Abstract

The Dicke subradiance and superradiance resulting from the interaction between surface plasmons of a nanosphere and an ensemble of quantum emitters have been investigated using a Green’s function approach. Based on such an investigation, we propose a scheme for a deterministic multi-qubit quantum phase gate. As an example, two-qubit, three-qubit, and four-qubit quantum phase gates have been designed and analyzed in detail. Phenomena due to the losses in the metal are discussed. Potential applications of these phenomena to quantum-information processing are anticipated.

© 2014 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  33. H. T. Dung, S. Scheel, D.-G. Welsch, and L. Knöll, “Atomic entanglement near a realistic microsphere,” J. Opt. B 4, S169–S175 (2002).
  34. D. Han, Y. Lai, J. Zi, Z. Zhang, and C. T. Chan, “Dirac spectra and edge states in honeycomb plasmonic lattices,” Phys. Rev. Lett. 102, 123904 (2009).
    [CrossRef]
  35. H. T. Dung, L. Knöll, and D.-G. Welsch, “Decay of an excited atom near an absorbing sphere,” Phys. Rev. A 64, 013804 (2001).
    [CrossRef]
  36. R. Ruppin, “Extinction due to surface modes near a spherical inclusion in a dispersive medium,” Phys. Status Solidi B 233, 331–338 (2002).
    [CrossRef]

2011 (1)

A. Gonzalez-Tudela, D. Martin-Cano, E. Moreno, L. Martin-Moreno, C. Tejedor, and F. J. Garcia-Vidal, “Entanglement of two qubits mediated by one-dimensional plasmonic waveguides,” Phys. Rev. Lett. 106, 020501 (2011).
[CrossRef]

2010 (1)

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

2009 (6)

Y. N. Pustovit and T. V. Shahbazyan, “Cooperative emission of light by an ensemble of dipoles near a metal nanoparticle: the plasmonic Dicke effect,” Phys. Rev. Lett. 102, 077401 (2009).
[CrossRef]

R. Kolesov, B. Grotz, G. Balasubramanian, R. J. Stohr, A. A. L. Nicolet, P. R. Hemmer, F. Jelezko, and J. Wrachtrup, “Wave-particle duality of single surface plasmon polaritons,” Nat. Phys. 5, 470–474 (2009).
[CrossRef]

D. Han, Y. Lai, J. Zi, Z. Zhang, and C. T. Chan, “Dirac spectra and edge states in honeycomb plasmonic lattices,” Phys. Rev. Lett. 102, 123904 (2009).
[CrossRef]

A. L. Falk, F. H. L. Koppens, C. L. Yu, K. Kang, N. D. Snapp, A. V. Akimov, M. H. Jo, M. D. Lukin, and H. Park, “Near-field electrical detection of optical plasmons and single-plasmon sources,” Nat. Phys. 5, 475–479 (2009).
[CrossRef]

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
[CrossRef]

R. F. Oulton, V. J. Sorger, T. Zentgraf, R. M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
[CrossRef]

2007 (2)

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

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

2006 (3)

S. Kühn, U. Håkanson, L. Rogobete, and V. Sandoghdar, “Enhancement of single-molecule fluorescence using a gold nanoparticle as an optical nanoantenna,” Phys. Rev. Lett. 97, 017402 (2006).
[CrossRef]

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

A. Joshi and M. Xiao, “Three-qubit quantum-gate operation in a cavity QED system,” Phys. Rev. A 74, 052318 (2006).
[CrossRef]

2005 (2)

C. P. Yang and S. Han, “n-qubit-controlled phase gate with superconducting quantum-interference devices coupled to a resonator,” Phys. Rev. A 72, 032311 (2005).
[CrossRef]

I. I. Smolyaninov, J. Elliott, A. V. Zayats, and C. C. Davis, “Far-field optical microscopy with a nanometer-scale resolution based on the in-plane image magnification by surface plasmon polaritons,” Phys. Rev. Lett. 94, 057401 (2005).
[CrossRef]

2004 (1)

H. Goto and K. Ichimura, “Multi-qubit controlled unitary gate by adiabatic passage with an optical cavity,” Phys. Rev. A 70, 012305 (2004).
[CrossRef]

2003 (3)

X. Li, Y. Wu, D. Steel, D. Gammon, T. H. Stievater, D. D. Katzer, D. Park, C. Piermarocchi, and L. J. Sham, “An all-optical quantum gate in a semiconductor quantum dot,” Science 301, 809–811 (2003).
[CrossRef]

T. Yamamoto, Y. A. Pashkin, O. Astafiev, Y. Nakamura, and J. S. Tsai, “Demonstration of conditional gate operation using superconducting charge qubits,” Nature 425, 941–944 (2003).
[CrossRef]

J. K. Pachos and P. L. Knight, “Quantum computation with a one-dimensional optical lattice,” Phys. Rev. Lett. 91, 107902 (2003).
[CrossRef]

2002 (3)

V. V. Klimov, M. Ducloy, and V. S. Letokhov, “A model of an apertureless scanning microscope with a prolate nanospheroid as a tip and an excited molecule as an object,” Chem. Phys. Lett. 358, 192–198 (2002).
[CrossRef]

H. T. Dung, S. Scheel, D.-G. Welsch, and L. Knöll, “Atomic entanglement near a realistic microsphere,” J. Opt. B 4, S169–S175 (2002).

R. Ruppin, “Extinction due to surface modes near a spherical inclusion in a dispersive medium,” Phys. Status Solidi B 233, 331–338 (2002).
[CrossRef]

2001 (1)

H. T. Dung, L. Knöll, and D.-G. Welsch, “Decay of an excited atom near an absorbing sphere,” Phys. Rev. A 64, 013804 (2001).
[CrossRef]

1999 (1)

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]

1998 (1)

J. A. Jones, M. Mosca, and R. H. Hansen, “Implementation of a quantum search algorithm on a quantum computer,” Nature 393, 344–346 (1998).
[CrossRef]

1997 (3)

K. Kneipp, Y. Wang, H. Kneipp, L. T. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, “Single molecule detection using surface-enhanced Raman scattering (SERS),” Phys. Rev. Lett. 78, 1667–1670 (1997).
[CrossRef]

S. Nie and S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science 275, 1102–1106 (1997).
[CrossRef]

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

1995 (3)

T. Sleator and H. Weinfurter, “Realizable universal quantum logic gates,” Phys. Rev. Lett. 74, 4087–4090 (1995).
[CrossRef]

C. Monroe, D. M. Meekhof, B. E. King, W. M. Itano, and D. J. Wineland, “Demonstration of a fundamental quantum logic gate,” Phys. Rev. Lett. 75, 4714–4717 (1995).
[CrossRef]

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]

Akimov, A. V.

A. L. Falk, F. H. L. Koppens, C. L. Yu, K. Kang, N. D. Snapp, A. V. Akimov, M. H. Jo, M. D. Lukin, and H. Park, “Near-field electrical detection of optical plasmons and single-plasmon sources,” Nat. Phys. 5, 475–479 (2009).
[CrossRef]

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

Astafiev, O.

T. Yamamoto, Y. A. Pashkin, O. Astafiev, Y. Nakamura, and J. S. Tsai, “Demonstration of conditional gate operation using superconducting charge qubits,” Nature 425, 941–944 (2003).
[CrossRef]

Bakker, R.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
[CrossRef]

Balasubramanian, G.

R. Kolesov, B. Grotz, G. Balasubramanian, R. J. Stohr, A. A. L. Nicolet, P. R. Hemmer, F. Jelezko, and J. Wrachtrup, “Wave-particle duality of single surface plasmon polaritons,” Nat. Phys. 5, 470–474 (2009).
[CrossRef]

Bartal, G.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R. M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
[CrossRef]

Belgrave, A. M.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
[CrossRef]

Benenti, G.

G. Benenti, G. Casati, and G. Strini, Principles of Quantum Computation and Information Volume I: Basic Concepts (World Scientific, 2004).

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]

Bouwmeester, D.

D. Bouwmeester, A. Ekert, and A. Zeilinger, The Physics of Quantum Information (Springer, 2000).

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]

Casati, G.

G. Benenti, G. Casati, and G. Strini, Principles of Quantum Computation and Information Volume I: Basic Concepts (World Scientific, 2004).

Chan, C. T.

D. Han, Y. Lai, J. Zi, Z. Zhang, and C. T. Chan, “Dirac spectra and edge states in honeycomb plasmonic lattices,” Phys. Rev. Lett. 102, 123904 (2009).
[CrossRef]

Chang, D. E.

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

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

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

Chuang, I. L.

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

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2000).

Dai, L.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R. M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
[CrossRef]

Dasari, R. R.

K. Kneipp, Y. Wang, H. Kneipp, L. T. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, “Single molecule detection using surface-enhanced Raman scattering (SERS),” Phys. Rev. Lett. 78, 1667–1670 (1997).
[CrossRef]

Davis, C. C.

I. I. Smolyaninov, J. Elliott, A. V. Zayats, and C. C. Davis, “Far-field optical microscopy with a nanometer-scale resolution based on the in-plane image magnification by surface plasmon polaritons,” Phys. Rev. Lett. 94, 057401 (2005).
[CrossRef]

Demler, E. A.

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

Ducloy, M.

V. V. Klimov, M. Ducloy, and V. S. Letokhov, “A model of an apertureless scanning microscope with a prolate nanospheroid as a tip and an excited molecule as an object,” Chem. Phys. Lett. 358, 192–198 (2002).
[CrossRef]

Dung, H. T.

H. T. Dung, S. Scheel, D.-G. Welsch, and L. Knöll, “Atomic entanglement near a realistic microsphere,” J. Opt. B 4, S169–S175 (2002).

H. T. Dung, L. Knöll, and D.-G. Welsch, “Decay of an excited atom near an absorbing sphere,” Phys. Rev. A 64, 013804 (2001).
[CrossRef]

Dzsotjan, D.

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

Ekert, A.

D. Bouwmeester, A. Ekert, and A. Zeilinger, The Physics of Quantum Information (Springer, 2000).

Elliott, J.

I. I. Smolyaninov, J. Elliott, A. V. Zayats, and C. C. Davis, “Far-field optical microscopy with a nanometer-scale resolution based on the in-plane image magnification by surface plasmon polaritons,” Phys. Rev. Lett. 94, 057401 (2005).
[CrossRef]

Emory, S. R.

S. Nie and S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science 275, 1102–1106 (1997).
[CrossRef]

Falk, A. L.

A. L. Falk, F. H. L. Koppens, C. L. Yu, K. Kang, N. D. Snapp, A. V. Akimov, M. H. Jo, M. D. Lukin, and H. Park, “Near-field electrical detection of optical plasmons and single-plasmon sources,” Nat. Phys. 5, 475–479 (2009).
[CrossRef]

Feld, M. S.

K. Kneipp, Y. Wang, H. Kneipp, L. T. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, “Single molecule detection using surface-enhanced Raman scattering (SERS),” Phys. Rev. Lett. 78, 1667–1670 (1997).
[CrossRef]

Fleischhauer, M.

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

Gammon, D.

X. Li, Y. Wu, D. Steel, D. Gammon, T. H. Stievater, D. D. Katzer, D. Park, C. Piermarocchi, and L. J. Sham, “An all-optical quantum gate in a semiconductor quantum dot,” Science 301, 809–811 (2003).
[CrossRef]

Garcia-Vidal, F. J.

A. Gonzalez-Tudela, D. Martin-Cano, E. Moreno, L. Martin-Moreno, C. Tejedor, and F. J. Garcia-Vidal, “Entanglement of two qubits mediated by one-dimensional plasmonic waveguides,” Phys. Rev. Lett. 106, 020501 (2011).
[CrossRef]

Gershenfeld, N. A.

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

Gladden, C.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R. M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
[CrossRef]

Gonzalez-Tudela, A.

A. Gonzalez-Tudela, D. Martin-Cano, E. Moreno, L. Martin-Moreno, C. Tejedor, and F. J. Garcia-Vidal, “Entanglement of two qubits mediated by one-dimensional plasmonic waveguides,” Phys. Rev. Lett. 106, 020501 (2011).
[CrossRef]

Goto, H.

H. Goto and K. Ichimura, “Multi-qubit controlled unitary gate by adiabatic passage with an optical cavity,” Phys. Rev. A 70, 012305 (2004).
[CrossRef]

Grotz, B.

R. Kolesov, B. Grotz, G. Balasubramanian, R. J. Stohr, A. A. L. Nicolet, P. R. Hemmer, F. Jelezko, and J. Wrachtrup, “Wave-particle duality of single surface plasmon polaritons,” Nat. Phys. 5, 470–474 (2009).
[CrossRef]

Håkanson, U.

S. Kühn, U. Håkanson, L. Rogobete, and V. Sandoghdar, “Enhancement of single-molecule fluorescence using a gold nanoparticle as an optical nanoantenna,” Phys. Rev. Lett. 97, 017402 (2006).
[CrossRef]

Han, D.

D. Han, Y. Lai, J. Zi, Z. Zhang, and C. T. Chan, “Dirac spectra and edge states in honeycomb plasmonic lattices,” Phys. Rev. Lett. 102, 123904 (2009).
[CrossRef]

Han, S.

C. P. Yang and S. Han, “n-qubit-controlled phase gate with superconducting quantum-interference devices coupled to a resonator,” Phys. Rev. A 72, 032311 (2005).
[CrossRef]

Hansen, R. H.

J. A. Jones, M. Mosca, and R. H. Hansen, “Implementation of a quantum search algorithm on a quantum computer,” Nature 393, 344–346 (1998).
[CrossRef]

Haroche, S.

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]

Hemmer, P. R.

R. Kolesov, B. Grotz, G. Balasubramanian, R. J. Stohr, A. A. L. Nicolet, P. R. Hemmer, F. Jelezko, and J. Wrachtrup, “Wave-particle duality of single surface plasmon polaritons,” Nat. Phys. 5, 470–474 (2009).
[CrossRef]

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

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X. Li, Y. Wu, D. Steel, D. Gammon, T. H. Stievater, D. D. Katzer, D. Park, C. Piermarocchi, and L. J. Sham, “An all-optical quantum gate in a semiconductor quantum dot,” Science 301, 809–811 (2003).
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R. F. Oulton, V. J. Sorger, T. Zentgraf, R. M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
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X. Li, Y. Wu, D. Steel, D. Gammon, T. H. Stievater, D. D. Katzer, D. Park, C. Piermarocchi, and L. J. Sham, “An all-optical quantum gate in a semiconductor quantum dot,” Science 301, 809–811 (2003).
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X. Li, Y. Wu, D. Steel, D. Gammon, T. H. Stievater, D. D. Katzer, D. Park, C. Piermarocchi, and L. J. Sham, “An all-optical quantum gate in a semiconductor quantum dot,” Science 301, 809–811 (2003).
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M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
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T. Yamamoto, Y. A. Pashkin, O. Astafiev, Y. Nakamura, and J. S. Tsai, “Demonstration of conditional gate operation using superconducting charge qubits,” Nature 425, 941–944 (2003).
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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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T. Sleator and H. Weinfurter, “Realizable universal quantum logic gates,” Phys. Rev. Lett. 74, 4087–4090 (1995).
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H. T. Dung, S. Scheel, D.-G. Welsch, and L. Knöll, “Atomic entanglement near a realistic microsphere,” J. Opt. B 4, S169–S175 (2002).

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M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
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C. Monroe, D. M. Meekhof, B. E. King, W. M. Itano, and D. J. Wineland, “Demonstration of a fundamental quantum logic gate,” Phys. Rev. Lett. 75, 4714–4717 (1995).
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X. Li, Y. Wu, D. Steel, D. Gammon, T. H. Stievater, D. D. Katzer, D. Park, C. Piermarocchi, and L. J. Sham, “An all-optical quantum gate in a semiconductor quantum dot,” Science 301, 809–811 (2003).
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Figures (9)

Fig. 1.
Fig. 1.

Plasmonic coupling of emitters near a metal nanoparticle.

Fig. 2.
Fig. 2.

(a) Dispersion curves of n=1, 2, 3 surface polaritons for a metal sphere with different radii. (b) Corresponding extinction spectrum with a=20nm and r=25nm. The dielectric constant of the metal we use is described by the Drude model, ωp=6.18eV and γ=0.05eV.

Fig. 3.
Fig. 3.

(a) Positions of two emitters (A and B) near a metal nanoparticle. (b) ΓAB/Γ as a function of the angle between two emitters around the sphere. The solid, dashed, and dotted lines correspond to the cases with ω=0.55ωp, 0.625ωp, and 0.652ωp, respectively. The other parameters are identical with those in Fig. 2.

Fig. 4.
Fig. 4.

Realization of a deterministic two-qubit quantum phase gate by applying external classical 2π pulses (a) using |gg, subradiant and superradiant states, and (b) for |sg, |se, |gs and |es states.

Fig. 5.
Fig. 5.

(a) ΓAB/Γ as a function of the angle between two emitters around the sphere. (b) Fidelity of a maximally entangled state, created by the phase gate, as a function of the distance between atoms and the sphere center. (c) Coated sphere with a=20nm, b=24nm, and the parameters of the core are identical with those in Fig. 2. (d) Positions of the two atoms and the sphere used in (b). In (a) and (b), the lines of the same color represent the same system. Red: no coating; blue: coated with ε=1.20.01i; green: coated with ε=1.20.05i. The other parameters are identical with those in Fig. 3.

Fig. 6.
Fig. 6.

Fidelity of a maximally entangled state created by the phase gate as a function of the distance between atoms and the center of a metal aluminum sphere. The radii of aluminum core and coated sphere are taken as a=20nm and b=24nm, respectively. The solid line: without coating at ω=0.59ωp, the dashed and dotted lines correspond to the aluminum sphere coated by ε=1.20.1i and ε=1.20.16i at ω=0.57ωp, respectively. The dielectric constant of aluminum is described by the Drude model, ωp=2.27×104THz and γ=0.05ωp.

Fig. 7.
Fig. 7.

Realization of a deterministic three-qubit quantum phase gate by applying external classical 2π pulses (a) using |ggg, subradiant, and superradiant states for (b) |gss, |sgs, |ssg states and for (c) |sgg states.

Fig. 8.
Fig. 8.

Realization of a deterministic four-qubit quantum phase gate by applying external classical 2π pulses (a) using |gggg, subradiant, and superradiant states for (b) |gsss, |sgss, |ssgs and |sssg states, for (c) |ssgg states, and for (d) |sggg states.

Fig. 9.
Fig. 9.

Fidelity of a maximally entangled state created by the phase gate as a function of the distance between the atoms and the sphere center. The inset shows the positions of atoms around the sphere for (a) the case with three atoms and (b) for the case with four atoms. The other parameters are identical with those in Fig. 2.

Equations (40)

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H^=d3r⃗0dωωf⃗^(r⃗,ω)f⃗^(r⃗,ω)+A12ωAσ^AzA[σ^AE⃗^(+)(r⃗A)d⃗A+H.c.],
E⃗^(+)(r⃗)=iπε00dωω2c2d3r⃗εI(r⃗,ω)G(r⃗,r⃗,ω)f⃗^(r⃗,ω),
|ψ(t)=ACUA(t)ei(ωAω¯)t|UA|{0}+d3r⃗0dω[CLi(r⃗,ω,t)ei(ωω¯)t|L|{1i(r⃗,ω)}],
CUA(t)=A0tdtKAA(t,t)CUA(t),
KAA(t,t)=1πε00dω[ω2c2ei(ωωA)tei(ωωA)td⃗AImG(r⃗A,r⃗A,ω)d⃗A].
|i=ANxA|UA.
Ci=ANxACUA,
Ci(t)=AN0tdt[ANxAKAA(tt)]CUA(t).
ANxAKAAxA=ANxAKABxB=ANxAKACxC==k,
|KkKBAKCAKNAKABKkKCBKNBKACKBCKkKNCKANKBNKCNKk|=0.
Ci(t)=0tkCi(t)dt.
Ci(t)=e(Γ/2+iδ)t,
δ=1xAANxAδAAandΓ=1xAANxAΓAA,
ΓABΓ0=6π{Ren=1m=nnn(n+1)hn(k1r)jn(k1r)(k1r)2Ynm(θA,φA)Ynm*(θA,φA)+Ren=1m=nnn(n+1)(hn(k1r)(k1r))2Ynm(θA,φA)Ynm*(θA,φA)bnm}.
ΓAB/Γ={6π[Ren=1m=nnn(n+1)hn(k1r)jn(k1r)(k1r)2Ynm(π2,0.0)Ynm*(π2,φ)+Ren=1m=nnn(n+1)(hn(k1r)(k1r))2Ynm(π2,0.0)Ynm*(π2,φ)bnm]Γ0}/Γ,
Γ=(1+32n=1n(n+1)(2n+1)Re{bn[hn(k1r)]2(k1r)2})Γ0.
|KkKABKBAKk|=0,
|1=1/2(|UA|UB)|2=1/2(|UA+|UB).
C1(t)=e(Γ1/2+iδ1)tC1(0)C2(t)=e(Γ2/2iδ1)tC1(0),
ΓΓAA,δδAAΓ1=ΓΓAB,δ1=δ+δABΓ2=Γ+ΓAB,δ2=δδAB.
Ω1=12(ΩA+ΩB),
Ω2=12(ΩAΩB).
|ψintinal=12(|ss+|sg+|gs+|gg),
|ss|ss;|sg|sg;|gs|gs;|gg|gg,
|ψideal=12(|ss+|sg+|gs|gg).
F=|ψideal|ψ|2=ψideal|ψψ|ψideal.
F=1Γ1Γ.
|3=13(|UA+|UB+|UC)|4=16(2|UA|UB|UC)|5=16(|UA+2|UB|UC),
Ci(t)=e(Γi/2+iδi)tCi(0);i=3,4,5,
Γ3=Γ+2ΓAB,δ3=δ+2δABΓ4(5)=ΓΓAB,δ4(5)=δδAB.
Ω3=13(ΩA+ΩB+ΩC),
Ω4=16(2ΩAΩBΩC),
Ω5=16(ΩA+2ΩBΩC).
|6=14(|UA+|UB+|UC+|UD)|7=112(3|UA|UB|UC|UD)|8=112(|UA+3|UB|UC|UD)|9=112(|UA|UB+3|UC|UD),
Ci(t)=e(Γi/2+iδi)tCi(0);i=6,7,8,9,
Γ6=Γ+3ΓAB,δ6=δ+3δABΓ7(8,9)=ΓΓAB,δ7(8,9)=δδAB.
Ω6=14(ΩA+ΩB+ΩC+ΩD),
Ω7=112(3ΩAΩBΩCΩD),
Ω8=112(ΩA+3ΩBΩCΩD),
Ω9=112(ΩAΩB+3ΩCΩD).

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