys. 5, 115–118 (2009).
[Crossref]

Buchler, H. P.

H. Weimer, M. Muller, H. P. Buchler, and I. Lesanovsky, “Digital quantum simulation with Rydberg atoms,” Quantum Inf. Process. 10, 885–906 (2011).
[Crossref]

H. Weimer, M. Muller, I. Lesanovsky, P. Zoller, and H. P. Buchler, “A Rydberg quantum simulator,” Nat. Phys. 6, 382–388 (2010).
[Crossref]

Butsch, A.

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

Cao, H.

Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent Perfect Absorbers: Time-Reversed Lasers,” Phys. Rev. Lett. 105, 053901 (2010).
[Crossref] [PubMed]

Carmele, A.

A. Carmele, B. Vogell, K. Stannigel, and P. Zoller, “Opto-nanomechanics strongly coupled to a Rydberg superatom: coherent versus incoherent dynamics,” New J. Phys. 16, 063042 (2014).
[Crossref]

Castellanos Munoz, M.

M. Castellanos Munoz, A. Y. Petrov, L. OFaolain, J. Li, T. F. Krauss, and M. Eich, “Optically Induced Indirect Photonic Transitions in a Slow Light Photonic Crystal Waveguide,” Phys. Rev. Lett. 112, 053904 (2014).
[Crossref] [PubMed]

Chang, Y.

Y. Chang, T. Shi, Y.-X. Liu, C.-P. Sun, and F. Nori, “Multistability of electromagnetically induced transparency in atom-assisted optomechanical cavities,” Phys. Rev. A 83, 063826 (2011).
[Crossref]

Chong, Y.

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljacic, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature (London) 461, 772–776 (2009).
[Crossref]

Chong, Y. D.

Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent Perfect Absorbers: Time-Reversed Lasers,” Phys. Rev. Lett. 105, 053901 (2010).
[Crossref] [PubMed]

Chotia, A.

A. Gaetan, A. Y. Miroshnychenko, T. Wilk, A. Chotia, M. Viteau, D. Comparat, P. Pillet, A. Browaeys, and P. Grangier, “Observation of collective excitation of two individual atoms in the Rydberg blockade regime,” Nat. Phys. 5, 115–118 (2009).
[Crossref]

Christodoulides, D. N.

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

Cirac, J. I.

D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Cote, and M. D. Lukin, “Fast Quantum Gates for Neutral Atoms,” Phys. Rev. Lett. 85, 2208 (2000).
[Crossref] [PubMed]

Ciuti, C.

J. Restrepo, I. Favero, and C. Ciuti, “Fully coupled hybrid cavity optomechanics: Quantum interferences and correlations,” Phys. Rev. A 95, 023832 (2017).
[Crossref]

Clerk, A. A.

Y. D. Wang and A. A. Clerk, “Using Interference for High Fidelity Quantum State Transfer in Optomechanics,” Phys. Rev. Lett. 108, 153603 (2012).
[Crossref] [PubMed]

Comparat, D.

A. Gaetan, A. Y. Miroshnychenko, T. Wilk, A. Chotia, M. Viteau, D. Comparat, P. Pillet, A. Browaeys, and P. Grangier, “Observation of collective excitation of two individual atoms in the Rydberg blockade regime,” Nat. Phys. 5, 115–118 (2009).
[Crossref]

Cote, R.

D. Tong, S. M. Farooqi, J. Stanojevic, S. Krishnan, Y. P. Zhang, R. Cote, E. E. Eyler, and P. L. Gould, “Local Blockade of Rydberg Excitation in an Ultracold Gas,” Phys. Rev. Lett. 93, 063001 (2004).
[Crossref] [PubMed]

D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Cote, and M. D. Lukin, “Fast Quantum Gates for Neutral Atoms,” Phys. Rev. Lett. 85, 2208 (2000).
[Crossref] [PubMed]

Cui, C. L.

X. B. Yan, C. L. Cui, K. H. Gu, X. D. Tian, C. B. Fu, and J. H. Wu, “Coherent perfect absorption, transmission, and synthesis in a double-cavity optomechanical system,” Opt. Express 22, 4886–4895 (2014).
[Crossref] [PubMed]

C. B. Fu, X. B. Yan, K. H. Gu, C. L. Cui, J. H. Wu, and T. D. Fu, “Steady-state solutions of a hybrid system involving atom-light and optomechanical interactions: Beyond the weak-cavity-field approximation,” Phys. Rev. A 87, 053841 (2013).
[Crossref]

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. Photonics 5, 758–762 (2011).
[Crossref]

Du, L.

L. Du, Y. Zhang, C.-H. Fan, Y.-M. Liu, F. Gao, and J.-H. Wu, “Enhanced nonlinear characteristics with the assistance of a PT-symmetric trimer system,” Sci. Rep. 8, 2933 (2018).
[Crossref]

Durr, S.

D. Tiarks, S. Baur, K. Schneider, S. Durr, and G. Rempe, “Single-Photon Transistor Using a Forster Resonance,” Phys. Rev. Lett. 113, 053602 (2014).
[Crossref]

Eggleton, B. J.

E. Li, B. J. Eggleton, K. Fang, and S. Fan, “Photonic Aharonov-Bohm effect in photon-phonon interactions,” Nat. Commun. 5, 3225 (2014).
[PubMed]

Eich, M.

M. Castellanos Munoz, A. Y. Petrov, L. OFaolain, J. Li, T. F. Krauss, and M. Eich, “Optically Induced Indirect Photonic Transitions in a Slow Light Photonic Crystal Waveguide,” Phys. Rev. Lett. 112, 053904 (2014).
[Crossref] [PubMed]

EI-Ganainy, R.

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

Esslinger, T.

C. Guerlin, E. Brion, T. Esslinger, and K. Molmer, “Cavity quantum electrodynamics with a Rydberg-blocked atomic ensemble,” Phys. Rev. A 82, 053832 (2010).
[Crossref]

Euter, C.

C. Euter, K. G. Makris, R. EI-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[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, 093901 (2013).
[Crossref] [PubMed]

Eyler, E. E.

D. Tong, S. M. Farooqi, J. Stanojevic, S. Krishnan, Y. P. Zhang, R. Cote, E. E. Eyler, and P. L. Gould, “Local Blockade of Rydberg Excitation in an Ultracold Gas,” Phys. Rev. Lett. 93, 063001 (2004).
[Crossref] [PubMed]

Fan, C.-H.

L. Du, Y. Zhang, C.-H. Fan, Y.-M. Liu, F. Gao, and J.-H. Wu, “Enhanced nonlinear characteristics with the assistance of a PT-symmetric trimer system,” Sci. Rep. 8, 2933 (2018).
[Crossref]

Fan, L.

L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. Qi, “An All-Silicon Passive Optical Diode,” Science 335, 447–450 (2012).
[Crossref]

Fan, S.

E. Li, B. J. Eggleton, K. Fang, and S. Fan, “Photonic Aharonov-Bohm effect in photon-phonon interactions,” Nat. Commun. 5, 3225 (2014).
[PubMed]

Fan, S. H.

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. H. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

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

Fang, K.

E. Li, B. J. Eggleton, K. Fang, and S. Fan, “Photonic Aharonov-Bohm effect in photon-phonon interactions,” Nat. Commun. 5, 3225 (2014).
[PubMed]

Farooqi, S. M.

D. Tong, S. M. Farooqi, J. Stanojevic, S. Krishnan, Y. P. Zhang, R. Cote, E. E. Eyler, and P. L. Gould, “Local Blockade of Rydberg Excitation in an Ultracold Gas,” Phys. Rev. Lett. 93, 063001 (2004).
[Crossref] [PubMed]

Favero, I.

J. Restrepo, I. Favero, and C. Ciuti, “Fully coupled hybrid cavity optomechanics: Quantum interferences and correlations,” Phys. Rev. A 95, 023832 (2017).
[Crossref]

Fedder, H.

H. Gorniaczyk, C. Tresp, J. Schmidt, H. Fedder, and S. Hofferberth, “Single-Photon Transistor Mediated by Interstate Rydberg Interactions,” Phys. Rev. Lett. 113, 053601 (2014).
[Crossref] [PubMed]

Fedotov, V. A.

I. V. Shadrivov, V. A. Fedotov, D. A. Powell, Y. S. Kivshar, and N. I. Zheludev, “Electromagnetic wave analogue of an electronic diode,” New J. Phys. 13, 033025 (2011).
[Crossref]

Fejer, M. M.

K. Gallo, G. Assanto, K. R. Parameswaran, and M. M. Fejer, “All-optical diode in a periodically poled lithium niobate waveguide,” Appl. Phys. Lett. 79, 314–316 (2001).
[Crossref]

Fleischhauer, M.

D. Petrosyan, J. Otterbach, and M. Fleischhauer, “Electromagnetically Induced Transparency with Rydberg Atoms,” Phys. Rev. Lett. 107, 213601 (2011).
[Crossref] [PubMed]

Fu, C. B.

X. B. Yan, C. L. Cui, K. H. Gu, X. D. Tian, C. B. Fu, and J. H. Wu, “Coherent perfect absorption, transmission, and synthesis in a double-cavity optomechanical system,” Opt. Express 22, 4886–4895 (2014).
[Crossref] [PubMed]

C. B. Fu, X. B. Yan, K. H. Gu, C. L. Cui, J. H. Wu, and T. D. Fu, “Steady-state solutions of a hybrid system involving atom-light and optomechanical interactions: Beyond the weak-cavity-field approximation,” Phys. Rev. A 87, 053841 (2013).
[Crossref]

Fu, T. D.

C. B. Fu, X. B. Yan, K. H. Gu, C. L. Cui, J. H. Wu, and T. D. Fu, “Steady-state solutions of a hybrid system involving atom-light and optomechanical interactions: Beyond the weak-cavity-field approximation,” Phys. Rev. A 87, 053841 (2013).
[Crossref]

Gaetan, A.

A. Gaetan, A. Y. Miroshnychenko, T. Wilk, A. Chotia, M. Viteau, D. Comparat, P. Pillet, A. Browaeys, and P. Grangier, “Observation of collective excitation of two individual atoms in the Rydberg blockade regime,” Nat. Phys. 5, 115–118 (2009).
[Crossref]

Gallo, K.

K. Gallo, G. Assanto, K. R. Parameswaran, and M. M. Fejer, “All-optical diode in a periodically poled lithium niobate waveguide,” Appl. Phys. Lett. 79, 314–316 (2001).
[Crossref]

Gao, F.

L. Du, Y. Zhang, C.-H. Fan, Y.-M. Liu, F. Gao, and J.-H. Wu, “Enhanced nonlinear characteristics with the assistance of a PT-symmetric trimer system,” Sci. Rep. 8, 2933 (2018).
[Crossref]

Gao, H.

D. Yan, Z.-H. Wang, C.-N. Ren, H. Gao, Y. Li, and J.-H. Wu, “Duality and bistability in an optomechanical cavity coupled to a Rydberg superatom,” Phys. Rev. A 91, 023813 (2015).
[Crossref]

Gauguet, A.

J. D. Pritchard, D. Maxwell, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Cooperative Atom-Light Interaction in a Blockaded Rydberg Ensemble,” Phys. Rev. Lett. 105, 193603 (2010).
[Crossref]

Ge, L.

Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent Perfect Absorbers: Time-Reversed Lasers,” Phys. Rev. Lett. 105, 053901 (2010).
[Crossref] [PubMed]

Genes, C.

C. Genes, D. Vitali, and P. Tombesi, “Emergence of atom-light-mirror entanglement inside an optical cavity,” Phys. Rev. A 77, 050307(R) (2008).
[Crossref]

Gianfreda, M.

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. H. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

Gill, A. T.

L. Isenhower, E. Urban, X. L. Zhang, A. T. Gill, T. Henage, T. A. Johnson, T. G. Walker, and M. Saffman, “Demonstration of a Neutral Atom Controlled-NOT Quantum Gate,” Phys. Rev. Lett. 104, 010503 (2010).
[Crossref] [PubMed]

Girvin, S. M.

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

Gong, Z. R.

H. Ian, Z. R. Gong, Y. X. Liu, C. P. Sun, and F. Nori, “Cavity optomechanical coupling assisted by an atomic gas,” Phys. Rev. A 78, 013824 (2008).
[Crossref]

Gorniaczyk, H.

H. Gorniaczyk, C. Tresp, J. Schmidt, H. Fedder, and S. Hofferberth, “Single-Photon Transistor Mediated by Interstate Rydberg Interactions,” Phys. Rev. Lett. 113, 053601 (2014).
[Crossref] [PubMed]

Gould, P. L.

D. Tong, S. M. Farooqi, J. Stanojevic, S. Krishnan, Y. P. Zhang, R. Cote, E. E. Eyler, and P. L. Gould, “Local Blockade of Rydberg Excitation in an Ultracold Gas,” Phys. Rev. Lett. 93, 063001 (2004).
[Crossref] [PubMed]

Grangier, P.

A. Gaetan, A. Y. Miroshnychenko, T. Wilk, A. Chotia, M. Viteau, D. Comparat, P. Pillet, A. Browaeys, and P. Grangier, “Observation of collective excitation of two individual atoms in the Rydberg blockade regime,” Nat. Phys. 5, 115–118 (2009).
[Crossref]

Grudinin, I. S.

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] [PubMed]

Gu, K. H.

X. B. Yan, C. L. Cui, K. H. Gu, X. D. Tian, C. B. Fu, and J. H. Wu, “Coherent perfect absorption, transmission, and synthesis in a double-cavity optomechanical system,” Opt. Express 22, 4886–4895 (2014).
[Crossref] [PubMed]

C. B. Fu, X. B. Yan, K. H. Gu, C. L. Cui, J. H. Wu, and T. D. Fu, “Steady-state solutions of a hybrid system involving atom-light and optomechanical interactions: Beyond the weak-cavity-field approximation,” Phys. Rev. A 87, 053841 (2013).
[Crossref]

Guerlin, C.

C. Guerlin, E. Brion, T. Esslinger, and K. Molmer, “Cavity quantum electrodynamics with a Rydberg-blocked atomic ensemble,” Phys. Rev. A 82, 053832 (2010).
[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, 093901 (2013).
[Crossref] [PubMed]

Habraken, S. J. M.

K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical Quantum Information Processing with Photons and Phonons,” Phys. Rev. Lett. 109, 013603 (2012).
[Crossref] [PubMed]

Hafezi, M.

Haldane, F. D. M.

F. D. M. Haldane and S. Raghu, “Possible Realization of Directional Optical Waveguides in Photonic Crystals with Broken Time-Reversal Symmetry,” Phys. Rev. Lett. 100, 013904 (2008).
[Crossref] [PubMed]

Henage, T.

L. Isenhower, E. Urban, X. L. Zhang, A. T. Gill, T. Henage, T. A. Johnson, T. G. Walker, and M. Saffman, “Demonstration of a Neutral Atom Controlled-NOT Quantum Gate,” Phys. Rev. Lett. 104, 010503 (2010).
[Crossref] [PubMed]

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L. Isenhower, E. Urban, X. L. Zhang, A. T. Gill, T. Henage, T. A. Johnson, T. G. Walker, and M. Saffman, “Demonstration of a Neutral Atom Controlled-NOT Quantum Gate,” Phys. Rev. Lett. 104, 010503 (2010).
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M. Saffman, T. G. Walker, and K. Molmer, “Quantum information with Rydberg atoms,” Rev. Mod. Phys. 82, 2313 (2010).
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E. Urban, T. A. Johnson, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker, and M. Saffman, “Observation of Rydberg blockade between two atoms,” Nat. Phys. 5, 110–114 (2009).
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M. Saffman and T. G. Walker, “Creating single-atom and single-photon sources from entangled atomic ensembles,” Phys. Rev. A 66, 065403 (2002).
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B. Anand, R. Podila, K. Lingam, S. R. Krishnan, S. S. S. Sai, R. Philip, and A. M. Rao, “Optical Diode Action from Axially Asymmetric Nonlinearity in an All-Carbon Solid-State Device,” Nano Lett. 13, 5771–5776 (2013).
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D. Tiarks, S. Baur, K. Schneider, S. Durr, and G. Rempe, “Single-Photon Transistor Using a Forster Resonance,” Phys. Rev. Lett. 113, 053602 (2014).
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C. Euter, K. G. Makris, R. EI-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
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L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. Qi, “An All-Silicon Passive Optical Diode,” Science 335, 447–450 (2012).
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H. Z. Shen, Y. H. Zhou, and X. X. Yi, “Quantum optical diode with semiconductor microcavities,” Phys. Rev. A 90, 023849 (2014).
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K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical Quantum Information Processing with Photons and Phonons,” Phys. Rev. Lett. 109, 013603 (2012).
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M. Saffman and T. G. Walker, “Creating single-atom and single-photon sources from entangled atomic ensembles,” Phys. Rev. A 66, 065403 (2002).
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D. W. Wang, H. T. Zhou, M. J. Guo, J. X. Zhang, J. Evers, and S. Y. Zhu, “Optical Diode Made from a Moving Photonic Crystal,” Phys. Rev. Lett. 110, 093901 (2013).
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K. Singer, M. Reetz-Lamour, T. Amthor, L. G. Marcassa, and M. Weidemuller, “Suppression of Excitation and Spectral Broadening Induced by Interactions in a Cold Gas of Rydberg Atoms,” Phys. Rev. Lett. 93, 163001 (2004).
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H. Weimer, M. Muller, I. Lesanovsky, P. Zoller, and H. P. Buchler, “A Rydberg quantum simulator,” Nat. Phys. 6, 382–388 (2010).
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L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. Qi, “An All-Silicon Passive Optical Diode,” Science 335, 447–450 (2012).
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A. Gaetan, A. Y. Miroshnychenko, T. Wilk, A. Chotia, M. Viteau, D. Comparat, P. Pillet, A. Browaeys, and P. Grangier, “Observation of collective excitation of two individual atoms in the Rydberg blockade regime,” Nat. Phys. 5, 115–118 (2009).
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X. B. Yan, C. L. Cui, K. H. Gu, X. D. Tian, C. B. Fu, and J. H. Wu, “Coherent perfect absorption, transmission, and synthesis in a double-cavity optomechanical system,” Opt. Express 22, 4886–4895 (2014).
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L. Du, Y. Zhang, C.-H. Fan, Y.-M. Liu, F. Gao, and J.-H. Wu, “Enhanced nonlinear characteristics with the assistance of a PT-symmetric trimer system,” Sci. Rep. 8, 2933 (2018).
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D. Yan, Z.-H. Wang, C.-N. Ren, H. Gao, Y. Li, and J.-H. Wu, “Duality and bistability in an optomechanical cavity coupled to a Rydberg superatom,” Phys. Rev. A 91, 023813 (2015).
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D. Yan, Z.-H. Wang, C.-N. Ren, H. Gao, Y. Li, and J.-H. Wu, “Duality and bistability in an optomechanical cavity coupled to a Rydberg superatom,” Phys. Rev. A 91, 023813 (2015).
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X. B. Yan, C. L. Cui, K. H. Gu, X. D. Tian, C. B. Fu, and J. H. Wu, “Coherent perfect absorption, transmission, and synthesis in a double-cavity optomechanical system,” Opt. Express 22, 4886–4895 (2014).
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C. B. Fu, X. B. Yan, K. H. Gu, C. L. Cui, J. H. Wu, and T. D. Fu, “Steady-state solutions of a hybrid system involving atom-light and optomechanical interactions: Beyond the weak-cavity-field approximation,” Phys. Rev. A 87, 053841 (2013).
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H. Wang, Z. X. Wang, J. Zhang, S. K. Ozdemir, L. Yang, and Y. X. Liu, “Phonon amplification in two coupled cavities containing one mechanical resonator,” Phys. Rev. A 90, 053814 (2014).
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B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. H. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
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E. Urban, T. A. Johnson, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker, and M. Saffman, “Observation of Rydberg blockade between two atoms,” Nat. Phys. 5, 110–114 (2009).
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H. Z. Shen, Y. H. Zhou, and X. X. Yi, “Quantum optical diode with semiconductor microcavities,” Phys. Rev. A 90, 023849 (2014).
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Z. F. Yu and S. H. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photonics 3, 91–94 (2009).
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Zhang, J.

H. Wang, Z. X. Wang, J. Zhang, S. K. Ozdemir, L. Yang, and Y. X. Liu, “Phonon amplification in two coupled cavities containing one mechanical resonator,” Phys. Rev. A 90, 053814 (2014).
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D. W. Wang, H. T. Zhou, M. J. Guo, J. X. Zhang, J. Evers, and S. Y. Zhu, “Optical Diode Made from a Moving Photonic Crystal,” Phys. Rev. Lett. 110, 093901 (2013).
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L. Isenhower, E. Urban, X. L. Zhang, A. T. Gill, T. Henage, T. A. Johnson, T. G. Walker, and M. Saffman, “Demonstration of a Neutral Atom Controlled-NOT Quantum Gate,” Phys. Rev. Lett. 104, 010503 (2010).
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L. Du, Y. Zhang, C.-H. Fan, Y.-M. Liu, F. Gao, and J.-H. Wu, “Enhanced nonlinear characteristics with the assistance of a PT-symmetric trimer system,” Sci. Rep. 8, 2933 (2018).
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D. Tong, S. M. Farooqi, J. Stanojevic, S. Krishnan, Y. P. Zhang, R. Cote, E. E. Eyler, and P. L. Gould, “Local Blockade of Rydberg Excitation in an Ultracold Gas,” Phys. Rev. Lett. 93, 063001 (2004).
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I. V. Shadrivov, V. A. Fedotov, D. A. Powell, Y. S. Kivshar, and N. I. Zheludev, “Electromagnetic wave analogue of an electronic diode,” New J. Phys. 13, 033025 (2011).
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D. W. Wang, H. T. Zhou, M. J. Guo, J. X. Zhang, J. Evers, and S. Y. Zhu, “Optical Diode Made from a Moving Photonic Crystal,” Phys. Rev. Lett. 110, 093901 (2013).
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D. W. Wang, H. T. Zhou, M. J. Guo, J. X. Zhang, J. Evers, and S. Y. Zhu, “Optical Diode Made from a Moving Photonic Crystal,” Phys. Rev. Lett. 110, 093901 (2013).
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A. Carmele, B. Vogell, K. Stannigel, and P. Zoller, “Opto-nanomechanics strongly coupled to a Rydberg superatom: coherent versus incoherent dynamics,” New J. Phys. 16, 063042 (2014).
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K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical Quantum Information Processing with Photons and Phonons,” Phys. Rev. Lett. 109, 013603 (2012).
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Appl. Phys. Lett. (2)

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Nature (London) (1)

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljacic, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature (London) 461, 772–776 (2009).
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I. V. Shadrivov, V. A. Fedotov, D. A. Powell, Y. S. Kivshar, and N. I. Zheludev, “Electromagnetic wave analogue of an electronic diode,” New J. Phys. 13, 033025 (2011).
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Opt. Express (4)

Opt. Lett. (1)

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Figures (6)

Fig. 1
Fig. 1 Schematic diagram of the hybrid system double optomechanical cavity coupled with Rydberg atoms. (a1) and (a2) shows energy level structures of the target Rydberg atom coupled with the left cavity and the gate one which dominates the target one by vdW potential VvdW between them. (b1) and (b2) show that the double-cavity decouple with atoms switching off the external control.
Fig. 2
Fig. 2 Energy level diagram (a) and external control timing flowchart (b). Standardization energy of the output probe field (c) Rr = |εoutR+/εR|2 (reflection coefficient of left cavity, blue) and Rl = |εoutL+/εL|2 (reflection coefficient of right cavity, red circle) as the function of x/κ, (c1) switching off the external control (coupled with atoms) and (c2) switching on that (decoupled with atoms), with G = κ, n = 1 and θ = π.
Fig. 3
Fig. 3 Transmission coefficient of left side Tl (in a1/b1) and right side Tr (in a2/b2) of the hybrid system vs. detuning x/κ and G, without (with) external control, when γ = 2κ and θ = with n = 1. c1 and c2 are the functional sketch maps.
Fig. 4
Fig. 4 Reflection coefficient of left side Rl (in a1(a2)) and Transmission coefficient of right side Tr (in b1(b2)) of the hybrid system vs. detuning x/κ and G, without (with) external control, when γ = 2κ, γm → 0 and θ = with εl ≠ 0, εr = 0. c1 and c2 are the functional sketch maps only with left side input probe field εL.
Fig. 5
Fig. 5 Reflection (Transmission) coefficient of left side Rl (Tr) of the hybrid system vs. detuning x/κ, without (with) external control (in a1, a2 (b1, b2)), when γ = 2κ, γmG → 0 and θ = (2n + 1)π/2 with n = 1. a2 (b2) is the functional sketch maps with both sides input probe field εL, εRe, corresponding to a1 (b1).
Fig. 6
Fig. 6 Standardization energy of the output probe field Tl = |εoutL+/εR|2 (transmission coefficient of left cavity, blue) and Tl = |εoutR+/εL|2 (transmission coefficient of right cavity, red circle) as the function of x/κ, (a1, a2) switching off the external control (coupled with atoms) and (b1, b2) switching on that (decoupled with atoms), with G = κ, 2κ, n = 1 and θ =

Equations (20)

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H = H 0 + H I .
H c = Δ cm [ c l c l + c r c r ] , H m = ω m b b , H a = [ δ e g σ e e + δ r g σ r r + δ r g σ r r ] , H c p = i ε cL [ c l c l ] + i [ ε l c l e i δ t ε l * c l e i δ t ] + i ε cR [ c r c r ] + i [ ε cR c r e i θ e i δ t ε cR * c r e i θ e i δ t ] , H m c = g 0 [ c r c r c l c l ] [ b + b ] , V a f = [ Ω r σ r e + g ac c l σ e g + Ω g σ g r ] + H . C . V v d W = C 6 R tg 6 σ r r σ r r ,
b ˙ = i ω m b i g 0 [ c l c l c r c r ] γ m 2 b + γ m b i n , c ˙ l = [ κ + i Δ c i g 0 ( b + b ) ] c l + ε cL + ε l e i δ t i g ac σ ^ e g + 2 κ c l in , c ˙ r = [ κ + i Δ c + i g 0 ( b + b ) ] c r + ε cR + ε r e i θ e i δ t + 2 κ c r in , σ ^ ˙ ge = [ i δ ge + γ e ] σ ^ ge + i Ω r * σ ^ gr + i g ac c l + f 1 ( t ) , σ ^ ˙ gr = [ i δ gr + i V t s + γ r ] σ ^ gr + i Ω r σ ^ ge + f 2 ( t ) , σ ^ ˙ g r = [ i δ g r + i V g s + γ r ] σ ^ g r + i Ω g * + f 3 ( t ) .
b ˙ s = i ω m b s i g 0 [ c ls * c ls c rs * c rs ] γ m 2 b s , c ˙ ls = [ κ + i Δ c i g 0 ( b s * + b s ) ] c ls + ε c L i g ac σ ^ ge s , c ˙ rs = [ κ + i Δ c + i g 0 ( b s * + b s ) ] c rs + ε c R , σ ^ ˙ ge s = [ i δ ge + γ e ] σ ^ ge s + i Ω r * σ ^ gr s + i g ac c ls , σ ^ ˙ gr s = [ i δ gr + i V t s + γ r ] σ ^ gr s + i Ω r σ ^ ge s , σ ^ ˙ g r s = [ i δ g r + i V g s + γ r ] σ ^ g r s + i Ω g * ,
δ b ˙ = i ω m δ b i g 0 [ c ls * δ c l + c ls δ c l c rs * δ c r c rs δ c r ] γ m 2 δ b + γ m b in , δ c ˙ l = [ κ + i Δ ˜ c ] δ c l i g 0 c ls ( δ b + δ b ) + ε l e i δ t i g ac δ σ ^ ge + 2 κ c l in , δ c ˙ r = [ κ + i Δ ˜ c ] δ c r + i g 0 c rs ( δ b + δ b ) + ε R e i θ e i δ t 2 κ c r in , δ σ ^ ˙ ge = [ i δ ge + γ e ] δ σ ^ ge + i Ω r * δ σ ^ gr + i g ac δ c l + f 1 ( t ) , δ σ ^ ˙ gr = [ i δ gr + i V t s + γ r ] δ σ ^ gr + i Ω r δ σ ^ ge + f 2 ( t ) , δ σ ^ ˙ g r = [ i δ g r + i V g s + γ r ] δ σ ^ g r + f 3 ( t ) ,
ϱ s = ϱ 2 σ r r s + ϱ 3 L [ 1 σ r r s ] .
ϱ 2 = i g ac c ls i δ ge + γ e , ϱ 3 L = i g ac [ i δ gr + i V t s + γ r ] c ls [ i δ ge + γ e ] [ i δ gr + i V t s + γ r ] + Ω r * Ω r .
σ r r s γ r 2 Ω g * Ω g σ r g s σ g r s γ r 2 γ r 2 + δ g r 2 .
b s = i g 0 ( c ls * c ls c rs * c rs ) i Δ m + γ m 2 , c ls = ε cL i g ac ϱ s κ + i Δ c i g 0 ( b s * + b s ) , c rs = ε cR κ + i Δ c + i g 0 ( b s * + b s ) .
δ ϱ 2 = i g ac δ c l i δ ge + γ e , δ ϱ 3 L = i g ac [ i δ gr + i V t s + γ r ] δ c l [ i δ ge + γ e ] [ i δ gr + i V t s + γ r ] + Ω r * Ω r .
δ b ˙ = i g 0 ( c ls * δ c l c rs * δ c r ) γ m 2 δ b + γ m b in , δ c ˙ l = κ δ c l i g 0 c ls δ b + ε L e i x t i g ac δ ϱ s + 2 κ c l in , δ c ˙ r = κ δ c r + i g 0 c rs δ b + ε R e i θ e i x t + 2 κ c r in ,
δ b ˙ = i g 0 ( c ls * δ c l c rs * δ c r ) γ m 2 δ b , δ c ˙ l = κ ( δ c l ) i g 0 c ls δ b + ε L e i x t i g ac δ ϱ s , δ c ˙ r = κ ( δ c r ) + i g 0 c rs δ b + ε R e i θ e i x t ,
δ b + = in G g ac ε r e i θ i G * κ ε l γ m κ g ac + G 2 κ + G 2 n 2 g ac , δ c l + = G n ε r e i θ t + [ G 2 n 2 + γ m κ ] ε l γ m κ g ac + G 2 κ + G 2 n 2 g ac , δ c r + = [ γ m g ac + G 2 ] ε r e i θ + G 2 n ε l γ m κ g ac + G 2 κ + G 2 n 2 g ac ,
ε outl + ε l e i x t = 2 κ δ c l , ε outr + ε r e i θ e i x t = 2 κ δ c r r ,
ε outl + = 2 κ δ c l + ε l , ε outr + = 2 κ δ c r + ε r e i θ ,
ε outl + = 2 κ n G 2 ε r e i θ + [ γ m κ ( 2 κ g ac ) + G 2 ( 2 κ n 2 κ n 2 g ac ) ] ε l γ m κ g ac + n 2 g ac G 2 + κ G 2 ε outr + = 2 κ n G 2 ε l [ ( i x 3 κ ) ( γ m g ac + G 2 ) + n 2 g ac G 2 ] ε r e i θ γ m κ g ac + n 2 g ac G 2 + κ G 2 .
R l = | 2 κ n 2 G 2 e i θ + [ γ m κ κ + 2 κ G 2 ( n 2 + 1 ) κ G 2 ] ε l / ε r γ m κ κ + ( n 2 + 1 ) κ G 2 | 2 , R r = | 2 κ n 2 G 2 + [ γ m κ κ + 2 κ G 2 ( n 2 + 1 ) κ G 2 ] ε r e i θ / ε l γ m κ κ + ( n 2 + 1 ) κ G 2 | 2 ,
T l | ( 3 κ i x ) ( γ m g ac + G 2 ) + n 2 g ac G 2 ] γ m κ g ac + n 2 g ac G 2 + κ G 2 | 2 1 .
T l | 2 κ n G 2 γ m κ g ac + n 2 g ac G 2 + κ G 2 | 2 0 , R l | [ ( i x 3 κ ) ( γ m g ac + G 2 ) + n 2 g ac G 2 ] γ m κ g ac + n 2 g ac G 2 + κ G 2 | 2 1 ,
T l ( 3 κ g ac G ) 2 + g ac 2 ( 1 + G ) 2 ( κ g ac G ) 2 + g ac 2 > 1 ,

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