Abstract

We investigate the scattering properties of a single surface plasmon in metal nanowire coupled to a nitrogen-vacancy (NV) center in diamond nanocrystal under optical excitation. We demonstrate that, by spatially modulating a classical control beam, alternating regions of high reflection and absorption as well as high transmission and absorption of a single plasmon can be created in the left- and right-going directions that act as a kind of scattering grating. Such approach to induce grating gets out the well investigating region in which the weak interactions between single atoms and light is often used. The proposal may be used for chip-integrated grating, switcher and multi-channel drop filter.

© 2011 OSA

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  1. H. Y. Ling, Y. Q. Li, and M. Xiao, “Electromagnetically induced grating: Homogeneously broadened medium,” Phys. Rev. A 57, 1338–1344 (1998) and references therein.
    [CrossRef]
  2. J. Wen, S. Du, H. Chen, and M. Xiao, “Electromagnetically induced Talbot effect,” Appl. Phys. Lett. 98, 081108 (2011).
    [CrossRef]
  3. M. Mitsunaga and N. Imoto, “Observation of an electromagnetically induced grating in cold sodium atoms,” Phys. Rev. A 59, 4773–4776 (1999).
    [CrossRef]
  4. A. André and M. D. Lukin, “Manipulating light pulses via dynamically controlled photonic band gap,” Phys. Rev. Lett. 89, 143602 (2002).
    [CrossRef] [PubMed]
  5. M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Stationary pulses of light in an atomic medium,” Nature (London) 426, 638–641 (2003).
    [CrossRef]
  6. A. W. Brown and M. Xiao, “All-optical switching and routing based on an electromagnetically induced absorption grating,” Opt. Lett. 30, 699–701 (2005).
    [CrossRef] [PubMed]
  7. J. W. Gao, J. H. Wu, N. Ba, C. L. Cui, and X. X. Tian, “Efficient all-optical routing using dynamically induced transparency windows and photonic band gaps,” Phys. Rev. A 81, 013804 (2010).
    [CrossRef]
  8. J. H. Wu, M. Artoni, and G. C. La Rocca, “All-optical light confinement in dynamic cavities in cold atoms,” Phys. Rev. Lett. 103, 133601 (2009).
    [CrossRef] [PubMed]
  9. A. V. Akimov, A. Mukherjee, C. L. Yu, D. E. Chang, A. S. Zibrov, P. R. Hemmer, H. Park, and M. D. Lukin, “Generation of single optical plasmons in metallic nanowires coupled to quantum dots,” Nature (London) 450, 402–406 (2007).
    [CrossRef]
  10. D. E. Chang, A. S. Sørensen, E. A. Demler, and M. D. Lukin, “A single-photon transistor using nanoscale surface plasmons,” Nat. Phys. 3, 807–812 (2007).
    [CrossRef]
  11. C. Ropers, C. C. Neacsu, T. Elsaesser, M. Albrecht, M. B. Raschke, and C. Lienau, “Grating-coupling of surface plasmons onto metallic tips: A nanoconfined light source,” Nano Lett. 7, 2784–2788 (2007).
    [CrossRef] [PubMed]
  12. G. Y. Chen, Y. N. Chen, and D. S. Chuu, “Spontaneous emission of quantum dot excitons into surface plasmons in a nanowire,” Opt. Lett. 33, 2212–2214 (2008).
    [CrossRef] [PubMed]
  13. A. Gonzalez-Tudela, F. J. Rodríguez, L. Quiroga, and C. Tejedor, “Dissipative dynamics of a solid-state qubit coupled to surface plasmons: From non-Markov to Markov regimes,” Phys. Rev. B 82, 115334 (2010).
    [CrossRef]
  14. D. Dzsotjan, A. S. Sørensen, and M. Fleischhauer, “Quantum emitters coupled to surface plasmons of a nanowire: A Greens function approach,” Phys. Rev. B 82, 075427 (2010).
    [CrossRef]
  15. W. Chen, G. Y. Chen, and Y. N. Chen, “Coherent transport of nanowire surface plasmons coupled to quantum dots,” Opt. Express 18, 10360–10368 (2010), http://www.opticsinfobase.org/abstract.cfm?URI=oe-18-10-10360 .
    [CrossRef]
  16. D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Quantum optics with surface plasmons,” Phys. Rev. Lett. 97, 053002 (2006).
    [CrossRef] [PubMed]
  17. D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Strong coupling of single emitters to surface plasmons,” Phys. Rev. B 76, 035420 (2007).
    [CrossRef]
  18. A. Huck, S. Kumar, A. Shakoor, and U. L. Andersen, “Controlled coupling of a single nitrogen-vacancy center to a silver nanowire,” Phys. Rev. Lett. 106, 096801 (2011).
    [CrossRef] [PubMed]
  19. 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 conditional quantum gate,” Phys. Rev. Lett. 93, 130501 (2004).
    [CrossRef] [PubMed]
  20. R. J. Epstein, F. M. Mendoza, Y. K. Kato, and D. D. Awschalom, “Anisotropic interactions of a single spin and dark-spin spectroscopy in diamond,” Nat. Phys. 1, 94–98 (2005).
    [CrossRef]
  21. T. Gaebel, M. Domhan, I. Popa, C. Wittmann, P. Neumann, F. Jelezko, J. R. Rabeau, N. Stavrias, A. D. Greentree, S. Prawer, J. Meijer, J. Twamley, P. R. Hemmer, and J. Wrachtrup, “Room-temperature coherent coupling of single spins in diamond,” Nat. Phys. 2, 408–413 (2006).
    [CrossRef]
  22. M. V. Gurudev 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,” Science 316, 1312–1316 (2007).
    [CrossRef]
  23. R. Hanson, V. V. Dobrovitski, A. E. Feiguin, O. Gywat, and D. D. Awschalom, “Coherent dynamics of a single spin interacting with an adjustable spin bath,” Science 320,, 352–355 (2008).
    [CrossRef] [PubMed]
  24. 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]
  25. 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]
  26. 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]
  27. N. B. Manson, J. P. Harrison, and M. J. Sellars, “Nitrogen-vacancy center in diamond: Model of the electronic structure and associated dynamics,” Phys. Rev. B 74, 104303 (2006).
    [CrossRef]
  28. 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).
    [CrossRef]
  29. P. Tamarat, N. B. Manson, J. P. Harrison, R. L. McMurtrie, A. Nizovtsev, C. Santori, R. G. Beausoleil, P. Neumann, T. Gaebel, F. Jelezko, P. Hemmer, and J. Wrachtrup, “Spin-flip and spin-conserving optical transitions of the nitrogen-vacancy centre in diamond,” N. J. Phys. 10, 045004 (2008).
    [CrossRef]
  30. J. T. Shen and S. Fan, “Theory of single-photon transport in a single-mode waveguide. I. Coupling to a cavity containing a two-level atom,” Phys. Rev. A 79, 023837 (2009).
    [CrossRef]
  31. J. T. Shen and S. Fan, “Coherent photon transport from spontaneous emission in one-dimensional waveguides,” Opt. Lett. 30, 2001–2003 (2005).
    [CrossRef] [PubMed]
  32. Y. Wu, “Effective Raman theory for a three-level atom in the Λ configuration,” Phys. Rev. A 54, 1586–1592 (1996).
    [CrossRef] [PubMed]

2011

A. Huck, S. Kumar, A. Shakoor, and U. L. Andersen, “Controlled coupling of a single nitrogen-vacancy center to a silver nanowire,” Phys. Rev. Lett. 106, 096801 (2011).
[CrossRef] [PubMed]

J. Wen, S. Du, H. Chen, and M. Xiao, “Electromagnetically induced Talbot effect,” Appl. Phys. Lett. 98, 081108 (2011).
[CrossRef]

2010

J. W. Gao, J. H. Wu, N. Ba, C. L. Cui, and X. X. Tian, “Efficient all-optical routing using dynamically induced transparency windows and photonic band gaps,” Phys. Rev. A 81, 013804 (2010).
[CrossRef]

A. Gonzalez-Tudela, F. J. Rodríguez, L. Quiroga, and C. Tejedor, “Dissipative dynamics of a solid-state qubit coupled to surface plasmons: From non-Markov to Markov regimes,” Phys. Rev. B 82, 115334 (2010).
[CrossRef]

D. Dzsotjan, A. S. Sørensen, and M. Fleischhauer, “Quantum emitters coupled to surface plasmons of a nanowire: A Greens function approach,” Phys. Rev. B 82, 075427 (2010).
[CrossRef]

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]

2009

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]

J. T. Shen and S. Fan, “Theory of single-photon transport in a single-mode waveguide. I. Coupling to a cavity containing a two-level atom,” Phys. Rev. A 79, 023837 (2009).
[CrossRef]

J. H. Wu, M. Artoni, and G. C. La Rocca, “All-optical light confinement in dynamic cavities in cold atoms,” Phys. Rev. Lett. 103, 133601 (2009).
[CrossRef] [PubMed]

2008

G. Y. Chen, Y. N. Chen, and D. S. Chuu, “Spontaneous emission of quantum dot excitons into surface plasmons in a nanowire,” Opt. Lett. 33, 2212–2214 (2008).
[CrossRef] [PubMed]

P. Tamarat, N. B. Manson, J. P. Harrison, R. L. McMurtrie, A. Nizovtsev, C. Santori, R. G. Beausoleil, P. Neumann, T. Gaebel, F. Jelezko, P. Hemmer, and J. Wrachtrup, “Spin-flip and spin-conserving optical transitions of the nitrogen-vacancy centre in diamond,” N. J. Phys. 10, 045004 (2008).
[CrossRef]

R. Hanson, V. V. Dobrovitski, A. E. Feiguin, O. Gywat, and D. D. Awschalom, “Coherent dynamics of a single spin interacting with an adjustable spin bath,” Science 320,, 352–355 (2008).
[CrossRef] [PubMed]

2007

D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Strong coupling of single emitters to surface plasmons,” Phys. Rev. B 76, 035420 (2007).
[CrossRef]

M. V. Gurudev 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,” Science 316, 1312–1316 (2007).
[CrossRef]

A. V. Akimov, A. Mukherjee, C. L. Yu, D. E. Chang, A. S. Zibrov, P. R. Hemmer, H. Park, and M. D. Lukin, “Generation of single optical plasmons in metallic nanowires coupled to quantum dots,” Nature (London) 450, 402–406 (2007).
[CrossRef]

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

C. Ropers, C. C. Neacsu, T. Elsaesser, M. Albrecht, M. B. Raschke, and C. Lienau, “Grating-coupling of surface plasmons onto metallic tips: A nanoconfined light source,” Nano Lett. 7, 2784–2788 (2007).
[CrossRef] [PubMed]

2006

T. Gaebel, M. Domhan, I. Popa, C. Wittmann, P. Neumann, F. Jelezko, J. R. Rabeau, N. Stavrias, A. D. Greentree, S. Prawer, J. Meijer, J. Twamley, P. R. Hemmer, and J. Wrachtrup, “Room-temperature coherent coupling of single spins in diamond,” Nat. Phys. 2, 408–413 (2006).
[CrossRef]

D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Quantum optics with surface plasmons,” Phys. Rev. Lett. 97, 053002 (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]

N. B. Manson, J. P. Harrison, and M. J. Sellars, “Nitrogen-vacancy center in diamond: Model of the electronic structure and associated dynamics,” Phys. Rev. B 74, 104303 (2006).
[CrossRef]

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).
[CrossRef]

2005

2004

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 conditional quantum gate,” Phys. Rev. Lett. 93, 130501 (2004).
[CrossRef] [PubMed]

2003

M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Stationary pulses of light in an atomic medium,” Nature (London) 426, 638–641 (2003).
[CrossRef]

2002

A. André and M. D. Lukin, “Manipulating light pulses via dynamically controlled photonic band gap,” Phys. Rev. Lett. 89, 143602 (2002).
[CrossRef] [PubMed]

1999

M. Mitsunaga and N. Imoto, “Observation of an electromagnetically induced grating in cold sodium atoms,” Phys. Rev. A 59, 4773–4776 (1999).
[CrossRef]

1998

H. Y. Ling, Y. Q. Li, and M. Xiao, “Electromagnetically induced grating: Homogeneously broadened medium,” Phys. Rev. A 57, 1338–1344 (1998) and references therein.
[CrossRef]

1996

Y. Wu, “Effective Raman theory for a three-level atom in the Λ configuration,” Phys. Rev. A 54, 1586–1592 (1996).
[CrossRef] [PubMed]

Akimov, A. V.

A. V. Akimov, A. Mukherjee, C. L. Yu, D. E. Chang, A. S. Zibrov, P. R. Hemmer, H. Park, and M. D. Lukin, “Generation of single optical plasmons in metallic nanowires coupled to quantum dots,” Nature (London) 450, 402–406 (2007).
[CrossRef]

Albrecht, M.

C. Ropers, C. C. Neacsu, T. Elsaesser, M. Albrecht, M. B. Raschke, and C. Lienau, “Grating-coupling of surface plasmons onto metallic tips: A nanoconfined light source,” Nano Lett. 7, 2784–2788 (2007).
[CrossRef] [PubMed]

Andersen, U. L.

A. Huck, S. Kumar, A. Shakoor, and U. L. Andersen, “Controlled coupling of a single nitrogen-vacancy center to a silver nanowire,” Phys. Rev. Lett. 106, 096801 (2011).
[CrossRef] [PubMed]

André, A.

A. André and M. D. Lukin, “Manipulating light pulses via dynamically controlled photonic band gap,” Phys. Rev. Lett. 89, 143602 (2002).
[CrossRef] [PubMed]

Artoni, M.

J. H. Wu, M. Artoni, and G. C. La Rocca, “All-optical light confinement in dynamic cavities in cold atoms,” Phys. Rev. Lett. 103, 133601 (2009).
[CrossRef] [PubMed]

Awschalom, D. D.

R. Hanson, V. V. Dobrovitski, A. E. Feiguin, O. Gywat, and D. D. Awschalom, “Coherent dynamics of a single spin interacting with an adjustable spin bath,” Science 320,, 352–355 (2008).
[CrossRef] [PubMed]

R. J. Epstein, F. M. Mendoza, Y. K. Kato, and D. D. Awschalom, “Anisotropic interactions of a single spin and dark-spin spectroscopy in diamond,” Nat. Phys. 1, 94–98 (2005).
[CrossRef]

Ba, N.

J. W. Gao, J. H. Wu, N. Ba, C. L. Cui, and X. X. Tian, “Efficient all-optical routing using dynamically induced transparency windows and photonic band gaps,” Phys. Rev. A 81, 013804 (2010).
[CrossRef]

Bajcsy, M.

M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Stationary pulses of light in an atomic medium,” Nature (London) 426, 638–641 (2003).
[CrossRef]

Beausoleil, R. G.

P. Tamarat, N. B. Manson, J. P. Harrison, R. L. McMurtrie, A. Nizovtsev, C. Santori, R. G. Beausoleil, P. Neumann, T. Gaebel, F. Jelezko, P. Hemmer, and J. Wrachtrup, “Spin-flip and spin-conserving optical transitions of the nitrogen-vacancy centre in diamond,” N. J. Phys. 10, 045004 (2008).
[CrossRef]

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).
[CrossRef]

Brown, A. W.

Chang, D. E.

A. V. Akimov, A. Mukherjee, C. L. Yu, D. E. Chang, A. S. Zibrov, P. R. Hemmer, H. Park, and M. D. Lukin, “Generation of single optical plasmons in metallic nanowires coupled to quantum dots,” Nature (London) 450, 402–406 (2007).
[CrossRef]

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

D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Strong coupling of single emitters to surface plasmons,” Phys. Rev. B 76, 035420 (2007).
[CrossRef]

D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Quantum optics with surface plasmons,” Phys. Rev. Lett. 97, 053002 (2006).
[CrossRef] [PubMed]

Chen, G. Y.

Chen, H.

J. Wen, S. Du, H. Chen, and M. Xiao, “Electromagnetically induced Talbot effect,” Appl. Phys. Lett. 98, 081108 (2011).
[CrossRef]

Chen, W.

Chen, Y. N.

Childress, L.

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]

M. V. Gurudev 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,” Science 316, 1312–1316 (2007).
[CrossRef]

Chu, Y.

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]

Chuu, D. S.

Cook, A. K.

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]

Cui, C. L.

J. W. Gao, J. H. Wu, N. Ba, C. L. Cui, and X. X. Tian, “Efficient all-optical routing using dynamically induced transparency windows and photonic band gaps,” Phys. Rev. A 81, 013804 (2010).
[CrossRef]

Demler, E. A.

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

Dinyari, K. N.

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]

Dobrovitski, V. V.

R. Hanson, V. V. Dobrovitski, A. E. Feiguin, O. Gywat, and D. D. Awschalom, “Coherent dynamics of a single spin interacting with an adjustable spin bath,” Science 320,, 352–355 (2008).
[CrossRef] [PubMed]

Domhan, M.

T. Gaebel, M. Domhan, I. Popa, C. Wittmann, P. Neumann, F. Jelezko, J. R. Rabeau, N. Stavrias, A. D. Greentree, S. Prawer, J. Meijer, J. Twamley, P. R. Hemmer, and J. Wrachtrup, “Room-temperature coherent coupling of single spins in diamond,” Nat. Phys. 2, 408–413 (2006).
[CrossRef]

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 conditional quantum gate,” Phys. Rev. Lett. 93, 130501 (2004).
[CrossRef] [PubMed]

Du, S.

J. Wen, S. Du, H. Chen, and M. Xiao, “Electromagnetically induced Talbot effect,” Appl. Phys. Lett. 98, 081108 (2011).
[CrossRef]

Dutt, M. V. G.

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]

Dzsotjan, D.

D. Dzsotjan, A. S. Sørensen, and M. Fleischhauer, “Quantum emitters coupled to surface plasmons of a nanowire: A Greens function approach,” Phys. Rev. B 82, 075427 (2010).
[CrossRef]

Elsaesser, T.

C. Ropers, C. C. Neacsu, T. Elsaesser, M. Albrecht, M. B. Raschke, and C. Lienau, “Grating-coupling of surface plasmons onto metallic tips: A nanoconfined light source,” Nano Lett. 7, 2784–2788 (2007).
[CrossRef] [PubMed]

Epstein, R. J.

R. J. Epstein, F. M. Mendoza, Y. K. Kato, and D. D. Awschalom, “Anisotropic interactions of a single spin and dark-spin spectroscopy in diamond,” Nat. Phys. 1, 94–98 (2005).
[CrossRef]

Fan, S.

J. T. Shen and S. Fan, “Theory of single-photon transport in a single-mode waveguide. I. Coupling to a cavity containing a two-level atom,” Phys. Rev. A 79, 023837 (2009).
[CrossRef]

J. T. Shen and S. Fan, “Coherent photon transport from spontaneous emission in one-dimensional waveguides,” Opt. Lett. 30, 2001–2003 (2005).
[CrossRef] [PubMed]

Fattal, D.

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).
[CrossRef]

Feiguin, A. E.

R. Hanson, V. V. Dobrovitski, A. E. Feiguin, O. Gywat, and D. D. Awschalom, “Coherent dynamics of a single spin interacting with an adjustable spin bath,” Science 320,, 352–355 (2008).
[CrossRef] [PubMed]

Fleischhauer, M.

D. Dzsotjan, A. S. Sørensen, and M. Fleischhauer, “Quantum emitters coupled to surface plasmons of a nanowire: A Greens function approach,” Phys. Rev. B 82, 075427 (2010).
[CrossRef]

Gaebel, T.

P. Tamarat, N. B. Manson, J. P. Harrison, R. L. McMurtrie, A. Nizovtsev, C. Santori, R. G. Beausoleil, P. Neumann, T. Gaebel, F. Jelezko, P. Hemmer, and J. Wrachtrup, “Spin-flip and spin-conserving optical transitions of the nitrogen-vacancy centre in diamond,” N. J. Phys. 10, 045004 (2008).
[CrossRef]

T. Gaebel, M. Domhan, I. Popa, C. Wittmann, P. Neumann, F. Jelezko, J. R. Rabeau, N. Stavrias, A. D. Greentree, S. Prawer, J. Meijer, J. Twamley, P. R. Hemmer, and J. Wrachtrup, “Room-temperature coherent coupling of single spins in diamond,” Nat. Phys. 2, 408–413 (2006).
[CrossRef]

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 conditional quantum gate,” Phys. Rev. Lett. 93, 130501 (2004).
[CrossRef] [PubMed]

Gao, J. W.

J. W. Gao, J. H. Wu, N. Ba, C. L. Cui, and X. X. Tian, “Efficient all-optical routing using dynamically induced transparency windows and photonic band gaps,” Phys. Rev. A 81, 013804 (2010).
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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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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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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 conditional quantum gate,” Phys. Rev. Lett. 93, 130501 (2004).
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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. Meijer, J. Twamley, P. R. Hemmer, and J. Wrachtrup, “Room-temperature coherent coupling of single spins in diamond,” Nat. Phys. 2, 408–413 (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. Meijer, J. Twamley, P. R. Hemmer, and J. Wrachtrup, “Room-temperature coherent coupling of single spins in diamond,” Nat. Phys. 2, 408–413 (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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N. B. Manson, J. P. Harrison, and M. J. Sellars, “Nitrogen-vacancy center in diamond: Model of the electronic structure and associated dynamics,” Phys. Rev. B 74, 104303 (2006).
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A. Huck, S. Kumar, A. Shakoor, and U. L. Andersen, “Controlled coupling of a single nitrogen-vacancy center to a silver nanowire,” Phys. Rev. Lett. 106, 096801 (2011).
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[CrossRef]

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A. Gonzalez-Tudela, F. J. Rodríguez, L. Quiroga, and C. Tejedor, “Dissipative dynamics of a solid-state qubit coupled to surface plasmons: From non-Markov to Markov regimes,” Phys. Rev. B 82, 115334 (2010).
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J. W. Gao, J. H. Wu, N. Ba, C. L. Cui, and X. X. Tian, “Efficient all-optical routing using dynamically induced transparency windows and photonic band gaps,” Phys. Rev. A 81, 013804 (2010).
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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. Gurudev 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,” Science 316, 1312–1316 (2007).
[CrossRef]

Trifonov, A. S.

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]

Twamley, J.

T. Gaebel, M. Domhan, I. Popa, C. Wittmann, P. Neumann, F. Jelezko, J. R. Rabeau, N. Stavrias, A. D. Greentree, S. Prawer, J. Meijer, J. Twamley, P. R. Hemmer, and J. Wrachtrup, “Room-temperature coherent coupling of single spins in diamond,” Nat. Phys. 2, 408–413 (2006).
[CrossRef]

Wang, H.

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]

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]

Wen, J.

J. Wen, S. Du, H. Chen, and M. Xiao, “Electromagnetically induced Talbot effect,” Appl. Phys. Lett. 98, 081108 (2011).
[CrossRef]

Wittmann, C.

T. Gaebel, M. Domhan, I. Popa, C. Wittmann, P. Neumann, F. Jelezko, J. R. Rabeau, N. Stavrias, A. D. Greentree, S. Prawer, J. Meijer, J. Twamley, P. R. Hemmer, and J. Wrachtrup, “Room-temperature coherent coupling of single spins in diamond,” Nat. Phys. 2, 408–413 (2006).
[CrossRef]

Wrachtrup, J.

P. Tamarat, N. B. Manson, J. P. Harrison, R. L. McMurtrie, A. Nizovtsev, C. Santori, R. G. Beausoleil, P. Neumann, T. Gaebel, F. Jelezko, P. Hemmer, and J. Wrachtrup, “Spin-flip and spin-conserving optical transitions of the nitrogen-vacancy centre in diamond,” N. J. Phys. 10, 045004 (2008).
[CrossRef]

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).
[CrossRef]

T. Gaebel, M. Domhan, I. Popa, C. Wittmann, P. Neumann, F. Jelezko, J. R. Rabeau, N. Stavrias, A. D. Greentree, S. Prawer, J. Meijer, J. Twamley, P. R. Hemmer, and J. Wrachtrup, “Room-temperature coherent coupling of single spins in diamond,” Nat. Phys. 2, 408–413 (2006).
[CrossRef]

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 conditional quantum gate,” Phys. Rev. Lett. 93, 130501 (2004).
[CrossRef] [PubMed]

Wu, J. H.

J. W. Gao, J. H. Wu, N. Ba, C. L. Cui, and X. X. Tian, “Efficient all-optical routing using dynamically induced transparency windows and photonic band gaps,” Phys. Rev. A 81, 013804 (2010).
[CrossRef]

J. H. Wu, M. Artoni, and G. C. La Rocca, “All-optical light confinement in dynamic cavities in cold atoms,” Phys. Rev. Lett. 103, 133601 (2009).
[CrossRef] [PubMed]

Wu, Y.

Y. Wu, “Effective Raman theory for a three-level atom in the Λ configuration,” Phys. Rev. A 54, 1586–1592 (1996).
[CrossRef] [PubMed]

Xiao, M.

J. Wen, S. Du, H. Chen, and M. Xiao, “Electromagnetically induced Talbot effect,” Appl. Phys. Lett. 98, 081108 (2011).
[CrossRef]

A. W. Brown and M. Xiao, “All-optical switching and routing based on an electromagnetically induced absorption grating,” Opt. Lett. 30, 699–701 (2005).
[CrossRef] [PubMed]

H. Y. Ling, Y. Q. Li, and M. Xiao, “Electromagnetically induced grating: Homogeneously broadened medium,” Phys. Rev. A 57, 1338–1344 (1998) and references therein.
[CrossRef]

Yu, C. L.

A. V. Akimov, A. Mukherjee, C. L. Yu, D. E. Chang, A. S. Zibrov, P. R. Hemmer, H. Park, and M. D. Lukin, “Generation of single optical plasmons in metallic nanowires coupled to quantum dots,” Nature (London) 450, 402–406 (2007).
[CrossRef]

Zibrov, A. S.

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]

M. V. Gurudev 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,” Science 316, 1312–1316 (2007).
[CrossRef]

A. V. Akimov, A. Mukherjee, C. L. Yu, D. E. Chang, A. S. Zibrov, P. R. Hemmer, H. Park, and M. D. Lukin, “Generation of single optical plasmons in metallic nanowires coupled to quantum dots,” Nature (London) 450, 402–406 (2007).
[CrossRef]

M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Stationary pulses of light in an atomic medium,” Nature (London) 426, 638–641 (2003).
[CrossRef]

Appl. Phys. Lett.

J. Wen, S. Du, H. Chen, and M. Xiao, “Electromagnetically induced Talbot effect,” Appl. Phys. Lett. 98, 081108 (2011).
[CrossRef]

N. J. Phys.

P. Tamarat, N. B. Manson, J. P. Harrison, R. L. McMurtrie, A. Nizovtsev, C. Santori, R. G. Beausoleil, P. Neumann, T. Gaebel, F. Jelezko, P. Hemmer, and J. Wrachtrup, “Spin-flip and spin-conserving optical transitions of the nitrogen-vacancy centre in diamond,” N. J. Phys. 10, 045004 (2008).
[CrossRef]

Nano Lett.

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]

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]

C. Ropers, C. C. Neacsu, T. Elsaesser, M. Albrecht, M. B. Raschke, and C. Lienau, “Grating-coupling of surface plasmons onto metallic tips: A nanoconfined light source,” Nano Lett. 7, 2784–2788 (2007).
[CrossRef] [PubMed]

Nat. Phys.

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

R. J. Epstein, F. M. Mendoza, Y. K. Kato, and D. D. Awschalom, “Anisotropic interactions of a single spin and dark-spin spectroscopy in diamond,” Nat. Phys. 1, 94–98 (2005).
[CrossRef]

T. Gaebel, M. Domhan, I. Popa, C. Wittmann, P. Neumann, F. Jelezko, J. R. Rabeau, N. Stavrias, A. D. Greentree, S. Prawer, J. Meijer, J. Twamley, P. R. Hemmer, and J. Wrachtrup, “Room-temperature coherent coupling of single spins in diamond,” Nat. Phys. 2, 408–413 (2006).
[CrossRef]

Nature (London)

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]

M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Stationary pulses of light in an atomic medium,” Nature (London) 426, 638–641 (2003).
[CrossRef]

A. V. Akimov, A. Mukherjee, C. L. Yu, D. E. Chang, A. S. Zibrov, P. R. Hemmer, H. Park, and M. D. Lukin, “Generation of single optical plasmons in metallic nanowires coupled to quantum dots,” Nature (London) 450, 402–406 (2007).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. A

Y. Wu, “Effective Raman theory for a three-level atom in the Λ configuration,” Phys. Rev. A 54, 1586–1592 (1996).
[CrossRef] [PubMed]

J. W. Gao, J. H. Wu, N. Ba, C. L. Cui, and X. X. Tian, “Efficient all-optical routing using dynamically induced transparency windows and photonic band gaps,” Phys. Rev. A 81, 013804 (2010).
[CrossRef]

M. Mitsunaga and N. Imoto, “Observation of an electromagnetically induced grating in cold sodium atoms,” Phys. Rev. A 59, 4773–4776 (1999).
[CrossRef]

J. T. Shen and S. Fan, “Theory of single-photon transport in a single-mode waveguide. I. Coupling to a cavity containing a two-level atom,” Phys. Rev. A 79, 023837 (2009).
[CrossRef]

H. Y. Ling, Y. Q. Li, and M. Xiao, “Electromagnetically induced grating: Homogeneously broadened medium,” Phys. Rev. A 57, 1338–1344 (1998) and references therein.
[CrossRef]

Phys. Rev. B

D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Strong coupling of single emitters to surface plasmons,” Phys. Rev. B 76, 035420 (2007).
[CrossRef]

N. B. Manson, J. P. Harrison, and M. J. Sellars, “Nitrogen-vacancy center in diamond: Model of the electronic structure and associated dynamics,” Phys. Rev. B 74, 104303 (2006).
[CrossRef]

A. Gonzalez-Tudela, F. J. Rodríguez, L. Quiroga, and C. Tejedor, “Dissipative dynamics of a solid-state qubit coupled to surface plasmons: From non-Markov to Markov regimes,” Phys. Rev. B 82, 115334 (2010).
[CrossRef]

D. Dzsotjan, A. S. Sørensen, and M. Fleischhauer, “Quantum emitters coupled to surface plasmons of a nanowire: A Greens function approach,” Phys. Rev. B 82, 075427 (2010).
[CrossRef]

Phys. Rev. Lett.

D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Quantum optics with surface plasmons,” Phys. Rev. Lett. 97, 053002 (2006).
[CrossRef] [PubMed]

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).
[CrossRef]

A. Huck, S. Kumar, A. Shakoor, and U. L. Andersen, “Controlled coupling of a single nitrogen-vacancy center to a silver nanowire,” Phys. Rev. Lett. 106, 096801 (2011).
[CrossRef] [PubMed]

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 conditional quantum gate,” Phys. Rev. Lett. 93, 130501 (2004).
[CrossRef] [PubMed]

A. André and M. D. Lukin, “Manipulating light pulses via dynamically controlled photonic band gap,” Phys. Rev. Lett. 89, 143602 (2002).
[CrossRef] [PubMed]

J. H. Wu, M. Artoni, and G. C. La Rocca, “All-optical light confinement in dynamic cavities in cold atoms,” Phys. Rev. Lett. 103, 133601 (2009).
[CrossRef] [PubMed]

Science

M. V. Gurudev 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,” Science 316, 1312–1316 (2007).
[CrossRef]

R. Hanson, V. V. Dobrovitski, A. E. Feiguin, O. Gywat, and D. D. Awschalom, “Coherent dynamics of a single spin interacting with an adjustable spin bath,” Science 320,, 352–355 (2008).
[CrossRef] [PubMed]

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Figures (4)

Fig. 1
Fig. 1

Schematic diagram of a metal nanowire coupled to a NV center in diamond nanocrystal. A single surface plasmon injected from the left is coherently scattered by the NV center. The bubble shows energy configuration of the NV center. gsp is the coupling strength between the NV center and metal nanowire, and Ωc (z) is the position-dependent coupling strength between the NV center and the standing-wave control field. Dgs = 2.88 GHz is the zero-field splitting between the ground state sublevels ms = 0 and ms = ±1 of the NV center (ms = ±1 are degenerate at zero magnetic field).

Fig. 2
Fig. 2

Reflection R, transmissions T and loss versus Δω. Panel (a) is produced without the control field [Ωc (z)/Γpl = 0]. Panel (b) is produced with the control field [Ωc (z)/Γpl = 0.5]. Note that, the intensity of the control field which is used for producing panels (a) and (b), corresponds to the peak intensity of a standing-wave field. The other system parameters are chosen as Γepl = 0.1 and Γspl = 0.001, respectively. The inset of (b) shows an enlarged view of the spectral line-shapes near resonant point Δω = 0.

Fig. 3
Fig. 3

Reflection R (solid line) and transmissions T (dashed line) versus the standing-wave control field Ωcpl for Δω = 0. The other system parameters are chosen as Γepl = 0.1 and Γspl = 0.001, respectively.

Fig. 4
Fig. 4

Reflection R (panel (a)) and transmissions T (panel (b)) a function of position z. The standing-wave control field: Ω c ( z ) = Ω c 0 sin ( π z Λ ), where Λ is the spatial period of the standing wave along the z direction. The system parameters are chosen as Ωc0pl = 0.5, Γepl = 0.1, Γspl = 0.001, and Λ = 1 μm, respectively.

Equations (24)

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^ = h ¯ ω e σ ^ ee + h ¯ ω s σ ^ ss h ¯ [ Ω c ( z ) e i ω c t σ ^ es + Ω c * ( z ) e i ω c t σ ^ se ] + + h ¯ ω a ^ R , ω a ^ R , ω d ω + + h ¯ ω a ^ L , ω a ^ L , ω d ω h ¯ g s p + ( a ^ R , ω σ ^ e g + a ^ R , ω σ ^ g e ) d ω h ¯ g s p + ( a ^ L , ω σ ^ e g + a ^ L , ω σ ^ g e ) d ω ,
^ int = e i ^ 0 t / h ¯ ^ res e i ^ 0 t / h ¯ ,
^ 0 / h ¯ = ω s σ ^ s s + ω e σ ^ e e + + ω e ( a ^ R , ω a ^ R , ω + a ^ L , ω a ^ L , ω ) d ω ,
^ res / h ¯ = + Δ ω ( a ^ R , ω a ^ R , ω + a ^ L , ω a ^ L , ω ) d ω [ Ω c ( z ) e i ω c t σ ^ e s + Ω c * ( z ) e i ω c t σ ^ s e ] g s p + [ ( a ^ R , ω + a ^ L , ω ) σ ^ e g + ( a ^ R , ω + a ^ L , ω ) σ ^ g e ] d ω ,
^ int / h ¯ = + Δ ω ( a ^ R , ω a ^ R , ω + a ^ L , ω a ^ L , ω ) d ω [ Ω c ( z ) σ ^ e s + Ω c * ( z ) σ ^ s e ] g s p + [ ( a ^ R , ω + a ^ L , ω ) σ ^ e g + ( a ^ R , ω + a ^ L , ω ) σ ^ g e ] d ω .
| Ψ IHS ( t ) = + [ α R , ω ( t ) a ^ R , ω | g , vac + α L , ω ( t ) a ^ L , ω | g , vac ] d ω + α e ( t ) | e , vac + α s ( t ) | s , vac .
α ˙ R , ω ( t ) = i Δ ω α R , ω ( t ) + i g s p α e ( t ) ,
α ˙ L , ω ( t ) = i Δ ω α L , ω ( t ) + i g s p α e ( t ) ,
α ˙ e ( t ) = Γ e 2 α e ( t ) + i Ω c ( z ) α s ( t ) + i g s p + [ α R , ω ( t ) + α L , ω ( t ) ] d ω ,
α ˙ s ( t ) = Γ s 2 α s ( t ) + i Ω c * ( z ) α e ( t ) ,
α R , ω ( t ) = α R , ω ( t 0 ) e i Δ ω ( t t 0 ) + i g s p t 0 t α e ( t ) e i Δ ω ( t t ) d t ,
α L , ω ( t ) = α L , ω ( t 0 ) e i Δ ω ( t t 0 ) + i g s p t 0 t α e ( t ) e i Δ ω ( t t ) d t ,
α ˙ e ( t ) = ( Γ e 2 + Γ pl ) α e ( t ) + i Ω c ( z ) α s ( t ) + i 2 π g s p [ α R , in ( t ) + α L , in ( t ) ] ,
α R , ω ( t ) = α R , ω ( t 1 ) e i Δ ω ( t t 1 ) i g s p t t 1 α e ( t ) e i Δ ω ( t t ) d t ,
α L , ω ( t ) = α L , ω ( t 1 ) e i Δ ω ( t t 1 ) i g s p t t 1 α e ( t ) e i Δ ω ( t t ) d t ,
α ˙ e ( t ) = ( Γ e 2 Γ p l ) α e ( t ) + i Ω c ( z ) α s ( t ) + i 2 π g s p [ α R , out ( t ) + α L , out ( t ) ] ,
α R , out α R , in = i 2 π g s p α e ,
α L , out α L , in = i 2 π g s p α e .
α e ( z ) = i 2 π g s p ( α R , in + α L , in ) i Δ ω ( Γ e 2 + Γ p l ) + | Ω c ( z ) | 2 i Δ ω Γ s 2 .
α R , out ( z ) = i Δ ω Γ e 2 + | Ω c ( z ) | 2 i Δ ω Γ s 2 i Δ ω ( Γ e 2 + Γ p l ) + | Ω c ( z ) | 2 i Δ ω Γ s 2 α R , in + Γ p l i Δ ω ( Γ e 2 + Γ p l ) + | Ω c ( z ) | 2 i Δ ω Γ s 2 α L , in ,
α L , out ( z ) = Γ p l i Δ ω ( Γ e 2 + Γ p l ) + | Ω c ( z ) | 2 i Δ ω Γ s 2 α R , in + i Δ ω Γ e 2 + | Ω c ( z ) | 2 i Δ ω Γ s 2 i Δ ω ( Γ e 2 + Γ p l ) + | Ω c ( z ) | 2 i Δ ω Γ s 2 α L , in .
r = α L , out α R , in = Γ p l i Δ ω ( Γ e 2 + Γ p l ) + S c ( z ) ,
t = α R , out α R , in = 1 + r = i Δ ω Γ e 2 + S c ( z ) i Δ ω ( Γ e 2 + Γ p l ) + S c ( z ) ,
R = | r | 2 and T = | t | 2 .

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