Abstract

We propose a scheme for the implementation of the quantum-controlled phase gate between distant atoms. Two special lasers adiabatically drive an atom trapped in a cavity that unidirectionally couples to the other cavity trapping an atom; then the phase gate between the two atoms is realized by introduction of measurement and local operations. The numerical simulations show that the quality factor of the gate operation is close to unity even if the atomic spontaneous emission is taken into account; the success probability approaches unity in principle.

© 2005 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  34. M. Hennrich, T. Legero, A. Kuhn, and G. Rempe, "Vacuum-stimulated Raman scattering based on adiabatic passage in a high-finesse optical cavity," Phys. Rev. Lett. 85, 4872-4875 (2000).
    [CrossRef] [PubMed]
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    [CrossRef]
  36. S. Dürr and G. Rempe, "Acceptance angle for Bragg reflection of atoms from a standing light wave," Phys. Rev. A 59, 1495-1499 (1999).
    [CrossRef]
  37. S. Dürr, T. Nonn, and G. Rempe, "Fringe visibility and which-way information in an atom interferometer," Phys. Rev. Lett. 81, 5705-5709 (1998).
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  38. T. Kuga, Y. Torii, N. Shiokawa, and T. Hirano, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713-4716 (1997).
    [CrossRef]
  39. Y. Torii, Y. Suzuki, M. Kozuma, T. Sugiura, T. Kuga, L. Deng, and E. W. Hagley, "Mach-Zehnder Bragg interferometer for a Bose-Einstein condensate," Phys. Rev. A 61, 041602 (2000).
    [CrossRef]
  40. Y. Shimizu, N. Shiokawa, N. Yamamoto, M. Kozuma, T. Kuga, L. Deng, and E. W. Hagley, "Control of light pulse propagation with only a few cold atoms in a high-finesse Microcavity," Phys. Rev. Lett. 89, 233001 (2002).
    [CrossRef] [PubMed]
  41. J. Ye, D. W. Vernooy, and H. J. Kimble, "Trapping of single atoms in cavity QED," Phys. Rev. Lett. 83, 4987-4990 (1999).
    [CrossRef]
  42. J. McKeever, A. Boca, A. D. Boozer, J. R. Buck, and H. J. Kimble, "Experimental realization of a one-atom laser in the regime of strong coupling," Nature 425, 268-271 (2003).
    [CrossRef] [PubMed]
  43. J. McKeever, J. R. Buck, A. D. Boozer, A. Kuzmich, H.-C. Nagerl, D. M. Stamper-Kurn, and H. J. Kimble, "State-insensitive cooling and trapping of single atoms in an optical cavity," Phys. Rev. Lett. 90, 133602 (2003).
    [CrossRef] [PubMed]
  44. J. McKeever, A. Boca, A. D. Boozer, R. Miller, J. R. Buck, A. Kuzmich, and H. J. Kimble, "One atom yields one photon," Science 303, 1992-1994 (2004).
    [CrossRef] [PubMed]
  45. L.-M. Duan, B. B. Blinov, D. L. Moehring, and C. Monroe, "Scalable trapped ion quantum computation with a probabilistic ion-photon mapping," http://xxx.lanl.gov/abs/quant-ph/0401020.

2004 (3)

J. K. Pachos and A. Beige, "Entangled-state preparation via dissipation-assisted adiabatic passages," Phys. Rev. A 69, 033817 (2004).
[CrossRef]

L.-M. Duan and H. J. Kimble, "Scalable photonic quantum computation through cavity-assisted interactions," Phys. Rev. Lett. 92, 127902 (2004).
[CrossRef] [PubMed]

J. McKeever, A. Boca, A. D. Boozer, R. Miller, J. R. Buck, A. Kuzmich, and H. J. Kimble, "One atom yields one photon," Science 303, 1992-1994 (2004).
[CrossRef] [PubMed]

2003 (7)

J. McKeever, A. Boca, A. D. Boozer, J. R. Buck, and H. J. Kimble, "Experimental realization of a one-atom laser in the regime of strong coupling," Nature 425, 268-271 (2003).
[CrossRef] [PubMed]

J. McKeever, J. R. Buck, A. D. Boozer, A. Kuzmich, H.-C. Nagerl, D. M. Stamper-Kurn, and H. J. Kimble, "State-insensitive cooling and trapping of single atoms in an optical cavity," Phys. Rev. Lett. 90, 133602 (2003).
[CrossRef] [PubMed]

S. Clark, A. Peng, M. Gu, and S. Parkins, "Unconditional preparation of entanglement between atoms in cascaded optical cavities," Phys. Rev. Lett. 91, 177901 (2003).
[CrossRef] [PubMed]

L.-M. Duan, A. Kuzmich, and H. J. Kimble, "Cavity QED and quantum-information processing with 'hot' trapped atoms," Phys. Rev. A 67, 032305 (2003).
[CrossRef]

L.-M. Duan and H. J. Kimble, "Efficient engineering of multiatom entanglement through single-photon detections," Phys. Rev. Lett. 90, 253601 (2003).
[CrossRef] [PubMed]

M. S. Zubairy, M. Kim, and M. O. Scully, "Cavity-QED-based quantum phase gate," Phys. Rev. A 68, 033820 (2003).
[CrossRef]

X. M. Lin, Z. W. Zhou, P. Xue, Y. J. Gu, and G. C. Guo, "Scheme for implementing quantum dense coding via cavity QED," Phys. Lett. A 313, 351-355 (2003).
[CrossRef]

2002 (5)

J. Pachos and H. Walther, "Quantum computation with trapped ions in an optical cavity," Phys. Rev. Lett. 89, 187903 (2002).
[CrossRef] [PubMed]

D. Kielpinski, C. Monroe, and D. J. Wineland, "Architecture for a large-scale ion-trap quantum computer," Nature 417, 709-711 (2002).
[CrossRef] [PubMed]

A. Kuhn, M. Hennrich, and G. Rempe, "Deterministic single-photon source for distributed quantum networking," Phys. Rev. Lett. 89, 067901 (2002).
[CrossRef] [PubMed]

T. Fischer, P. Maunz, P. W. H. Pinkse, T. Puppe, and G. Rempe, "Feedback on the motion of a single atom in an optical cavity," Phys. Rev. Lett. 88, 163002 (2002).
[CrossRef] [PubMed]

Y. Shimizu, N. Shiokawa, N. Yamamoto, M. Kozuma, T. Kuga, L. Deng, and E. W. Hagley, "Control of light pulse propagation with only a few cold atoms in a high-finesse Microcavity," Phys. Rev. Lett. 89, 233001 (2002).
[CrossRef] [PubMed]

2001 (1)

E. Knill, R. Laflamme, and G. J. Milburn, "A scheme for efficient quantum computation with linear optics," Nature 409, 46-52 (2001).
[CrossRef] [PubMed]

2000 (6)

V. Giovannetti, D. Vitali, P. Tombesi, and A. Ekert, "Scalable quantum computation with cavity QED systems," Phys. Rev. A 62, 032306 (2000).
[CrossRef]

A. Steane, C. F. Roos, D. Stevens, A. Mundt, D. Leibfried, F. Schmidt-Kaler, and R. Blatt, "Speed of ion-trap quantum-information processors," Phys. Rev. A 62, 042305 (2000).
[CrossRef]

P. Münstermann, T. Fischer, P. Maunz, P. W. H. Pinkse, and G. Rempe, "Observation of cavity-mediated long-range light forces between strongly coupled atoms," Phys. Rev. Lett. 84, 4068-4071 (2000).
[CrossRef] [PubMed]

M. Hennrich, T. Legero, A. Kuhn, and G. Rempe, "Vacuum-stimulated Raman scattering based on adiabatic passage in a high-finesse optical cavity," Phys. Rev. Lett. 85, 4872-4875 (2000).
[CrossRef] [PubMed]

M. Fleischhauer, S. F. Yelin, and M. D. Lukin, "How to trap photons? Storing single-photon quantum states in collective atomic excitations," Opt. Commun. 179, 395-410 (2000).
[CrossRef]

Y. Torii, Y. Suzuki, M. Kozuma, T. Sugiura, T. Kuga, L. Deng, and E. W. Hagley, "Mach-Zehnder Bragg interferometer for a Bose-Einstein condensate," Phys. Rev. A 61, 041602 (2000).
[CrossRef]

1999 (5)

Y. Makhlin, G. Scohn, and A. Shnirman, "Josephson-junction qubits with controlled couplings," Nature 398, 305-307 (1999).
[CrossRef]

P. Münstermann, T. Fischer, P. Maunz, P. W. H. Pinkse, and G. Rempe, "Dynamics of single-atom motion observed in a high-finesse cavity," Phys. Rev. Lett. 82, 3791-3794 (1999).
[CrossRef]

S. Dürr and G. Rempe, "Acceptance angle for Bragg reflection of atoms from a standing light wave," Phys. Rev. A 59, 1495-1499 (1999).
[CrossRef]

A. Rauschenbeutel, G. Nogues, S. Osnaghi, P. Bertet, M. Brune, J. M. Raimond, and S. Haroche, "Coherent operation of a tunable quantum phase gate in cavity QED," Phys. Rev. Lett. 83, 5166-5169 (1999).
[CrossRef]

J. Ye, D. W. Vernooy, and H. J. Kimble, "Trapping of single atoms in cavity QED," Phys. Rev. Lett. 83, 4987-4990 (1999).
[CrossRef]

1998 (3)

D. Loss and D. P. DiVincenzo, "Quantum computation with quantum dots," Phys. Rev. A 57, 120-126 (1998).
[CrossRef]

B. E. Kane, "A silicon-based nuclear spin quantum computer," Nature 393, 133-137 (1998).
[CrossRef]

S. Dürr, T. Nonn, and G. Rempe, "Fringe visibility and which-way information in an atom interferometer," Phys. Rev. Lett. 81, 5705-5709 (1998).
[CrossRef]

1997 (4)

T. Kuga, Y. Torii, N. Shiokawa, and T. Hirano, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

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).
[CrossRef]

N. A. Gerschenfeld and I. L. Chuang, "Bulk spin-resonance quantum computation," Science 275, 350-356 (1997).
[CrossRef]

L. K. Grover, "Quantum mechanics helps in searching for a needle in a haystack," Phys. Rev. Lett. 79, 325-328 (1997).
[CrossRef]

1996 (1)

C. K. Law and J. H. Eberly, "Arbitrary control of a quantum electromagnetic field," Phys. Rev. Lett. 76, 1055-1058 (1996).
[CrossRef] [PubMed]

1995 (4)

J. I. Cirac and P. Zoller, "Quantum computations with cold trapped ions," Phys. Rev. Lett. 74, 4091-4094 (1995).
[CrossRef] [PubMed]

T. Pellizzari, S. A. Gardiner, J. I. Cirac, and P. Zoller, "Decoherence, continuous observation, and quantum computing: a cavity QED model," Phys. Rev. Lett. 75, 3788-3791 (1995).
[CrossRef] [PubMed]

Q. A. Turchette, C. J. Hood, W. Lange, H. Mabuchi, and H. J. Kimble, "Measurement of conditional phase shifts for quantum logic," Phys. Rev. Lett. 75, 4710-4713 (1995).
[CrossRef] [PubMed]

A. S. Parkins, P. Marte, P. Zoller, O. Carnal, and H. J. Kimble, "Quantum-state mapping between multilevel atoms and cavity light fields," Phys. Rev. A 51, 1578-1596 (1995).
[CrossRef] [PubMed]

1994 (1)

L. Davidovich, N. Zagury, M. Brune, J. M. Raimond, and S. Haroche, "Teleportation of an atomic state between two cavities using nonlocal microwave fields," Phys. Rev. A 50, R895-R898 (1994).
[CrossRef] [PubMed]

Beige, A.

J. K. Pachos and A. Beige, "Entangled-state preparation via dissipation-assisted adiabatic passages," Phys. Rev. A 69, 033817 (2004).
[CrossRef]

Bertet, P.

A. Rauschenbeutel, G. Nogues, S. Osnaghi, P. Bertet, M. Brune, J. M. Raimond, and S. Haroche, "Coherent operation of a tunable quantum phase gate in cavity QED," Phys. Rev. Lett. 83, 5166-5169 (1999).
[CrossRef]

Blatt, R.

A. Steane, C. F. Roos, D. Stevens, A. Mundt, D. Leibfried, F. Schmidt-Kaler, and R. Blatt, "Speed of ion-trap quantum-information processors," Phys. Rev. A 62, 042305 (2000).
[CrossRef]

Blinov, B. B.

L.-M. Duan, B. B. Blinov, D. L. Moehring, and C. Monroe, "Scalable trapped ion quantum computation with a probabilistic ion-photon mapping," http://xxx.lanl.gov/abs/quant-ph/0401020.

Boca, A.

J. McKeever, A. Boca, A. D. Boozer, R. Miller, J. R. Buck, A. Kuzmich, and H. J. Kimble, "One atom yields one photon," Science 303, 1992-1994 (2004).
[CrossRef] [PubMed]

J. McKeever, A. Boca, A. D. Boozer, J. R. Buck, and H. J. Kimble, "Experimental realization of a one-atom laser in the regime of strong coupling," Nature 425, 268-271 (2003).
[CrossRef] [PubMed]

Boozer, A. D.

J. McKeever, A. Boca, A. D. Boozer, R. Miller, J. R. Buck, A. Kuzmich, and H. J. Kimble, "One atom yields one photon," Science 303, 1992-1994 (2004).
[CrossRef] [PubMed]

J. McKeever, A. Boca, A. D. Boozer, J. R. Buck, and H. J. Kimble, "Experimental realization of a one-atom laser in the regime of strong coupling," Nature 425, 268-271 (2003).
[CrossRef] [PubMed]

J. McKeever, J. R. Buck, A. D. Boozer, A. Kuzmich, H.-C. Nagerl, D. M. Stamper-Kurn, and H. J. Kimble, "State-insensitive cooling and trapping of single atoms in an optical cavity," Phys. Rev. Lett. 90, 133602 (2003).
[CrossRef] [PubMed]

Brune, M.

A. Rauschenbeutel, G. Nogues, S. Osnaghi, P. Bertet, M. Brune, J. M. Raimond, and S. Haroche, "Coherent operation of a tunable quantum phase gate in cavity QED," Phys. Rev. Lett. 83, 5166-5169 (1999).
[CrossRef]

L. Davidovich, N. Zagury, M. Brune, J. M. Raimond, and S. Haroche, "Teleportation of an atomic state between two cavities using nonlocal microwave fields," Phys. Rev. A 50, R895-R898 (1994).
[CrossRef] [PubMed]

Buck, J. R.

J. McKeever, A. Boca, A. D. Boozer, R. Miller, J. R. Buck, A. Kuzmich, and H. J. Kimble, "One atom yields one photon," Science 303, 1992-1994 (2004).
[CrossRef] [PubMed]

J. McKeever, J. R. Buck, A. D. Boozer, A. Kuzmich, H.-C. Nagerl, D. M. Stamper-Kurn, and H. J. Kimble, "State-insensitive cooling and trapping of single atoms in an optical cavity," Phys. Rev. Lett. 90, 133602 (2003).
[CrossRef] [PubMed]

J. McKeever, A. Boca, A. D. Boozer, J. R. Buck, and H. J. Kimble, "Experimental realization of a one-atom laser in the regime of strong coupling," Nature 425, 268-271 (2003).
[CrossRef] [PubMed]

Carnal, O.

A. S. Parkins, P. Marte, P. Zoller, O. Carnal, and H. J. Kimble, "Quantum-state mapping between multilevel atoms and cavity light fields," Phys. Rev. A 51, 1578-1596 (1995).
[CrossRef] [PubMed]

Chuang, I. L.

N. A. Gerschenfeld and I. L. Chuang, "Bulk spin-resonance quantum computation," Science 275, 350-356 (1997).
[CrossRef]

Cirac, J. I.

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).
[CrossRef]

T. Pellizzari, S. A. Gardiner, J. I. Cirac, and P. Zoller, "Decoherence, continuous observation, and quantum computing: a cavity QED model," Phys. Rev. Lett. 75, 3788-3791 (1995).
[CrossRef] [PubMed]

J. I. Cirac and P. Zoller, "Quantum computations with cold trapped ions," Phys. Rev. Lett. 74, 4091-4094 (1995).
[CrossRef] [PubMed]

Clark, S.

S. Clark, A. Peng, M. Gu, and S. Parkins, "Unconditional preparation of entanglement between atoms in cascaded optical cavities," Phys. Rev. Lett. 91, 177901 (2003).
[CrossRef] [PubMed]

Davidovich, L.

L. Davidovich, N. Zagury, M. Brune, J. M. Raimond, and S. Haroche, "Teleportation of an atomic state between two cavities using nonlocal microwave fields," Phys. Rev. A 50, R895-R898 (1994).
[CrossRef] [PubMed]

Deng, L.

Y. Shimizu, N. Shiokawa, N. Yamamoto, M. Kozuma, T. Kuga, L. Deng, and E. W. Hagley, "Control of light pulse propagation with only a few cold atoms in a high-finesse Microcavity," Phys. Rev. Lett. 89, 233001 (2002).
[CrossRef] [PubMed]

Y. Torii, Y. Suzuki, M. Kozuma, T. Sugiura, T. Kuga, L. Deng, and E. W. Hagley, "Mach-Zehnder Bragg interferometer for a Bose-Einstein condensate," Phys. Rev. A 61, 041602 (2000).
[CrossRef]

DiVincenzo, D. P.

D. Loss and D. P. DiVincenzo, "Quantum computation with quantum dots," Phys. Rev. A 57, 120-126 (1998).
[CrossRef]

Duan, L.-M.

L.-M. Duan and H. J. Kimble, "Scalable photonic quantum computation through cavity-assisted interactions," Phys. Rev. Lett. 92, 127902 (2004).
[CrossRef] [PubMed]

L.-M. Duan, A. Kuzmich, and H. J. Kimble, "Cavity QED and quantum-information processing with 'hot' trapped atoms," Phys. Rev. A 67, 032305 (2003).
[CrossRef]

L.-M. Duan and H. J. Kimble, "Efficient engineering of multiatom entanglement through single-photon detections," Phys. Rev. Lett. 90, 253601 (2003).
[CrossRef] [PubMed]

L.-M. Duan, B. B. Blinov, D. L. Moehring, and C. Monroe, "Scalable trapped ion quantum computation with a probabilistic ion-photon mapping," http://xxx.lanl.gov/abs/quant-ph/0401020.

Dürr, S.

S. Dürr and G. Rempe, "Acceptance angle for Bragg reflection of atoms from a standing light wave," Phys. Rev. A 59, 1495-1499 (1999).
[CrossRef]

S. Dürr, T. Nonn, and G. Rempe, "Fringe visibility and which-way information in an atom interferometer," Phys. Rev. Lett. 81, 5705-5709 (1998).
[CrossRef]

Eberly, J. H.

C. K. Law and J. H. Eberly, "Arbitrary control of a quantum electromagnetic field," Phys. Rev. Lett. 76, 1055-1058 (1996).
[CrossRef] [PubMed]

Ekert, A.

V. Giovannetti, D. Vitali, P. Tombesi, and A. Ekert, "Scalable quantum computation with cavity QED systems," Phys. Rev. A 62, 032306 (2000).
[CrossRef]

Fischer, T.

T. Fischer, P. Maunz, P. W. H. Pinkse, T. Puppe, and G. Rempe, "Feedback on the motion of a single atom in an optical cavity," Phys. Rev. Lett. 88, 163002 (2002).
[CrossRef] [PubMed]

P. Münstermann, T. Fischer, P. Maunz, P. W. H. Pinkse, and G. Rempe, "Observation of cavity-mediated long-range light forces between strongly coupled atoms," Phys. Rev. Lett. 84, 4068-4071 (2000).
[CrossRef] [PubMed]

P. Münstermann, T. Fischer, P. Maunz, P. W. H. Pinkse, and G. Rempe, "Dynamics of single-atom motion observed in a high-finesse cavity," Phys. Rev. Lett. 82, 3791-3794 (1999).
[CrossRef]

Fleischhauer, M.

M. Fleischhauer, S. F. Yelin, and M. D. Lukin, "How to trap photons? Storing single-photon quantum states in collective atomic excitations," Opt. Commun. 179, 395-410 (2000).
[CrossRef]

Gardiner, C. W.

C. W. Gardiner, Quantum Noise (Springer-Verlag, New York, 2000).
[CrossRef]

Gardiner, S. A.

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T. Pellizzari, S. A. Gardiner, J. I. Cirac, and P. Zoller, "Decoherence, continuous observation, and quantum computing: a cavity QED model," Phys. Rev. Lett. 75, 3788-3791 (1995).
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T. Fischer, P. Maunz, P. W. H. Pinkse, T. Puppe, and G. Rempe, "Feedback on the motion of a single atom in an optical cavity," Phys. Rev. Lett. 88, 163002 (2002).
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T. Fischer, P. Maunz, P. W. H. Pinkse, T. Puppe, and G. Rempe, "Feedback on the motion of a single atom in an optical cavity," Phys. Rev. Lett. 88, 163002 (2002).
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J. McKeever, J. R. Buck, A. D. Boozer, A. Kuzmich, H.-C. Nagerl, D. M. Stamper-Kurn, and H. J. Kimble, "State-insensitive cooling and trapping of single atoms in an optical cavity," Phys. Rev. Lett. 90, 133602 (2003).
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Y. Torii, Y. Suzuki, M. Kozuma, T. Sugiura, T. Kuga, L. Deng, and E. W. Hagley, "Mach-Zehnder Bragg interferometer for a Bose-Einstein condensate," Phys. Rev. A 61, 041602 (2000).
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Q. A. Turchette, C. J. Hood, W. Lange, H. Mabuchi, and H. J. Kimble, "Measurement of conditional phase shifts for quantum logic," Phys. Rev. Lett. 75, 4710-4713 (1995).
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J. Ye, D. W. Vernooy, and H. J. Kimble, "Trapping of single atoms in cavity QED," Phys. Rev. Lett. 83, 4987-4990 (1999).
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V. Giovannetti, D. Vitali, P. Tombesi, and A. Ekert, "Scalable quantum computation with cavity QED systems," Phys. Rev. A 62, 032306 (2000).
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Figures (6)

Fig. 1
Fig. 1

Schematic setup to implement the CPF gate between two distant atoms. Where two curve arrows denote two driving laser pulses with the same frequency but different polarizations, the filter is frequency selective to filter the driving lases; QWP1 and QWP2 signify quarter-wave plates, PBS1 and PBS2 symbolize polarization beam splitters, HWP1 and HWP2 mark 135° half-wave plates, M represents a reflecting mirror, D1 and D2 are single-photon detectors, and F denotes a Faraday isolator. The optical paths from PBS1 to M to PBS2 and from cavity B to PBS2 are assumed to be equal.

Fig. 2
Fig. 2

Relevant atomic level structure and transitions of atom 1. The states ∣0⟩ ( f ) , r (∣1⟩) correspond to the Zeeman sublevels of the Rb 85 atom ground hyperfine level 5 S 1 2 F = 3 ( F = 2 ) , and e 0 , e 1 correspond to the Zeeman sublevels of the Rb 85 atom excited hyperfine level 5 P 3 2 F = 3 . The transition 0 e 0 ( r e 1 ) is driven resonantly by a clockwise (counterclockwise) R ( L ) polarized classical laser pulse with Rabi frequency Ω, e 0 f ( e 1 1 ) is resonantly coupled to the cavity A mode a L ( a R ) with coupling constant g.

Fig. 3
Fig. 3

Relevant atomic level structure and transition of atom 2. The transition 1 e 1 is resonantly coupled to the cavity B mode a R with coupling constant g .

Fig. 4
Fig. 4

Shape functions f i ( t ) for the input pulse (solid curve) and the reflected pulse by cavity B with the atom 2 in the state 0 2 (dashed curve) and 1 2 (dotted curve), respectively. The dotted and solid curves have only tiny mismatch. The dashed and solid curves closely overlap and are hardly distinguishable in the figure. Here, we have taken g = 4 κ , κ = γ s = 30 MHz , and κ T = 150 .

Fig. 5
Fig. 5

Phase variation δ ψ between input pulse and output pulse of cavity B with the atom 2 in the state 0 2 (dotted curve) and 1 2 (solid curve), respectively. The parameters are the same as in Fig. 4.

Fig. 6
Fig. 6

The relation between the fidelity F i of the CPF gate and the states of two atoms qubits. The parameters are the same as in Fig. 4.

Equations (35)

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H A = Ω ( r , t ) ( 0 1 e 0 + r 1 e 1 ) + g ( r ) ( a L e 0 1 f + a R e 1 1 1 ) + H.c. ,
D 0 = g ( g 2 + Ω 2 ) 1 2 0 1 00 A + Ω ( g 2 + Ω 2 ) 1 2 f 1 10 A ,
D 1 = g ( g 2 + Ω 2 ) 1 2 r 1 00 A + Ω ( g 2 + Ω 2 ) 1 2 1 1 01 A .
H B = g ( a R e 1 2 1 + a R + 1 2 e 1 ) .
Ψ 0 = ( α 1 0 1 + β 1 1 1 ) ( α 2 0 2 + β 2 1 2 ) 00 A 00 B vac f ,
Ψ 1 = ( α 1 0 1 + β 1 r 1 ) ( α 2 0 2 + β 2 1 2 ) 00 A 00 B vac f .
Ψ 2 = ( α 1 f 1 10 A + β 1 1 1 01 A ) ( α 2 0 2 + β 2 1 2 ) 00 B vac f ,
f A ( t ) = κ sin θ ( t ) exp [ κ 2 0 t sin 2 θ ( τ ) d τ ] ,
Ψ 3 = ( α 1 f 1 v f + β 1 1 1 h f ) ( α 2 0 2 + β 2 1 2 ) 00 A 00 B .
Ψ 4 = ( α 1 f 1 v a + β 1 1 1 h b ) ( α 2 0 2 + β 2 1 2 ) 00 A 00 B .
Ψ 5 = α 1 f 1 v a ( α 2 0 2 + β 2 1 2 ) + β 1 1 1 h b ( α 2 0 2 + β 2 1 2 ) .
Ψ 6 = 1 2 [ α 1 f 1 ( v a + h a ) ( α 2 0 2 + β 2 1 2 ) + β 1 1 1 ( v b h b ) ( α 2 0 2 + β 2 1 2 ) ] .
Ψ 7 = α 1 α 2 f 1 0 2 + α 1 β 2 f 1 1 2 + α 2 β 1 1 1 0 2 β 1 β 2 1 1 1 2 ,
Ψ 7 = α 1 α 2 f 1 0 2 + α 1 β 2 f 1 1 2 α 2 β 1 1 1 0 2 + β 1 β 2 1 1 1 2 .
Ψ 8 = α 1 α 2 0 1 0 2 + α 1 β 2 0 1 1 2 + α 2 β 1 1 1 0 2 β 1 β 2 1 1 1 2 ,
H A = i γ s 2 ( e 0 1 e 0 + e 1 1 e 1 ) + H A + λ = L , R { ω b + ω b ω b λ + ( ω ) b λ ( ω ) d ω + i ( κ 2 π ) 1 2 ω b + ω b [ a λ + b λ ( ω ) a λ b λ + ( ω ) ] d ω } ,
sin θ ( t ) = Ω ( r , t ) [ g 2 ( r ) + Ω 2 ( r , t ) ] 1 2 = C Ω E ̃ ( t ) [ C g 2 + C Ω 2 E ̃ 2 ( t ) ] 1 2 ,
H B = i γ s 2 e 1 2 e 1 + H B + ω b + ω b ω b R + ( ω ) b R ( ω ) d ω + i ( κ 2 π ) 1 2 ω b + ω b [ a R + b R ( ω ) a R b R + ( ω ) ] d ω ,
Φ ( t ) 20 = 0 2 00 B ϕ ( t ) f + d ( t ) 0 2 01 B vac f ,
ϕ ( t ) f = ω b + ω b c ( ω , t ) b R + ( ω ) vac f d ω
i t Φ ( t ) 20 = H B Φ ( t ) 20 ,
c ̇ ( ω , t ) = i ω c ( ω , t ) ( κ 2 π ) 1 2 d ( t ) ,
d ̇ ( t ) = ( κ 2 π ) 1 2 ω b + ω b c ( ω , t ) d ω .
f b ( t ) = 1 2 π ω b + ω b c ( ω , T ) exp [ i ω ( t T ) ] d ω ,
ϕ ( t ) f = j = 1 N c j ( ω j , t ) b j + vac f ,
c ̇ j = i ω j c j ( κ δ ω 2 π ) 1 2 d ( t ) ,
d ̇ = ( κ δ ω 2 π ) 1 2 j = 1 N c j .
Φ ( t ) 21 = 1 2 00 B ϕ ( t ) f + p ( t ) 1 2 01 B vac f + q ( t ) e 1 2 00 B vac f ,
ϕ ( t ) f = j = 1 N c j ( ω j , t ) b j + vac f .
c ̇ j = i ω j c j ( κ δ ω 2 π ) 1 2 p ,
p ̇ = ( κ δ ω 2 π ) 1 2 j = 1 N c j i g ( r ) q ,
q ̇ = i g ( r ) p γ s 2 q ,
E ̃ ( t ) = 2 C g C Ω exp [ ( t T 2 ) 2 ( T 5 ) 2 ] ,
F i = Ψ ideal Ψ real 2 = α 1 2 ( 1 α 1 2 ) α 2 2 ϕ 0 ( T ) ϕ ( T ) + ( 1 α 1 2 ) ( 1 α 2 2 ) ϕ 0 ( T ) ϕ ( T ) 2 ,
P spon = 1 p ( T ) 2 q ( T ) 2 j = 1 N c j ( ω j , T ) 2 = 0.060592 .

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