## Entanglement purification based on hybrid entangled state using quantum-dot and microcavity coupled system |

Optics Express, Vol. 19, Issue 25, pp. 25685-25695 (2011)

http://dx.doi.org/10.1364/OE.19.025685

Acrobat PDF (769 KB)

### Abstract

We theoretically investigate an entanglement purification protocol with photon and electron hybrid entangled state resorting to quantum-dot spin and microcavity coupled system. The present system is used to construct the parity check gate which allows a quantum nonde-molition measurement on the spin parity. The cavity-spin coupled system provides a novel experimental platform of quantum information processing with photon and solid qubit.

© 2011 OSA

## 1. Introduction

24. C. Bonato, F. Haupt, S. S. R. Oemrawsingh, J. Gudat, D. -P. Ding, M. P. van Exter, and D. Bouwmeester, “CNOT and Bell-state analysis in the weak-coupling cavity QED regime,” Phys. Rev. Lett. **104**, 160503 (2010). [CrossRef] [PubMed]

27. C. Y. Hu and J. G. Rarity, “Loss-resistant state teleportation and entanglement swapping using a quantum-dot spin in an optical microcavity,”, Phys. Rev. B **83**, 115303 (2011). [CrossRef]

*è*ves-Garnier

*et al.*presented that the neutral QD-cavity system behaves like a beam splitter in the limit of weak incoming field [28

28. A. Auffèves-Garnier, C. Simon, J. M. Gérard, and J. P. Poizat, “Giant optical nonlinearity induced by a single two-level system interacting with a cavity in the Purcell regime,”, Phys. Rev. A **75**, 053823 (2007). [CrossRef]

29. P. van Loock, T. D. Ladd, K. Sanaka, F. Yamaguchi, K. Nemoto, W. J. Munro, and Y. Yamamoto, “Hybrid quantum repeater using bright coherent light,” Phys. Rev. Lett. **96**, 240501 (2006). [CrossRef] [PubMed]

*et al.*proposed a realistic protocol to generate entanglement between quantum memories at neighboring nodes in hybrid quantum repeaters [30

30. K. Azuma, N. Sota, R. Namiki, S. K. Özdemir, T. Yamamoto, M. Koashi, and N. Imoto, “Optimal entanglement generation for efficient hybrid quantum repeaters,” Phys. Rev. A **80**, 060303(R) (2009). [CrossRef]

31. E. Waks and C. Monroe, “Protocol for hybrid entanglement between a trapped atom and a quantum dot,” Phys. Rev. A **80**, 062330 (2009). [CrossRef]

*et al.*proposed a hybrid quantum repeater protocol for long-distance entanglement distribution [32

32. J. B. Brask, I. Rigas, E. S. Polzik, U. L. Andersen, and A. S. Sørensen, “Hybrid long-distance entanglement distribution protocol,” Phys. Rev. Lett. **105**, 160501 (2010). [CrossRef]

26. C. Y. Hu, W. J. Munro, and J. G. Rarity, “Deterministic photon entangler using a charged quantum dot inside a microcavity,” Phys. Rev. B **78**, 125318 (2008). [CrossRef]

27. C. Y. Hu and J. G. Rarity, “Loss-resistant state teleportation and entanglement swapping using a quantum-dot spin in an optical microcavity,”, Phys. Rev. B **83**, 115303 (2011). [CrossRef]

33. C. Y. Hu, A. Young, J. L. O’Brien, W. J. Munro, and J. G. Rarity, “Giant optical Faraday rotation induced by a single-electron spin in a quantum dot: Applications to entangling remote spins via a single photon,” Phys. Rev. B **78**, 085307 (2008). [CrossRef]

34. A. B. Young, R. Oulton, C. Y. Hu, A. C. T. Thijssen, C. Schneider, S. Reitzenstein, M. Kamp, S. Höfling, L. Worschech, A. Forchel, and J. G. Rarity, “Quantum-dot-induced phase shift in a pillar microcavity,” Phys. Rev. A **84**, 011803(R) (2011). [CrossRef]

## 2. Theoretical model of hybrid entanglement purification using QD and micro-cavity coupled system

*X*that consists of two electrons bound to one hole. As illustrated by Pauli’s exclusion principle, the exciton can be created by the optical excitation of the system, e.g., when a photon passes through the cavity and interacts with the electron in the coupling cavity, the left circularly polarized photon only couples with the cavity when the electron in the spin up state |↑〉 and generate the exciton

*X*

^{−}in the state |↑↓⇑〉; the right circularly polarized photon only couples with the cavity when the electron of spin is in the spin down state |↓〉 and generate the exciton in the state |↓↑⇓〉. Here |⇑〉 and |⇓〉 represent the spin direction of the heavy hole spin state.

25. C. Y. Hu, W. J. Munro, J. L. O’Brien, and J. G. Rarity, “Proposed entanglement beam splitter using a quantum-dot spin in a double-sided optical microcavity,” Phys. Rev. B **80**, 205326 (2009). [CrossRef]

*â*and

_{in}*â*′

*are the input field operators from different side of the cavity,*

_{in}*â*and

_{r}*â*are the reflected and transmitted operators, respectively.

_{t}25. C. Y. Hu, W. J. Munro, J. L. O’Brien, and J. G. Rarity, “Proposed entanglement beam splitter using a quantum-dot spin in a double-sided optical microcavity,” Phys. Rev. B **80**, 205326 (2009). [CrossRef]

*ω*=

_{c}*ω*

_{X−}=

*ω*

_{0}, by taking

*g*= 0, the the reflection and transmission coefficients

*r*and

*t*for the uncoupled cavity system can be written as For example, the spin of QD in microcavity is in the up direction with s=1/2, if a photon in the left circular polarized state is passed through the cavity, it couples with the cavity and the exciton can be generated. Thus the reflected coefficient and transmitted coefficient are described by |

*r*(

*ω*)| and |

*t*(

*ω*)|, respectively. However, the right circular polarized photon will not couple with the cavity as the right circular polarized photon only couple the cavity with the spin of QD in the down direction (s=−1/2), the reflected coefficient and transmitted coefficient are described by |

*r*

_{0}(

*ω*)| and |

*t*

_{0}(

*ω*)|, respectively. So the left circular polarized photon feels a coupled cavity, the reflection coefficient in Eq. (4) is equal to unity. The right circular polarized photon feels an uncoupled cavity with the coupling constant g=0, the transmission coefficient in Eq. (5) is equal to 1. This process has been developed in the quantum information processes, such as entanglement generation [24

24. C. Bonato, F. Haupt, S. S. R. Oemrawsingh, J. Gudat, D. -P. Ding, M. P. van Exter, and D. Bouwmeester, “CNOT and Bell-state analysis in the weak-coupling cavity QED regime,” Phys. Rev. Lett. **104**, 160503 (2010). [CrossRef] [PubMed]

26. C. Y. Hu, W. J. Munro, and J. G. Rarity, “Deterministic photon entangler using a charged quantum dot inside a microcavity,” Phys. Rev. B **78**, 125318 (2008). [CrossRef]

27. C. Y. Hu and J. G. Rarity, “Loss-resistant state teleportation and entanglement swapping using a quantum-dot spin in an optical microcavity,”, Phys. Rev. B **83**, 115303 (2011). [CrossRef]

*L*〉 and |

*R*〉 represent the state of the left and right circular polarized photons, respectively. The superscript arrow in the photon state indicates the propagation direction along the

*z*axis and the arrows represent the direction of the electrons. When a circularly polarized photon passes through the cavity and interacts with QD, the photon will either be transmitted or be reflected by the cavity according to the spin direction of the QD, e.g., a right circularly polarized photon passes through the cavity with QD in spin up state, the photon will be reflected by the cavity and the polarized state of the photon and the direction of propagation will both changed. Thus the photon polarization and electron spin may get entangled if the initial electron state is superposed. The principles of photon and electron entangler is shown in Fig. 1. The entangler works as follows: when a left circularly polarized photon |

*L*〉 passes along the −

*z*direction through the cavity with QD spin prepared in the spin up and down superposed state

*z*direction. If the spin of the two electrons are not in the same direction, the photon will leave the cavity in the upper mode and trigger the detector which indicates that the parity of the particles is odd. Otherwise, if the two electrons are prepared in the state |↑〉|↑〉 or |↓〉|↓〉, the output photon will trigger the lower mode detector which indicates that the parity of the two particles is even.

*L*〉 and |

*R*〉 represent the left and right circularly polarized state. The generic error modes on photons can be described in three forms as follows: the bit-flip error mode, the phase-flip error mode and both the two error mode. The corresponding states with errors can be written as:

*F*) indicates the probability that the photonic qubit takes place a bit-flip operation.

*ρ*randomly. With probability

*F*

^{2}, the two pairs are in the state

*L*〉. Then the ancillary photon passes through the two cavities and Bob performs the PCG operation on his two photons. The evolution of the composite system can be described as follows:

_{1}|↑〉

_{2}or |↓〉

_{1}|↓〉

_{2}), the mode information of the input photon will not changed and trigger the detector in the lower mode (D2). The evolution of the composite system can be written as: Otherwise, if the two electron spins are not in the same state, the mode of the photon will be changed and detected by the detector in the upper mode (D1): By detecting the output mode of the ancillary photon, one can distinguish the spin states of electron systems in the even number parity {|↑〉

_{1}|↑〉

_{2}, |↓〉

_{1}|↓〉

_{2}} or in the odd number parity {|↑〉

_{1}|↓〉

_{2}, |↓〉

_{1}|↑〉

_{2}}. If the photon reveals the detector D2, the state will collapse to

*X*〉, the remaining state is

*ϕ*

^{−}〉

_{1,2}|

*ϕ*

^{−}〉

_{3,4}with probability (1 −

*F*)

^{2}, the setup will evolve the composite system as

*R*

^{↓}〉. Under this condition, the state can not be distinguished by Alice and Bob with the case of |

*ψ*

^{−}〉

_{1,2}|

*ψ*

^{−}〉

_{3,4}and the two pairs are preserved.

*ϕ*

^{−}〉

_{1,2}|

*ψ*

^{−}〉

_{3,4}and |

*ψ*

^{−}〉

_{1,2}|

*ϕ*

^{−}〉

_{3,4}are described as and

*F*(1–

*F*) and can be distinguished by the case that Alice’s ancillary photon is detected in the lower spatial mode and Bob’s parity check reveals two photon event or none photon event on his detector D3. These two cases are discarded after EPP and we can get a new ensemble

*ρ*′ with the fidelity of which is larger than F when

*F*> 1/2.

*ψ*

^{−}〉 and |

*ψ*

^{+}〉 under Hadamard operations can be described as follows: and It is obvious that the state with phase flip errors |

*ψ*

^{+}〉 can be changed into the state |

*ϕ*

^{−}〉 by simply Hadamard operations acting on the two particles. Then the two users can perform the EPP on the bit-flip errors to purify the ensemble efficiently.

## 3. Efficiency and experiment feasibilities

36. Y. -F. Xiao, S.K. Özdemir, V. Gaddam, C. H. Dong, N. Imoto, and L. Yang, “Quantum nondemolition measurement of photon number via optical Kerr effect in an ultra-high-Q microtoroid cavity,” Opt. Exp. **16**, 21462–21475 (2008). [CrossRef]

*η*can be modeled by placing the transmission coefficients

*η*in front of ideal detectors. The performance of the QD and microcavity coupled system relative to the frequency detuning and the normalized coupling strength, we employed the coupling constant

*g*to calculate the fidelity in the coupled system to test the performance of our proposed protocol.

*g*/

_{A}*κ*= 1.2,

*g*/

_{B}*κ*= 0.3 and

*g*/

_{C}*κ*= 0.1, respectively. As show in this figure, weak coupling rapidly decrease the EPP fidelity lines. It is obvious that if the coupling strength is large than 1, the results of EPP in realistic systems is similarly with ideal conditions. If the cavity side leakage

*κ*is taken into account, the entanglement fidelity after EPP with respect to the initial fidelity is described by the dotted line in Fig. 4. And the solid line represents the EPP fidelity with no leaky. The dotted line is the fidelity with cavity leaky rate

_{s}*κ*= 0.05. As shown in this figure, the cavity leakage which generally decrease the entanglement fidelity in EPP.

_{s}37. T. H. Stievater, X. Q. Li, D. G. Steel, D. Gammon, D. S. Katzer, D. Park, C. Piermarocchi, and L. J. Sham, “Rabi oscillations of excitons in single quantum dots,”, Phys. Rev. Lett. **87**, 133603 (2001). [CrossRef] [PubMed]

38. H. Kamada, H. Gotoh, J. Temmyo, T. Takagahara, and H. Ando, “Exciton rabi oscillation in a single quantum dot,” Phys. Rev. Lett. **87**, 246401 (2001). [CrossRef] [PubMed]

39. J. Berezovsky, M. H. Mikkelsen, N. G. Stoltz, L. A. Coldren, and D. D. Awschalom, “Picosecond coherent optical manipulation of a single electron spin in a quantum dot,” Science **320**, 349–352 (2008). [CrossRef] [PubMed]

41. A. Greilich, S. E. Economou, S. Spatzek, D. R. Yakovlev, D. Reuter, A. D. Wieck, T. L. Reinecke, and M. Bayer, “Ultrafast optical rotations of electron spins in quantum dots,” Nature Physics **5**, 262–266 (2009). [CrossRef]

42. X. D. Xu, W. Yao, B. Sun, D. G. Steel, A. S. Bracker, D. Gammon, and L. J. Sham, “Optically controlled locking of the nuclear field via coherent dark-state spectroscopy,” Nature **459**, 1105–1109 (2009). [CrossRef] [PubMed]

43. J. P. Reithmaier, G. Sek, A. Löffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. V. Keldysh, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, “Strong coupling in a single quantum dot-semiconductor microcavity system,” Nature **432**, 197–200 (2004). [CrossRef] [PubMed]

46. E. Abe, H. Wu, A. Ardavan, and J.J. L. Morton, “Electron spin ensemble strongly coupled to a three-dimensional microwave cavity,” App. Phys. Lett. **98**, 251108 (2011). [CrossRef]

43. J. P. Reithmaier, G. Sek, A. Löffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. V. Keldysh, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, “Strong coupling in a single quantum dot-semiconductor microcavity system,” Nature **432**, 197–200 (2004). [CrossRef] [PubMed]

*g*= 0.5 in microcavity with diameter

*d*= 1.5

*μm*with the cavity leakage acceptable. The quality factors for the micropillars of the same size were increased to 4 × 10

^{4}for diameters below

*d*= 2

*μm*, corresponding to

*g*/(

*κ*+

*κ*) ≈ 2.4 by improving the sample designs and fabrication [47

_{s}47. S. Reitzenstein, C. Hofmann, A. Gorbunov, M. Strauβ, S. H. Kwon, C. Schneider, A. Löffler, S. Höfling, M. Kamp, and A. Forchel, “AlAs/GaAs micropillar cavities with quality factors exceeding 150.000,” App. Phys. Lett. **90**, 251109 (2007). [CrossRef]

*T*is the spin coherence time and

^{e}*t*is the cavity photon lifetime. The spin coherence time could be extended to

*μ*s using spin echo techniques [48

48. S. M. Clark, K.-M. C. Fu, Q. Zhang, T. D. Ladd, C. Stanley, and Y. Yamamoto, “Ultrafast optical spin echo for electron spins in semiconductors,” Phys. Rev. Lett. **102**, 247601 (2009). [CrossRef] [PubMed]

49. D. Press, K. De Greve, P. L. McMahon, T. D. Ladd, B. Friess, C. Schneider, M. Kamp, S. Höfling, A. Forchel, and Y. Yamamoto, “Ultrafast optical spin echo in a single quantum dot,” Nature Photonics **4**, 367–370 (2010). [CrossRef]

## 4. Summary

## Acknowledgments

## References and links

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24. | C. Bonato, F. Haupt, S. S. R. Oemrawsingh, J. Gudat, D. -P. Ding, M. P. van Exter, and D. Bouwmeester, “CNOT and Bell-state analysis in the weak-coupling cavity QED regime,” Phys. Rev. Lett. |

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37. | T. H. Stievater, X. Q. Li, D. G. Steel, D. Gammon, D. S. Katzer, D. Park, C. Piermarocchi, and L. J. Sham, “Rabi oscillations of excitons in single quantum dots,”, Phys. Rev. Lett. |

38. | H. Kamada, H. Gotoh, J. Temmyo, T. Takagahara, and H. Ando, “Exciton rabi oscillation in a single quantum dot,” Phys. Rev. Lett. |

39. | J. Berezovsky, M. H. Mikkelsen, N. G. Stoltz, L. A. Coldren, and D. D. Awschalom, “Picosecond coherent optical manipulation of a single electron spin in a quantum dot,” Science |

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42. | X. D. Xu, W. Yao, B. Sun, D. G. Steel, A. S. Bracker, D. Gammon, and L. J. Sham, “Optically controlled locking of the nuclear field via coherent dark-state spectroscopy,” Nature |

43. | J. P. Reithmaier, G. Sek, A. Löffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. V. Keldysh, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, “Strong coupling in a single quantum dot-semiconductor microcavity system,” Nature |

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47. | S. Reitzenstein, C. Hofmann, A. Gorbunov, M. Strauβ, S. H. Kwon, C. Schneider, A. Löffler, S. Höfling, M. Kamp, and A. Forchel, “AlAs/GaAs micropillar cavities with quality factors exceeding 150.000,” App. Phys. Lett. |

48. | S. M. Clark, K.-M. C. Fu, Q. Zhang, T. D. Ladd, C. Stanley, and Y. Yamamoto, “Ultrafast optical spin echo for electron spins in semiconductors,” Phys. Rev. Lett. |

49. | D. Press, K. De Greve, P. L. McMahon, T. D. Ladd, B. Friess, C. Schneider, M. Kamp, S. Höfling, A. Forchel, and Y. Yamamoto, “Ultrafast optical spin echo in a single quantum dot,” Nature Photonics |

**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: August 25, 2011

Revised Manuscript: October 25, 2011

Manuscript Accepted: October 27, 2011

Published: December 1, 2011

**Citation**

Chuan Wang, Yong Zhang, and Ru Zhang, "Entanglement purification based on hybrid entangled state using quantum-dot and microcavity coupled system," Opt. Express **19**, 25685-25695 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-25-25685

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### References

- A. K. Ekert, “Quantum cryptography based on Bells theorem,” Phys. Rev. Lett. 67, 661–663 (1991). [CrossRef] [PubMed]
- C. H. Bennett, G. Brassard, and N. D. Mermin, “Quantum cryptography without Bells theorem,” Phys. Rev. Lett. 68, 557–559 (1992). [CrossRef] [PubMed]
- C. H. Bennett and S. J. Wiesner, “Communication via one-and two-particle operators on Einstein-Podolsky-Rosen states,” Phys. Rev. Lett. 69, 2881–2884 (1992). [CrossRef] [PubMed]
- D. Bouwmeester, J. W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature 390, 575–579 (1997). [CrossRef]
- L. M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature 414, 413–418 (2001). [CrossRef] [PubMed]
- B. Zhao, Z. B. Chen, Y. A. Chen, J. Schmiedmayer, and J. W. Pan, “Robust creation of entanglement between remote memory qubits,” Phys. Rev. Lett. 98, 240502 (2007). [CrossRef] [PubMed]
- N. Sangouard, C. Simon, H. Riedmatten, and N. Gisin, “Quantum repeaters based on atomic ensembles and linear optics,” Rev. Mod. Phys. 83, 33–80 (2011). [CrossRef]
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