Abstract

We investigate the entanglement generation between two nitrogen-vacancy (NV) centers in diamond nanocrystal coupled to a high-Q counterpropagating twin whispering-gallery modes (WGMs) of a microtoroidal resonator. For looking into the degree and dynamics of the entanglement, we calculate the concurrence using the microscopic master equation approach. The influences of the coupling strength between the WGMs (or the size of the two spherical NV centers), the distance between two NV centers, the frequency detuning between the NV center and microresonator, and the initial state of the system on the dynamics of concurrence are discussed in detail. It is found that the maximum entanglement between the two NV centers can be created by properly adjusting these controllable system parameters. Our results may provide further insight into future solid-state cavity quantum electrodynamics (CQED) system for quantum information engineering.

© 2013 OSA

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2012 (4)

T. van der Sar, Z. H. Wang, M. S. Blok, H. Bernien, T. H. Taminiau, D. M. Toyli, D. A. Lidar, D. D. Awschalom, R. Hanson, and V. V. Dobrovitski, “Decoherence-protected quantum gates for a hybrid solid-state spin register,” Nature (London)484, 82–86 (2012).
[CrossRef]

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. A85, 042306 (2012).
[CrossRef]

Y. F. Xiao, Y. C. Liu, B. B. Li, Y. L. Chen, Y. Li, and Q. H. Gong, “Strongly enhanced light-matter interaction in a hybrid photonic-plasmonic resonator,” Phys. Rev. A85, 031805(R) (2012).
[CrossRef]

G. Y. Chen, C. M. Li, and Y. N. Chen, “Generating maximum entanglement under asymmetric couplings to surface plasmons,” Opt. Lett.37, 1337–1339 (2012).
[CrossRef] [PubMed]

2011 (10)

G. Y. Chen, N. Lambert, C. H. Chou, Y. N. Chen, and F. Nori, “Surface plasmons in a metal nanowire coupled to colloidal quantum dots: scattering properties and quantum entanglement,” Phys. Rev. B84, 045310 (2011).
[CrossRef]

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. A84, 011805(R) (2011).
[CrossRef]

Q. Chen, W. Yang, M. Feng, and J. Du, “Entangling separate nitrogen-vacancy centers in a scalable fashion via coupling to microtoroidal resonators,” Phys. Rev. A83, 054305 (2011).
[CrossRef]

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. A84, 043849 (2011).
[CrossRef]

P. E. Barclay, K. M. C. Fu, C. Santori, A. Faraon, and R. G. Beausoleil, “Hybrid nanocavity resonant enhancement of color center emission in diamond,” Phys. Rev. X1, 011007 (2011).
[CrossRef]

K. M. C. Fu, P. E. Barclay, C. Santori, A. Faraon, and R. G. Beausoleil, “Low-temperature tapered-fiber probing of diamond nitrogen-vacancy ensembles coupled to GaP microcavities,” New J. Phys.13, 055023 (2011).
[CrossRef]

X. Yi, Y. F. Xiao, Y. C. Liu, B. B. Li, Y. L. Chen, Y. Li, and Q. Gong, “Multiple-Rayleigh-scatterer-induced mode splitting in a high-Q whispering-gallery-mode microresonator,” Phys. Rev. A83, 023803 (2011).
[CrossRef]

D. Ratchford, F. Shafiei, S. Kim, S. K. Gray, and X. Li, “Manipulating coupling between a single semiconductor quantum dot and single gold nanoparticle,” Nano. Lett.11, 1049–1054 (2011).
[CrossRef] [PubMed]

V. Montenegro and M. Orszag, “Creation of entanglement of two atoms coupled to two distant cavities with losses,” J. Phys. B: At. Mol. Opt. Phys.44, 154019 (2011).
[CrossRef]

G. D. Fuchs, G. Burkard, P. V. Klimov, and D. D. Awschalom, “A quantum memory intrinsic to single nitrogen-vacancy centres in diamond,” Nat. Phys.7, 789–793 (2011).
[CrossRef]

2010 (7)

F. Shi, X. Rong, N. Xu, Y. Wang, J. Wu, B. Chong, X. Peng, J. Kniepert, R. S. Schoenfeld, W. Harneit, M. Feng, and J. Du, “Room-temperature implementation of the Deutsch-Jozsa algorithm with a single electronic spin in diamond,” Phys. Rev. Lett.105, 040504 (2010).
[CrossRef] [PubMed]

E. Togan, Y. Chu, A. S. Trifonov, L. Jiang, J. Maze, L. Childress, M. V. G. 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).
[CrossRef]

W. L. Yang, Z. Q. Xu, M. Feng, and J. F. Du, “Entanglement of separate nitrogen-vacancy centers coupled to a whispering-gallery mode cavity,” New J. Phys.12, 113039 (2010).
[CrossRef]

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)
[CrossRef]

P. Xue, “A controlled phase gate with nitrogen-vacancy centers in nanocrystal coupled to a silica microsphere cavity,” Chin. Phys. Lett.27, 060301 (2010).
[CrossRef]

J. S. Jin, C. S. Yu, P. Pei, and H. S. Song, “Positive effect of scattering strength of a microtoroidal cavity on atomic entanglement evolution,” Phys. Rev. A81, 042309 (2010).
[CrossRef]

M. Orszag and M. Hernandez, “Coherence and entanglement in a two-qubit system,” Adv. Opt. Photon.2, 229–286 (2010).
[CrossRef]

2009 (7)

P. E. Barclay, C. Santori, K. M. Fu, R. G. Beausoleil, and O. Painter, “Coherent interference effects in a nano-assembled diamond NV center cavity-QED system,” Opt. Express17, 8081–8097 (2009).
[CrossRef] [PubMed]

M. Larsson, K. N. Dinyari, and H. Wang, “Composite optical microcavity of diamond nanopillar and silica microsphere,” Nano. Lett.9, 1447–1450 (2009).
[CrossRef] [PubMed]

P. E. Barclay, K. M. C. Fu, C. Santori, and R. G. Beausoleil, “Chip-based microcavities coupled to nitrogen-vacancy centers in single crystal diamond,” Appl. Phys. Lett.95, 191115 (2009).
[CrossRef]

S. Schietinger and O. Benson, “Coupling single NV-centres to high-Q whispering gallery modes of a preselected frequency-matched microresonator,” J. Phys. B: At. Mol. Opt. Phys.42, 114001 (2009).
[CrossRef]

M. Wilczewski and M. Czachor, “Theory versus experiment for vacuum rabi oscillations in lossy cavities,” Phys. Rev. A79, 033836 (2009).
[CrossRef]

S. C. Benjamin, B. W. Lovett, and J. M. Smith, “Prospects for measurement-based quantum computing with solid state spins,” Laser & Photon. Rev.3, 556–574 (2009).
[CrossRef]

A. M. Stoneham, “Is a room-temperature, solid-state quantum computer mere fantasy?” Physics2, 34 (2009).
[CrossRef]

2008 (5)

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,” Science319, 1062–1065 (2008).
[CrossRef] [PubMed]

H. J. Kimble, “The quantum internet,” Nature (London)453, 1023–1030 (2008).
[CrossRef]

R. Blatt and D. Wineland, “Entangled states of trapped atomic ions,” Nature (London)453, 1008–1015 (2008).
[CrossRef]

J. Merlein, M. Kah, A. Zuschlag, A. Sell, A. Halm, J. Boneberg, P. Leiderer, A. Leitenstorfer, and R. Bratschitsch, “Nanomechanical control of an optical antenna,” Nat. Photonics2, 230–233 (2008).
[CrossRef]

S. Schietinger, T. 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).
[CrossRef] [PubMed]

2007 (3)

M. Scala, B. Militello, A. Messina, J. Piilo, and S. Maniscalco, “Microscopic derivation of the Jaynes-Cummings model with cavity losses,” Phys. Rev. A75, 013811 (2007).
[CrossRef]

A. D. Boozer, A. Boca, R. Miller, T. E. Northup, and H. J. Kimble, “Reversible state transfer between light and a single trapped atom,” Phys. Rev. Lett.98, 193601 (2007).
[CrossRef] [PubMed]

M. V. G. Dutt, L. Childress, L. Jiang, E. Togan, J. Maze, F. Jelezko, A. S. Zibrov, P. R. Hemmer, and M. D. Lukin, “Quantum register based on individual electronic and nuclear spin qubits in diamond,” Science316, 1312–1316 (2007).
[CrossRef] [PubMed]

2006 (4)

R. Hanson, O. Gywat, and D. D. Awschalom, “Room-temperature manipulation and decoherence of a single spin in diamond,” Phys. Rev. B74, 161203(R) (2006).
[CrossRef]

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature (London)443, 671–674 (2006).
[CrossRef]

M. Steffen, M. Ansmann, R. C. Bialczak, N. Katz, E. Lucero, R. McDermott, M. Neeley, E. M. Weig, A. N. Cleland, and J. M. Martinis, “Measurement of the entanglement of two superconducting qubits via state tomography,” Science313, 1423–1425 (2006).
[CrossRef] [PubMed]

Y. S. Park, A. K. Cook, and H. Wang, “Cavity QED with diamond nanocrystals and silica microspheres,” Nano Lett.6, 2075–2079 (2006).
[CrossRef] [PubMed]

2005 (5)

C. W. Chou, H. de Riedmatten, D. Felinto, S. V. Polyakov, S. J. van Enk, and H. J. Kimble, “Measurement-induced entanglement for excitation stored in remote atomic ensembles,” Nature (London)438, 828–832 (2005).
[CrossRef]

T. Di, A. Muthukrishnan, M. O. Scully, and M. S. Zubairy, “Quantum teleportation of an arbitrary superposition of atomic Dicke states,” Phys. Rev. A71, 062308 (2005).
[CrossRef]

S. B. Zheng, “Nongeometric conditional phase shift via adiabatic evolution of dark eigenstates: a new approach to quantum computation,” Phys. Rev. Lett.95, 080502 (2005).
[CrossRef] [PubMed]

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A71, 013817 (2005).
[CrossRef]

R. Miller, T. E. Northup, K. M. Birnbaum, A. Boca, A. D. Boozer, and H. J. Kimble, “Trapped atoms in cavity QED: coupling quantized light and matter,” J. Phys. B: At. Mol. Opt. Phys.38, S551–S565 (2005).
[CrossRef]

2004 (1)

F. Jelezko, T. Gaebel, I. Popa, M. Domhan, A. Gruber, and J. Wrachtrup, “Observation of coherent oscillation of a single nuclear spin and realization of a two-qubit conditoinal quantum gate,” Phys. Rev. Lett.93, 130501 (2004).
[CrossRef] [PubMed]

2003 (3)

T. A. Kennedy, J. S. Colton, J. E. Butler, R. C. Linares, and P. J. Doering, “Long coherence times at 300 K for nitrogen-vacancy center spins in diamond grown by chemical vapor deposition,” Appl. Phys. Lett.83, 4190–4192 (2003).
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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).
[CrossRef]

K. J. Vahala, “Optical microcavities,” Nature (London)424, 839–846 (2003).
[CrossRef]

2002 (1)

H. Mabuchi and A. C. Doherty, “Cavity quantum electrodynamics: coherence in context,” Science298, 1372–1377 (2002).
[CrossRef] [PubMed]

2001 (1)

L. M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature (London)414, 413–418 (2001).
[CrossRef]

2000 (2)

S. B. Zheng and G. C. Guo, “Efficient scheme for two-atom entanglement and quantum information processing in cavity QED,” Phys. Rev. Lett.85, 2392–2395 (2000).
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C. H. Bennett and D. P. Divincenzo, “Quantum information and computation,” Nature (London)404, 247–255 (2000).
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1999 (2)

A. Sørensen and K. Mølmer, “Quantum computation with ions in thermal motion,” Phys. Rev. Lett.82, 1971–1974 (1999).
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A. Aspect, “Bell’s inequality test: more ideal than ever,” Nature (London)398, 189–190 (1999).
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1998 (2)

Q. A. Turchette, C. S. Wood, B. E. King, C. J. Myatt, D. Leibfried, W. M. Itano, C. Monroe, and D. J. Wineland, “Deterministic entanglement of two trapped ions,” Phys. Rev. Lett.81, 3631–3634 (1998).
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W. K. Wootters, “Entanglement of formation of an arbitrary state of two qubits,” Phys. Rev. Lett.80, 2245–2248 (1998).
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1997 (4)

J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi, “Quantum state transfer and entanglement distribution among distant nodes in a quantum network,” Phys. Rev. Lett.78, 3221–3224 (1997).
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T. Pellizzari, “Quantum networking with optical fibres,” Phys. Rev. Lett.79, 5242–5245 (1997).
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D. Bouwmeester, J. W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature (London)390, 575–579 (1997).
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A. Gruber, A. Dräbenstedt, C. Tietz, L. Fleury, J. Wrachtrup, and C. Borczyskowski, “Scanning confocal optical microscopy and magnetic resonance of single defect centers,” Science276, 2012–2014 (1997).
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1995 (2)

1993 (1)

C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett.70, 1895–1899 (1993).
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1974 (1)

E. B. Davies, “Markovian master equations,” Commun. Math. Phys.39, 91–110 (1974).
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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,” Science319, 1062–1065 (2008).
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T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature (London)443, 671–674 (2006).
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Armani, D. K.

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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Aspect, A.

A. Aspect, “Bell’s inequality test: more ideal than ever,” Nature (London)398, 189–190 (1999).
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Awschalom, D. D.

T. van der Sar, Z. H. Wang, M. S. Blok, H. Bernien, T. H. Taminiau, D. M. Toyli, D. A. Lidar, D. D. Awschalom, R. Hanson, and V. V. Dobrovitski, “Decoherence-protected quantum gates for a hybrid solid-state spin register,” Nature (London)484, 82–86 (2012).
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Barclay, P. E.

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K. M. C. Fu, P. E. Barclay, C. Santori, A. Faraon, and R. G. Beausoleil, “Low-temperature tapered-fiber probing of diamond nitrogen-vacancy ensembles coupled to GaP microcavities,” New J. Phys.13, 055023 (2011).
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P. E. Barclay, K. M. C. Fu, C. Santori, and R. G. Beausoleil, “Chip-based microcavities coupled to nitrogen-vacancy centers in single crystal diamond,” Appl. Phys. Lett.95, 191115 (2009).
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P. E. Barclay, K. M. C. Fu, C. Santori, A. Faraon, and R. G. Beausoleil, “Hybrid nanocavity resonant enhancement of color center emission in diamond,” Phys. Rev. X1, 011007 (2011).
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K. M. C. Fu, P. E. Barclay, C. Santori, A. Faraon, and R. G. Beausoleil, “Low-temperature tapered-fiber probing of diamond nitrogen-vacancy ensembles coupled to GaP microcavities,” New J. Phys.13, 055023 (2011).
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P. E. Barclay, K. M. C. Fu, C. Santori, and R. G. Beausoleil, “Chip-based microcavities coupled to nitrogen-vacancy centers in single crystal diamond,” Appl. Phys. Lett.95, 191115 (2009).
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P. E. Barclay, C. Santori, K. M. Fu, R. G. Beausoleil, and O. Painter, “Coherent interference effects in a nano-assembled diamond NV center cavity-QED system,” Opt. Express17, 8081–8097 (2009).
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C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett.70, 1895–1899 (1993).
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M. Steffen, M. Ansmann, R. C. Bialczak, N. Katz, E. Lucero, R. McDermott, M. Neeley, E. M. Weig, A. N. Cleland, and J. M. Martinis, “Measurement of the entanglement of two superconducting qubits via state tomography,” Science313, 1423–1425 (2006).
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R. Miller, T. E. Northup, K. M. Birnbaum, A. Boca, A. D. Boozer, and H. J. Kimble, “Trapped atoms in cavity QED: coupling quantized light and matter,” J. Phys. B: At. Mol. Opt. Phys.38, S551–S565 (2005).
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A. D. Boozer, A. Boca, R. Miller, T. E. Northup, and H. J. Kimble, “Reversible state transfer between light and a single trapped atom,” Phys. Rev. Lett.98, 193601 (2007).
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R. Miller, T. E. Northup, K. M. Birnbaum, A. Boca, A. D. Boozer, and H. J. Kimble, “Trapped atoms in cavity QED: coupling quantized light and matter,” J. Phys. B: At. Mol. Opt. Phys.38, S551–S565 (2005).
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A. D. Boozer, A. Boca, R. Miller, T. E. Northup, and H. J. Kimble, “Reversible state transfer between light and a single trapped atom,” Phys. Rev. Lett.98, 193601 (2007).
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R. Miller, T. E. Northup, K. M. Birnbaum, A. Boca, A. D. Boozer, and H. J. Kimble, “Trapped atoms in cavity QED: coupling quantized light and matter,” J. Phys. B: At. Mol. Opt. Phys.38, S551–S565 (2005).
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Borczyskowski, C.

A. Gruber, A. Dräbenstedt, C. Tietz, L. Fleury, J. Wrachtrup, and C. Borczyskowski, “Scanning confocal optical microscopy and magnetic resonance of single defect centers,” Science276, 2012–2014 (1997).
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Bouwmeester, D.

D. Bouwmeester, J. W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature (London)390, 575–579 (1997).
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Bowen, W. P.

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature (London)443, 671–674 (2006).
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Brassard, G.

C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett.70, 1895–1899 (1993).
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G. D. Fuchs, G. Burkard, P. V. Klimov, and D. D. Awschalom, “A quantum memory intrinsic to single nitrogen-vacancy centres in diamond,” Nat. Phys.7, 789–793 (2011).
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Butler, J. E.

T. A. Kennedy, J. S. Colton, J. E. Butler, R. C. Linares, and P. J. Doering, “Long coherence times at 300 K for nitrogen-vacancy center spins in diamond grown by chemical vapor deposition,” Appl. Phys. Lett.83, 4190–4192 (2003).
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G. Y. Chen, C. M. Li, and Y. N. Chen, “Generating maximum entanglement under asymmetric couplings to surface plasmons,” Opt. Lett.37, 1337–1339 (2012).
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G. Y. Chen, N. Lambert, C. H. Chou, Y. N. Chen, and F. Nori, “Surface plasmons in a metal nanowire coupled to colloidal quantum dots: scattering properties and quantum entanglement,” Phys. Rev. B84, 045310 (2011).
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Chen, Q.

Q. Chen, W. Yang, M. Feng, and J. Du, “Entangling separate nitrogen-vacancy centers in a scalable fashion via coupling to microtoroidal resonators,” Phys. Rev. A83, 054305 (2011).
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Chen, Y. L.

Y. F. Xiao, Y. C. Liu, B. B. Li, Y. L. Chen, Y. Li, and Q. H. Gong, “Strongly enhanced light-matter interaction in a hybrid photonic-plasmonic resonator,” Phys. Rev. A85, 031805(R) (2012).
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X. Yi, Y. F. Xiao, Y. C. Liu, B. B. Li, Y. L. Chen, Y. Li, and Q. Gong, “Multiple-Rayleigh-scatterer-induced mode splitting in a high-Q whispering-gallery-mode microresonator,” Phys. Rev. A83, 023803 (2011).
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Chen, Y. N.

G. Y. Chen, C. M. Li, and Y. N. Chen, “Generating maximum entanglement under asymmetric couplings to surface plasmons,” Opt. Lett.37, 1337–1339 (2012).
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E. Togan, Y. Chu, A. S. Trifonov, L. Jiang, J. Maze, L. Childress, M. V. G. 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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M. V. G. Dutt, L. Childress, L. Jiang, E. Togan, J. Maze, F. Jelezko, A. S. Zibrov, P. R. Hemmer, and M. D. Lukin, “Quantum register based on individual electronic and nuclear spin qubits in diamond,” Science316, 1312–1316 (2007).
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Chong, B.

F. Shi, X. Rong, N. Xu, Y. Wang, J. Wu, B. Chong, X. Peng, J. Kniepert, R. S. Schoenfeld, W. Harneit, M. Feng, and J. 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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G. Y. Chen, N. Lambert, C. H. Chou, Y. N. Chen, and F. Nori, “Surface plasmons in a metal nanowire coupled to colloidal quantum dots: scattering properties and quantum entanglement,” Phys. Rev. B84, 045310 (2011).
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T. A. Kennedy, J. S. Colton, J. E. Butler, R. C. Linares, and P. J. Doering, “Long coherence times at 300 K for nitrogen-vacancy center spins in diamond grown by chemical vapor deposition,” Appl. Phys. Lett.83, 4190–4192 (2003).
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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,” Science319, 1062–1065 (2008).
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T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature (London)443, 671–674 (2006).
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C. W. Chou, H. de Riedmatten, D. Felinto, S. V. Polyakov, S. J. van Enk, and H. J. Kimble, “Measurement-induced entanglement for excitation stored in remote atomic ensembles,” Nature (London)438, 828–832 (2005).
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T. van der Sar, Z. H. Wang, M. S. Blok, H. Bernien, T. H. Taminiau, D. M. Toyli, D. A. Lidar, D. D. Awschalom, R. Hanson, and V. V. Dobrovitski, “Decoherence-protected quantum gates for a hybrid solid-state spin register,” Nature (London)484, 82–86 (2012).
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T. A. Kennedy, J. S. Colton, J. E. Butler, R. C. Linares, and P. J. Doering, “Long coherence times at 300 K for nitrogen-vacancy center spins in diamond grown by chemical vapor deposition,” Appl. Phys. Lett.83, 4190–4192 (2003).
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H. Mabuchi and A. C. Doherty, “Cavity quantum electrodynamics: coherence in context,” Science298, 1372–1377 (2002).
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F. Jelezko, T. Gaebel, I. Popa, M. Domhan, A. Gruber, and J. Wrachtrup, “Observation of coherent oscillation of a single nuclear spin and realization of a two-qubit conditoinal quantum gate,” Phys. Rev. Lett.93, 130501 (2004).
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A. Gruber, A. Dräbenstedt, C. Tietz, L. Fleury, J. Wrachtrup, and C. Borczyskowski, “Scanning confocal optical microscopy and magnetic resonance of single defect centers,” Science276, 2012–2014 (1997).
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Du, J.

Q. Chen, W. Yang, M. Feng, and J. Du, “Entangling separate nitrogen-vacancy centers in a scalable fashion via coupling to microtoroidal resonators,” Phys. Rev. A83, 054305 (2011).
[CrossRef]

F. Shi, X. Rong, N. Xu, Y. Wang, J. Wu, B. Chong, X. Peng, J. Kniepert, R. S. Schoenfeld, W. Harneit, M. Feng, and J. 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. Xu, M. Feng, and J. F. Du, “Entanglement of separate nitrogen-vacancy centers coupled to a whispering-gallery mode cavity,” New J. Phys.12, 113039 (2010).
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L. M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature (London)414, 413–418 (2001).
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E. Togan, Y. Chu, A. S. Trifonov, L. Jiang, J. Maze, L. Childress, M. V. G. 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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M. V. G. Dutt, L. Childress, L. Jiang, E. Togan, J. Maze, F. Jelezko, A. S. Zibrov, P. R. Hemmer, and M. D. Lukin, “Quantum register based on individual electronic and nuclear spin qubits in diamond,” Science316, 1312–1316 (2007).
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D. Bouwmeester, J. W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature (London)390, 575–579 (1997).
[CrossRef]

Faraon, A.

P. E. Barclay, K. M. C. Fu, C. Santori, A. Faraon, and R. G. Beausoleil, “Hybrid nanocavity resonant enhancement of color center emission in diamond,” Phys. Rev. X1, 011007 (2011).
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K. M. C. Fu, P. E. Barclay, C. Santori, A. Faraon, and R. G. Beausoleil, “Low-temperature tapered-fiber probing of diamond nitrogen-vacancy ensembles coupled to GaP microcavities,” New J. Phys.13, 055023 (2011).
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C. W. Chou, H. de Riedmatten, D. Felinto, S. V. Polyakov, S. J. van Enk, and H. J. Kimble, “Measurement-induced entanglement for excitation stored in remote atomic ensembles,” Nature (London)438, 828–832 (2005).
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Figures (7)

Fig. 1
Fig. 1

Schematic illustration of the coupling system composed of a microtoroidal resonator and two two-level NV centers. The microtoroidal cavity supports two counter-propagating WGMs, denoted as acw and accw. In the presence of NV1, one of the modes, say cw couples to NV1. The scattered light will couple-back to either the cw or the ccw mode. The same is true when the ccw couples to NV2. The distance between two NV centers is d. The distance d is far enough, so the interaction of two NV centers can be neglected. Experimentally, we can use atomic force microscope manipulation to controllably position the NV centers [58, 59]. The bubble shows energy configuration for each of the NV centers.

Fig. 2
Fig. 2

Concurrence dynamics of the two NV centers versus the dimensionless time Gt for different phases (a1)–(c1) kd = (2n + 1)π/2 and (a2)–(c2) kd = under the initial state ρ(0) = |3〉〈3| = |gg10〉〈gg10| and the detuning Δ = 0. (a1) and (a2): g = G/10; (b1) and (b2): g = G; (c1) and (c2): g = 2G. The blue solid curves represent the damping rate γ = 0 of the WGMs and the red dashed curves correspond to γ = G/30, respectively.

Fig. 3
Fig. 3

Concurrence dynamics of the two NV centers versus the dimensionless time Gt for three different coupling strengths: (a) g = G/10, (b) g = G, and (c) g = 2G. The other system parameters are chosen as ρ(0) = |1〉〈1| = |eg00〉〈eg00|, kd = , and Δ = 0, respectively. The blue solid curves represent the damping rate γ = 0 of the WGMs and the red dashed curves correspond to γ = G/30.

Fig. 4
Fig. 4

Concurrence dynamics of the two NV centers versus the dimensionless time Gt for three different initial states ρ(0). In this case, the mixed state is chosen as ρ(0) = ε|1〉 〈1| + (1 − ε)|2〉 〈2| = ε|eg00〉 〈eg00| + (1 − ε)|ge00〉 〈ge00| with ε being a real number. The other system parameters are chosen as Δ = 0, kd = , g = 2G, and γ = 0, respectively.

Fig. 5
Fig. 5

Concurrence dynamics of the two NV centers versus the dimensionless time Gt for three different detunings Δ. The other system parameters are chosen as ρ(0) = |3〉 〈3| = |gg10〉 〈gg10|, kd = , g = G, and γ = 0, respectively.

Fig. 6
Fig. 6

Concurrence dynamics of the two NV centers versus the dimensionless time Gt for three different detunings Δ. The other system parameters are the same as Fig. 5 except for the initial state ρ(0) = |1〉 〈1| = |eg00〉 〈eg00|.

Fig. 7
Fig. 7

Concurrence dynamics of the two NV centers versus the dimensionless time Gt for three different proportional coefficients s. The other system parameters are the same as Fig. 2(a1) for γ = 0.

Equations (57)

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H = H 0 + H 1 + H 2 ,
H 0 = j = 1 , 2 ω a 2 σ j z + m ω c a m a m ,
H 1 = m = c w , c c w [ ( G 1 σ 1 + a m + G 2 σ 2 + e i k d a m ) + H . c . ] ,
H 2 = m , m = c w , c c w ( g 1 m m a m a m + g 2 m m a m a m ) = 2 g ( a c w + a c w + a c c w + a c c w ) + [ g ( a c c w + a c w + e 2 i k d a c c w + a c w ) + H . c . ] ,
H = ( 0 0 G 1 G 2 0 0 0 G 2 e i k d G 2 e i k d 0 G 1 G 2 e i k d Δ + 2 g g ( 1 + e 2 i k d ) 0 G 1 G 2 e i k d g ( 1 + e 2 i k d ) Δ + 2 g 0 0 0 0 0 ω a ) ,
ρ ˙ ( t ) = i [ H , ρ ] + ω ¯ > 0 , m = { c w , c c w } γ m ( ω ¯ ) × [ A m ( ω ¯ ) ρ ( t ) A m ( ω ¯ ) 1 2 { A m ( ω ¯ ) A m ( ω ¯ ) , ρ ( t ) } ] ,
| ϕ 1 = c 11 | 1 + c 12 | 2 + c 13 | 3 + c 14 | 4 ,
| ϕ 2 = c 21 | 1 + c 22 | 2 + c 23 | 3 + c 24 | 4 ,
| ϕ 3 = c 31 | 1 + c 32 | 2 + c 33 | 3 + c 34 | 4 ,
| ϕ 4 = c 41 | 1 + c 42 | 2 + c 43 | 3 + c 44 | 4 ,
| ϕ 5 = | 5 ,
A m ( ω ¯ i j ) = | ϕ i ϕ i | a m | ϕ j ϕ j | ,
γ 1 = | c 13 | 2 γ c w ( ω ¯ 51 ) + | c 14 | 2 γ c c w ( ω ¯ 51 ) ,
γ 2 = | c 23 | 2 γ c w ( ω ¯ 52 ) + | c 24 | 2 γ c c w ( ω ¯ 52 ) ,
γ 3 = | c 33 | 2 γ c w ( ω ¯ 53 ) + | c 34 | 2 γ c c w ( ω ¯ 53 ) ,
γ 4 = | c 43 | 2 γ c w ( ω ¯ 54 ) + | c 44 | 2 γ c c w ( ω ¯ 54 ) ,
ρ ˙ ( t ) = i [ H , ρ ( t ) ] + i = 1 4 γ i [ | ϕ 5 ϕ i | ρ ( t ) | ϕ i ϕ 5 | 1 2 { | ϕ i ϕ i | , ρ } ] ,
ρ ( 0 ) = i , j = 1 4 i | ρ ( 0 ) | j | i j | .
ρ ( 0 ) = i , j = 1 4 i | ρ ( 0 ) | j | i j | = i , j = 1 4 ϕ i | ρ ( 0 ) | ϕ j | ϕ i ϕ j | .
( | 1 | 2 | 3 | 4 | 5 ) = ( f 1 f 2 f 3 f 4 0 f 5 f 6 f 7 f 8 0 f 9 f 10 f 11 f 12 0 f 13 f 14 f 15 f 16 0 0 0 0 0 1 ) ( | ϕ 1 | ϕ 2 | ϕ 3 | ϕ 4 | ϕ 5 ) = F ( | ϕ 1 | ϕ 2 | ϕ 3 | ϕ 4 | ϕ 5 ) ,
ρ ( t ) = i , j = 1 4 ϕ i | ρ ( t ) | ϕ j | ϕ i ϕ j | ,
ρ a ( t ) = 00 | ρ ( t ) | 00 = i , j = 1 4 ϕ i | ρ ( t ) | ϕ j 00 | ϕ i ϕ j | 00 ,
00 | ϕ 1 = c 11 | e g + c 12 | g e ,
00 | ϕ 2 = c 21 | e g + c 22 | g e ,
00 | ϕ 3 = c 31 | e g + c 32 | g e ,
00 | ϕ 4 = c 41 | e g + c 42 | g e ,
00 | ϕ 5 = | g g .
ρ a ( t ) = ρ e g , e g | e g e g | + ρ g e , g e | g e g e | + ρ g g , g g | g g g g | + ρ e g , g e | e g g e | + ρ e g , g e * | g e e g | .
C ( t ) = C ( ρ a ) = 2 max { 0 , μ 1 μ 2 μ 3 μ 4 } ,
C ( t ) = C ( ρ a ) = 2 | ρ e g , g e | ,
A c w ( ω ¯ 51 = λ 1 λ 5 ) = c 13 | ϕ 5 ϕ 1 | ,
A c w ( ω ¯ 52 = λ 2 λ 5 ) = c 23 | ϕ 5 ϕ 2 | ,
A c w ( ω ¯ 53 = λ 3 λ 5 ) = c 33 | ϕ 5 ϕ 3 | ,
A c w ( ω ¯ 54 = λ 4 λ 5 ) = c 43 | ϕ 5 ϕ 4 | .
A c c w ( ω ¯ 51 = λ 1 λ 5 ) = c 14 | ϕ 5 ϕ 1 | ,
A c c w ( ω ¯ 52 = λ 2 λ 5 ) = c 24 | ϕ 5 ϕ 2 | ,
A c c w ( ω ¯ 53 = λ 3 λ 5 ) = c 34 | ϕ 5 ϕ 3 | ,
A c c w ( ω ¯ 54 = λ 4 λ 5 ) = c 44 | ϕ 5 ϕ 4 | .
ρ 11 ( t ) = ρ 11 ( 0 ) e γ 1 t ,
ρ 22 ( t ) = ρ 22 ( 0 ) e γ 2 t ,
ρ 33 t = ρ 33 ( 0 ) e γ 3 t ,
ρ 44 ( t ) = ρ 44 ( 0 ) e γ 4 t ,
ρ 55 ( t ) = ρ 55 ( 0 ) + ρ 11 ( 0 ) ( 1 e γ 1 t ) + ρ 22 ( 0 ) ( 1 e γ 2 t ) + ρ 33 ( 0 ) ( 1 e γ 3 t ) + ρ 44 ( 0 ) ( 1 e γ 4 t ) ,
ρ 21 ( t ) = ρ 21 ( 0 ) e ( i ( λ 1 λ 2 ) 1 2 ( γ 1 + γ 2 ) ) t ,
ρ 31 ( t ) = ρ 31 ( 0 ) e ( i ( λ 1 λ 3 ) 1 2 ( γ 1 + γ 3 ) ) t ,
ρ 41 ( t ) = ρ 41 ( 0 ) e ( i ( λ 1 λ 4 ) 1 2 ( γ 1 + γ 4 ) ) t ,
ρ 51 ( t ) = ρ 51 ( 0 ) e ( i ( λ 1 λ 5 ) 1 2 γ 1 ) t ,
ρ 32 ( t ) = ρ 32 ( 0 ) e ( i ( λ 2 λ 3 ) 1 2 ( γ 2 + γ 3 ) ) t ,
ρ 42 ( t ) = ρ 42 ( 0 ) e ( i ( λ 2 λ 4 ) 1 2 ( γ 2 + γ 4 ) ) t ,
ρ 52 ( t ) = ρ 52 ( 0 ) e ( i ( λ 2 λ 5 ) 1 2 γ 2 ) t ,
ρ 43 ( t ) = ρ 43 ( 0 ) e ( i ( λ 3 λ 4 ) 1 2 ( γ 3 + γ 4 ) ) t ,
ρ 53 ( t ) = ρ 53 ( 0 ) e ( i ( λ 3 λ 5 ) 1 2 γ 3 ) t ,
ρ 54 ( t ) = ρ 54 ( 0 ) e ( i ( λ 4 λ 5 ) 1 2 γ 4 ) t .
ρ e g , e g = ρ 11 ( t ) | c 11 | 2 + ρ 22 ( t ) | c 21 | 2 + ρ 33 ( t ) | c 31 | 2 + ρ 44 ( t ) | c 41 | 2 + [ ( ρ 21 ( t ) c 21 c 11 * + ρ 31 ( t ) c 31 c 11 * + ρ 41 ( t ) c 41 c 11 * + ρ 32 ( t ) c 31 c 21 * + ρ 42 ( t ) c 41 c 21 * + ρ 43 ( t ) c 41 c 31 * ) + c . c . ] ,
ρ g e , g e = ρ 11 ( t ) | c 12 | 2 + ρ 22 ( t ) | c 22 | 2 + ρ 33 ( t ) | c 32 | 2 + ρ 44 ( t ) | c 42 | 2 + [ ( ρ 21 ( t ) c 22 c 12 * + ρ 31 ( t ) c 32 c 12 * + ρ 41 ( t ) c 42 c 12 * + ρ 32 ( t ) c 32 c 22 * + ρ 42 ( t ) c 42 c 22 * + ρ 43 ( t ) c 42 c 32 * ) + c . c . ] ,
ρ g g , g g = ρ 55 ( 0 ) + ρ 11 ( 0 ) ( 1 e γ 1 t ) + ρ 22 ( 0 ) ( 1 e γ 2 t ) + ρ 33 ( 0 ) ( 1 e γ 3 t ) + ρ 44 ( 0 ) ( 1 e γ 4 t ) ,
ρ e g , g e = ρ 11 ( t ) c 11 c 12 * + ρ 22 ( t ) c 21 c 22 * + ρ 33 ( t ) c 31 c 32 * + ρ 44 ( t ) c 41 c 42 * + ρ 21 ( t ) c 21 c 12 * + ρ 21 * ( t ) c 11 c 22 * + ρ 31 ( t ) c 31 c 12 * + ρ 31 * ( t ) c 11 c 32 * + ρ 41 ( t ) c 41 c 12 * + ρ 41 * ( t ) c 11 c 42 * + ρ 32 ( t ) c 31 c 22 * + ρ 32 * ( t ) c 21 c 32 * , + ρ 42 ( t ) c 41 c 22 * + ρ 42 * ( t ) c 21 c 42 * + ρ 43 ( t ) c 41 c 32 * + ρ 43 * ( t ) c 31 c 42 * ,

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