Abstract

The solid-state qubits based on diamond nitrogen-vacancy centers (NVC) are promising for future quantum information processing. We investigate the dynamics of entanglement among three NVCs coupled to a microtoroidal cavity supporting two counter-propagating whispering-gallery-modes (WGMs) in the presence of Rayleigh scattering. Our results indicate that the maximal entanglement among all the NVCs could be achieved through adjusting several key parameters, such as the scattering-induced coupling between the WGMs, the distance between the NVCs, and the NVC-WGM coupling strengths, as well as the frequency detuning between the NVC and cavity. We show that entanglement of the NVCs displays a series of damped oscillations under various experimental situations, which reflects the intricate interplay and competition between the Rayleigh scattering and the NVC-WGM coupling. The quantum dynamics of the system is obtained via solutions to the corresponding microscopic master equation, which agrees well with the numerical simulation results using the phenomenological master equation. The feasibility of our proposal is supported by the application of currently available experimental techniques.

© 2015 Optical Society of America

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2015 (1)

S. Armstrong, M. Wang, R. Y. Teh, Q. H. Gong, Q. Y. He, J. Janousek, H. A. Bachor, M. D. Reid, and P. K. Lam, “Multipartite Einstein–Podolsky–Rosen steering and genuine tripartite entanglement with optical networks,” Nature Phys. 11, 167–172 (2015).
[Crossref]

2014 (11)

Q. Y. He and Z. Ficek, “Einstein-Podolsky-Rosen paradox and quantum steering in a three-mode optomechanical system,” Phys. Rev. A 89, 022332 (2014).
[Crossref]

M. Wang, Q. H. Gong, Z. Ficek, and Q. Y. He, “Role of thermal noise in tripartite quantum steering,” Phys. Rev. A 90, 023801 (2014).
[Crossref]

C. Eltschka, D. Braun, and J. Siewert, “Heat bath can generate all classes of three-qubit entanglement,” Phys. Rev. A 89, 062307 (2014).
[Crossref]

B. B. Li, W. R. Clements, X. C. Yu, K. Shi, Q. H. Gong, and Y. F. Xiao, “Single nanoparticle detection using split-mode microcavity Raman lasers,” PNAS 111, 14657–14662 (2014).
[Crossref] [PubMed]

J. Wolters, J. Kabuss, A. Knorr, and O. Benson, “Deterministic and robust entanglement of nitrogen-vacancy centers using low-Q photonic-crystal cavities,” Phys. Rev. A 89, 060303 (2014).
[Crossref]

I. M. Georgescu, S. Ashhab, and F. Nori, “Quantum simulation,” Rev. Mod. Phys. 86, 153–185 (2014).
[Crossref]

B. Peng, S. K. Özdemir, 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,” Nature Phys. 10, 394–398 (2014).
[Crossref]

B. Peng, S. K. Özdemir, W. J. Chen, F. Nori, and L. Yang, “What is and what is not electromagnetically induced transparency in whispering-gallery microcavities,” Nature Comms. 10, 1038 (2014).

B. Peng, S. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346, 328–332 (2014).
[Crossref] [PubMed]

H. Jing, S. K. Özdemir, X. Y. Lü, J. Zhang, L. Yang, and F. Nori, “PT -Symmetric Phonon Laser,” Phys. Rev. Lett. 113, 053604 (2014).
[Crossref]

R. Coto and M. Orszag, “Determination of the maximum global quantum discord via measurements of excitations in a cavity QED network,” J. Phys. B 47, 095501 (2014).
[Crossref]

2013 (5)

S. P. Liu, J. H. Li, R. Yu, and Y. Wu, “Achieving maximum entanglement between two nitrogen-vacancy centers coupling to a whispering-gallery-mode microresonator,” Opt. Express 21, 3501–3515 (2013).
[Crossref] [PubMed]

Z. L. Xiang, X. Y. Lü, T. F. Li, J. Q. You, and F. Nori, “Hybrid quantum circuit consisting of a superconducting flux qubit coupled to a spin ensemble and a transmission-line resonator,” Phys. Rev. B 87, 144516 (2013).
[Crossref]

Z. L. Xiang, S. Ashhab, J. Q. You, and F. Nori, “Hybrid quantum circuits: superconducting circuits interacting with other quantum systems,” Rev. Mod. Phys. 85, 623–653 (2013).
[Crossref]

X. Y. Lü, Z. L. Xiang, W. Cui, J. Q. You, and F. Nori, “Quantum memory using a hybrid circuit with flux qubits and nitrogen-vacancy centers,” Phys. Rev. A 88, 012329 (2013).
[Crossref]

W. L. Yang, J. H. An, C. J. Zhang, M. Feng, and C. H. Oh, “Preservation of quantum correlation between separated nitrogen-vacancy centers embedded in photonic-crystal cavities,” Phys. Rev. A 87, 022312 (2013).
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2012 (5)

P. B. Li, S. Y. Gao, H. R. Li, S. L. Ma, and F. L. Li, “Dissipative preparation of entangled states between two spatially separated nitrogen-vacancy centers,” Phys. Rev. A 85, 042306 (2012).
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B. J. M. Hausmann, B. Shields, Q. M. Quan, P. Maletinsky, M. McCutcheon, J. F. Choy, T. M. Babinec, A. Kubanek, A. Yacoby, M. D. Lukin, and M. Lončar, “Integrated diamond networks for quantum nanophotonics,” Nano Lett. 12, 1578–1582 (2012).
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J. I. de Vicente, T. Carle, C. Streitberger, and B. Kraus, “Complete set of operational measures for the characterization of three-qubit entanglement,” Phys. Rev. Lett. 108, 060501 (2012).
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X. C. Yu, Y. C. Liu, M. Y. Yan, W. L. Jin, and Y. F. Xiao, “Coupling of diamond nanocrystals to a high-Q whispering-gallery microresonator,” Phys. Rev. A 86, 043833 (2012).
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V. Eremeev, V. Montenegro, and M. Orszag, “Thermally generated long-lived quantum correlations for two atoms trapped in fiber-coupled cavities,” Phys. Rev. A 85, 032315 (2012).
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2011 (10)

D. Ratchford, F. Shafiei, S. Kim, S. K. Gray, and X. Q. Li, “Manipulating coupling between a single semiconductor quantum dot and single cold nanoparticle,” Nano Lett. 11, 1049–1054 (2011).
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X. Yi, Y. F. Xiao, Y. C. Liu, B. B. Li, Y. L. Chen, Y. Li, and Q. H. Gong, “Multiple-Rayleigh-scatterer-induced mode splitting in a high-Q whispering-gallery-mode microresonator,” Phys. Rev. A 83, 023803 (2011).
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Y. C. Liu, Y. F. Xiao, B. B. Li, X. F. Jiang, Y. Li, and Q. H. Gong, “Coupling of a single diamond nanocrystal to a whispering-gallery microcavity: photon transport benefitting from Rayleigh scattering,” Phys. Rev. A 84, 011805 (2011).
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Q. Chen, W. L. Yang, M. Feng, and J. F. Du, “Entangling separate nitrogen-vacancy centers in a scalable fashion via coupling to microtoroidal resonators,” Phys. Rev. A 83, 054305 (2011).
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W. L. Yang, Z. Q. Yin, Z. Y. Xu, M. Feng, and C. H. Oh, “Quantum dynamics and quantum state transfer between separated nitrogen-vacancy centers embedded in photonic crystal cavities,” Phys. Rev. A 84, 043849 (2011).
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2010 (10)

P. Neumann, R. Kolesov, B. Naydenov, J. Beck, F. Rempp, M. Steiner, V. Jacques, G. Balasubramanian, M. L. Markham, D. J. Twitchen, S. Pezzagna, J. Meijer, J. Twamley, F. Jelezko, and J. Wrachtrup, “Quantum register based on coupled electron spins in a room-temperature solid,” Nat. Phys. 6, 249–253 (2010).
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E. Togan, Y. Chu, A. S. Trifonov, L. Jiang, J. Maze, L. Childress, M. V. Gurudev Dutt, A. S. Sørensen, P. R. Hemmer, A. S. Zibrov, and M. D. Lukin, “Quantum entanglement between an optical photon and a solid-state spin qubit,” Nature (London) 466, 730–734 (2010).
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Y. F. Xiao, C. L. Zhou, B. B. Li, Y. Li, C. H. Dong, Z. F. Han, and Q. H. Gong, “High-Q exterior whispering-gallery modes in a metal-coated microresonator,” Phys. Rev. Lett. 105, 153902 (2010).
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F. Shi, X. Rong, N. Y. Xu, Y. Wang, J. Wu, B. Chong, X. H. Peng, J. Kniepert, R. S. Schoenfeld, W. Harneit, M. Feng, and J. F. Du, “Room-temperature implementation of the Deutsch-Jozsa algorithm with a single electronic spin in diamond,” Phys. Rev. Lett. 105, 040504 (2010).
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W. L. Yang, Z. Q. Yin, Z. Y. Xu, M. Feng, and J. F. Du, “One-step implementation of multiqubit conditional phase gating with nitrogen-vacancy centers coupled to a high-Q silica microsphere cavity,” Appl. Phys. Lett. 96, 241113 (2010).
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2009 (16)

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M. Wilczewski and M. Czachor, “Theory versus experiment for vacuum Rabi oscillations in lossy cavities,” Phys. Rev. A 79, 033836 (2009).
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G. Balasubramanian, P. Neumann, D. Twitchen, M. Markham, R. Kolesov, N. Mizuochi, J. Isoya, J. Achard, J. Beck, J. Tissler, V. Jacques, P. R. Hemmer, F. Jelezko, and J. Wrachtrup, “Ultralong spin coherence time in isotopically engineered diamond,” Nat. Mater. 8, 383–387 (2009).
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S. C. Benjamin, B. W. Lovett, and J. M. Smith, “Prospects for measurement-based quantum computing with solid state spins,” Laser Photonics Rev. 3, 556–574 (2009).
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P. E. Barclay, C. Santori, K. M. Fu, R. G. Beausoleil, and O. Painter, “Coherent interference effects in a nanoassembled diamond NV center cavity-QED system,” Opt. Express 17, 8081–8097 (2009).
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V. Jacques, P. Neumann, J. Beck, M. Markham, D. Twitchen, J. Meijer, F. Kaiser, G. Balasubramanian, F. Jelezko, and J. Wrachtrup, “Dynamic polarization of single nuclear spins by optical pumping of nitrogen-vacancy color centers in diamond at room temperature,” Phys. Rev. Lett. 102, 057403 (2009).
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B. Min, E. Ostby, V. Sorger, E. U. Avila, L. Yang, X. Zhang, and K. Vahala, “High-Q surface-plasmon-polariton whispering-gallery microcavity,” Nature (London) 457, 455–458 (2009).
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2008 (4)

S. Schietinger, Y. Schröder, and O. Benson, “One-by-one coupling of single defect centers in nanodiamonds to high-Q modes of an optical microresonator,” Nano Lett. 8, 3911–3915 (2008).
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C. H. Su, A. D. Greentree, and L. C. L. Hollenberg, “Towards a picosecond transform-limited nitrogen-vacancy based single photon,” Opt. Express 16, 6240–6250 (2008).
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C. H. Su, A. D. Greentree, W. J. Munro, K. Nemoto, and L. C. L. Hollenberg, “High-speed quantum gates with cavity quantum electrodynamics,” Phys. Rev. A 78, 062336 (2008).
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J. Merlein, M. Kahl, A. Zuschag, A. Sell, A. Halm, J. Boneberg, P. Leiderer, A. Leitenstorfer, and R. Bratschisch, “Nanomechanical control of an optical antenna,” Nature Photonics 2, 230–233 (2008).
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2007 (6)

M. Scala, B. Militello, A. Messina, J. Piilo, and S. Maniscalco, “Microscopic derivation of the Jaynes-Cummings model with cavity losses,” Phys. Rev. A 75, 013811 (2007).
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A. mazzei, S. Gotzinger, L. de S. Menezes, G. Zumofen, O. Benson, and V. Sandoghdar, “Controlled coupling of counterpropagating whispering-gallery modes by a single Rayleigh scatterer: a classical problem in a quantum optical light,” Phys. Rev. Lett. 99, 173603 (2007).
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A. M. Zagoskin, J. R. Johansson, S. Ashhav, and F. Nori, “Quantum information processing using frequency control of impurity spins in diamond,” Phys. Rev. B 76, 014122 (2007).
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K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot–cavity system,” Nature (London) 445, 896–899 (2007).
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Y. D. Yang, Y. Z. Huang, and Q. Chen, “High-Q TM whispering-gallery modes in three-dimensional microcylinders,” Phys. Rev. A 75, 013817 (2007).
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K. Srinivasan and O. Painter, “Mode coupling and cavity–quantum-dot interactions in a fiber-coupled microdisk cavity,” Phys. Rev. A 75, 023814 (2007).
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2006 (7)

R. Hanson, F. M. Mendoza, R. J. Epstein, and D. D. Awschalom, “Polarization and readout of coupled single spins in diamond,” Phys. Rev. Lett. 97, 087601 (2006).
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L. Childress, M. V. GurudevDutt, J. M. Taylor, A. S. Zibrov, F. Jelezko, J. Wrachtrup, P. R. Hemmer, and M. D. Lukin, “Coherent dynamics of coupled electron and nuclear spin qubits in diamond,” Science 314, 281–285 (2006).
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C. Santori, P. Tamarat, P. Neumann, J. Wrachtrup, D. Fattal, R. G. Beausoleil, J. Rabeau, P. Olivero, A. D. Greentree, S. Prawer, F. Jelezko, and P. Hemmer, “Coherent population trapping of single spins in diamond under optical excitation,” Phys. Rev. Lett. 97, 247401 (2006).
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T. Gaebel, M. Domhan, I. Popa, C. Wittmann, P. Neumann, F. Jelezko, J. R. Rabeau, N. Stavrias, A. D. Greentree, S. Prawer, J. Twamley, P. Hemmer, and J. Wrachtrup, “Room-temperature coherent coupling of single spins in diamond,” Nat. Phys. 2, 408–413 (2006).
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X. H. Gao, S. M. Fei, and K. Wu, “Lower bounds of concurrence for tripartite quantum systems,” Phys. Rev. A 74, 050303 (2006).
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2005 (2)

H. Jing, X. J. Liu, M. L. Ge, and M. S. Zhan, “Correlated quantum memory: Manipulating atomic entanglement via electromagnetically induced transparency,” Phys. Rev. A 71, 062336 (2005).
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A. Badolato, K. Hennessy, M. Atatüre, J. Dreiser, E. Hu, P. M. Petroff, and A. Imamoğlu, “Deterministic coupling of single quantum dots to single nanocavity modes,” Science 308, 1158–1161 (2005).
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2004 (1)

F. Jelezko, T. Gaebel, I. Popa, A. Gruber, and J. Wrachtrup, “Observation of coherent oscillations in a single electron spin,” Phys. Rev. Lett. 92, 076401 (2004).
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2003 (3)

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature (London) 421, 925–928 (2003).
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T. Schröder, A. W. Schell, G. Kewes, T. Aichele, and O. Benson, “Fiber-integrated diamond-based single photon source,” Nano Lett. 11, 198–202 (2011).
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An, J. H.

W. L. Yang, J. H. An, C. J. Zhang, M. Feng, and C. H. Oh, “Preservation of quantum correlation between separated nitrogen-vacancy centers embedded in photonic-crystal cavities,” Phys. Rev. A 87, 022312 (2013).
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N. B. An, J. Kim, and K. Kim, “Entanglement dynamics of three interacting two-level atoms within a common structured environment,” Phys. Rev. A 84, 022329 (2011).
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D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature (London) 421, 925–928 (2003).
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S. Armstrong, M. Wang, R. Y. Teh, Q. H. Gong, Q. Y. He, J. Janousek, H. A. Bachor, M. D. Reid, and P. K. Lam, “Multipartite Einstein–Podolsky–Rosen steering and genuine tripartite entanglement with optical networks,” Nature Phys. 11, 167–172 (2015).
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I. M. Georgescu, S. Ashhab, and F. Nori, “Quantum simulation,” Rev. Mod. Phys. 86, 153–185 (2014).
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Z. L. Xiang, S. Ashhab, J. Q. You, and F. Nori, “Hybrid quantum circuits: superconducting circuits interacting with other quantum systems,” Rev. Mod. Phys. 85, 623–653 (2013).
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I. Buluta, S. Ashhab, and F. Nori, “Natural and artificial atoms for quantum computation,” Rep. Prog. Phys. 74, 104401 (2011).
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A. M. Zagoskin, J. R. Johansson, S. Ashhav, and F. Nori, “Quantum information processing using frequency control of impurity spins in diamond,” Phys. Rev. B 76, 014122 (2007).
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K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot–cavity system,” Nature (London) 445, 896–899 (2007).
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B. Min, E. Ostby, V. Sorger, E. U. Avila, L. Yang, X. Zhang, and K. Vahala, “High-Q surface-plasmon-polariton whispering-gallery microcavity,” Nature (London) 457, 455–458 (2009).
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Figures (8)

Fig. 1
Fig. 1 (a) Schematic of the system consisting of three NVCs and a microtoroidal cavity supporting two counter-propagating WGMs acw and accw. (b) Level diagram for the j-th NVC, where Δj is the frequency detuning of the NVC from the cavity. Deg = γeB0 is the level splitting induced by an external magnetic field B0 with γe the electron gyromagnetic ratio.
Fig. 2
Fig. 2 The LBC vs time t and the phase difference θ13 when the system is initially prepared in the state ψ(0). (a) θ 12 = π 2 + n π, (b) θ12 = . The time parameter is dimensionless and we set G = 1. Other parameters are Δ = 0, g = 0.1, Γ = 0.05, γ = 0.01 and κ = 0.03.
Fig. 3
Fig. 3 (a) The maximal values of the LBC in the parameter plane of {θ12, θ13} with Δ = 0, G = 1, g = 0.1, Γ = 0.05, γ = 0.01 and κ = 0.03, where the system is initially prepared in the state ψ(0). (b) The zooming-in plot of (a) with one period.
Fig. 4
Fig. 4 The LBC vs time t and the coupling strength g for different initial states of the system (a) ψ(0) and (b) Ψ(0). Other parameters are θ 12 = π 4 + n π, θ 13 = π 8 + n π, Δ = 0, G = 1, Γ = 0.05, γ = 0.01 and κ = 0.03. In the bottom six subgraphs, the curves of the LBC dynamics are plotted in (a1,b1) g = 0.1, (a2,b2) g = 1, (a3,b3) g = 5.
Fig. 5
Fig. 5 The LBC vs time t and the phase difference θ13 under different detunings (a) Δ = 5 and (c) Δ = 10, where θ12 = , G = 1, g = 0.1, Γ = 0.05, γ = 0.01 and κ = 0.03. The LBC vs time t and the coupling strength g under different detunings (b) Δ = 5 and (d) Δ = 10, where θ 12 = π 4 π, θ 13 = π 8 + n π, G = 1, Γ = 0.05, γ = 0.01 and κ = 0.03.
Fig. 6
Fig. 6 The LBC vs time t and the detuning Δ for different scattering strengths (a) g = 0.1, (b) g = 1, (c) g = 5. The other parameters are θ 12 = π 4 + n π, θ 13 = π 8 + n π, G = 1, Γ = 0.05, γ = 0.01 and κ = 0.03.
Fig. 7
Fig. 7 The maximal LBC vs the coupling strength g and the detuning Δ in (a) θ12 = θ13 = , (b) θ 12 = π 2 + n π and θ 13 = π 4 + n π, (c) θ 12 = π 4 + n π and θ 13 = π 8 + n π, (d) θ 12 = 3 π 4 + n π and θ 13 = 3 π 8 + n π. Other parameters are G = 1, Γ = 0.05, γ = 0.01 and κ = 0.03.
Fig. 8
Fig. 8 (a) The LBC vs time t under the condition θ 12 = π 4, θ 13 = π 8, Δ = 0, g = 2, where the red-solid (blue-dashed) lines denote the analytical (numerical) results, respectively. (b) The slight difference between the LBC dynamics using these two different methods.

Equations (24)

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H = H 0 + H 1 + H 2 , H 0 = j = 1 3 ω 0 2 σ j z + ω c a c w a c w + ω c a c c w a c c w , H 1 = j = 1 3 G j ( e i k d 1 j σ j + a c w + e i k d 1 j σ j + a c c w ) + H . C . , H 2 = j = 1 3 g j ( a c w a c w + a c c w a c c w ) + j = 1 3 g j ( e 2 i k d 1 j a c c w a c w + H . C ) . ,
C 3 ( | φ ) = 1 3 [ 3 Tr ( ρ 1 2 ) Tr ( ρ 2 2 ) Tr ( ρ 3 2 ) ] ,
C 3 ( ρ ) = min { p i , φ i } i p i C 3 ( | φ i )
C ¯ 3 ( ρ ) = 1 3 j = 1 6 [ ( C j 12 | 3 ( ρ ) ) 2 + ( C j 31 | 2 ( ρ ) ) 2 + ( C j 23 | 1 ( ρ ) ) 2 ] ,
C j k l | m ( ρ ) = max { 0 , λ j , 1 k l | m n > 1 λ j , n k l | m } ,
C ¯ 3 ( ρ ) a n a = max { 0 , 8 5 ( ρ e g g , e g g ρ g e g , g e g + ρ e g g , e g g ρ g g e , g g e + ρ g e g , g e g ρ g g e , g g e ) } .
H = j = 1 3 [ ω 0 σ e j e j + δ 1 a 1 a 1 + δ 2 a 2 a 2 + G ˜ 1 j ( a 1 σ j + a 1 σ j + ) ] + j = 2 3 G ˜ 2 j ( a 2 σ j + a 2 σ j + ) + g ˜ ( a 1 a 2 a 2 a 1 )
1 { | 1 = | e 1 g 2 g 3 0 c w 0 c c w , | 2 = | g 1 e 2 g 3 0 c w 0 c c w , | 3 = | g 1 g 2 e 3 0 c w 0 c c w , | 4 = | g 1 g 2 g 3 1 c w 0 c c w , | 5 = | g 1 g 2 g 3 0 c w 1 c c w } ,
2 { | 6 = | g 1 g 2 g 3 0 c w 0 c c w } ,
H = ( ω 0 2 0 0 G G 0 0 ω 0 2 0 e i k d 12 G e i k d 12 G 0 0 0 ω 0 2 e i k d 13 G e i k d 13 G 0 G e i k d 12 G e i k d 13 G 3 ω 0 2 + ω c + 3 g g ( 1 + e 2 i k d 12 + e 2 i k d 13 ) 0 G e i k d 12 G e i k d 13 G g ( 1 + e 2 i k d 12 + e 2 i k d 13 ) 3 ω 0 2 + ω c + 3 g 0 0 0 0 0 0 3 ω 0 2 )
ρ ˙ ( t ) = i [ H , ρ ] + ω ¯ > 0 , m = { c w , c c w } κ m ( ω ¯ ) × [ A m ( ω ¯ ) ρ ( t ) A m ( ω ¯ ) 1 2 { A m ( ω ¯ ) A m ( ω ¯ ) , ρ ( t ) } ] + ω ¯ > 0 , n = { 1 , 2 , 3 } Γ n ( ω ¯ ) × [ n ( ω ¯ ) ρ ( t ) n + ( ω ¯ ) 1 2 { n + ( ω ¯ ) n ( ω ¯ ) , ρ ( t ) } ] + ω ¯ > 0 , n = { 1 , 2 , 3 } γ n ( ω ¯ ) × [ n z ( ω ¯ ) ρ ( t ) n z ( ω ¯ ) 1 2 { n z ( ω ¯ ) n z ( ω ¯ ) , ρ ( t ) } ] ,
ρ ˙ ( t ) = i [ H , ρ ] + i = 1 5 Γ ˜ i [ | ϕ 6 ϕ i | ρ ( t ) | ϕ i ϕ 6 | 1 2 { | ϕ i ϕ i | , ρ ( t ) } ] + i = 1 5 γ ˜ i [ ( | ϕ i ϕ i | | ϕ 6 ϕ 6 | ) ρ ( t ) ( | ϕ i ϕ i | | ϕ 6 ϕ 6 | ) 1 2 { ( | ϕ i ϕ i | + | ϕ 6 ϕ 6 | ) , ρ ( t ) } ] ,
U ˜ = ( c 11 c 12 c 13 c 14 c 15 0 c 21 c 22 c 23 c 24 c 25 0 c 31 c 32 c 33 c 34 c 35 0 c 41 c 42 c 43 c 44 c 45 0 c 51 c 52 c 53 c 54 c 55 0 0 0 0 0 0 1 ) .
ρ ˙ = i [ H e f f , ρ ] + ρ ,
ρ = m = c w , c c w κ m ( a m ρ a m + 1 2 a m + a m ρ 1 2 ρ a m + a m ) + j = 1 3 Γ j ( σ j ρ σ j + 1 2 σ j + σ j ρ 1 2 ρ σ j + σ j ) + j = 1 3 γ j ( σ j z ρ σ j z 1 2 σ j z σ j z ρ 1 2 ρ σ j z σ j z ) .
H e f f = j = 1 3 [ e i Δ t G j ( e i k d 1 j σ j + a c w + e i k d 1 j σ j + a c c w ) + g j e 2 i k d 1 j a c c w a c c w + H . C ] + j = 1 3 g j ( a c w a c w + a c c w a c c w ) ,
c 11 = e i k d 13 ( e 2 i k d 13 e 2 i k d 12 ) / ( e 2 i k d 12 1 ) , c 12 = e i k d 12 i k d 13 ( 1 e 2 i k d 13 ) / ( e 2 i k d 12 1 ) , c i 1 = G { ξ [ ( e 2 i k d 12 + e 2 i k d 13 2 ) η + Δ A i 1 ] + A i 1 2 } q A i 1 , c i 2 = e i k d 12 G { ξ [ ( e 2 i k d 12 + e 2 i k d 13 2 ) η + Δ A i 1 ] + A i 1 2 } q A i 1 , c i 3 = e i k d 13 G { ξ [ ( e 2 i k d 13 + e 2 i k d 12 2 ) η + Δ A i 1 ] + A i 1 2 } q A i 1 , c i 4 = { ξ [ 3 η Δ A i 1 ] A i 1 2 } / ξ q , c 14 = c 15 = c 16 = c i 6 = 0 , c 13 = c i 5 = 1 ,
x 4 + a x 3 + b x 2 + c x + d = 0 ,
A c w ( λ i λ 6 ) = c i 4 | ϕ 6 ϕ i | , A c c w ( λ i λ 6 ) = c i 5 | ϕ 6 ϕ i | ,
1 ( λ i λ 6 ) = c i 1 | ϕ 6 ϕ i | , 2 ( λ i λ 6 ) = c i 2 | ϕ 6 ϕ i | , 3 ( λ i λ 6 ) = c i 3 | ϕ 6 ϕ i | .
Γ ˜ i = j = 1 3 | c i j | 2 Γ j ( λ i λ 6 ) + | c i 4 | 2 γ c w ( λ i λ 6 ) + | c i 5 | 2 γ c c w ( λ i λ 6 ) , γ ˜ i = j = 1 3 | c i j | 4 γ j ( λ j λ 6 ) .
ρ i i ( t ) = ρ i i ( 0 ) e Γ ˜ i t , ρ i j ( t ) = ρ i j ( 0 ) e [ 2 i ( λ i λ j ) ( Γ ˜ i + Γ ˜ j + γ ˜ i + γ ˜ j ) ] t / 2 , ρ i 6 ( t ) = ρ i 6 ( 0 ) e [ i ( λ i λ 6 ) Γ ˜ i 2 j = 1 5 ( γ ˜ j + 3 γ ˜ i ) 2 ] t , ρ 66 ( t ) = j = 1 5 ρ j j ( 0 ) ( 1 e Γ ˜ j t ) + ρ 66 ( 0 ) .
ρ a n a ( t ) = i , j = 1 6 ϕ i | ρ ( t ) | ϕ j 00 | ϕ i ϕ j | 00 + 10 | ϕ i ϕ j | 10 + 01 | ϕ i ϕ j | 01 + 11 | ϕ i ϕ j | 11 ] ,
ρ a n a ( t ) = ρ e g g , g e g | e 1 g 2 g 3 g 1 e 2 g 3 | + ρ e g g , g g e | e 1 g 2 g 3 g 1 g 2 e 3 | + ρ e g g , g g g | e 1 g 2 g 3 g 1 g 2 g 3 | + ρ g e g , e g g | g 1 e 2 g 3 e 1 g 2 g 3 | + ρ g e g , g g e | g 1 e 2 g 3 g 1 g 2 e 3 | + ρ g e g , g g g | g 1 e 2 g 3 g 1 g 2 g 3 | + ρ g g e , e g g | g 1 g 2 e 3 e 1 g 2 g 3 | + ρ g g e , g e g | g 1 g 2 e 3 g 1 e 2 g 3 | + ρ g g e , g g g | g 1 g 2 e 3 g 1 g 2 g 3 | + ρ g g g , e g g | g 1 g 2 g 3 e 1 g 2 g 3 | + ρ g g g , g e g | g 1 g 2 g 3 g 1 e 2 g 3 | + ρ g g g , g g e | g 1 g 2 g 3 g 1 g 2 e 3 | + ρ e g g , e g g | e 1 g 2 g 3 e 1 g 2 g 3 | + ρ g e g , g e g | g 1 e 2 g 3 g 1 e 2 g 3 | + ρ g g e , g g e | g 1 g 2 e 3 g 1 g 2 e 3 | + ρ g g g , g g g | g 1 g 2 g 3 g 1 g 2 g 3 | ,

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