## Concentration of entangled nitrogen-vacancy centers in decoherence free subspace |

Optics Express, Vol. 22, Issue 2, pp. 1551-1559 (2014)

http://dx.doi.org/10.1364/OE.22.001551

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### Abstract

Exploiting the input-output process of low-Q cavities confining nitrogen-vacancy centers, we present an efficient entanglement concentration protocol on electron spin state in decoherence free subspace. Less entangled state can be concentrated to maximally entangled state with the assistance of single photon detection. With its robustness and scalability, the present protocol is immune to dephasing and can be further applied to quantum repeaters and distributed quantum computation.

© 2014 Optical Society of America

## 1. Introduction

1. M. Saffman, T. G. Walker, and K. Mølmer, “Quantum information with Rydberg atoms,” Rev. Mod. Phys. **82**, 2313–2363 (2010). [CrossRef]

*et al.*[2

2. 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]

*et al.*[3

3. L. Robledo, L. Childress, H. Bernien, B. Hensen, P. F. A. Alkemade, and R. Hanson, “High-fidelity projective read-out of a solid-state spin quantum register,” Nature **477**, 574–578(2011). [CrossRef] [PubMed]

*et al.*[4

4. D. M. Toyli, C. D. Weis, G. D. Fuchs, T. Schenkel, and D. D. Awschalom, “Chip-scale nanofabrication of single spins and spin arrays in diamond,” Nano Lett. **10**(8), 3168–3172(2010). [CrossRef] [PubMed]

*et al.*realized the optical diamond nanostructures containing a single-color center. Also Fuchs

*et al.*[5

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

*et al.*[6

6. P. C. Maurer, G. Kucsko, C. Latta, L. Jiang, N. Y. Yao, S. D. Bennett, F. Pastawski, D. Hunger, N. Chisholm, M. Markham, D. J. Twitchen, J. I. Cirac, and M. D. Lukin, “Room-temperature quantum bit memory exceeding one second,” Science **336**, 1283–1286(2012). [CrossRef] [PubMed]

*et al.*[7

7. B. J. M. Hausmann, T. M. Babinec, J. T. Choy, J. S. Hodges, S. Hong, I. Bulu, A. Yacoby, M. D. Lukin, and M. Lonc̆ar, “Single-color centers implanted in diamond nanostructures,” New J. Phys. **13**, 045004(2011). [CrossRef]

## 2. Single DFS qubit assisted entanglement concentration

### 2.1. The model of single photon input-output relation

*a*and

*a*

^{†}for the annihilation and creation operators of the cavity mode with the frequency

*ω*, the Hamiltonian of the composite system can be described as where

_{c}*g*denote the coupling strength between the

_{j}*j*th N-V centers and microcavity. The operators

*σ*and

_{z}*σ*

_{+(−)}represent the inversion and raising(lowering) operators of the N-V center with frequency

*ω*

_{0}.

*γ*/2 denotes the decay rate of the

_{j}*j*th N-V center.

*κ*and

*κ*are the cavity decay rate and the cavity leaky rate, respectively.

_{s}*ω*,

*ω*and

_{c}*ω*

_{0}denote the frequencies of the input photon, cavity mode and the atomic level transition, respectively. For a simple solution, we assume that the coupling strength between the two N-V centers and the resonator are identical, as

*g*

_{1}=

*g*

_{2}=

*g*and

*γ*

_{1}=

*γ*

_{2}=

*γ*. The above Heisenberg equations of motion can be solved and the reflection coefficient of this system can be obtained: here we omit the cavity loss as

*κ*= 0. On the resonant condition of the system with

_{s}*ω*=

_{c}*ω*=

_{p}*ω*

_{0}, the reflection coefficient

*r*for the uncoupled cavity system can be written as So the definition of the reflection coefficient on uncoupled condition shows unity reflectance |

*r*

_{0}(

*ω*)| = 1 as

*g*= 0.

*L*〉. The output photon pulse acquires a relative phase shift

*ϕ*determined by the input-output relation: |Φ

*〉 =*

_{out}*r*(

*ω*)|

*L*〉 =

*e*|

^{iϕ}*L*〉. On the other hand, if the input photon is prepared in the right circularly polarized state |

*R*〉, the output photon will evolve as |Φ

*〉 =*

_{out}*r*(

*ω*)|

*R*〉 =

*e*|

^{i}^{ψ}*R*〉. In all, the dynamics of the photon input-output process on the logic qubits can be described as: The relative phase shifts are defined by the reflection coefficient

*r*(

*ω*) during the input-output process which can be set as:

*ϕ*

_{0}=

*π*/2 and

*ϕ*

_{2}= −

*π*/2 in our further application. Recently, in Ref. [25

25. Q. Chen and M. Feng, “Quantum-information processing in decoherence-free subspace with low-Q cavities,” Phys. Rev. A **82**, 052329 (2010). [CrossRef]

*et al.*combines the idea of input-output process and the DFS encoding theory, and presents a high quality QIP protocol. And in Ref. [26

26. A. P. Liu, L. -Y. Cheng, L. Chen, S. -L. Su, H. -F. Wang, and S. Zhang, “Quantum information processing in decoherence-free subspace with nitrogen-vacancy centers coupled to a whispering-gallery mode microresonator,” Opt. Comm. **313**, 180–185 (2014). [CrossRef]

*et al.*a hybrid controlled phase flip gate based on the inputCoutput process of the DFS qubit and microcavity system.

### 2.2. Single DFS qubit assisted entanglement concentration

_{1}|1〉

_{2}and |1̃〉 = |1〉

_{1}|0〉

_{2}. The generated entangled state in DFS requires local operations

*σ*on one atom to flip one spin direction of the logical qubit, and the entangled state can be described as

_{x}*α*|0̃〉

_{1}|0̃〉

_{2}+

*β*|1̃〉

_{1}|1̃〉

_{2}), here the probability amplitude of the state obeys |

*α*|

^{2}+ |

*β*|

^{2}= 1.

*α*|0̃〉

*|0̃〉*

_{A}*+*

_{B}*β*|1̃〉

*|1̃〉*

_{A}*. One party, say Alice, prepares a single logic qubit in the state*

_{B}*α*|0̃〉

*+*

_{a}*β*|1̃〉

*. At first, a local operation*

_{a}*σ*on Alice’s side is needed to flip the single qubit state in the DFS. Exploiting the setup shown in Fig. 2, the evolution of the system can be described as:

_{x}*c*

_{1}(

*α*

^{2}|0̃〉|0̃〉|0̃〉 +

*β*

^{2}|1̃〉|1̃〉|1̃〉) and

*αβ*|

^{2}that the state has been concentrated to the maximally entanglement.

*ψ*

_{2}〉 =

*α*

^{2}|1̃〉 +

*β*

^{2}|0̃〉. By iterating the process exploiting the setup shown in Fig. 2, the composite system state |

*ϕ*

_{2}〉|

*ψ*

_{2}〉 can be concentrated to the maximally entangled state with the yield 2

*Z*|

*αβ*|

^{4}. Here

*Z*represents the normalized constant. Moreover, the system without been concentrated are remaining in the state

*α*in ideal conditions. By comparing the yield with different iteration times, the total yield of the protocol is efficiently improved. As the iteration times increase to three, the yield values increase from 0.4998 to 0.812 if the probability amplitude

*α*= 0.7.

### 2.3. Experiment feasibilities

*et al.*[27

27. J. Zhu, S. K. Ozdemir, Y. F. Xiao, L. Li, L. He, D. R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nature Photonics **4**, 46–49 (2010) [CrossRef]

*et al.*[28

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

^{8}. Secondly, Bernien

*et al.*[29

29. H. Bernien, L. Childress, L. Robledo, M. Markham, D. Twitchen, and R. Hanson, “Two-photon quantum interference from separate nitrogen vacancy centers in diamond,” Phys. Rev. Lett. **108**, 043604 (2012). [CrossRef] [PubMed]

9. 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,” Nature Physics **2**, 408–413(2006). [CrossRef]

30. D. K. Armani, T. J. Kippenberg, S. M. Spillance, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature **421**, 925–928 (2003). [CrossRef] [PubMed]

31. J. Zhu, S. K. Ozdemir, L. He, and L. Yang, “Controlled manipulation of mode splitting in an optical microcavity by two Rayleigh scatterers,” Optics Express **18**, 23535–23543 (2010). [CrossRef] [PubMed]

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

*Q*> 25, 000 and the coupling strength between a single microdisk photon and the N-V center zero photon line (ZPL) is

*g*/2

*π*= 0.3

*GHz*. The total spontaneous emission rates of the N-V center is

*γ*/2

*π*= 0.013

*GHz*. As illustrated in Ref. [33

33. A. Faraon, C. Santori, Z. Huang, V. M. Acosta, and R. G. Beausoleil, “Coupling of nitrogen-vacancy centers to photonic crystal cavities in monocrystalline diamond,” Phys. Rev. Lett. **109**, 033604 (2012). [CrossRef] [PubMed]

## 3. Discussion and summary

*g/κ*= 0.8 in the dephasing error mode. In realistic experiment, suppose the coefficient

^{−4}. We can simply estimate the success probability of the protocol to be

*p*= 2 × (1 – 2%) × (1 – 10

^{−4}) × 95% × 4/9 = 82.7%. On condition that there are dephasing process, the success probability reduces to (1 +

*cos*

^{2}

*δθ*)

*p*/2 where

*δθ*represents the relative phase shift induced by the noise on the orthogonal states. Moreover, we can improve the success probabilities of our protocol by repeating our operations.

*T*

_{1}and dephasing time

*T*

_{2}. In Ref. [34

34. A. Jarmola, V. M. Acosta, K. Jensen, S. Chemerisov, and D. Budke, “Temperature- and magnetic-field-dependent longitudinal spin relaxation in nitrogen-vacancy ensembles in diamond,” Phys. Rev. Lett. **108**, 197601 (2012). [CrossRef] [PubMed]

*T*

_{1}approaches to microsecond at room temperature, and the longest

*T*

_{1}observed was on the order of minutes at 10 K for the sample. Also in Ref. [35

35. P. Neumann, N. Mizuochi, F. Rempp, P. Hemmer, H. Watanabe, S. Yamasaki, V. Jacques, T. Gaebel, F. Jelezko, and J. Wrachtrup, “Multipartite entanglement among single spins in diamond,” Science **320**, 1326–1329 (2008). [CrossRef] [PubMed]

*T*

_{1}of N-V centers in diamond scales from microseconds to seconds at low temperature. Moreover, the dephasing time

*T*

_{2}of N-V center is about 2ms in a isotopically pure diamond [36

36. 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,” Nature Material **8**, 383–387 (2009). [CrossRef]

## Acknowledgments

## References and links

1. | M. Saffman, T. G. Walker, and K. Mølmer, “Quantum information with Rydberg atoms,” Rev. Mod. Phys. |

2. | 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 |

3. | L. Robledo, L. Childress, H. Bernien, B. Hensen, P. F. A. Alkemade, and R. Hanson, “High-fidelity projective read-out of a solid-state spin quantum register,” Nature |

4. | D. M. Toyli, C. D. Weis, G. D. Fuchs, T. Schenkel, and D. D. Awschalom, “Chip-scale nanofabrication of single spins and spin arrays in diamond,” Nano Lett. |

5. | G. D. Fuchs, G. Burkard, P. V. Klimov, and D. D. Awschalom, “A quantum memory intrinsic to single nitrogenC-vacancy centres in diamond,” Nature Physics |

6. | P. C. Maurer, G. Kucsko, C. Latta, L. Jiang, N. Y. Yao, S. D. Bennett, F. Pastawski, D. Hunger, N. Chisholm, M. Markham, D. J. Twitchen, J. I. Cirac, and M. D. Lukin, “Room-temperature quantum bit memory exceeding one second,” Science |

7. | B. J. M. Hausmann, T. M. Babinec, J. T. Choy, J. S. Hodges, S. Hong, I. Bulu, A. Yacoby, M. D. Lukin, and M. Lonc̆ar, “Single-color centers implanted in diamond nanostructures,” New J. Phys. |

8. | J. H. An, M. Feng, and C. H. Oh, “Quantum-information processing with a single photon by an input-output process with respect to low-Q cavities,” Phys. Rev. A |

9. | 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,” Nature Physics |

10. | W. L. Yang, Z. Y. Xu, M. Feng, and J. F. Du, “Entanglement of separate nitrogen-vacancy centers coupled to a whispering-gallery mode cavity,” New J. Phys. |

11. | Q. Chen, W. L. Yang, M. Fang, and J. F. Du, “Entangling separate nitrogen-vacancy centers in a scalable fashion via coupling to microtoroidal resonators,” Phys. Rev. A |

12. | B. Dayan, A. S. Parkins, T. Aoki, E. P. Ostby, K. J. Vahala, and H. J. Kimble, “A photon turnstile dynamically regulated by one atom,” Science |

13. | E. Togan, Y. Chu, A. S. Trifonov, L. Jiang, J. Maze, L. Childress, M. Dutt, A. S. Sorensen, P. R. Hemmer, A. S. Zibrov, and M. D. Lukin, “Quantum entanglement between an optical photon and a solid-state spin qubit,” Nature |

14. | C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, “Concentrating partial entanglement by local operations,” Phys. Rev. A |

15. | Z. Zhao, T. Yang, Y. A. Chen, A. N. Zhang, and J. W. Pan, “Experimental realization of entanglement concentration and a quantum repeater,” Phys. Rev. Lett. |

16. | T. Yamamoto, M. Koashi, S. K. Ozdemir, and N. Imoto, “Experimental extraction of an entangled photon pair from two identically decohered pairs,” Nature |

17. | Y. B. Sheng, F. G. Deng, and H. Y. Zhou, “Nonlocal entanglement concentration scheme for partially entangled multipartite systems with nonlinear optics,” Phys. Rev. A |

18. | C. Wang, Y. Zhang, and G. S. Jin, “Entanglement purification and concentration of electron-spin entangled states using quantum-dot spins in optical microcavities,” Phys. Rev. A |

19. | Y. B. Sheng, F. G. Deng, and H. Y. Zhou, “Single-photon entanglement concentration for long-distance quantum communication.,” Quantum Inform. Comput. |

20. | D. A. Lidar, I. L. Chuang, and K. B. Whaley, “Decoherence-free subspaces for quantum computation,” Phys. Rev. Lett. |

21. | M. Mohseni, J. S. Lundeen, K. J. Resch, and A. M. Steinberg, “Experimental application of decoherence-free subspaces in an optical quantum-computing algorithm,” Phys. Rev. Lett. |

22. | L. M. Duan and G. C. Guo, “Preserving coherence in quantum computation by pairing quantum bits,” Phys. Rev. Lett. |

23. | D. Kielpinski, C. Monroe, and D.J. Wineland, “Architecture for a large-scale ion-trap quantum computer,” Nature |

24. | D. F. Walls and G. J. Milburn, “ |

25. | Q. Chen and M. Feng, “Quantum-information processing in decoherence-free subspace with low-Q cavities,” Phys. Rev. A |

26. | A. P. Liu, L. -Y. Cheng, L. Chen, S. -L. Su, H. -F. Wang, and S. Zhang, “Quantum information processing in decoherence-free subspace with nitrogen-vacancy centers coupled to a whispering-gallery mode microresonator,” Opt. Comm. |

27. | J. Zhu, S. K. Ozdemir, Y. F. Xiao, L. Li, L. He, D. R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nature Photonics |

28. | Y.-S. Park, A. K. Cook, and H. Wang, “Cavity QED with diamond nanocrystals and silica microspheres,” Nano Letters |

29. | H. Bernien, L. Childress, L. Robledo, M. Markham, D. Twitchen, and R. Hanson, “Two-photon quantum interference from separate nitrogen vacancy centers in diamond,” Phys. Rev. Lett. |

30. | D. K. Armani, T. J. Kippenberg, S. M. Spillance, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature |

31. | J. Zhu, S. K. Ozdemir, L. He, and L. Yang, “Controlled manipulation of mode splitting in an optical microcavity by two Rayleigh scatterers,” Optics Express |

32. | P. E. Barclay, F. M. C. Fu, C. Santori, and R. G. Beausoleil, “Chip-based microcavities coupled to nitrogen-vacancy centers in single crystal diamond,” App. Phys. Lett. |

33. | A. Faraon, C. Santori, Z. Huang, V. M. Acosta, and R. G. Beausoleil, “Coupling of nitrogen-vacancy centers to photonic crystal cavities in monocrystalline diamond,” Phys. Rev. Lett. |

34. | A. Jarmola, V. M. Acosta, K. Jensen, S. Chemerisov, and D. Budke, “Temperature- and magnetic-field-dependent longitudinal spin relaxation in nitrogen-vacancy ensembles in diamond,” Phys. Rev. Lett. |

35. | P. Neumann, N. Mizuochi, F. Rempp, P. Hemmer, H. Watanabe, S. Yamasaki, V. Jacques, T. Gaebel, F. Jelezko, and J. Wrachtrup, “Multipartite entanglement among single spins in diamond,” Science |

36. | 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,” Nature Material |

**OCIS Codes**

(270.0270) Quantum optics : Quantum optics

(270.5568) Quantum optics : Quantum cryptography

(270.5585) Quantum optics : Quantum information and processing

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: November 18, 2013

Revised Manuscript: December 22, 2013

Manuscript Accepted: December 26, 2013

Published: January 15, 2014

**Citation**

Chuan Wang, Tie-Jun Wang, Yong Zhang, Rong-zhen Jiao, and Guang-sheng Jin, "Concentration of entangled nitrogen-vacancy centers in decoherence free subspace," Opt. Express **22**, 1551-1559 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-2-1551

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### References

- M. Saffman, T. G. Walker, K. Mølmer, “Quantum information with Rydberg atoms,” Rev. Mod. Phys. 82, 2313–2363 (2010). [CrossRef]
- M. V. Gurudev Dutt, L. Childress, L. Jiang, E. Togan, J. Maze, F. Jelezko, A. S. Zibrov, P. R. Hemmer, M. D. Lukin, “Quantum register based on individual electronic and nuclear spin qubits in diamond,” Science 316, 1312–1316(2007). [CrossRef]
- L. Robledo, L. Childress, H. Bernien, B. Hensen, P. F. A. Alkemade, R. Hanson, “High-fidelity projective read-out of a solid-state spin quantum register,” Nature 477, 574–578(2011). [CrossRef] [PubMed]
- D. M. Toyli, C. D. Weis, G. D. Fuchs, T. Schenkel, D. D. Awschalom, “Chip-scale nanofabrication of single spins and spin arrays in diamond,” Nano Lett. 10(8), 3168–3172(2010). [CrossRef] [PubMed]
- G. D. Fuchs, G. Burkard, P. V. Klimov, D. D. Awschalom, “A quantum memory intrinsic to single nitrogenC-vacancy centres in diamond,” Nature Physics 7, 789–793(2011). [CrossRef]
- P. C. Maurer, G. Kucsko, C. Latta, L. Jiang, N. Y. Yao, S. D. Bennett, F. Pastawski, D. Hunger, N. Chisholm, M. Markham, D. J. Twitchen, J. I. Cirac, M. D. Lukin, “Room-temperature quantum bit memory exceeding one second,” Science 336, 1283–1286(2012). [CrossRef] [PubMed]
- B. J. M. Hausmann, T. M. Babinec, J. T. Choy, J. S. Hodges, S. Hong, I. Bulu, A. Yacoby, M. D. Lukin, M. Lonc̆ar, “Single-color centers implanted in diamond nanostructures,” New J. Phys. 13, 045004(2011). [CrossRef]
- J. H. An, M. Feng, C. H. Oh, “Quantum-information processing with a single photon by an input-output process with respect to low-Q cavities,” Phys. Rev. A 79, 032303 (2009). [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, J. Wrachtrup, “Room-temperature coherent coupling of single spins in diamond,” Nature Physics 2, 408–413(2006). [CrossRef]
- W. L. Yang, Z. Y. Xu, M. Feng, J. F. Du, “Entanglement of separate nitrogen-vacancy centers coupled to a whispering-gallery mode cavity,” New J. Phys. 12, 113039 (2010). [CrossRef]
- Q. Chen, W. L. Yang, M. Fang, J. F. Du, “Entangling separate nitrogen-vacancy centers in a scalable fashion via coupling to microtoroidal resonators,” Phys. Rev. A 83, 054305 (2011). [CrossRef]
- B. Dayan, A. S. Parkins, T. Aoki, E. P. Ostby, K. J. Vahala, H. J. Kimble, “A photon turnstile dynamically regulated by one atom,” Science 319, 1062–1065 (2008). [CrossRef] [PubMed]
- E. Togan, Y. Chu, A. S. Trifonov, L. Jiang, J. Maze, L. Childress, M. Dutt, A. S. Sorensen, P. R. Hemmer, A. S. Zibrov, M. D. Lukin, “Quantum entanglement between an optical photon and a solid-state spin qubit,” Nature 466, 730–734 (2010). [CrossRef] [PubMed]
- C. H. Bennett, H. J. Bernstein, S. Popescu, B. Schumacher, “Concentrating partial entanglement by local operations,” Phys. Rev. A 53, 2046–2052(1996). [CrossRef] [PubMed]
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- Y. B. Sheng, F. G. Deng, H. Y. Zhou, “Nonlocal entanglement concentration scheme for partially entangled multipartite systems with nonlinear optics,” Phys. Rev. A 77, 062325 (2008). [CrossRef]
- C. Wang, Y. Zhang, G. S. Jin, “Entanglement purification and concentration of electron-spin entangled states using quantum-dot spins in optical microcavities,” Phys. Rev. A 84, 032307 (2011). [CrossRef]
- Y. B. Sheng, F. G. Deng, H. Y. Zhou, “Single-photon entanglement concentration for long-distance quantum communication.,” Quantum Inform. Comput. 10, 272–281 (2010).
- D. A. Lidar, I. L. Chuang, K. B. Whaley, “Decoherence-free subspaces for quantum computation,” Phys. Rev. Lett. 81, 2594–2597 (1998). [CrossRef]
- M. Mohseni, J. S. Lundeen, K. J. Resch, A. M. Steinberg, “Experimental application of decoherence-free subspaces in an optical quantum-computing algorithm,” Phys. Rev. Lett. 91, 187903 (2003). [CrossRef] [PubMed]
- L. M. Duan, G. C. Guo, “Preserving coherence in quantum computation by pairing quantum bits,” Phys. Rev. Lett. 79, 1953–1956 (1997). [CrossRef]
- D. Kielpinski, C. Monroe, D.J. Wineland, “Architecture for a large-scale ion-trap quantum computer,” Nature 417, 709–711 (2002). [CrossRef] [PubMed]
- D. F. Walls, G. J. Milburn, “Quantum Optics,” (Springer-Verlag, Berlin Heidelberg, 1994).
- Q. Chen, M. Feng, “Quantum-information processing in decoherence-free subspace with low-Q cavities,” Phys. Rev. A 82, 052329 (2010). [CrossRef]
- A. P. Liu, L. -Y. Cheng, L. Chen, S. -L. Su, H. -F. Wang, S. Zhang, “Quantum information processing in decoherence-free subspace with nitrogen-vacancy centers coupled to a whispering-gallery mode microresonator,” Opt. Comm. 313, 180–185 (2014). [CrossRef]
- J. Zhu, S. K. Ozdemir, Y. F. Xiao, L. Li, L. He, D. R. Chen, L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nature Photonics 4, 46–49 (2010) [CrossRef]
- Y.-S. Park, A. K. Cook, H. Wang, “Cavity QED with diamond nanocrystals and silica microspheres,” Nano Letters 6, 2075–2079(2006). [CrossRef] [PubMed]
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