Abstract

We propose a scheme of photon blockade in a system comprising of coupled cavities embedded in Kerr nonlinear material, where two cavities are driven and dissipated. We analytically derive the exact optimal conditions for strong photon antibunching, which are in good agreement with those obtained by numerical simulations. We find that conventional and unconventional photon blockades have controllable flexibilities by tuning the strength ratio and relative phase between two complex driving fields. Such unconventional photon-blockade effects are ascribed to the quantum interference effect to avoid two-photon excitation of the coupled cavities. We also discuss the statistical properties of the photons under given optimal conditions. Our results provide a promising platform for the coherent manipulation of photon blockade, which has potential applications for quantum information processing and quantum optical devices.

© 2015 Optical Society of America

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2015 (7)

J. Tang, W. D. Geng, and X. L. Xu, “Quantum interference induced photon blockade in a coupled single quantum dot-cavity system,” Sci. Rep. 5, 9252 (2015).
[Crossref] [PubMed]

X. N. Xu, M. Gullans, and J. M. Taylor, “Quantum nonlinear optics near optomechanical instabilities,” Phys. Rev. A 91, 013818 (2015).
[Crossref]

W. Z. Zhang, J. Cheng, J. Y. Liu, and L. Zhou, “Controlling photon transport in the single-photon weak-coupling regime of cavity optomechanics,” Phys. Rev. A 91, 063836 (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] [PubMed]

H. Wang, X. Gu, Y. X. Liu, A. Miranowicz, and F. Nori, “Tunable photon blockade in a hybrid system consisting of an optomechanical device coupled to a two-level system,” Phys. Rev. A 92, 033806 (2015).
[Crossref]

H. Z. Shen, Y. H. Zhou, and X. X. Yi, “Tunable photon blockade in coupled semiconductor cavities,” Phys. Rev. A 91, 063808 (2015).
[Crossref]

Y. H. Zhou, H. Z. Shen, and X. X. Yi, “Unconventional photon blockade with second-order nonlinearity,” Phys. Rev. A 92, 023838 (2015).
[Crossref]

2014 (13)

O. Kyriienko and T. C. H. Liew, “Triggered single-photon emitters based on stimulated parametric scattering in weakly nonlinear systems,” Phys. Rev. A 90, 063805 (2014).
[Crossref]

M. A. Lemonde, N. Didier, and A. A. Clerk, “Antibunching and unconventional photon blockade with Gaussian squeezed states,” Phys. Rev. A 90, 063824 (2014).
[Crossref]

W. Zhang, Z. Y. Yu, Y. M. Liu, and Y. W. Peng, “Optimal photon antibunching in a quantum-dot-bimodal-cavity system,” Phys. Rev. A 89, 043832 (2014).
[Crossref]

O. Kyriienko, I. A. Shelykh, and T. C. H. Liew, “Tunable single-photon emission from dipolaritons,” Phys. Rev. A 90, 033807 (2014).
[Crossref]

D. Gerace and V. Savona, “Unconventional photon blockade in doubly resonant microcavities with second-order nonlinearity,” Phys. Rev. A 89, 031803 (2014).
[Crossref]

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

F. Fratini, E. Mascarenhas, L. Safari, J-Ph. Poizat, D. Valente, A. Auffèves, D. Gerace, and M. F. Santos, “Fabry-Perot interferometer with quantum mirrors: nonlinear light transport and rectification,” Phys. Rev. Lett. 113, 243601 (2014).
[Crossref] [PubMed]

E. Mascarenhas, D. Gerace, D. Valente, S. Montangero, A. Auffovès, and M. F. Santos, “A quantum optical valve in a nonlinear-linear resonators junction,” Europhys. Lett. 106, 54003 (2014).
[Crossref]

X. W. Xu and Y. Li, “Tunable photon statistics in weakly nonlinear photonic molecules,” Phys. Rev. A 90, 043822 (2014).
[Crossref]

X. W. Xu and Y. Li, “Strongly correlated two-photon transport in a one-dimensional waveguide coupled to a weakly nonlinear cavity,” Phys. Rev. A 90, 033832 (2014).
[Crossref]

X. W. Xu and Y. Li, “Strong photon antibunching of symmetric and antisymmetric modes in weakly nonlinear photonic molecules,” Phys. Rev. A 90, 033809 (2014).
[Crossref]

Y. X. Liu, X. W. Xu, A. Miranowicz, and F. Nori, “From blockade to transparency: controllable photon transmission through a circuit-QED system,” Phys. Rev. A 89, 043818 (2014).
[Crossref]

M. Peiris, K. Konthasinghe, Y. Yu, Z. C. Niu, and A. Muller, “Bichromatic resonant light scattering from a quantum dot,” Phys. Rev. B 89, 155305 (2014).
[Crossref]

2013 (13)

A. Majumdar and D. Gerace, “Single-photon blockade in doubly resonant nanocavities with second-order non-linearity,” Phys. Rev. B 87, 235319 (2013).
[Crossref]

J. Q. Liao and F. Nori, “Photon blockade in quadratically coupled optomechanical systems,” Phys. Rev. A 88, 023853 (2013).
[Crossref]

P. Kómár, S. D. Bennett, K. Stannigel, S. J. M. Habraken, P. Rabl, P. Zoller, and M. D. Lukin, “Single-photon nonlinearities in two-mode optomechanics,” Phys. Rev. A 87, 013839 (2013).
[Crossref]

A. Miranowicz, M. Paprzycka, Y. X. Liu, J. Bajer, and F. Nori, “Two-photon and three-photon blockades in driven nonlinear systems,” Phys. Rev. A 87, 023809 (2013).
[Crossref]

X. Y Lü, W. M. Zhang, S. Ashhab, Y. Wu, and F. Nori, “Quantum-criticality-induced strong Kerr nonlinearities in optomechanical systems,” Sci. Rep. 3, 2943 (2013).
[Crossref] [PubMed]

L. Qiu, L. Gan, W. Ding, and Z. Y. Li, “Single-photon generation by pulsed laser in optomechanical system via photon blockade effect,” J. Opt. Soc. Am. B 30, 1683–1687 (2013).
[Crossref]

M. Bajcsy, A. Majumdar, A. Rundquist, and J. Vučković, “Photon blockade with a four-level quantum emitter coupled to a photonic-crystal nanocavity,” New J. Phys. 15, 025014 (2013).
[Crossref]

T. C. H. Liew and V. Savona, “Multimode entanglement in coupled cavity arrays,” New J. Phys. 15, 025015 (2013).
[Crossref]

H. Flayac and V. Savona, “Input-output theory of the unconventional photon blockade,” Phys. Rev. A 88, 033836 (2013).
[Crossref]

I. Carusotto and C. Ciuti, “Quantum fluids of light,” Rev. Mod. Phys. 85, 299 (2013).
[Crossref]

A. D. Cimmarusti, C. A. Schroeder, B. D. Patterson, L. A. Orozco, P. Barberis-Blostein, and H. J. Carmichael, “Control of conditional quantum beats in cavity QED: amplitude decoherence and phase shifts,” New J. Phys. 15, 013017 (2013).
[Crossref]

X. W. Xu and Y. J. Li, “Antibunching photons in a cavity coupled to an optomechanical system,” J. Opt. B: At. Mol. Opt. Phys. 46, 035502 (2013).

S. Ferretti, V. Savona, and D. Gerace, “Optimal antibunching in passive photonic devices based on coupled nonlinear resonators,” New J. Phys. 15, 025012 (2013).
[Crossref]

2012 (3)

A. Majumdar, M. Bajcsy, A. Rundquist, and J. Vučković, “Loss-enabled sub-Poissonian light generation in a bimodal nanocavity,” Phys. Rev. Lett. 108, 183601 (2012).
[Crossref] [PubMed]

T. C. H. Liew and V. Savona, “Quantum entanglement in nanocavity arrays,” Phys. Rev. A 85, 050301 (2012).
[Crossref]

S. Ferretti and D. Gerace, “Single-photon nonlinear optics with Kerr-type nanostructured materials,” Phys. Rev. B 85, 033303 (2012).
[Crossref]

2011 (7)

M. Bamba and C. Ciuti, “Counter-polarized single-photon generation from the auxiliary cavity of a weakly nonlinear photonic molecule,” Appl. Phys. Lett. 99, 171111 (2011).
[Crossref]

M. Bamba, A. Imamoğlu, I. Carusotto, and C. Ciuti, “Origin of strong photon antibunching in weakly nonlinear photonic molecules,” Phys. Rev. A 83, 021802 (2011).
[Crossref]

C. Lang, D. Bozyigit, C. Eichler, L. Steffen, J. M. Fink, A. A. Abdumalikov, M. Baur, S. Filipp, M. P. da Silva, A. Blais, and A. Wallraff, “Observation of resonant photon blockade at microwave frequencies using correlation function measurements,” Phys. Rev. Lett. 106, 243601 (2011).
[Crossref] [PubMed]

A. J. Hoffman, S. J. Srinivasan, S. Schmidt, L. Spietz, J. Aumentado, H. E. Türeci, and A. A. Houck, “Dispersive photon blockade in a superconducting circuit,” Phys. Rev. Lett. 107, 053602 (2011).
[Crossref] [PubMed]

P. Rabl, “Photon blockade effect in optomechanical systems,” Phys. Rev. Lett. 107, 063601 (2011).
[Crossref] [PubMed]

A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett. 107, 063602 (2011).
[Crossref] [PubMed]

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5, 222–229 (2011).
[Crossref]

2010 (6)

J. Q. Liao and C. K. Law, “Correlated two-photon transport in a one-dimensional waveguide side-coupled to a nonlinear cavity,” Phys. Rev. A 82, 053836 (2010).
[Crossref]

F. Dubin, C. Russo, H. G. Barros, A. Stute, C. Becher, P. O. Schmidt, and R. Blatt, “Quantum to classical transition in a single-ion laser,” Nat. Phys. 6, 350–353 (2010).
[Crossref]

M. Leib and M. J. Hartmann, “Bose-Hubbard dynamics of polaritons in a chain of circuit quantum electrodynamics cavities,” New J. Phys. 12, 093031 (2010).
[Crossref]

Y. P. Rakovich and J. F. Donegan, “Photonic atoms and molecules,” Laser Photonics Rev. 4, 179–191 (2010).
[Crossref]

T. C. H. Liew and V. Savona, “Single photons from coupled quantum modes,” Phys. Rev. Lett. 104, 183601 (2010).
[Crossref] [PubMed]

S. Ferretti, L. C. Andreani, H. E. Türeci, and D. Gerace, “Photon correlations in a two-site nonlinear cavity system under coherent drive and dissipation,” Phys. Rev. A 82, 013841 (2010).
[Crossref]

2009 (4)

J. Wiersig, C. Gies, F. Jahnke, M. Asmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460, 245–249 (2009).
[Crossref] [PubMed]

D. Gerace, H. E. Tureci, A. Imamoḡlu, V. Giovannetti, and R. Fazio, “The quantum-optical Josephson interferometer,” Nat. Phys. 5, 281–284 (2009).
[Crossref]

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81, 1301–1350 (2009).
[Crossref]

J. L. ÓBrien, A. Furusawa, and J. Vučković, “Photonic quantum technologies,” Nat. Photonics 3, 687–695 (2009).
[Crossref]

2008 (2)

B. Dayan, A. S. Parkins, T. Aoki, E. P. Ostby, K. J. Vahala, and H. J. Kimble, “A photon turnstile dynamically regulated by one atom,” Science 319, 1062–1065 (2008).
[Crossref] [PubMed]

A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vučković, “Coherent generation of non-classical light on a chip via photon-induced tunnelling and blockade,” Nat. Phys. 4, 859–863 (2008).
[Crossref]

2007 (5)

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[Crossref]

A. J. Shields, “Semiconductor quantum light sources,” Nat. Photonics 1, 215–223 (2007).
[Crossref]

D. G. Angelakis, M. F. Santos, and S. Bose, “Photon-blockade-induced Mott transitions and XY spin models in coupled cavity arrays,” Phys. Rev. A 76, 031805 (2007).
[Crossref]

D. E. Chang, A. S. Sorensen, E. A. Demler, and M. D. Lukin, “A single-photon transistor using nanoscale surface plasmons,” Nat. Phys. 3, 807–812 (2007).
[Crossref]

J. T. Shen and S. H. Fan, “Strongly correlated two-photon transport in a one-dimensional waveguide coupled to a two-level system,” Phys. Rev. Lett. 98, 153003 (2007).
[Crossref] [PubMed]

2006 (1)

A. Verger, C. Ciuti, and I. Carusotto, “Polariton quantum blockade in a photonic dot,” Phys. Rev. B 73, 193306 (2006).
[Crossref]

2005 (1)

K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, “Photon blockade in an optical cavity with one trapped atom,” Nature 436, 87–90 (2005).
[Crossref] [PubMed]

2002 (2)

I. I. Smolyaninov, A. V. Zayats, A. Gungor, and C. C. Davis, “Single-photon tunneling via localized surface plasmons,” Phys. Rev. Lett. 88, 187402 (2002).
[Crossref] [PubMed]

S. Rebić, A. S. Parkins, and S. M. Tan, “Photon statistics of a single-atom intracavity system involving electromagnetically induced transparency,” Phys. Rev. A 65, 063804 (2002).
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2001 (1)

E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409, 46–52 (2001).
[Crossref] [PubMed]

1999 (3)

R. J. Brecha, P. R. Rice, and M. Xiao, “N two-level atoms in a driven optical cavity: Quantum dynamics of forward photon scattering for weak incident fields,” Phys. Rev. A 59, 2392 (1999).
[Crossref]

J. Kim, O. Bensen, H. Kan, and Y. Yamamoto, “A single-photon turnstile device,” Nature 397, 500–503 (1999).
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M. J. Werner and A. Imamoḡlu, “Photon-photon interactions in cavity electromagnetically induced transparency,” Phys. Rev. A 61, 011801 (1999).
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1998 (1)

M. Bayer, T. Gutbrod, J. P. Reithmaier, A. Forchel, T. L. Reinecke, P. A. Knipp, A. A. Dremin, and V. D. Kulakovskii, “Optical modes in photonic molecules,” Phys. Rev. Lett. 81, 2582 (1998).
[Crossref]

1997 (1)

A. Imamoğlu, H. Schmidt, G. Woods, and M. Deutsch, “Strongly interacting photons in a nonlinear cavity,” Phys. Rev. Lett. 79, 1467 (1997).
[Crossref]

1996 (1)

L. Davidovich, “Sub-Poissonian processes in quantum optics,” Rev. Mod. Phys. 68, 127–173 (1996).
[Crossref]

1992 (1)

L. Tian and H. J. Carmichael, “Quantum trajectory simulations of two-state behavior in an optical cavity containing one atom,” Phys. Rev. A 46, R6801 (1992).
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1991 (1)

H. J. Carmichael, R. J. Brecha, and P. R. Rice, “Quantum interference and collapse of the wavefunction in cavity QED,” Opt. Commun. 82, 73 (1991).
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1988 (1)

Z. Y. Ou and L. Mandel, “Observation of spatial quantum beating with separated photodetectors,” Phys. Rev. Lett. 61, 54 (1988).
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C. Lang, D. Bozyigit, C. Eichler, L. Steffen, J. M. Fink, A. A. Abdumalikov, M. Baur, S. Filipp, M. P. da Silva, A. Blais, and A. Wallraff, “Observation of resonant photon blockade at microwave frequencies using correlation function measurements,” Phys. Rev. Lett. 106, 243601 (2011).
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Andreani, L. C.

S. Ferretti, L. C. Andreani, H. E. Türeci, and D. Gerace, “Photon correlations in a two-site nonlinear cavity system under coherent drive and dissipation,” Phys. Rev. A 82, 013841 (2010).
[Crossref]

Angelakis, D. G.

D. G. Angelakis, M. F. Santos, and S. Bose, “Photon-blockade-induced Mott transitions and XY spin models in coupled cavity arrays,” Phys. Rev. A 76, 031805 (2007).
[Crossref]

Aoki, T.

B. Dayan, A. S. Parkins, T. Aoki, E. P. Ostby, K. J. Vahala, and H. J. Kimble, “A photon turnstile dynamically regulated by one atom,” Science 319, 1062–1065 (2008).
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Ashhab, S.

X. Y Lü, W. M. Zhang, S. Ashhab, Y. Wu, and F. Nori, “Quantum-criticality-induced strong Kerr nonlinearities in optomechanical systems,” Sci. Rep. 3, 2943 (2013).
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Asmann, M.

J. Wiersig, C. Gies, F. Jahnke, M. Asmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460, 245–249 (2009).
[Crossref] [PubMed]

Auffèves, A.

F. Fratini, E. Mascarenhas, L. Safari, J-Ph. Poizat, D. Valente, A. Auffèves, D. Gerace, and M. F. Santos, “Fabry-Perot interferometer with quantum mirrors: nonlinear light transport and rectification,” Phys. Rev. Lett. 113, 243601 (2014).
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Auffovès, A.

E. Mascarenhas, D. Gerace, D. Valente, S. Montangero, A. Auffovès, and M. F. Santos, “A quantum optical valve in a nonlinear-linear resonators junction,” Europhys. Lett. 106, 54003 (2014).
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Aumentado, J.

A. J. Hoffman, S. J. Srinivasan, S. Schmidt, L. Spietz, J. Aumentado, H. E. Türeci, and A. A. Houck, “Dispersive photon blockade in a superconducting circuit,” Phys. Rev. Lett. 107, 053602 (2011).
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Bachor, H. A.

H. A. Bachor and T. C. Ralph, A Guide to Experiments in Quantum Optics (Wiley VCH, 2004).

Bajcsy, M.

M. Bajcsy, A. Majumdar, A. Rundquist, and J. Vučković, “Photon blockade with a four-level quantum emitter coupled to a photonic-crystal nanocavity,” New J. Phys. 15, 025014 (2013).
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A. Majumdar, M. Bajcsy, A. Rundquist, and J. Vučković, “Loss-enabled sub-Poissonian light generation in a bimodal nanocavity,” Phys. Rev. Lett. 108, 183601 (2012).
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Bajer, J.

A. Miranowicz, M. Paprzycka, Y. X. Liu, J. Bajer, and F. Nori, “Two-photon and three-photon blockades in driven nonlinear systems,” Phys. Rev. A 87, 023809 (2013).
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Bamba, M.

M. Bamba and C. Ciuti, “Counter-polarized single-photon generation from the auxiliary cavity of a weakly nonlinear photonic molecule,” Appl. Phys. Lett. 99, 171111 (2011).
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M. Bamba, A. Imamoğlu, I. Carusotto, and C. Ciuti, “Origin of strong photon antibunching in weakly nonlinear photonic molecules,” Phys. Rev. A 83, 021802 (2011).
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Barberis-Blostein, P.

A. D. Cimmarusti, C. A. Schroeder, B. D. Patterson, L. A. Orozco, P. Barberis-Blostein, and H. J. Carmichael, “Control of conditional quantum beats in cavity QED: amplitude decoherence and phase shifts,” New J. Phys. 15, 013017 (2013).
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Barros, H. G.

F. Dubin, C. Russo, H. G. Barros, A. Stute, C. Becher, P. O. Schmidt, and R. Blatt, “Quantum to classical transition in a single-ion laser,” Nat. Phys. 6, 350–353 (2010).
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Baur, M.

C. Lang, D. Bozyigit, C. Eichler, L. Steffen, J. M. Fink, A. A. Abdumalikov, M. Baur, S. Filipp, M. P. da Silva, A. Blais, and A. Wallraff, “Observation of resonant photon blockade at microwave frequencies using correlation function measurements,” Phys. Rev. Lett. 106, 243601 (2011).
[Crossref] [PubMed]

Bayer, M.

J. Wiersig, C. Gies, F. Jahnke, M. Asmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460, 245–249 (2009).
[Crossref] [PubMed]

M. Bayer, T. Gutbrod, J. P. Reithmaier, A. Forchel, T. L. Reinecke, P. A. Knipp, A. A. Dremin, and V. D. Kulakovskii, “Optical modes in photonic molecules,” Phys. Rev. Lett. 81, 2582 (1998).
[Crossref]

Becher, C.

F. Dubin, C. Russo, H. G. Barros, A. Stute, C. Becher, P. O. Schmidt, and R. Blatt, “Quantum to classical transition in a single-ion laser,” Nat. Phys. 6, 350–353 (2010).
[Crossref]

Bechmann-Pasquinucci, H.

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81, 1301–1350 (2009).
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Bennett, S. D.

P. Kómár, S. D. Bennett, K. Stannigel, S. J. M. Habraken, P. Rabl, P. Zoller, and M. D. Lukin, “Single-photon nonlinearities in two-mode optomechanics,” Phys. Rev. A 87, 013839 (2013).
[Crossref]

Bensen, O.

J. Kim, O. Bensen, H. Kan, and Y. Yamamoto, “A single-photon turnstile device,” Nature 397, 500–503 (1999).
[Crossref]

Berstermann, T.

J. Wiersig, C. Gies, F. Jahnke, M. Asmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460, 245–249 (2009).
[Crossref] [PubMed]

Birnbaum, K. M.

K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, “Photon blockade in an optical cavity with one trapped atom,” Nature 436, 87–90 (2005).
[Crossref] [PubMed]

Blais, A.

C. Lang, D. Bozyigit, C. Eichler, L. Steffen, J. M. Fink, A. A. Abdumalikov, M. Baur, S. Filipp, M. P. da Silva, A. Blais, and A. Wallraff, “Observation of resonant photon blockade at microwave frequencies using correlation function measurements,” Phys. Rev. Lett. 106, 243601 (2011).
[Crossref] [PubMed]

Blatt, R.

F. Dubin, C. Russo, H. G. Barros, A. Stute, C. Becher, P. O. Schmidt, and R. Blatt, “Quantum to classical transition in a single-ion laser,” Nat. Phys. 6, 350–353 (2010).
[Crossref]

Boca, A.

K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, “Photon blockade in an optical cavity with one trapped atom,” Nature 436, 87–90 (2005).
[Crossref] [PubMed]

Boozer, A. D.

K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, “Photon blockade in an optical cavity with one trapped atom,” Nature 436, 87–90 (2005).
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Børkje, K.

A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett. 107, 063602 (2011).
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Bose, S.

D. G. Angelakis, M. F. Santos, and S. Bose, “Photon-blockade-induced Mott transitions and XY spin models in coupled cavity arrays,” Phys. Rev. A 76, 031805 (2007).
[Crossref]

Bozyigit, D.

C. Lang, D. Bozyigit, C. Eichler, L. Steffen, J. M. Fink, A. A. Abdumalikov, M. Baur, S. Filipp, M. P. da Silva, A. Blais, and A. Wallraff, “Observation of resonant photon blockade at microwave frequencies using correlation function measurements,” Phys. Rev. Lett. 106, 243601 (2011).
[Crossref] [PubMed]

Brecha, R. J.

R. J. Brecha, P. R. Rice, and M. Xiao, “N two-level atoms in a driven optical cavity: Quantum dynamics of forward photon scattering for weak incident fields,” Phys. Rev. A 59, 2392 (1999).
[Crossref]

H. J. Carmichael, R. J. Brecha, and P. R. Rice, “Quantum interference and collapse of the wavefunction in cavity QED,” Opt. Commun. 82, 73 (1991).
[Crossref]

Carmichael, H. J.

A. D. Cimmarusti, C. A. Schroeder, B. D. Patterson, L. A. Orozco, P. Barberis-Blostein, and H. J. Carmichael, “Control of conditional quantum beats in cavity QED: amplitude decoherence and phase shifts,” New J. Phys. 15, 013017 (2013).
[Crossref]

L. Tian and H. J. Carmichael, “Quantum trajectory simulations of two-state behavior in an optical cavity containing one atom,” Phys. Rev. A 46, R6801 (1992).
[Crossref] [PubMed]

H. J. Carmichael, R. J. Brecha, and P. R. Rice, “Quantum interference and collapse of the wavefunction in cavity QED,” Opt. Commun. 82, 73 (1991).
[Crossref]

Carusotto, I.

I. Carusotto and C. Ciuti, “Quantum fluids of light,” Rev. Mod. Phys. 85, 299 (2013).
[Crossref]

M. Bamba, A. Imamoğlu, I. Carusotto, and C. Ciuti, “Origin of strong photon antibunching in weakly nonlinear photonic molecules,” Phys. Rev. A 83, 021802 (2011).
[Crossref]

A. Verger, C. Ciuti, and I. Carusotto, “Polariton quantum blockade in a photonic dot,” Phys. Rev. B 73, 193306 (2006).
[Crossref]

Cerf, N. J.

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81, 1301–1350 (2009).
[Crossref]

Chang, D. E.

D. E. Chang, A. S. Sorensen, E. A. Demler, and M. D. Lukin, “A single-photon transistor using nanoscale surface plasmons,” Nat. Phys. 3, 807–812 (2007).
[Crossref]

Cheng, J.

W. Z. Zhang, J. Cheng, J. Y. Liu, and L. Zhou, “Controlling photon transport in the single-photon weak-coupling regime of cavity optomechanics,” Phys. Rev. A 91, 063836 (2015).
[Crossref]

Cimmarusti, A. D.

A. D. Cimmarusti, C. A. Schroeder, B. D. Patterson, L. A. Orozco, P. Barberis-Blostein, and H. J. Carmichael, “Control of conditional quantum beats in cavity QED: amplitude decoherence and phase shifts,” New J. Phys. 15, 013017 (2013).
[Crossref]

Ciuti, C.

I. Carusotto and C. Ciuti, “Quantum fluids of light,” Rev. Mod. Phys. 85, 299 (2013).
[Crossref]

M. Bamba, A. Imamoğlu, I. Carusotto, and C. Ciuti, “Origin of strong photon antibunching in weakly nonlinear photonic molecules,” Phys. Rev. A 83, 021802 (2011).
[Crossref]

M. Bamba and C. Ciuti, “Counter-polarized single-photon generation from the auxiliary cavity of a weakly nonlinear photonic molecule,” Appl. Phys. Lett. 99, 171111 (2011).
[Crossref]

A. Verger, C. Ciuti, and I. Carusotto, “Polariton quantum blockade in a photonic dot,” Phys. Rev. B 73, 193306 (2006).
[Crossref]

Clerk, A. A.

M. A. Lemonde, N. Didier, and A. A. Clerk, “Antibunching and unconventional photon blockade with Gaussian squeezed states,” Phys. Rev. A 90, 063824 (2014).
[Crossref]

da Silva, M. P.

C. Lang, D. Bozyigit, C. Eichler, L. Steffen, J. M. Fink, A. A. Abdumalikov, M. Baur, S. Filipp, M. P. da Silva, A. Blais, and A. Wallraff, “Observation of resonant photon blockade at microwave frequencies using correlation function measurements,” Phys. Rev. Lett. 106, 243601 (2011).
[Crossref] [PubMed]

Davidovich, L.

L. Davidovich, “Sub-Poissonian processes in quantum optics,” Rev. Mod. Phys. 68, 127–173 (1996).
[Crossref]

Davis, C. C.

I. I. Smolyaninov, A. V. Zayats, A. Gungor, and C. C. Davis, “Single-photon tunneling via localized surface plasmons,” Phys. Rev. Lett. 88, 187402 (2002).
[Crossref] [PubMed]

Dayan, B.

B. Dayan, A. S. Parkins, T. Aoki, E. P. Ostby, K. J. Vahala, and H. J. Kimble, “A photon turnstile dynamically regulated by one atom,” Science 319, 1062–1065 (2008).
[Crossref] [PubMed]

Demler, E. A.

D. E. Chang, A. S. Sorensen, E. A. Demler, and M. D. Lukin, “A single-photon transistor using nanoscale surface plasmons,” Nat. Phys. 3, 807–812 (2007).
[Crossref]

Deutsch, M.

A. Imamoğlu, H. Schmidt, G. Woods, and M. Deutsch, “Strongly interacting photons in a nonlinear cavity,” Phys. Rev. Lett. 79, 1467 (1997).
[Crossref]

Didier, N.

M. A. Lemonde, N. Didier, and A. A. Clerk, “Antibunching and unconventional photon blockade with Gaussian squeezed states,” Phys. Rev. A 90, 063824 (2014).
[Crossref]

Ding, W.

Donegan, J. F.

Y. P. Rakovich and J. F. Donegan, “Photonic atoms and molecules,” Laser Photonics Rev. 4, 179–191 (2010).
[Crossref]

Dowling, J. P.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[Crossref]

Dremin, A. A.

M. Bayer, T. Gutbrod, J. P. Reithmaier, A. Forchel, T. L. Reinecke, P. A. Knipp, A. A. Dremin, and V. D. Kulakovskii, “Optical modes in photonic molecules,” Phys. Rev. Lett. 81, 2582 (1998).
[Crossref]

Dubin, F.

F. Dubin, C. Russo, H. G. Barros, A. Stute, C. Becher, P. O. Schmidt, and R. Blatt, “Quantum to classical transition in a single-ion laser,” Nat. Phys. 6, 350–353 (2010).
[Crossref]

Dušek, M.

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81, 1301–1350 (2009).
[Crossref]

Eichler, C.

C. Lang, D. Bozyigit, C. Eichler, L. Steffen, J. M. Fink, A. A. Abdumalikov, M. Baur, S. Filipp, M. P. da Silva, A. Blais, and A. Wallraff, “Observation of resonant photon blockade at microwave frequencies using correlation function measurements,” Phys. Rev. Lett. 106, 243601 (2011).
[Crossref] [PubMed]

Englund, D.

A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vučković, “Coherent generation of non-classical light on a chip via photon-induced tunnelling and blockade,” Nat. Phys. 4, 859–863 (2008).
[Crossref]

Fan, S. H.

J. T. Shen and S. H. Fan, “Strongly correlated two-photon transport in a one-dimensional waveguide coupled to a two-level system,” Phys. Rev. Lett. 98, 153003 (2007).
[Crossref] [PubMed]

Faraon, A.

A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vučković, “Coherent generation of non-classical light on a chip via photon-induced tunnelling and blockade,” Nat. Phys. 4, 859–863 (2008).
[Crossref]

Fazio, R.

D. Gerace, H. E. Tureci, A. Imamoḡlu, V. Giovannetti, and R. Fazio, “The quantum-optical Josephson interferometer,” Nat. Phys. 5, 281–284 (2009).
[Crossref]

Ferretti, S.

S. Ferretti, V. Savona, and D. Gerace, “Optimal antibunching in passive photonic devices based on coupled nonlinear resonators,” New J. Phys. 15, 025012 (2013).
[Crossref]

S. Ferretti and D. Gerace, “Single-photon nonlinear optics with Kerr-type nanostructured materials,” Phys. Rev. B 85, 033303 (2012).
[Crossref]

S. Ferretti, L. C. Andreani, H. E. Türeci, and D. Gerace, “Photon correlations in a two-site nonlinear cavity system under coherent drive and dissipation,” Phys. Rev. A 82, 013841 (2010).
[Crossref]

Filipp, S.

C. Lang, D. Bozyigit, C. Eichler, L. Steffen, J. M. Fink, A. A. Abdumalikov, M. Baur, S. Filipp, M. P. da Silva, A. Blais, and A. Wallraff, “Observation of resonant photon blockade at microwave frequencies using correlation function measurements,” Phys. Rev. Lett. 106, 243601 (2011).
[Crossref] [PubMed]

Fink, J. M.

C. Lang, D. Bozyigit, C. Eichler, L. Steffen, J. M. Fink, A. A. Abdumalikov, M. Baur, S. Filipp, M. P. da Silva, A. Blais, and A. Wallraff, “Observation of resonant photon blockade at microwave frequencies using correlation function measurements,” Phys. Rev. Lett. 106, 243601 (2011).
[Crossref] [PubMed]

Flayac, H.

H. Flayac and V. Savona, “Input-output theory of the unconventional photon blockade,” Phys. Rev. A 88, 033836 (2013).
[Crossref]

Forchel, A.

J. Wiersig, C. Gies, F. Jahnke, M. Asmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460, 245–249 (2009).
[Crossref] [PubMed]

M. Bayer, T. Gutbrod, J. P. Reithmaier, A. Forchel, T. L. Reinecke, P. A. Knipp, A. A. Dremin, and V. D. Kulakovskii, “Optical modes in photonic molecules,” Phys. Rev. Lett. 81, 2582 (1998).
[Crossref]

Fratini, F.

F. Fratini, E. Mascarenhas, L. Safari, J-Ph. Poizat, D. Valente, A. Auffèves, D. Gerace, and M. F. Santos, “Fabry-Perot interferometer with quantum mirrors: nonlinear light transport and rectification,” Phys. Rev. Lett. 113, 243601 (2014).
[Crossref] [PubMed]

Furusawa, A.

J. L. ÓBrien, A. Furusawa, and J. Vučković, “Photonic quantum technologies,” Nat. Photonics 3, 687–695 (2009).
[Crossref]

Fushman, I.

A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vučković, “Coherent generation of non-classical light on a chip via photon-induced tunnelling and blockade,” Nat. Phys. 4, 859–863 (2008).
[Crossref]

Gan, L.

Geng, W. D.

J. Tang, W. D. Geng, and X. L. Xu, “Quantum interference induced photon blockade in a coupled single quantum dot-cavity system,” Sci. Rep. 5, 9252 (2015).
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Figures (15)

Fig. 1
Fig. 1 (a) Scheme diagram of the setup. Two cavities are driven by external fields with the complex Rabi frequency F a e i φ a and F b e i φ b, respectively. (b) The equal-time second-order correlation function g a ( 2 ) ( 0 ) is plotted by numerically solving the master Eq. (4). The nearly perfect antibunching can be obtained at U = 1.593 × 10−3κ with φ = 1.6 rad. The other parameters are chosen as Δa = Δb = 0.1539κ, J = 7κ, Ua = 0, and Fa = 0.7κ, Fb = 0.028κ, φ = 1.6rad for blue-bold line, φ = 1.8rad for dashed-green line, φ = 1.4rad for red-dashed-dotted line. (c) The first (second) number in |mn〉 denotes the photon number in cavity a (cavity b). The transition paths (two dashed lines with arrows) lead to the quantum interference (indicated by the ellipse) for the strong photon antibunching.
Fig. 2
Fig. 2 Dependence of logarithmic plot for g a ( 2 ) ( 0 ) on U (UUb) and Δ by numerically solving master Eq. (4). Defining FbrFa. Parameters chosen are J = 7κ, Fa = 0.7κ, Ua = 0, and r = 0.02, φ = 2rad for (a), and r = 0.2, φ = 1rad for(b)–(d). We have an interesting observation that there are two and four minimum values for two-order correlation function for (a) [see points P1 and P2 denoted by blue-star] and (b)–(d) [see points P3-P6 denoted by blue-star], respectively. Therefore points P1-P4 correspond to unconventional single-photon blockade, which is smaller than the mode broadening κ, while points P5 and P6 indicate conventional single-photon blockade, which is larger than the mode broadening κ.
Fig. 3
Fig. 3 Dependence of logarithmic plot of g a ( 2 ) ( 0 ) on U and φ (in units of π). The detuning Δ is given by Eq. (21). In this case, the discriminant δ = m3 − 27n2 in Eq. (20) is always less than zero when phase φ varies. We set Δ o p t = Δ o p t ( 1 ) for (a), whereas Δ o p t = Δ o p t ( 2 ) for (b). The red-dashed lines in (a)–(b) are plotted with U given by Eq. (22). Other parameters are J = 7κ, Fa = 0.3κ, r = 0.02.
Fig. 4
Fig. 4 Dependence of logarithmic plot of g a ( 2 ) ( 0 ) on U and φ (in units of π), where the detuning Δ is given by Eqs. (21), (25) and (26). The discriminant δ = m3 − 27n2 in Eq. (20) changes its sign when phase φ varies, this results in the changes of solutions to Eq. (19). Figure (a)–(d) are for different Δopt. (a) Δ o p t = Δ o p t ( 1 ) when δ < 0, whereas Δ o p t = Δ o p t ( 3 ) when δ > 0. The same for (b), (c), and (d). (b) Δ o p t ( 1 ) and Δ o p t ( 4 ), (c) Δ o p t ( 1 ) and Δ o p t ( 5 ), (d) Δ o p t ( 1 ) and Δ o p t ( 6 ). The red-dashed lines in (a)–(d) are plotted with U given by Eqs. (22) and (27). Other parameters are J = 7κ, r = 0.5, and Fa = 0.2κ.
Fig. 5
Fig. 5 The parameters of this figure are the same as Fig. 4, but Δopt is different. (a) Δ o p t ( 2 ) and Δ o p t ( 3 ), (b) Δ o p t ( 2 ) and Δ o p t ( 4 ), (c) Δ o p t ( 2 ) and Δ o p t ( 5 ), (d) Δ o p t ( 2 ) and Δ o p t ( 6 ). The red-dashed lines in (a) and (b) are plotted with U given by Eqs. (22) and (27).
Fig. 6
Fig. 6 Logarithmic plot for g a ( 2 ) ( 0 ) by numerically solving master Eq. (4) as a function of U and Δ at φ = 0. Parameters chosen are r = 0.1, J = κ for (a), and J = 20κ for (b) and (c). Points r1r4 (denoted by red-star) are given by Eqs. (21) and (22). while points s1 and s2 (denoted by red-circle) are given by Eqs. (30) and (31). In (d), J = κ, red-dashed line: Uopt = −0.424κ given by Δ o p t ( 2 ) in Eq. (22), blue-solid line: Uopt = 0.496κ given by Δ o p t ( 1 ) in Eq. (22); pink-dashed-dotted line: Uopt = 0.051κ given by Eq. (31).
Fig. 7
Fig. 7 Dependence of logarithmic plot of g a ( 2 ) ( 0 ) given by numerically solving the master Eq. (4) on U and r at φ = 0. The detuning Δ satisfies Δ o p t ( 1 ) in Eq. (21) in (a), Δ o p t ( 2 ) in (b), and satisfies Eq. (30) in (c). The red-dashed lines in (a)–(b) are plotted with U given by Eq. (22), while in (c) U takes Eq. (31). Parameters chosen are J = 0.8κ, Fa = 0.1κ. In (d), blue-solid line: the parameters take ( U o p t , Δ o p t ( 1 ) ) = ( 2.244 κ , 0.291 κ ) given by substituting r = 0.303 into Eqs. (21) and (22); simliarly red-dashed line: ( U o p t , Δ o p t ( 2 ) ) = ( 0.853 κ , 0.092 κ ); green-dashed-dotted line: (Uoptopt) = (0.208κ,0.242κ) given by substituting r = 0.303 into Eqs. (30) and (31).
Fig. 8
Fig. 8 Dependence of logarithmic plot g a ( 2 ) ( 0 ) given by numerically solving the master Eq. (4) on U and the ratio r. The detuning Δ is given by Eq. (21): (a) Δ o p t = Δ o p t ( 1 ), whereas (b) Δ o p t = Δ o p t ( 2 ) due to the discriminant δ < 0 when r varies. The red-dashed lines in (a)–(b) are plotted with U given by Eqs. (22) and (27). Other parameters are J = 0.8κ, Fa = 0.008κ, φ = 2.5rad.
Fig. 9
Fig. 9 Dependence of logarithmic plot of g a ( 2 ) ( 0 ) given by numerically solving the master Eq. (4) on U and r at φ = 0. The detuning Δ satisfies Δ o p t ( 1 ) in Eq. (21) in (a), Δ o p t ( 2 ) in (b), and satisfies Eq. (30) in (c). The black-dashed lines in (a)–(b) are plotted with U given by Eq. (22), while in (c) U takes Eq. (31). Parameters chosen are J = 10κ, Fa = 0.1κ. In (d), blue-solid line: the parameters take ( U o p t , Δ o p t ( 2 ) ) = ( 1.955 κ , 3.360 κ ) given by substituting r = 0.803 into (21) and (22); similarly green-dashed-dotted line: ( U o p t , Δ o p t ( 1 ) ) = ( 0.115 κ , 8.045 κ ); red-dashed line: (Uoptopt) = (0.113κ,8.030κ) given by substituting r = 0.803 into Eqs. (30) and (31).
Fig. 10
Fig. 10 Dependence of logarithmic plot of g a ( 2 ) ( 0 ) given by numerically solving the master Eq. (4) on U and J at φ = 0. The detuning Δ satisfies Δ o p t ( 1 ) in Eq. (21) in (a) and Δ o p t ( 2 ) in (b) or satisfies Eq. (30) in (c). The red-dashed lines in (a)–(b) are plotted with U given by Eq. (22), while in (c) U take Eq. (31). Parameters chosen are r = 0.2, Fa = 0.1κ. In (d), blue-dashed line: the parameters take ( U o p t , Δ o p t ( 1 ) ) = ( 0.0269 κ , 1.1055 κ ) given by substituting J = 5κ into Eqs. (21) and (22); similarly, red-solid line: ( U o p t , Δ o p t ( 2 ) ) = ( 0.0369 κ , 0.247 κ ); black-dashed-dotted line: (Uoptopt) = (0.0208κ,κ) given by substituting J = 5κ into Eqs. (30) and (31).
Fig. 11
Fig. 11 Dependence of logarithmic plot g a ( 2 ) ( 0 ) given by numerically solving the master Eq. (4) on U and J. The detuning Δ is given by Eqs. (21), (25) and (26). The discriminant δ = m3 − 27n2 in Eq. (20) changes its sign when phase φ varies, this results in the changes of solutions to Eq. (19). Figure (a)–(d) are for different Δopt. (a) Δ o p t = Δ o p t ( 1 ) when δ < 0, whereas Δ o p t = Δ o p t ( 3 ) when δ > 0. The same for (b), (c), and (d). (b) Δ o p t ( 1 ) and Δ o p t ( 4 ), (c) Δ o p t ( 1 ) and Δ o p t ( 5 ), (d) Δ o p t ( 1 ) and Δ o p t ( 6 ). The white-dashed lines in (a)–(d) are plotted with U given by Eqs. (22) and (27). Other parameters are r = 0.2, φ = 2.5rad, and Fa = 0.1κ.
Fig. 12
Fig. 12 The parameters of this figure are the same as Fig. 11, but Δopt is different. (a) Δ o p t ( 2 ) and Δ o p t ( 3 ), (b) Δ o p t ( 2 ) and Δ o p t ( 4 ), (c) Δ o p t ( 2 ) and Δ o p t ( 5 ), (b) Δ o p t ( 2 ) and Δ o p t ( 6 ). The red-dashed lines in (a) and (b) are plotted with U given by Eqs. (22) and (27).
Fig. 13
Fig. 13 g a ( 2 ) ( 0 ) versus Δ. The solid lines and dotted lines denote the analytical expression Eq. (32) and the numerical simulation, respectively. The parameters chosen are J = 3κ, r = 0.5, and Fa = 0.03κ, φ = 4.2rad, U = 0.409κ for (a), φ = 4.2rad, U = 1.387κ for (b), φ = 0.8rad, U = 0.221κ for (c), φ = 0.8rad, U = 14.766κ for (d), φ = 0.8rad, U = 0.408κ for (e), φ = 0.8rad, U = −8.963κ for (f). We find the minimum values of the correlation functions corresponding to completely single-photon blockade are consistent with these optimal values Eqs. (21)(27), i.e., the minimum value of (a) versus Δ o p t ( 1 ), (b) versus Δ o p t ( 2 ),(c) versus Δ o p t ( 3 ), (d) versus Δ o p t ( 4 ), (e) versus Δ o p t ( 5 ), (f) versus Δ o p t ( 6 ).
Fig. 14
Fig. 14 g a ( 2 ) ( τ ) versus τ. U and Δ take its optimal values, i.e., ( Δ , U ) = ( Δ o p t ( 1 ) , U o p t ( 1 ) ) or ( Δ o p t ( 4 ) , U o p t ( 4 ) ). The solid lines and dotted lines denote the analytical expression (34) and the numerical simulation respectively. The parameters chosen are J = 7κ, φ = 2rad, Δopt = −3.086κ, Uopt = 7.020κ, r = 5, and Fa = 0.084κ, for (a), J = 7κ, φ = 2rad, Δopt = −2.074κ, Uopt = 6.965κ, r = 2, and Fa = 0.21κ for (b), J = 0.4κ, φ = 2rad, Δopt = −0.26κ, Uopt = 44.83κ, r = 5, and Fa = 0.0167κ for (c), J = 0.4κ, φ = 2rad, Δopt = −0.0416κ, Uopt = −0.201κ, r = 10, and Fa = 0.0832κ for (d), J = 7κ, φ = 2.7rad, Δopt = −4.52κ, Uopt = 12.247κ, r = 5, and Fa = 0.05κ for (e), J = 7κ, φ = 3.1rad, Δopt = −11.892κ, Uopt = −1.162κ, r = 5, and Fa = 0.2κ for (c).
Fig. 15
Fig. 15 g a ( 2 ) ( τ ) versus τ. The parameters chosen are φ = 3rad, r = 2, U = 2.667κ, Fa = 0.386κ, J = 10κ, Δ = 0.2κ for (a), J = 10κ, Fa = 0.278κ, Δ = 0.5κ for (b) J = 0.53κ, Fa = 0.98κ, Δ = 10κ for (d). In (c), J = 10κ, Fa = 0.0015κ, Δ = 10κ, and U = 2.667κ for L1, U = 0.1κ for L2, U = 0.02κ for L3. Interestingly, we find beats frequency for photon are observed, whose period is equal to π Δ or π J given by Eqs. (36) and (37).

Tables (1)

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Table 1 Comparison of regimes of exact and approximate optimal parameters.

Equations (52)

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H ^ 0 = h ¯ ω a a ^ a ^ + h ¯ ω b b ^ b ^ + h ¯ J ( a ^ b ^ + b ^ a ^ ) + U a a ^ a ^ a ^ a ^ + U b b ^ b ^ b ^ b ^ ,
H ^ d r ( t ) = H ^ 0 + ( h ¯ F a a ^ e i φ a i ω L t + h ¯ F b b ^ e i φ b i ω L t + H . c . ) ,
H ^ = h ¯ Δ a a ^ a ^ + h ¯ Δ b b ^ b ^ + h ¯ J ( a ^ b ^ + a ^ b ^ ) + U a a ^ a ^ a ^ a ^ + U b b ^ b ^ b ^ b ^ + ( h ¯ F a e i φ a a ^ + h ¯ F b e i φ b b ^ + H . c . ) ,
ρ ˙ = i [ H ^ , ρ ] + κ a D ( a ^ ) ρ + κ b D ( b ^ ) ρ ,
g a ( 2 ) ( 0 ) = a ^ 2 a ^ 2 a ^ a ^ 2 .
| Ψ = A 00 | 00 + A 10 | 10 + A 01 | 01 + A 20 | 20 + A 02 | 02 + A 11 | 11 ,
i t A 00 = F a e i φ a A 10 + F b e i φ b A 01 ,
i t A 10 = δ a A 10 + J A 01 + F a e i φ a A 00 + A b e i φ b A 11 + 2 F a e i φ a A 20 ,
i t A 01 = δ b A 01 + J A 10 + F b e i φ b A 00 + F a e i φ a A 11 + 2 F b e i φ b A 02 ,
i t A 11 = ( δ a + δ b ) A 11 + 2 J ( A 20 + A 02 ) + F b e i φ b A 10 + F a e i φ a A 01 ,
i t A 20 = 2 ( δ a + U a ) A 20 + 2 J A 11 + 2 F a e i φ a A 10 ,
i t A 02 = 2 ( δ b + U b ) A 02 + 2 J A 11 + 2 F b e i φ b A 01 ,
δ a A 10 + J A 01 = F a e i φ a A 00 , J A 10 + δ b A 01 = F b e i φ b A 00 ,
0 = 2 ( δ a + U a ) A 20 + 2 J A 11 + 2 F a e i φ a A 10 , 0 = 2 ( δ b + U b ) A 02 + 2 J A 11 + 2 F b e i φ b A 01 , 0 = ( δ a + δ b ) A 11 + 2 J ( A 20 + A 02 ) + F b e i φ b A 10 + F a e i φ a A 01 .
A 01 A 10 = e i φ a F a J e i φ b F b δ a e i φ b F b J e i φ a F a δ b ζ .
g a ( 2 ) ( 0 ) 2 | A 20 | 2 | A 10 | 4 .
0 = 2 J A 11 + 2 F a e i φ a A 10 , 0 = 2 ( δ b + U b ) A 02 + 2 J A 11 + 2 F b e i φ b ζ A 10 , 0 = ( δ a + δ b ) A 11 + 2 J A 02 + F b e i φ b A 10 + F a e i φ a ζ A 10 .
0 = 4 i r 2 J 2 [ κ + i ( U + 2 Δ ) ] + e 2 i φ [ ( κ + 2 i Δ ) 2 ( 2 U + 2 Δ i κ ) 4 J 2 U ] 4 r e i φ J ( κ + 2 i Δ ) [ κ + 2 i ( U + Δ ) ] .
Δ 4 + 4 b Δ 3 + 6 c Δ 2 + 4 d Δ + e = 0 ,
δ = m 3 27 n 2 < 0
Δ o p t ( 1 ) , ( 2 ) = [ b sgn ( g ) ] λ ± | g | / λ λ + 3 h ,
U o p t = 4 r J v 2 + 2 Δ ( 3 κ 2 4 Δ 2 ) sin ( 2 φ ) + v 3 4 cos ( φ ) v 1 8 Δ [ κ cos ( 2 φ ) + 2 r J sin ( φ ) ] ,
δ = m 3 27 n 2 > 0 ,
h > 0 , η > 0 ,
Δ o p t ( 3 ) , ( 4 ) = b + s γ 1 ± γ 2 ± γ 3 ,
Δ o p t ( 5 ) , ( 6 ) = b s γ 1 ± γ 2 γ 3 .
U o p t = 4 r J v 2 + 2 Δ ( 3 κ 2 4 Δ 2 ) sin ( 2 φ ) + v 3 4 cos ( φ ) v 1 8 Δ [ κ cos ( 2 φ ) + 2 r J sin ( φ ) ] ,
φ = 0 ,
J κ ,
Δ o p t J r ,
U o p t κ 2 2 J r 1 r 2 .
g a ( 2 ) ( 0 ) = q 2 + [ 4 r J ( κ 2 4 Δ U 4 Δ 2 ) cos φ + q 3 ] 2 q 1 1 t 2 [ t 1 + κ 2 ( 4 J 2 + κ 2 8 Δ U 12 Δ 2 ) 2 ] ,
g a ( 2 ) ( τ ) = a ^ ( t ) a ^ ( t 1 ) a ^ ( t 1 ) a ^ ( t ) a ^ ( t ) a ^ ( t ) 2 ,
g a ( 2 ) ( τ ) = 1 c 3 { ς 3 + e τ κ 2 ( ς 1 sin [ τ ( J Δ ) + β 1 ] + ς 2 sin [ τ ( J + Δ ) + β 2 ] ) } 2 + 1 c 3 { ς 4 + e τ κ 2 ( ς 1 sin [ τ ( J Δ ) + β 3 ] + ς 2 sin [ τ ( J + Δ ) + β 4 ] ) } 2 ,
ς sin [ τ ( J Δ ) + β 1 ] + ς sin [ τ ( J + Δ ) + β 2 ] = 2 ς sin ( τ J + β ¯ 1 ) cos ( τ Δ + β ¯ 2 ) ,
T b e a t = π Δ .
T b e a t = π J .
0 = 4 e i φ i J r [ 2 i ( U + Δ b ) + κ b ] [ 2 i ( Δ a + Δ b ) + κ a + κ b ] + 4 r 2 J 2 [ 2 i ( U + Δ a + Δ b ) + κ a + κ b ] + e 2 i φ { 8 i J 2 U ( 2 i Δ b + κ b ) [ 2 i ( U + Δ b ) + κ b ] [ 2 i ( Δ a + Δ b ) + κ a + κ b ] } .
η 1 Δ a 2 + η 2 Δ a + η 3 = 0 ,
Δ a , o p t ( 1 ) , ( 2 ) = η 2 ± η 2 2 4 η 1 η 3 2 η 1 .
U a , o p t = { 8 J 2 r 2 ( Δ a + Δ b ) + 2 cos ( 2 φ ) [ 4 Δ b 2 ( Δ a + Δ b ) + 2 Δ b κ a κ b + ( Δ a + 3 Δ b ) κ b 2 ] 4 J r cos ( φ ) [ 4 Δ b ( Δ a + Δ b ) + κ b ( κ a + κ b ) ] + 8 J r sin ( φ ) [ Δ a κ b + Δ b ( κ a + 2 κ b ) ] + sin ( 2 φ ) [ 8 Δ a Δ b κ b + κ b 2 ( κ a + κ b ) 4 Δ b 2 ( κ a + 3 κ b ) ] } / { 8 J 2 r 2 16 J r cos ( φ ) ( Δ a + Δ b ) 8 J r sin ( φ ) ( κ a + κ b ) + cos ( 2 φ ) [ 8 J 2 + 8 Δ b ( Δ a + Δ b ) 2 κ b ( κ a + κ b ) ] 4 sin ( 2 φ ) [ Δ a κ b + Δ b ( κ a + 2 κ b ) ] } ,
Δ b 4 + b 1 Δ b 3 + c 1 Δ b 2 + d 1 Δ b + e 1 = 0 ,
δ = J 4 ( r 2 1 ) 2 [ N 2 J 8 + N 1 + 16 J 2 κ 6 ( 1 + 3 r 2 ) ] 4096 , h = 1 12 [ 3 ( r 2 1 ) J 2 κ 2 ] , η = J 2 16 ( r 2 1 ) [ 9 ( r 2 1 ) J 2 8 κ 2 ] ,
A ¯ 10 = 2 e i φ b F a [ 2 r J + i e i φ ( κ + 2 i Δ ) ] 4 J 2 + ( κ + 2 i Δ ) 2 , A ¯ 20 = 2 2 e 2 i φ b F a 2 ( 4 i r J w 1 e i φ + w 2 + e 2 i φ w 3 ) w 4 [ ( κ + 2 i Δ ) 2 w 5 + 4 J 2 κ + 4 J 2 ( U + 2 Δ ) i ] , A ¯ 11 = 4 e 2 i φ b F a 2 ( 2 J e i φ + i r κ 2 r Δ ) ( i w 1 e i φ + w 2 2 r J ) w 4 [ ( κ + 2 i Δ ) 2 w 5 + 4 J 2 κ + 4 J 2 ( U + 2 Δ ) i ] ,
g a ( 2 ) ( τ ) = Tr S [ A ^ A ^ ρ ˜ ( τ ) ] ,
ρ ˜ ( τ ) = Tr E [ U T ( τ ) ρ ˜ ( 0 ) ρ E U T ( τ ) ] ,
| Ψ ˜ ( 0 ) = a ^ | Ψ ¯ ,
g a ( 2 ) ( τ ) = | A 10 ( τ ) | 2 | A ¯ 10 | 4 .
| Ψ ˜ ( 0 ) = A ¯ 10 | 00 + 2 A ¯ 20 | 10 + A ¯ 11 | 01 .
A 10 ( τ ) + A 01 ( τ ) = x ( τ ) , A 10 ( τ ) A 01 ( τ ) = y ( τ ) ,
x τ + u 1 x ( τ ) = λ 1 , y τ + u 2 y ( τ ) = λ 2 ,
x ( τ ) = x ( 0 ) e u 1 τ + λ 1 u 1 ( 1 e u 1 τ ) , y ( τ ) = y ( 0 ) e u 2 τ + λ 2 u 2 ( 1 e u 2 τ ) .

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