Abstract

We solve the Jaynes–Cummings Hamiltonian with time-dependent coupling parameters under the dipole and rotating-wave approximations for a three-dimensional photonic crystal (PC) single-mode cavity with a sufficiently high-quality Q factor. We then exploit the results to show how to create a maximally entangled state of two atoms and how to implement several quantum logic gates: a dual-rail Hadamard gate, a dual-rail NOT gate, and a SWAP gate. The atoms in all of these operations are syncronized, which is not the case in previous studies of PCs [J. Mod. Opt. 48, 1495 (2001) ; Eur. Phys. J. D 10, 285 (2000) ; Eur. Phys. J. D 18, 247 (2002) ]. Our method has the potential for extension to N-atom entanglement, universal quantum logic operations, and the implementation of other useful, cavity QED-based quantum information processing tasks.

© 2007 Optical Society of America

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References

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  1. M. A. Nielsen and I. L. Chuang, Quantum Information and Quantum Computing (Cambridge U. Press, 2000).
  2. C. M. Soukoulis, "The history and a review of the modelling and fabrication of photonic crystals," Nanotechnology 13, 420-423 (2002).
    [CrossRef]
  3. E. Yablonovitch, "Photonic crystals," J. Mod. Opt. 41, 173-194 (1994).
    [CrossRef]
  4. E. Özbay, Nanotechnology Research Center, Bilkent University, Ankara, Turkey, 06800 (personal communication, 2006).
  5. M. Qi, E. Lidorikis, P. T. Rakich, S. G. Johnson, J. D. Joannopoulos, E. P. Ippen, and H. I. Smith, "A three-dimensional optical photonic crystal with designed point defects," Nature 429, 538-542 (2004).
    [CrossRef] [PubMed]
  6. N. Vats and T. Rudolph, "Quantum information processing in localized modes of light within a photonic bandgap material," J. Mod. Opt. 48, 1495-1502 (2001).
    [CrossRef]
  7. M. Konopka and V. Buzek, "Entangling atoms in photonic crystals," Eur. Phys. J. D 10, 285-293 (2000).
    [CrossRef]
  8. S. John and J. Wang, "Quantum optics of localized light in a photonic band gap," Phys. Rev. B 43, 12772-12789 (1991).
    [CrossRef]
  9. J. J. Sakurai, Modern Quantum Mechanics (Addison-Wesley, 1994).
  10. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).
  11. M. Tavis and F. W. Cummings, "Exact solution for N-molecule-radiation field Hamiltonian," Phys. Rev. 170, 379-384 (1968).
    [CrossRef]
  12. T. B. Pittman and J. D. Franson, "Cyclical quantum memory for photonic qubits," Phys. Rev. A 66, 062302-062305 (2002).
    [CrossRef]
  13. D. G. Angelakis and P. L. Knight, "Testing Bell inequalities in photonic crystals," Eur. Phys. J. D 18, 247-250 (2002).
    [CrossRef]
  14. E. Hagley, X. Maitre, G. Nogues, C. Wunderlich, M. Brune, J. M. Raimond, and S. Haroche, "Generation of Einstein-Podolsky-Rosen pairs of atoms," Phys. Rev. Lett. 79, 1-5 (1997).
    [CrossRef]
  15. P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, "Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency," Phys. Rev. B 54, 7837-7842 (1996).
    [CrossRef]
  16. J. Vuckovic, Ginzton Laboratory, Stanford University, Stanford, Calif. 94305 (personal communication, 2004).
  17. H. J. Kimble, "Strong interactions of single atoms and photons in cavity QED," Phys. Scr. T76, 127-137 (1998).
    [CrossRef]
  18. S. Haroche and J. M. Raimond, "Manipulation of nonclassical field states," in Cavity Electrodynamics, P.Berman, ed. (Academic, 1994), pp. 126-127.
  19. Y. Yamamoto and A. Imamoglu, Mesoscopic Quantum Optics (Wiley-Interscience, 1999).
  20. R. J. Glauber and M. Lewenstein, "Quantum optics of dielectric media," Phys. Rev. A 43, 467-491 (1991).
    [CrossRef] [PubMed]
  21. S. G. Johnson and J. D. Joannopoulos, "Block-iterative frequency-domain methods for Maxwell's equations in a plane-wave basis," Opt. Express 8, 173-190 (2001).
    [CrossRef] [PubMed]
  22. S. G. Johnson and J. D. Joannopoulos, "Three-dimensionally periodic dielectric layered structure with omnidirectional photonic band gap," Appl. Phys. Lett. 77, 3490-3492 (2000).
    [CrossRef]
  23. M. L. Povinelli, S. G. Johnson, S. Fan, and J. D. Joannopoulos, "Emulation of two-dimensional photonic crystal defect modes in a photonic crystal with a three-dimensional photonic band gap," Phys. Rev. B 64, 075313-075320 (2001).
    [CrossRef]
  24. S. Fan, P. R. Villeneuve, R. D. Meade, and J. D. Joannopoulos, "Design of three-dimensional photonic crystals at submicron lengthscales," Appl. Phys. Lett. 65, 1466-1468 (1994).
    [CrossRef]
  25. K. H. Dridi, "Intrinsic eigenstate spectrum of planar multilayer stacks of two-dimensional photonic crystals," Opt. Express 11, 1156-1165 (2003).
    [CrossRef] [PubMed]
  26. K. H. Dridi, "Mode dispersion and photonic storage in planar defects within Bragg stacks of photonic crystal slabs," J. Opt. Soc. Am. B 21, 522-530 (2004).
    [CrossRef]
  27. J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, "Design of photonic crystal microcavities for cavity QED," Phys. Rev. E 65, 016608-016618 (2001).
    [CrossRef]

2004

M. Qi, E. Lidorikis, P. T. Rakich, S. G. Johnson, J. D. Joannopoulos, E. P. Ippen, and H. I. Smith, "A three-dimensional optical photonic crystal with designed point defects," Nature 429, 538-542 (2004).
[CrossRef] [PubMed]

K. H. Dridi, "Mode dispersion and photonic storage in planar defects within Bragg stacks of photonic crystal slabs," J. Opt. Soc. Am. B 21, 522-530 (2004).
[CrossRef]

2003

2002

T. B. Pittman and J. D. Franson, "Cyclical quantum memory for photonic qubits," Phys. Rev. A 66, 062302-062305 (2002).
[CrossRef]

D. G. Angelakis and P. L. Knight, "Testing Bell inequalities in photonic crystals," Eur. Phys. J. D 18, 247-250 (2002).
[CrossRef]

C. M. Soukoulis, "The history and a review of the modelling and fabrication of photonic crystals," Nanotechnology 13, 420-423 (2002).
[CrossRef]

2001

M. L. Povinelli, S. G. Johnson, S. Fan, and J. D. Joannopoulos, "Emulation of two-dimensional photonic crystal defect modes in a photonic crystal with a three-dimensional photonic band gap," Phys. Rev. B 64, 075313-075320 (2001).
[CrossRef]

J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, "Design of photonic crystal microcavities for cavity QED," Phys. Rev. E 65, 016608-016618 (2001).
[CrossRef]

S. G. Johnson and J. D. Joannopoulos, "Block-iterative frequency-domain methods for Maxwell's equations in a plane-wave basis," Opt. Express 8, 173-190 (2001).
[CrossRef] [PubMed]

N. Vats and T. Rudolph, "Quantum information processing in localized modes of light within a photonic bandgap material," J. Mod. Opt. 48, 1495-1502 (2001).
[CrossRef]

2000

M. Konopka and V. Buzek, "Entangling atoms in photonic crystals," Eur. Phys. J. D 10, 285-293 (2000).
[CrossRef]

S. G. Johnson and J. D. Joannopoulos, "Three-dimensionally periodic dielectric layered structure with omnidirectional photonic band gap," Appl. Phys. Lett. 77, 3490-3492 (2000).
[CrossRef]

1998

H. J. Kimble, "Strong interactions of single atoms and photons in cavity QED," Phys. Scr. T76, 127-137 (1998).
[CrossRef]

1997

E. Hagley, X. Maitre, G. Nogues, C. Wunderlich, M. Brune, J. M. Raimond, and S. Haroche, "Generation of Einstein-Podolsky-Rosen pairs of atoms," Phys. Rev. Lett. 79, 1-5 (1997).
[CrossRef]

1996

P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, "Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency," Phys. Rev. B 54, 7837-7842 (1996).
[CrossRef]

1994

E. Yablonovitch, "Photonic crystals," J. Mod. Opt. 41, 173-194 (1994).
[CrossRef]

S. Fan, P. R. Villeneuve, R. D. Meade, and J. D. Joannopoulos, "Design of three-dimensional photonic crystals at submicron lengthscales," Appl. Phys. Lett. 65, 1466-1468 (1994).
[CrossRef]

1991

R. J. Glauber and M. Lewenstein, "Quantum optics of dielectric media," Phys. Rev. A 43, 467-491 (1991).
[CrossRef] [PubMed]

S. John and J. Wang, "Quantum optics of localized light in a photonic band gap," Phys. Rev. B 43, 12772-12789 (1991).
[CrossRef]

1968

M. Tavis and F. W. Cummings, "Exact solution for N-molecule-radiation field Hamiltonian," Phys. Rev. 170, 379-384 (1968).
[CrossRef]

Angelakis, D. G.

D. G. Angelakis and P. L. Knight, "Testing Bell inequalities in photonic crystals," Eur. Phys. J. D 18, 247-250 (2002).
[CrossRef]

Brune, M.

E. Hagley, X. Maitre, G. Nogues, C. Wunderlich, M. Brune, J. M. Raimond, and S. Haroche, "Generation of Einstein-Podolsky-Rosen pairs of atoms," Phys. Rev. Lett. 79, 1-5 (1997).
[CrossRef]

Buzek, V.

M. Konopka and V. Buzek, "Entangling atoms in photonic crystals," Eur. Phys. J. D 10, 285-293 (2000).
[CrossRef]

Chuang, I. L.

M. A. Nielsen and I. L. Chuang, Quantum Information and Quantum Computing (Cambridge U. Press, 2000).

Cummings, F. W.

M. Tavis and F. W. Cummings, "Exact solution for N-molecule-radiation field Hamiltonian," Phys. Rev. 170, 379-384 (1968).
[CrossRef]

Dridi, K. H.

Fan, S.

M. L. Povinelli, S. G. Johnson, S. Fan, and J. D. Joannopoulos, "Emulation of two-dimensional photonic crystal defect modes in a photonic crystal with a three-dimensional photonic band gap," Phys. Rev. B 64, 075313-075320 (2001).
[CrossRef]

P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, "Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency," Phys. Rev. B 54, 7837-7842 (1996).
[CrossRef]

S. Fan, P. R. Villeneuve, R. D. Meade, and J. D. Joannopoulos, "Design of three-dimensional photonic crystals at submicron lengthscales," Appl. Phys. Lett. 65, 1466-1468 (1994).
[CrossRef]

Franson, J. D.

T. B. Pittman and J. D. Franson, "Cyclical quantum memory for photonic qubits," Phys. Rev. A 66, 062302-062305 (2002).
[CrossRef]

Glauber, R. J.

R. J. Glauber and M. Lewenstein, "Quantum optics of dielectric media," Phys. Rev. A 43, 467-491 (1991).
[CrossRef] [PubMed]

Hagley, E.

E. Hagley, X. Maitre, G. Nogues, C. Wunderlich, M. Brune, J. M. Raimond, and S. Haroche, "Generation of Einstein-Podolsky-Rosen pairs of atoms," Phys. Rev. Lett. 79, 1-5 (1997).
[CrossRef]

Haroche, S.

E. Hagley, X. Maitre, G. Nogues, C. Wunderlich, M. Brune, J. M. Raimond, and S. Haroche, "Generation of Einstein-Podolsky-Rosen pairs of atoms," Phys. Rev. Lett. 79, 1-5 (1997).
[CrossRef]

S. Haroche and J. M. Raimond, "Manipulation of nonclassical field states," in Cavity Electrodynamics, P.Berman, ed. (Academic, 1994), pp. 126-127.

Imamoglu, A.

Y. Yamamoto and A. Imamoglu, Mesoscopic Quantum Optics (Wiley-Interscience, 1999).

Ippen, E. P.

M. Qi, E. Lidorikis, P. T. Rakich, S. G. Johnson, J. D. Joannopoulos, E. P. Ippen, and H. I. Smith, "A three-dimensional optical photonic crystal with designed point defects," Nature 429, 538-542 (2004).
[CrossRef] [PubMed]

Joannopoulos, J. D.

M. Qi, E. Lidorikis, P. T. Rakich, S. G. Johnson, J. D. Joannopoulos, E. P. Ippen, and H. I. Smith, "A three-dimensional optical photonic crystal with designed point defects," Nature 429, 538-542 (2004).
[CrossRef] [PubMed]

M. L. Povinelli, S. G. Johnson, S. Fan, and J. D. Joannopoulos, "Emulation of two-dimensional photonic crystal defect modes in a photonic crystal with a three-dimensional photonic band gap," Phys. Rev. B 64, 075313-075320 (2001).
[CrossRef]

S. G. Johnson and J. D. Joannopoulos, "Block-iterative frequency-domain methods for Maxwell's equations in a plane-wave basis," Opt. Express 8, 173-190 (2001).
[CrossRef] [PubMed]

S. G. Johnson and J. D. Joannopoulos, "Three-dimensionally periodic dielectric layered structure with omnidirectional photonic band gap," Appl. Phys. Lett. 77, 3490-3492 (2000).
[CrossRef]

P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, "Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency," Phys. Rev. B 54, 7837-7842 (1996).
[CrossRef]

S. Fan, P. R. Villeneuve, R. D. Meade, and J. D. Joannopoulos, "Design of three-dimensional photonic crystals at submicron lengthscales," Appl. Phys. Lett. 65, 1466-1468 (1994).
[CrossRef]

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

John, S.

S. John and J. Wang, "Quantum optics of localized light in a photonic band gap," Phys. Rev. B 43, 12772-12789 (1991).
[CrossRef]

Johnson, S. G.

M. Qi, E. Lidorikis, P. T. Rakich, S. G. Johnson, J. D. Joannopoulos, E. P. Ippen, and H. I. Smith, "A three-dimensional optical photonic crystal with designed point defects," Nature 429, 538-542 (2004).
[CrossRef] [PubMed]

S. G. Johnson and J. D. Joannopoulos, "Block-iterative frequency-domain methods for Maxwell's equations in a plane-wave basis," Opt. Express 8, 173-190 (2001).
[CrossRef] [PubMed]

M. L. Povinelli, S. G. Johnson, S. Fan, and J. D. Joannopoulos, "Emulation of two-dimensional photonic crystal defect modes in a photonic crystal with a three-dimensional photonic band gap," Phys. Rev. B 64, 075313-075320 (2001).
[CrossRef]

S. G. Johnson and J. D. Joannopoulos, "Three-dimensionally periodic dielectric layered structure with omnidirectional photonic band gap," Appl. Phys. Lett. 77, 3490-3492 (2000).
[CrossRef]

Kimble, H. J.

H. J. Kimble, "Strong interactions of single atoms and photons in cavity QED," Phys. Scr. T76, 127-137 (1998).
[CrossRef]

Knight, P. L.

D. G. Angelakis and P. L. Knight, "Testing Bell inequalities in photonic crystals," Eur. Phys. J. D 18, 247-250 (2002).
[CrossRef]

Konopka, M.

M. Konopka and V. Buzek, "Entangling atoms in photonic crystals," Eur. Phys. J. D 10, 285-293 (2000).
[CrossRef]

Lewenstein, M.

R. J. Glauber and M. Lewenstein, "Quantum optics of dielectric media," Phys. Rev. A 43, 467-491 (1991).
[CrossRef] [PubMed]

Lidorikis, E.

M. Qi, E. Lidorikis, P. T. Rakich, S. G. Johnson, J. D. Joannopoulos, E. P. Ippen, and H. I. Smith, "A three-dimensional optical photonic crystal with designed point defects," Nature 429, 538-542 (2004).
[CrossRef] [PubMed]

Loncar, M.

J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, "Design of photonic crystal microcavities for cavity QED," Phys. Rev. E 65, 016608-016618 (2001).
[CrossRef]

Mabuchi, H.

J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, "Design of photonic crystal microcavities for cavity QED," Phys. Rev. E 65, 016608-016618 (2001).
[CrossRef]

Maitre, X.

E. Hagley, X. Maitre, G. Nogues, C. Wunderlich, M. Brune, J. M. Raimond, and S. Haroche, "Generation of Einstein-Podolsky-Rosen pairs of atoms," Phys. Rev. Lett. 79, 1-5 (1997).
[CrossRef]

Meade, R. D.

S. Fan, P. R. Villeneuve, R. D. Meade, and J. D. Joannopoulos, "Design of three-dimensional photonic crystals at submicron lengthscales," Appl. Phys. Lett. 65, 1466-1468 (1994).
[CrossRef]

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

Nielsen, M. A.

M. A. Nielsen and I. L. Chuang, Quantum Information and Quantum Computing (Cambridge U. Press, 2000).

Nogues, G.

E. Hagley, X. Maitre, G. Nogues, C. Wunderlich, M. Brune, J. M. Raimond, and S. Haroche, "Generation of Einstein-Podolsky-Rosen pairs of atoms," Phys. Rev. Lett. 79, 1-5 (1997).
[CrossRef]

Özbay, E.

E. Özbay, Nanotechnology Research Center, Bilkent University, Ankara, Turkey, 06800 (personal communication, 2006).

Pittman, T. B.

T. B. Pittman and J. D. Franson, "Cyclical quantum memory for photonic qubits," Phys. Rev. A 66, 062302-062305 (2002).
[CrossRef]

Povinelli, M. L.

M. L. Povinelli, S. G. Johnson, S. Fan, and J. D. Joannopoulos, "Emulation of two-dimensional photonic crystal defect modes in a photonic crystal with a three-dimensional photonic band gap," Phys. Rev. B 64, 075313-075320 (2001).
[CrossRef]

Qi, M.

M. Qi, E. Lidorikis, P. T. Rakich, S. G. Johnson, J. D. Joannopoulos, E. P. Ippen, and H. I. Smith, "A three-dimensional optical photonic crystal with designed point defects," Nature 429, 538-542 (2004).
[CrossRef] [PubMed]

Raimond, J. M.

E. Hagley, X. Maitre, G. Nogues, C. Wunderlich, M. Brune, J. M. Raimond, and S. Haroche, "Generation of Einstein-Podolsky-Rosen pairs of atoms," Phys. Rev. Lett. 79, 1-5 (1997).
[CrossRef]

S. Haroche and J. M. Raimond, "Manipulation of nonclassical field states," in Cavity Electrodynamics, P.Berman, ed. (Academic, 1994), pp. 126-127.

Rakich, P. T.

M. Qi, E. Lidorikis, P. T. Rakich, S. G. Johnson, J. D. Joannopoulos, E. P. Ippen, and H. I. Smith, "A three-dimensional optical photonic crystal with designed point defects," Nature 429, 538-542 (2004).
[CrossRef] [PubMed]

Rudolph, T.

N. Vats and T. Rudolph, "Quantum information processing in localized modes of light within a photonic bandgap material," J. Mod. Opt. 48, 1495-1502 (2001).
[CrossRef]

Sakurai, J. J.

J. J. Sakurai, Modern Quantum Mechanics (Addison-Wesley, 1994).

Scherer, A.

J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, "Design of photonic crystal microcavities for cavity QED," Phys. Rev. E 65, 016608-016618 (2001).
[CrossRef]

Smith, H. I.

M. Qi, E. Lidorikis, P. T. Rakich, S. G. Johnson, J. D. Joannopoulos, E. P. Ippen, and H. I. Smith, "A three-dimensional optical photonic crystal with designed point defects," Nature 429, 538-542 (2004).
[CrossRef] [PubMed]

Soukoulis, C. M.

C. M. Soukoulis, "The history and a review of the modelling and fabrication of photonic crystals," Nanotechnology 13, 420-423 (2002).
[CrossRef]

Tavis, M.

M. Tavis and F. W. Cummings, "Exact solution for N-molecule-radiation field Hamiltonian," Phys. Rev. 170, 379-384 (1968).
[CrossRef]

Vats, N.

N. Vats and T. Rudolph, "Quantum information processing in localized modes of light within a photonic bandgap material," J. Mod. Opt. 48, 1495-1502 (2001).
[CrossRef]

Villeneuve, P. R.

P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, "Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency," Phys. Rev. B 54, 7837-7842 (1996).
[CrossRef]

S. Fan, P. R. Villeneuve, R. D. Meade, and J. D. Joannopoulos, "Design of three-dimensional photonic crystals at submicron lengthscales," Appl. Phys. Lett. 65, 1466-1468 (1994).
[CrossRef]

Vuckovic, J.

J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, "Design of photonic crystal microcavities for cavity QED," Phys. Rev. E 65, 016608-016618 (2001).
[CrossRef]

Wang, J.

S. John and J. Wang, "Quantum optics of localized light in a photonic band gap," Phys. Rev. B 43, 12772-12789 (1991).
[CrossRef]

Winn, J. N.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

Wunderlich, C.

E. Hagley, X. Maitre, G. Nogues, C. Wunderlich, M. Brune, J. M. Raimond, and S. Haroche, "Generation of Einstein-Podolsky-Rosen pairs of atoms," Phys. Rev. Lett. 79, 1-5 (1997).
[CrossRef]

Yablonovitch, E.

E. Yablonovitch, "Photonic crystals," J. Mod. Opt. 41, 173-194 (1994).
[CrossRef]

Yamamoto, Y.

Y. Yamamoto and A. Imamoglu, Mesoscopic Quantum Optics (Wiley-Interscience, 1999).

Appl. Phys. Lett.

S. G. Johnson and J. D. Joannopoulos, "Three-dimensionally periodic dielectric layered structure with omnidirectional photonic band gap," Appl. Phys. Lett. 77, 3490-3492 (2000).
[CrossRef]

S. Fan, P. R. Villeneuve, R. D. Meade, and J. D. Joannopoulos, "Design of three-dimensional photonic crystals at submicron lengthscales," Appl. Phys. Lett. 65, 1466-1468 (1994).
[CrossRef]

Eur. Phys. J. D

M. Konopka and V. Buzek, "Entangling atoms in photonic crystals," Eur. Phys. J. D 10, 285-293 (2000).
[CrossRef]

D. G. Angelakis and P. L. Knight, "Testing Bell inequalities in photonic crystals," Eur. Phys. J. D 18, 247-250 (2002).
[CrossRef]

J. Mod. Opt.

N. Vats and T. Rudolph, "Quantum information processing in localized modes of light within a photonic bandgap material," J. Mod. Opt. 48, 1495-1502 (2001).
[CrossRef]

E. Yablonovitch, "Photonic crystals," J. Mod. Opt. 41, 173-194 (1994).
[CrossRef]

J. Opt. Soc. Am. B

Nanotechnology

C. M. Soukoulis, "The history and a review of the modelling and fabrication of photonic crystals," Nanotechnology 13, 420-423 (2002).
[CrossRef]

Nature

M. Qi, E. Lidorikis, P. T. Rakich, S. G. Johnson, J. D. Joannopoulos, E. P. Ippen, and H. I. Smith, "A three-dimensional optical photonic crystal with designed point defects," Nature 429, 538-542 (2004).
[CrossRef] [PubMed]

Opt. Express

Phys. Rev.

M. Tavis and F. W. Cummings, "Exact solution for N-molecule-radiation field Hamiltonian," Phys. Rev. 170, 379-384 (1968).
[CrossRef]

Phys. Rev. A

T. B. Pittman and J. D. Franson, "Cyclical quantum memory for photonic qubits," Phys. Rev. A 66, 062302-062305 (2002).
[CrossRef]

R. J. Glauber and M. Lewenstein, "Quantum optics of dielectric media," Phys. Rev. A 43, 467-491 (1991).
[CrossRef] [PubMed]

Phys. Rev. B

M. L. Povinelli, S. G. Johnson, S. Fan, and J. D. Joannopoulos, "Emulation of two-dimensional photonic crystal defect modes in a photonic crystal with a three-dimensional photonic band gap," Phys. Rev. B 64, 075313-075320 (2001).
[CrossRef]

P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, "Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency," Phys. Rev. B 54, 7837-7842 (1996).
[CrossRef]

S. John and J. Wang, "Quantum optics of localized light in a photonic band gap," Phys. Rev. B 43, 12772-12789 (1991).
[CrossRef]

Phys. Rev. E

J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, "Design of photonic crystal microcavities for cavity QED," Phys. Rev. E 65, 016608-016618 (2001).
[CrossRef]

Phys. Rev. Lett.

E. Hagley, X. Maitre, G. Nogues, C. Wunderlich, M. Brune, J. M. Raimond, and S. Haroche, "Generation of Einstein-Podolsky-Rosen pairs of atoms," Phys. Rev. Lett. 79, 1-5 (1997).
[CrossRef]

Phys. Scr.

H. J. Kimble, "Strong interactions of single atoms and photons in cavity QED," Phys. Scr. T76, 127-137 (1998).
[CrossRef]

Other

S. Haroche and J. M. Raimond, "Manipulation of nonclassical field states," in Cavity Electrodynamics, P.Berman, ed. (Academic, 1994), pp. 126-127.

Y. Yamamoto and A. Imamoglu, Mesoscopic Quantum Optics (Wiley-Interscience, 1999).

J. Vuckovic, Ginzton Laboratory, Stanford University, Stanford, Calif. 94305 (personal communication, 2004).

J. J. Sakurai, Modern Quantum Mechanics (Addison-Wesley, 1994).

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

M. A. Nielsen and I. L. Chuang, Quantum Information and Quantum Computing (Cambridge U. Press, 2000).

E. Özbay, Nanotechnology Research Center, Bilkent University, Ankara, Turkey, 06800 (personal communication, 2006).

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Figures (13)

Fig. 1
Fig. 1

Probability amplitudes as a function of the velocity and p. Surfaces (a) a ( V , p ) and (b) b ( V , p ) [or a ( V , p ) ] if the initial state is 100 (or 010 ). (c) Surface b ( V , p ) , if the system is initially prepared in the 010 state.

Fig. 2
Fig. 2

Slices from each surface in Fig. 1 (a) Probability amplitudes a ( V , p ) —solid curve—and b ( V , p ) —dashed curve—with V = 433 m s . The entangled state, Eq. (19), is obtained at p = 0.414 . (b) Probability amplitudes a ( V , p ) —dashed curve—and b ( V , p ) —solid curve—with the same velocity, V = 433 m s . The entangled state, Eq. (20), is observed at the same value, p = 0.414 .

Fig. 3
Fig. 3

Coupling parameters in the reference frame of moving atoms with velocities V = 433 m s at p = 0.414 . Ω 0 = 11 × 10 9 Hz , ω = 2.4 × 10 15 Hz , l = 1.6 π c ω , L = 10 l , R def = l .[6] Atom A experiences the coupling parameter shown with the solid curve and atom B the one shown with the dashed curve.

Fig. 4
Fig. 4

Time evolution of the probabilities showing the final entanglement when (a) the initial state is 100 and (b) 010 .

Fig. 5
Fig. 5

Time evolution of the probabilities that leads to a dual-rail NOT (Pauli σ x ) logic operation when the initial state is (a) 100 and (b) 010 .

Fig. 6
Fig. 6

(a) Single-mode microcavity in a 2D photonic crystal with a triangular lattice (see the text for details). (b) Corresponding electric field spatial profile for the TM mode allowed in the cavity.

Fig. 7
Fig. 7

Normalized coupling parameters (i.e., divided by g 0 ) in the reference frame of moving atoms with velocities V = 374 m s at p = 0.414 , where g 0 is found to be 2.765 MHz . The black curve and the gray curve correspond to normalized coupling strengths for atoms A and B, respectively.

Fig. 8
Fig. 8

Probability amplitudes for entangled atoms created by a dual-rail Hadamard operation in the 2D photonic crystal [see Fig. 6a] when (a) atom A is initially in the excited state and (b) when atom B is initially in the excited state. a ( t ) , b ( t ) , and γ ( t ) are the probability amplitudes for the states 100 , 010 and 001 , respectively.

Fig. 9
Fig. 9

Probability amplitudes for atoms A and B under the dual-rail NOT operation in the 2D photonic crystal when initially (a) only atom A is excited and (b) only atom B is excited.

Fig. 10
Fig. 10

(a) Top view of the 3D photonic crystal with fcc lattice. It consists of alternating layers of a triangular lattice of air holes and a triangular lattice of dielectric rods (for details of the structure see Refs. [5, 22, 23]). The nearest-neighbor spacing within either a hole or rod layer is ( 1 2 ) l , where l is the fcc lattice constant. The hole and rod radii are 0.293 l and 0.124 l , respectively. The thicknesses of a hole layer and a rod layer are taken to be 0.225 l and 0.354 l , respectively. Silicon is assumed as the high-index material of dielectric constant 12. (b) Horizontal cross section of the crystal. Dashed lines show the obvious paths atoms can travel. In our simulations we assumed the path shown by the dashed horizontal line. A defect is introduced by reducing the radius of the middle rod down to 0.050 l to hold a single mode in the cavity. The (c) real part and (d) imaginary part of the electric field of the TM mode allowed in the cavity at a particular instant in time with the frequency of 0.539 c l . The imaginary part is one-half a period later.

Fig. 11
Fig. 11

Coupling parameters in the reference frame of the moving atoms with both velocities V = 353 m s , and p = 0.414 , where g 0 is found to be 2.899 MHz . The black curve and the gray curve correspond to normalized coupling strengths for atoms A and B, respectively.

Fig. 12
Fig. 12

Probability amplitudes for the entangled atoms under dual-rail Hadamard operation in the 3D photonic crystal (see Fig. 10) when (a) atom A is initially in the excited state and (b) atom B is initially in the excited state. a ( t ) , b ( t ) , and γ ( t ) are the probability amplitudes for the states 100 , 010 and 001 , respectively.

Fig. 13
Fig. 13

Probability amplitudes for atoms A and B under the dual-rail NOT operation in the 3D photonic crystal when initially (a) only atom A is excited and (b) only atom B is excited.

Equations (32)

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H ( t ) = ϖ 2 j σ z j + ϖ α α + j [ G j ( t ) σ + j α + H.c. ] ,
G j ( t ) = Ω 0 f j ( t ) cos ( ζ j ) ,
Ψ ( 0 ) = 100 ,
Ψ ( t ) = a ( t ) 100 + b ( t ) 010 + γ ( t ) 001
i t U ( t , t 0 ) = H ( t ) U ( t , t 0 ) .
H 11 = H 12 = H 21 = H 22 = H 33 = 0 ,
H 13 = H 31 = G A ( t ) ,
H 23 = H 32 = G B ( t ) .
U ( t , t 0 ) = exp [ i t 0 t d τ H ( τ ) ] = exp ( i I ) ,
U ( t , t 0 ) = 1 + ( i ) I + 1 2 ! ( i ) 2 I 2 + + ( 1 n ! ) ( i ) n I n + .
a ( t ) = 1 + G A 2 n = 1 ( 1 ) n 1 2 ! ( G A 2 + G B 2 ) n 1 ,
b ( t ) = G A G B n = 1 ( 1 ) n 1 2 n ! ( G A 2 + G B 2 ) n 1 ,
γ ( t ) = i G A n = 1 ( 1 ) n 1 ( 2 n 1 ) ! ( G A 2 + G B 2 ) n 1 ,
G j = t 0 t G j ( τ ) d τ .
a ( t ) = 1 + G A 2 G A 2 + G B 2 [ cos ( G A 2 + G B 2 ) 1 2 1 ] ,
b ( t ) = G A G B G A 2 + G B 2 [ cos ( G A 2 + G B 2 ) 1 2 1 ] ,
γ ( t ) = i G A ( G A 2 + G B 2 ) 1 2 sin ( G A 2 + G B 2 ) 1 2 .
f i = exp ( V j t L R def ) cos [ π l ( V j t L ) ] ,
Ψ 10 10 + 01 2 ,
Ψ 01 10 01 2 ,
10 01 ,
01 10 .
H I = g ( r ) ( α σ + α σ + )
g ( r j ) = g 0 Ψ ( r j ) cos ( ζ j ) ,
g 0 = μ e g ( ω 2 ε 0 ε m V mode ) 1 2 ,
Ψ ( r j ) E ( r j ) E ( r m ) .
V mode = ε ( r ) E ( r ) 2 d r ε m E ( r m ) 2 .
P = d 2 r E z ( ω ; r ) 2 d 2 r E ( ω ; r ) 2 .
00 00 ,
01 10 ,
10 01 ,
11 11 .

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