Abstract

We investigate the viability of coupled waveguides as basic units of quantum circuits. We study entanglement when the waveguides are fed in by light produced by a down-converter working either in low gain limit or under large gain. We present explicit analytical results for the measure of entanglement in terms of the logarithmic negativity for a variety of input states. We also address the effect of loss on entanglement dynamics of waveguide modes. Our results indicate that the waveguide structures are reasonably robust against the effect of loss and thus quite appropriate for quantum architectures as well as for the study of coherent phenomena like random walks. Our analysis is based on realistic structures used currently.

© 2010 Optical Society of America

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  1. R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, "Experimental Observation of Linear and Nonlinear Optical Bloch Oscillations," Phys. Rev. Lett. 83, 4756-4759 (1999).
    [CrossRef]
  2. D. N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices," Nature (London) 424, 817-823 (2003).
    [CrossRef]
  3. S. Longhi, "Optical analog of population trapping in the continuum: Classical and quantum interference effects," Phys. Rev. A 79, 023811 (2009).
    [CrossRef]
  4. H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, and Y. Silberberg, "Realization of Quantum Walks with Negligible Decoherence in Waveguide Lattices," Phys. Rev. Lett. 100, 170506 (2008).
    [CrossRef] [PubMed]
  5. Y. Bromberg, Y. Lahini, R. Morandotti, and Y. Silberberg, "Quantum and Classical Correlations in Waveguide Lattices," Phys. Rev. Lett. 102, 253904 (2009).
    [CrossRef] [PubMed]
  6. U. Peschel, T. Pertsch, and F. Lederer, "Optical Bloch oscillations in waveguide arrays," Opt. Lett. 23, 1701-1703 (1998).
    [CrossRef]
  7. T. Pertsch, P. Dannberg,W. Elflein, A. Brauer, and F. Lederer, "Optical Bloch Oscillations in Temperature Tuned Waveguide Arrays," Phys. Rev. Lett. 83, 4752-4755 (1999).
    [CrossRef]
  8. R. Iwanow, D. A. May-Arrioja, D. N. Christodoulides, G. I. Stegeman, Y. Min, and W. Sohler, "Discrete Talbot Effect in Waveguide Arrays," Phys. Rev. Lett. 95, 053902 (2005).
    [CrossRef] [PubMed]
  9. S. Longhi, "Optical Bloch Oscillations and Zener Tunneling with Nonclassical Light," Phys. Rev. Lett. 101, 193902 (2008).
    [CrossRef] [PubMed]
  10. A. Rai, G. S. Agarwal, and J. H. H. Perk, "Transport and quantum walk of nonclassical light in coupled waveguides," Phys. Rev. A 78, 042304 (2008).
    [CrossRef]
  11. A. Rai and G. S. Agarwal, "Possibility of coherent phenomena such as Bloch oscillations with single photons via W states," Phys. Rev. A 79, 053849 (2009).
    [CrossRef]
  12. A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, "Silica-on-Silicon Waveguide Quantum Circuits," Science 320, 646-649 (2008).
    [CrossRef] [PubMed]
  13. J. C. F. Matthews, A. Politi, A. Stefanov, and J. L. O’Brien, "Manipulation of multiphoton entanglement in waveguide quantum circuits," Nature Photonics 3, 346-350 (2009).
    [CrossRef]
  14. A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, "Quantum Interferometric Optical Lithography: Exploiting Entanglement to Beat the Diffraction Limit," Phys. Rev. Lett. 85, 2733-2736 (2000).
    [CrossRef] [PubMed]
  15. C. H. Bennett and D. P. DiVincenzo, "Quantum information and computation," Nature 404, 247-255 (2000).
    [CrossRef] [PubMed]
  16. W. H. Zurek, "Decoherence, einselection, and the quantum origins of the classical, " Rev. Mod. Phys. 75, 715-775 (2003).
    [CrossRef]
  17. B. Do, M. L. Stohler, S. Balasubramanian, D. S. Elliott, C. Eash, E. Fischbach, M. A. Fischbach, A. Mills, and B. Zwickl, "Experimental realization of a quantum quincunx by use of linear optical elements," J. Opt. Soc. Am. B 22, 499-504 (2005).
    [CrossRef]
  18. P. K. Pathak and G. S. Agarwal, "Quantum random walk of two photons in separable and entangled states," Phys. Rev. A 75, 032351 (2007).
    [CrossRef]
  19. A. Politi, J. C. F. Matthews, and J. L. O’Brien, "Shors Quantum Factoring Algorithm on a Photonic Chip," Science 325, 1221 (2009).
    [CrossRef] [PubMed]
  20. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd Edition, (Wiley, New York 2007), p. 319.
  21. W. K. Lai, V. Buˇzek, and P. L. Knight, "Nonclassical fields in a linear directional coupler, " Phys. Rev. A 43, 6323-6336 (1991).
    [CrossRef] [PubMed]
  22. R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, "Observation of Discrete Quadratic Solitons, " Phys. Rev. Lett. 93, 113902 (2004).
    [CrossRef] [PubMed]
  23. U. Peschel, R. Morandotti, J. M. Arnold, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, T. Pertsch, and F. Lederer, "Optical Discrete Solitons in Waveguide Arrays. 2. Dynamic Properties," J. Opt. Soc. Am. B 19, 2637- 2644 (2002).
    [CrossRef]
  24. K. B. Mogensen, F. Eriksson, O. Gustafsson, R. P. H. Nikolajsen, and J. P. Kutter, "Pure-silica optical waveguides, fiber couplers, and high-aspect ratio submicrometer channels for electrokinetic separation devices," Electrophoresis 25, 3788-3795 (2004).
    [CrossRef]
  25. G. Vidal and R. F. Werner, "Computable measure of entanglement," Phys. Rev. A 65, 032314 (2002).
    [CrossRef]
  26. C. K. Hong, Z. Y. Ou, and L. Mandel, "Measurement of subpicosecond time intervals between two photons by interference," Phys. Rev. Lett. 59, 2044-2046 (1987).
    [CrossRef] [PubMed]
  27. G. S. Agarwal, "Entropy, the Wigner Distribution Function, and the Approach to Equilibrium of a System of Coupled Harmonic Oscillators, " Phys. Rev. A 3, 828-831 (1971).
    [CrossRef]
  28. P. J. Dodd and J. J. Halliwell, "Disentanglement and decoherence by open system dynamics, " Phys. Rev. A 69, 052105 (2004).
    [CrossRef]
  29. L. -M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, "Inseparability Criterion for Continuous Variable Systems," Phys. Rev. Lett. 84, 2722-2725 (2000).
    [CrossRef] [PubMed]
  30. R. Simon, "Peres-Horodecki Separability Criterion for Continuous Variable Systems," Phys. Rev. Lett. 84, 2726- 2729 (2000).
    [CrossRef] [PubMed]
  31. G. Adesso, A. Serafini, and F. Illuminati, "Extremal entanglement and mixedness in continuous variable systems," Phys. Rev. A 70, 022318 (2004).
    [CrossRef]
  32. S. M. Barnett and P. Radmore, Methods in Theoretical Quantum Optics (Oxford University Press, 2002), p. 168.
  33. Sumanta Das and G. S. Agarwal, "Bright and dark periods in the entanglement dynamics of interacting qubits in contact with the environment," J. Phys. B 42, 141003 (2009).
    [CrossRef]
  34. Sumanta Das and G. S. Agarwal, "Decoherence effects in interacting qubits under the influence of various environments," J. Phys. B 42, 205502 (2009).
    [CrossRef]
  35. S. Longhi, "Transfer of light waves in optical waveguides via a continuum," Phys. Rev. A 78, 013815 (2008).
    [CrossRef]

2009 (7)

S. Longhi, "Optical analog of population trapping in the continuum: Classical and quantum interference effects," Phys. Rev. A 79, 023811 (2009).
[CrossRef]

Y. Bromberg, Y. Lahini, R. Morandotti, and Y. Silberberg, "Quantum and Classical Correlations in Waveguide Lattices," Phys. Rev. Lett. 102, 253904 (2009).
[CrossRef] [PubMed]

A. Rai and G. S. Agarwal, "Possibility of coherent phenomena such as Bloch oscillations with single photons via W states," Phys. Rev. A 79, 053849 (2009).
[CrossRef]

J. C. F. Matthews, A. Politi, A. Stefanov, and J. L. O’Brien, "Manipulation of multiphoton entanglement in waveguide quantum circuits," Nature Photonics 3, 346-350 (2009).
[CrossRef]

A. Politi, J. C. F. Matthews, and J. L. O’Brien, "Shors Quantum Factoring Algorithm on a Photonic Chip," Science 325, 1221 (2009).
[CrossRef] [PubMed]

Sumanta Das and G. S. Agarwal, "Bright and dark periods in the entanglement dynamics of interacting qubits in contact with the environment," J. Phys. B 42, 141003 (2009).
[CrossRef]

Sumanta Das and G. S. Agarwal, "Decoherence effects in interacting qubits under the influence of various environments," J. Phys. B 42, 205502 (2009).
[CrossRef]

2008 (5)

S. Longhi, "Transfer of light waves in optical waveguides via a continuum," Phys. Rev. A 78, 013815 (2008).
[CrossRef]

A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, "Silica-on-Silicon Waveguide Quantum Circuits," Science 320, 646-649 (2008).
[CrossRef] [PubMed]

S. Longhi, "Optical Bloch Oscillations and Zener Tunneling with Nonclassical Light," Phys. Rev. Lett. 101, 193902 (2008).
[CrossRef] [PubMed]

A. Rai, G. S. Agarwal, and J. H. H. Perk, "Transport and quantum walk of nonclassical light in coupled waveguides," Phys. Rev. A 78, 042304 (2008).
[CrossRef]

H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, and Y. Silberberg, "Realization of Quantum Walks with Negligible Decoherence in Waveguide Lattices," Phys. Rev. Lett. 100, 170506 (2008).
[CrossRef] [PubMed]

2007 (1)

P. K. Pathak and G. S. Agarwal, "Quantum random walk of two photons in separable and entangled states," Phys. Rev. A 75, 032351 (2007).
[CrossRef]

2005 (2)

2004 (4)

G. Adesso, A. Serafini, and F. Illuminati, "Extremal entanglement and mixedness in continuous variable systems," Phys. Rev. A 70, 022318 (2004).
[CrossRef]

P. J. Dodd and J. J. Halliwell, "Disentanglement and decoherence by open system dynamics, " Phys. Rev. A 69, 052105 (2004).
[CrossRef]

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, "Observation of Discrete Quadratic Solitons, " Phys. Rev. Lett. 93, 113902 (2004).
[CrossRef] [PubMed]

K. B. Mogensen, F. Eriksson, O. Gustafsson, R. P. H. Nikolajsen, and J. P. Kutter, "Pure-silica optical waveguides, fiber couplers, and high-aspect ratio submicrometer channels for electrokinetic separation devices," Electrophoresis 25, 3788-3795 (2004).
[CrossRef]

2003 (2)

W. H. Zurek, "Decoherence, einselection, and the quantum origins of the classical, " Rev. Mod. Phys. 75, 715-775 (2003).
[CrossRef]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices," Nature (London) 424, 817-823 (2003).
[CrossRef]

2002 (2)

2000 (4)

L. -M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, "Inseparability Criterion for Continuous Variable Systems," Phys. Rev. Lett. 84, 2722-2725 (2000).
[CrossRef] [PubMed]

R. Simon, "Peres-Horodecki Separability Criterion for Continuous Variable Systems," Phys. Rev. Lett. 84, 2726- 2729 (2000).
[CrossRef] [PubMed]

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, "Quantum Interferometric Optical Lithography: Exploiting Entanglement to Beat the Diffraction Limit," Phys. Rev. Lett. 85, 2733-2736 (2000).
[CrossRef] [PubMed]

C. H. Bennett and D. P. DiVincenzo, "Quantum information and computation," Nature 404, 247-255 (2000).
[CrossRef] [PubMed]

1999 (2)

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, "Experimental Observation of Linear and Nonlinear Optical Bloch Oscillations," Phys. Rev. Lett. 83, 4756-4759 (1999).
[CrossRef]

T. Pertsch, P. Dannberg,W. Elflein, A. Brauer, and F. Lederer, "Optical Bloch Oscillations in Temperature Tuned Waveguide Arrays," Phys. Rev. Lett. 83, 4752-4755 (1999).
[CrossRef]

1998 (1)

1991 (1)

W. K. Lai, V. Buˇzek, and P. L. Knight, "Nonclassical fields in a linear directional coupler, " Phys. Rev. A 43, 6323-6336 (1991).
[CrossRef] [PubMed]

1987 (1)

C. K. Hong, Z. Y. Ou, and L. Mandel, "Measurement of subpicosecond time intervals between two photons by interference," Phys. Rev. Lett. 59, 2044-2046 (1987).
[CrossRef] [PubMed]

1971 (1)

G. S. Agarwal, "Entropy, the Wigner Distribution Function, and the Approach to Equilibrium of a System of Coupled Harmonic Oscillators, " Phys. Rev. A 3, 828-831 (1971).
[CrossRef]

Abrams, D. S.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, "Quantum Interferometric Optical Lithography: Exploiting Entanglement to Beat the Diffraction Limit," Phys. Rev. Lett. 85, 2733-2736 (2000).
[CrossRef] [PubMed]

Adesso, G.

G. Adesso, A. Serafini, and F. Illuminati, "Extremal entanglement and mixedness in continuous variable systems," Phys. Rev. A 70, 022318 (2004).
[CrossRef]

Agarwal, G. S.

A. Rai and G. S. Agarwal, "Possibility of coherent phenomena such as Bloch oscillations with single photons via W states," Phys. Rev. A 79, 053849 (2009).
[CrossRef]

A. Rai, G. S. Agarwal, and J. H. H. Perk, "Transport and quantum walk of nonclassical light in coupled waveguides," Phys. Rev. A 78, 042304 (2008).
[CrossRef]

P. K. Pathak and G. S. Agarwal, "Quantum random walk of two photons in separable and entangled states," Phys. Rev. A 75, 032351 (2007).
[CrossRef]

G. S. Agarwal, "Entropy, the Wigner Distribution Function, and the Approach to Equilibrium of a System of Coupled Harmonic Oscillators, " Phys. Rev. A 3, 828-831 (1971).
[CrossRef]

Aitchison, J. S.

U. Peschel, R. Morandotti, J. M. Arnold, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, T. Pertsch, and F. Lederer, "Optical Discrete Solitons in Waveguide Arrays. 2. Dynamic Properties," J. Opt. Soc. Am. B 19, 2637- 2644 (2002).
[CrossRef]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, "Experimental Observation of Linear and Nonlinear Optical Bloch Oscillations," Phys. Rev. Lett. 83, 4756-4759 (1999).
[CrossRef]

Arnold, J. M.

Balasubramanian, S.

Bennett, C. H.

C. H. Bennett and D. P. DiVincenzo, "Quantum information and computation," Nature 404, 247-255 (2000).
[CrossRef] [PubMed]

Boto, A. N.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, "Quantum Interferometric Optical Lithography: Exploiting Entanglement to Beat the Diffraction Limit," Phys. Rev. Lett. 85, 2733-2736 (2000).
[CrossRef] [PubMed]

Brauer, A.

T. Pertsch, P. Dannberg,W. Elflein, A. Brauer, and F. Lederer, "Optical Bloch Oscillations in Temperature Tuned Waveguide Arrays," Phys. Rev. Lett. 83, 4752-4755 (1999).
[CrossRef]

Braunstein, S. L.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, "Quantum Interferometric Optical Lithography: Exploiting Entanglement to Beat the Diffraction Limit," Phys. Rev. Lett. 85, 2733-2736 (2000).
[CrossRef] [PubMed]

Bromberg, Y.

Y. Bromberg, Y. Lahini, R. Morandotti, and Y. Silberberg, "Quantum and Classical Correlations in Waveguide Lattices," Phys. Rev. Lett. 102, 253904 (2009).
[CrossRef] [PubMed]

Bu?zek, V.

W. K. Lai, V. Buˇzek, and P. L. Knight, "Nonclassical fields in a linear directional coupler, " Phys. Rev. A 43, 6323-6336 (1991).
[CrossRef] [PubMed]

Christodoulides, D. N.

R. Iwanow, D. A. May-Arrioja, D. N. Christodoulides, G. I. Stegeman, Y. Min, and W. Sohler, "Discrete Talbot Effect in Waveguide Arrays," Phys. Rev. Lett. 95, 053902 (2005).
[CrossRef] [PubMed]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices," Nature (London) 424, 817-823 (2003).
[CrossRef]

Cirac, J. I.

L. -M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, "Inseparability Criterion for Continuous Variable Systems," Phys. Rev. Lett. 84, 2722-2725 (2000).
[CrossRef] [PubMed]

Cryan, M. J.

A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, "Silica-on-Silicon Waveguide Quantum Circuits," Science 320, 646-649 (2008).
[CrossRef] [PubMed]

Dannberg, P.

T. Pertsch, P. Dannberg,W. Elflein, A. Brauer, and F. Lederer, "Optical Bloch Oscillations in Temperature Tuned Waveguide Arrays," Phys. Rev. Lett. 83, 4752-4755 (1999).
[CrossRef]

DiVincenzo, D. P.

C. H. Bennett and D. P. DiVincenzo, "Quantum information and computation," Nature 404, 247-255 (2000).
[CrossRef] [PubMed]

Do, B.

Dodd, P. J.

P. J. Dodd and J. J. Halliwell, "Disentanglement and decoherence by open system dynamics, " Phys. Rev. A 69, 052105 (2004).
[CrossRef]

Dowling, J. P.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, "Quantum Interferometric Optical Lithography: Exploiting Entanglement to Beat the Diffraction Limit," Phys. Rev. Lett. 85, 2733-2736 (2000).
[CrossRef] [PubMed]

Duan, L. -M.

L. -M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, "Inseparability Criterion for Continuous Variable Systems," Phys. Rev. Lett. 84, 2722-2725 (2000).
[CrossRef] [PubMed]

Eash, C.

Eisenberg, H. S.

U. Peschel, R. Morandotti, J. M. Arnold, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, T. Pertsch, and F. Lederer, "Optical Discrete Solitons in Waveguide Arrays. 2. Dynamic Properties," J. Opt. Soc. Am. B 19, 2637- 2644 (2002).
[CrossRef]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, "Experimental Observation of Linear and Nonlinear Optical Bloch Oscillations," Phys. Rev. Lett. 83, 4756-4759 (1999).
[CrossRef]

Elflein, W.

T. Pertsch, P. Dannberg,W. Elflein, A. Brauer, and F. Lederer, "Optical Bloch Oscillations in Temperature Tuned Waveguide Arrays," Phys. Rev. Lett. 83, 4752-4755 (1999).
[CrossRef]

Elliott, D. S.

Eriksson, F.

K. B. Mogensen, F. Eriksson, O. Gustafsson, R. P. H. Nikolajsen, and J. P. Kutter, "Pure-silica optical waveguides, fiber couplers, and high-aspect ratio submicrometer channels for electrokinetic separation devices," Electrophoresis 25, 3788-3795 (2004).
[CrossRef]

Fischbach, E.

Fischbach, M. A.

Giedke, G.

L. -M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, "Inseparability Criterion for Continuous Variable Systems," Phys. Rev. Lett. 84, 2722-2725 (2000).
[CrossRef] [PubMed]

Gustafsson, O.

K. B. Mogensen, F. Eriksson, O. Gustafsson, R. P. H. Nikolajsen, and J. P. Kutter, "Pure-silica optical waveguides, fiber couplers, and high-aspect ratio submicrometer channels for electrokinetic separation devices," Electrophoresis 25, 3788-3795 (2004).
[CrossRef]

Halliwell, J. J.

P. J. Dodd and J. J. Halliwell, "Disentanglement and decoherence by open system dynamics, " Phys. Rev. A 69, 052105 (2004).
[CrossRef]

Hong, C. K.

C. K. Hong, Z. Y. Ou, and L. Mandel, "Measurement of subpicosecond time intervals between two photons by interference," Phys. Rev. Lett. 59, 2044-2046 (1987).
[CrossRef] [PubMed]

Illuminati, F.

G. Adesso, A. Serafini, and F. Illuminati, "Extremal entanglement and mixedness in continuous variable systems," Phys. Rev. A 70, 022318 (2004).
[CrossRef]

Iwanow, R.

R. Iwanow, D. A. May-Arrioja, D. N. Christodoulides, G. I. Stegeman, Y. Min, and W. Sohler, "Discrete Talbot Effect in Waveguide Arrays," Phys. Rev. Lett. 95, 053902 (2005).
[CrossRef] [PubMed]

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, "Observation of Discrete Quadratic Solitons, " Phys. Rev. Lett. 93, 113902 (2004).
[CrossRef] [PubMed]

Knight, P. L.

W. K. Lai, V. Buˇzek, and P. L. Knight, "Nonclassical fields in a linear directional coupler, " Phys. Rev. A 43, 6323-6336 (1991).
[CrossRef] [PubMed]

Kok, P.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, "Quantum Interferometric Optical Lithography: Exploiting Entanglement to Beat the Diffraction Limit," Phys. Rev. Lett. 85, 2733-2736 (2000).
[CrossRef] [PubMed]

Kutter, J. P.

K. B. Mogensen, F. Eriksson, O. Gustafsson, R. P. H. Nikolajsen, and J. P. Kutter, "Pure-silica optical waveguides, fiber couplers, and high-aspect ratio submicrometer channels for electrokinetic separation devices," Electrophoresis 25, 3788-3795 (2004).
[CrossRef]

Lahini, Y.

Y. Bromberg, Y. Lahini, R. Morandotti, and Y. Silberberg, "Quantum and Classical Correlations in Waveguide Lattices," Phys. Rev. Lett. 102, 253904 (2009).
[CrossRef] [PubMed]

H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, and Y. Silberberg, "Realization of Quantum Walks with Negligible Decoherence in Waveguide Lattices," Phys. Rev. Lett. 100, 170506 (2008).
[CrossRef] [PubMed]

Lai, W. K.

W. K. Lai, V. Buˇzek, and P. L. Knight, "Nonclassical fields in a linear directional coupler, " Phys. Rev. A 43, 6323-6336 (1991).
[CrossRef] [PubMed]

Lederer, F.

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, "Observation of Discrete Quadratic Solitons, " Phys. Rev. Lett. 93, 113902 (2004).
[CrossRef] [PubMed]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices," Nature (London) 424, 817-823 (2003).
[CrossRef]

U. Peschel, R. Morandotti, J. M. Arnold, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, T. Pertsch, and F. Lederer, "Optical Discrete Solitons in Waveguide Arrays. 2. Dynamic Properties," J. Opt. Soc. Am. B 19, 2637- 2644 (2002).
[CrossRef]

T. Pertsch, P. Dannberg,W. Elflein, A. Brauer, and F. Lederer, "Optical Bloch Oscillations in Temperature Tuned Waveguide Arrays," Phys. Rev. Lett. 83, 4752-4755 (1999).
[CrossRef]

U. Peschel, T. Pertsch, and F. Lederer, "Optical Bloch oscillations in waveguide arrays," Opt. Lett. 23, 1701-1703 (1998).
[CrossRef]

Longhi, S.

S. Longhi, "Optical analog of population trapping in the continuum: Classical and quantum interference effects," Phys. Rev. A 79, 023811 (2009).
[CrossRef]

S. Longhi, "Optical Bloch Oscillations and Zener Tunneling with Nonclassical Light," Phys. Rev. Lett. 101, 193902 (2008).
[CrossRef] [PubMed]

S. Longhi, "Transfer of light waves in optical waveguides via a continuum," Phys. Rev. A 78, 013815 (2008).
[CrossRef]

Mandel, L.

C. K. Hong, Z. Y. Ou, and L. Mandel, "Measurement of subpicosecond time intervals between two photons by interference," Phys. Rev. Lett. 59, 2044-2046 (1987).
[CrossRef] [PubMed]

Matthews, J. C. F.

J. C. F. Matthews, A. Politi, A. Stefanov, and J. L. O’Brien, "Manipulation of multiphoton entanglement in waveguide quantum circuits," Nature Photonics 3, 346-350 (2009).
[CrossRef]

A. Politi, J. C. F. Matthews, and J. L. O’Brien, "Shors Quantum Factoring Algorithm on a Photonic Chip," Science 325, 1221 (2009).
[CrossRef] [PubMed]

May-Arrioja, D. A.

R. Iwanow, D. A. May-Arrioja, D. N. Christodoulides, G. I. Stegeman, Y. Min, and W. Sohler, "Discrete Talbot Effect in Waveguide Arrays," Phys. Rev. Lett. 95, 053902 (2005).
[CrossRef] [PubMed]

Mills, A.

Min, Y.

R. Iwanow, D. A. May-Arrioja, D. N. Christodoulides, G. I. Stegeman, Y. Min, and W. Sohler, "Discrete Talbot Effect in Waveguide Arrays," Phys. Rev. Lett. 95, 053902 (2005).
[CrossRef] [PubMed]

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, "Observation of Discrete Quadratic Solitons, " Phys. Rev. Lett. 93, 113902 (2004).
[CrossRef] [PubMed]

Mogensen, K. B.

K. B. Mogensen, F. Eriksson, O. Gustafsson, R. P. H. Nikolajsen, and J. P. Kutter, "Pure-silica optical waveguides, fiber couplers, and high-aspect ratio submicrometer channels for electrokinetic separation devices," Electrophoresis 25, 3788-3795 (2004).
[CrossRef]

Morandotti, R.

Y. Bromberg, Y. Lahini, R. Morandotti, and Y. Silberberg, "Quantum and Classical Correlations in Waveguide Lattices," Phys. Rev. Lett. 102, 253904 (2009).
[CrossRef] [PubMed]

H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, and Y. Silberberg, "Realization of Quantum Walks with Negligible Decoherence in Waveguide Lattices," Phys. Rev. Lett. 100, 170506 (2008).
[CrossRef] [PubMed]

U. Peschel, R. Morandotti, J. M. Arnold, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, T. Pertsch, and F. Lederer, "Optical Discrete Solitons in Waveguide Arrays. 2. Dynamic Properties," J. Opt. Soc. Am. B 19, 2637- 2644 (2002).
[CrossRef]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, "Experimental Observation of Linear and Nonlinear Optical Bloch Oscillations," Phys. Rev. Lett. 83, 4756-4759 (1999).
[CrossRef]

Nikolajsen, R. P. H.

K. B. Mogensen, F. Eriksson, O. Gustafsson, R. P. H. Nikolajsen, and J. P. Kutter, "Pure-silica optical waveguides, fiber couplers, and high-aspect ratio submicrometer channels for electrokinetic separation devices," Electrophoresis 25, 3788-3795 (2004).
[CrossRef]

O’Brien, J. L.

A. Politi, J. C. F. Matthews, and J. L. O’Brien, "Shors Quantum Factoring Algorithm on a Photonic Chip," Science 325, 1221 (2009).
[CrossRef] [PubMed]

J. C. F. Matthews, A. Politi, A. Stefanov, and J. L. O’Brien, "Manipulation of multiphoton entanglement in waveguide quantum circuits," Nature Photonics 3, 346-350 (2009).
[CrossRef]

A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, "Silica-on-Silicon Waveguide Quantum Circuits," Science 320, 646-649 (2008).
[CrossRef] [PubMed]

Ou, Z. Y.

C. K. Hong, Z. Y. Ou, and L. Mandel, "Measurement of subpicosecond time intervals between two photons by interference," Phys. Rev. Lett. 59, 2044-2046 (1987).
[CrossRef] [PubMed]

Pathak, P. K.

P. K. Pathak and G. S. Agarwal, "Quantum random walk of two photons in separable and entangled states," Phys. Rev. A 75, 032351 (2007).
[CrossRef]

Perets, H. B.

H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, and Y. Silberberg, "Realization of Quantum Walks with Negligible Decoherence in Waveguide Lattices," Phys. Rev. Lett. 100, 170506 (2008).
[CrossRef] [PubMed]

Perk, J. H. H.

A. Rai, G. S. Agarwal, and J. H. H. Perk, "Transport and quantum walk of nonclassical light in coupled waveguides," Phys. Rev. A 78, 042304 (2008).
[CrossRef]

Pertsch, T.

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, "Observation of Discrete Quadratic Solitons, " Phys. Rev. Lett. 93, 113902 (2004).
[CrossRef] [PubMed]

U. Peschel, R. Morandotti, J. M. Arnold, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, T. Pertsch, and F. Lederer, "Optical Discrete Solitons in Waveguide Arrays. 2. Dynamic Properties," J. Opt. Soc. Am. B 19, 2637- 2644 (2002).
[CrossRef]

T. Pertsch, P. Dannberg,W. Elflein, A. Brauer, and F. Lederer, "Optical Bloch Oscillations in Temperature Tuned Waveguide Arrays," Phys. Rev. Lett. 83, 4752-4755 (1999).
[CrossRef]

U. Peschel, T. Pertsch, and F. Lederer, "Optical Bloch oscillations in waveguide arrays," Opt. Lett. 23, 1701-1703 (1998).
[CrossRef]

Peschel, U.

Politi, A.

J. C. F. Matthews, A. Politi, A. Stefanov, and J. L. O’Brien, "Manipulation of multiphoton entanglement in waveguide quantum circuits," Nature Photonics 3, 346-350 (2009).
[CrossRef]

A. Politi, J. C. F. Matthews, and J. L. O’Brien, "Shors Quantum Factoring Algorithm on a Photonic Chip," Science 325, 1221 (2009).
[CrossRef] [PubMed]

A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, "Silica-on-Silicon Waveguide Quantum Circuits," Science 320, 646-649 (2008).
[CrossRef] [PubMed]

Pozzi, F.

H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, and Y. Silberberg, "Realization of Quantum Walks with Negligible Decoherence in Waveguide Lattices," Phys. Rev. Lett. 100, 170506 (2008).
[CrossRef] [PubMed]

Rai, A.

A. Rai and G. S. Agarwal, "Possibility of coherent phenomena such as Bloch oscillations with single photons via W states," Phys. Rev. A 79, 053849 (2009).
[CrossRef]

A. Rai, G. S. Agarwal, and J. H. H. Perk, "Transport and quantum walk of nonclassical light in coupled waveguides," Phys. Rev. A 78, 042304 (2008).
[CrossRef]

Rarity, J. G.

A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, "Silica-on-Silicon Waveguide Quantum Circuits," Science 320, 646-649 (2008).
[CrossRef] [PubMed]

Schiek, R.

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, "Observation of Discrete Quadratic Solitons, " Phys. Rev. Lett. 93, 113902 (2004).
[CrossRef] [PubMed]

Serafini, A.

G. Adesso, A. Serafini, and F. Illuminati, "Extremal entanglement and mixedness in continuous variable systems," Phys. Rev. A 70, 022318 (2004).
[CrossRef]

Silberberg, Y.

Y. Bromberg, Y. Lahini, R. Morandotti, and Y. Silberberg, "Quantum and Classical Correlations in Waveguide Lattices," Phys. Rev. Lett. 102, 253904 (2009).
[CrossRef] [PubMed]

H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, and Y. Silberberg, "Realization of Quantum Walks with Negligible Decoherence in Waveguide Lattices," Phys. Rev. Lett. 100, 170506 (2008).
[CrossRef] [PubMed]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices," Nature (London) 424, 817-823 (2003).
[CrossRef]

U. Peschel, R. Morandotti, J. M. Arnold, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, T. Pertsch, and F. Lederer, "Optical Discrete Solitons in Waveguide Arrays. 2. Dynamic Properties," J. Opt. Soc. Am. B 19, 2637- 2644 (2002).
[CrossRef]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, "Experimental Observation of Linear and Nonlinear Optical Bloch Oscillations," Phys. Rev. Lett. 83, 4756-4759 (1999).
[CrossRef]

Simon, R.

R. Simon, "Peres-Horodecki Separability Criterion for Continuous Variable Systems," Phys. Rev. Lett. 84, 2726- 2729 (2000).
[CrossRef] [PubMed]

Sohler, W.

R. Iwanow, D. A. May-Arrioja, D. N. Christodoulides, G. I. Stegeman, Y. Min, and W. Sohler, "Discrete Talbot Effect in Waveguide Arrays," Phys. Rev. Lett. 95, 053902 (2005).
[CrossRef] [PubMed]

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, "Observation of Discrete Quadratic Solitons, " Phys. Rev. Lett. 93, 113902 (2004).
[CrossRef] [PubMed]

Sorel, M.

H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, and Y. Silberberg, "Realization of Quantum Walks with Negligible Decoherence in Waveguide Lattices," Phys. Rev. Lett. 100, 170506 (2008).
[CrossRef] [PubMed]

Stefanov, A.

J. C. F. Matthews, A. Politi, A. Stefanov, and J. L. O’Brien, "Manipulation of multiphoton entanglement in waveguide quantum circuits," Nature Photonics 3, 346-350 (2009).
[CrossRef]

Stegeman, G. I.

R. Iwanow, D. A. May-Arrioja, D. N. Christodoulides, G. I. Stegeman, Y. Min, and W. Sohler, "Discrete Talbot Effect in Waveguide Arrays," Phys. Rev. Lett. 95, 053902 (2005).
[CrossRef] [PubMed]

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, "Observation of Discrete Quadratic Solitons, " Phys. Rev. Lett. 93, 113902 (2004).
[CrossRef] [PubMed]

Stohler, M. L.

Vidal, G.

G. Vidal and R. F. Werner, "Computable measure of entanglement," Phys. Rev. A 65, 032314 (2002).
[CrossRef]

Werner, R. F.

G. Vidal and R. F. Werner, "Computable measure of entanglement," Phys. Rev. A 65, 032314 (2002).
[CrossRef]

Williams, C. P.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, "Quantum Interferometric Optical Lithography: Exploiting Entanglement to Beat the Diffraction Limit," Phys. Rev. Lett. 85, 2733-2736 (2000).
[CrossRef] [PubMed]

Yu, S.

A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, "Silica-on-Silicon Waveguide Quantum Circuits," Science 320, 646-649 (2008).
[CrossRef] [PubMed]

Zoller, P.

L. -M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, "Inseparability Criterion for Continuous Variable Systems," Phys. Rev. Lett. 84, 2722-2725 (2000).
[CrossRef] [PubMed]

Zurek, W. H.

W. H. Zurek, "Decoherence, einselection, and the quantum origins of the classical, " Rev. Mod. Phys. 75, 715-775 (2003).
[CrossRef]

Zwickl, B.

Electrophoresis (1)

K. B. Mogensen, F. Eriksson, O. Gustafsson, R. P. H. Nikolajsen, and J. P. Kutter, "Pure-silica optical waveguides, fiber couplers, and high-aspect ratio submicrometer channels for electrokinetic separation devices," Electrophoresis 25, 3788-3795 (2004).
[CrossRef]

J. Opt. Soc. Am. B (2)

J. Phys. B (2)

Sumanta Das and G. S. Agarwal, "Bright and dark periods in the entanglement dynamics of interacting qubits in contact with the environment," J. Phys. B 42, 141003 (2009).
[CrossRef]

Sumanta Das and G. S. Agarwal, "Decoherence effects in interacting qubits under the influence of various environments," J. Phys. B 42, 205502 (2009).
[CrossRef]

Nature (1)

C. H. Bennett and D. P. DiVincenzo, "Quantum information and computation," Nature 404, 247-255 (2000).
[CrossRef] [PubMed]

Nature (London) (1)

D. N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices," Nature (London) 424, 817-823 (2003).
[CrossRef]

Nature Photonics (1)

J. C. F. Matthews, A. Politi, A. Stefanov, and J. L. O’Brien, "Manipulation of multiphoton entanglement in waveguide quantum circuits," Nature Photonics 3, 346-350 (2009).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (10)

S. Longhi, "Optical analog of population trapping in the continuum: Classical and quantum interference effects," Phys. Rev. A 79, 023811 (2009).
[CrossRef]

A. Rai, G. S. Agarwal, and J. H. H. Perk, "Transport and quantum walk of nonclassical light in coupled waveguides," Phys. Rev. A 78, 042304 (2008).
[CrossRef]

A. Rai and G. S. Agarwal, "Possibility of coherent phenomena such as Bloch oscillations with single photons via W states," Phys. Rev. A 79, 053849 (2009).
[CrossRef]

P. K. Pathak and G. S. Agarwal, "Quantum random walk of two photons in separable and entangled states," Phys. Rev. A 75, 032351 (2007).
[CrossRef]

W. K. Lai, V. Buˇzek, and P. L. Knight, "Nonclassical fields in a linear directional coupler, " Phys. Rev. A 43, 6323-6336 (1991).
[CrossRef] [PubMed]

G. Vidal and R. F. Werner, "Computable measure of entanglement," Phys. Rev. A 65, 032314 (2002).
[CrossRef]

S. Longhi, "Transfer of light waves in optical waveguides via a continuum," Phys. Rev. A 78, 013815 (2008).
[CrossRef]

G. Adesso, A. Serafini, and F. Illuminati, "Extremal entanglement and mixedness in continuous variable systems," Phys. Rev. A 70, 022318 (2004).
[CrossRef]

G. S. Agarwal, "Entropy, the Wigner Distribution Function, and the Approach to Equilibrium of a System of Coupled Harmonic Oscillators, " Phys. Rev. A 3, 828-831 (1971).
[CrossRef]

P. J. Dodd and J. J. Halliwell, "Disentanglement and decoherence by open system dynamics, " Phys. Rev. A 69, 052105 (2004).
[CrossRef]

Phys. Rev. Lett. (11)

L. -M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, "Inseparability Criterion for Continuous Variable Systems," Phys. Rev. Lett. 84, 2722-2725 (2000).
[CrossRef] [PubMed]

R. Simon, "Peres-Horodecki Separability Criterion for Continuous Variable Systems," Phys. Rev. Lett. 84, 2726- 2729 (2000).
[CrossRef] [PubMed]

C. K. Hong, Z. Y. Ou, and L. Mandel, "Measurement of subpicosecond time intervals between two photons by interference," Phys. Rev. Lett. 59, 2044-2046 (1987).
[CrossRef] [PubMed]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, "Experimental Observation of Linear and Nonlinear Optical Bloch Oscillations," Phys. Rev. Lett. 83, 4756-4759 (1999).
[CrossRef]

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, "Observation of Discrete Quadratic Solitons, " Phys. Rev. Lett. 93, 113902 (2004).
[CrossRef] [PubMed]

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, "Quantum Interferometric Optical Lithography: Exploiting Entanglement to Beat the Diffraction Limit," Phys. Rev. Lett. 85, 2733-2736 (2000).
[CrossRef] [PubMed]

H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, and Y. Silberberg, "Realization of Quantum Walks with Negligible Decoherence in Waveguide Lattices," Phys. Rev. Lett. 100, 170506 (2008).
[CrossRef] [PubMed]

Y. Bromberg, Y. Lahini, R. Morandotti, and Y. Silberberg, "Quantum and Classical Correlations in Waveguide Lattices," Phys. Rev. Lett. 102, 253904 (2009).
[CrossRef] [PubMed]

T. Pertsch, P. Dannberg,W. Elflein, A. Brauer, and F. Lederer, "Optical Bloch Oscillations in Temperature Tuned Waveguide Arrays," Phys. Rev. Lett. 83, 4752-4755 (1999).
[CrossRef]

R. Iwanow, D. A. May-Arrioja, D. N. Christodoulides, G. I. Stegeman, Y. Min, and W. Sohler, "Discrete Talbot Effect in Waveguide Arrays," Phys. Rev. Lett. 95, 053902 (2005).
[CrossRef] [PubMed]

S. Longhi, "Optical Bloch Oscillations and Zener Tunneling with Nonclassical Light," Phys. Rev. Lett. 101, 193902 (2008).
[CrossRef] [PubMed]

Rev. Mod. Phys. (1)

W. H. Zurek, "Decoherence, einselection, and the quantum origins of the classical, " Rev. Mod. Phys. 75, 715-775 (2003).
[CrossRef]

Science (2)

A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, "Silica-on-Silicon Waveguide Quantum Circuits," Science 320, 646-649 (2008).
[CrossRef] [PubMed]

A. Politi, J. C. F. Matthews, and J. L. O’Brien, "Shors Quantum Factoring Algorithm on a Photonic Chip," Science 325, 1221 (2009).
[CrossRef] [PubMed]

Other (2)

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd Edition, (Wiley, New York 2007), p. 319.

S. M. Barnett and P. Radmore, Methods in Theoretical Quantum Optics (Oxford University Press, 2002), p. 168.

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Figures (5)

Fig. 1.
Fig. 1.

(Color online) Schematic diagram of a coupled waveguide system. The parameter J gives the coupling between the waveguide modes.

Fig. 2.
Fig. 2.

(a) Time evolution of log negatively for the separable input state. The red curve shows the result for ∣1,1〉 state while the black curve shows the result for the ∣2,0〉 state. (b) The behavior of log negatively for the NOON state. The black curve shows the result for two photon NOON state while the red curve shows the result (En - 1) for the four photon NOON state

Fig. 3.
Fig. 3.

Plot of the time dependent logarithmic negativity E�� for the state ∣ζ〉. Here amount of squeezing is taken to be r = 0.9.

Fig. 4.
Fig. 4.

Time evolution of the logarithmic negativity E�� in presence of loss of the waveguide modes for the initial separable input state ∣1,1〉. The decay rates of the modes are given by γ/J = 0.1 (solid black), γ/J = 0.2 (broken black) and γ/J = 0.3 (red).

Fig. 5.
Fig. 5.

(a)Time evolution of the logarithmic negativity E�� in presence of loss of the waveguide modes for the input state ∣ζ〉. The decay rates of the modes are given by γ/J = 0.1 (solid black), γ/J = 0.2 (broken black) and γ/J = 0.3 (red). Here the squeezing is taken to be r = 0.9. The loss leads to new behavior in the entanglement. (b) Long time behavior of the logarithmic negativity in presence of loss of the waveguide modes for the the state ∣ζ〉.

Tables (1)

Tables Icon

Table 1. Approximate values of some of the parameters used in waveguide structures [22, 23, 24]. The loss, usually quoted in dB/cm, for different waveguides is converted to frequency units used in this paper by using the formula, 10 Log ( P out P in ) 10 Log ( e 2 γ / c ) , where Pin is the input power, Pout is the power after traveling unit length.

Equations (36)

Equations on this page are rendered with MathJax. Learn more.

H = h ̅ ω ( a a + b b ) + h ̅ J ( a b + b a ) ,
𝓛 ρ = γ ( a ̂ a ̂ ρ 2 a ̂ ρ a ̂ + ρ a ̂ a ̂ ) γ ( b ̂ b ̂ ρ 2 b ̂ ρ b ̂ + ρ b ̂ b ̂ ) ,
ρ ˙ = i h ̅ [ H , ρ ] + ρ
a ( t ) = a ( 0 ) cos ( J t ) i b ( 0 ) sin ( J t ) ; ( a b ) .
E 𝒩 ( t ) = log 2 ρ T , ρ T = ( 2 N ( ρ ) + 1 ) ,
α 1 i sin ( 2 J t ) / 2 , β 1 cos ( 2 J t ) , δ 1 i sin ( 2 J t ) / 2 .
E 𝒩 = log 2 ( 1 + 2 N ( ρ ) ) = log 2 ( 1 + 2 ( α 1 β 1 + α 1 δ 1 + δ 1 β 1 ) ) .
α 2 cos ( J t ) 2 , β 2 2 i cos ( J t ) sin ( J t ) , δ 2 sin ( J t ) 2 .
ψ in = ( N , 0 + 0 , N ) / 2 = ( ( a ( 0 ) ) N + ( b ( 0 ) ) N ) 0,0 ) / 2 N !
ψ out = ( ( ( a ( t ) ) N + ( b ( t ) ) N ) 0,0 ) / 2 N ! ,
ψ out = ( Σ β k k , N k ) , β k = α k + α N k , α k = ( C ( N , k ) ) 1 2 ( cos ( J t ) ) k ( i sin ( J t ) ) N k
E 𝒩 = log 2 ( 1 + 2 N ( ρ ) ) = log 2 ( 1 + 2 k m β k β m ) .
σ = [ α μ μ T β ] ;
α = [ x 1 2 x 1 p 1 + p 1 x 1 2 x 1 p 1 + p 1 x 1 2 p 1 2 ] = β ( 1 2 ) ; μ = [ x 1 x 2 + x 2 x 1 2 x 1 p 2 + p 2 x 1 2 x 2 p 1 + p 1 x 2 2 p 1 p 2 + p 2 p 1 2 ] .
x 1 = ( a + a ) 2 ( x 2 : a b ) , p 1 = ( a a ) 2 i ( p 2 ; a b ) ;
ν ̃ < < 1 2 ; ν ̃ < = min [ ν ̃ + , ν ̃ ] ;
ν ̃ ± = Δ ̃ ( σ ) ± Δ ( σ ) 2 ̃ 4 Det σ 2 ;
E 𝒩 ( t ) = max [ 0 , In { 2 ν ̃ < ( t ) } ] ;
α = β = [ c 0 0 d ] ; μ = [ 0 e e 0 ] ;
c = 1 2 { cosh ( 2 r ) + sinh ( 2 r ) cos ( 2 Jt ) }
d = 1 2 { cosh ( 2 r ) sinh ( 2 r ) cos ( 2 J t ) } ; e = 1 2 sinh ( 2 r ) sin ( 2 J t ) .
ν ̃ ± = c d ± e .
σ = [ f g h 0 g f 0 h h 0 f g 0 h g f ]
f = 1 2 cosh ( 2 r ) ; g = 1 2 sinh ( 2 r ) sin ( 2 J t ) ; h = 1 3 sinh ( 2 r ) cos ( 2 J t ) .
ν ̃ ± = ( f + g ) ( f g ) ± h ;
ρ ̃ ( t ) t = γ ( a ̃ a ̃ ρ ̃ 2 a ̃ ρ ̃ a ̃ + ρ ̃ a ̃ a ̃ ) γ ( b ̃ b ̃ ρ ̃ 2 b ̃ ρ ̃ b ̃ + ρ ̃ b ̃ b ̃ ) ,
a ̃ = a cos ( J t ) i b sin ( J t ) ( b ̃ : a b ) .
ρ ̃ ( t ) t = γ ( a a ρ ̃ 2 a ρ ̃ a + ρ ̃ a a ) γ ( b b ρ ̃ 2 b ρ ̃ b + ρ ̃ b b ) .
ρ ̃ ( t ) = e 4 γt { ( e 2 γt 1 ) 2 0,0 0,0 + ( e 2 γt 1 ) ( 1,0 1,0 + 0,1 0,1 ) + 1,1 1,1 } .
ρ ( t ) = e i J t ( a b + b a ) ρ ̃ ( t ) e i J t ( a b + b a ) .
σ = [ c 0 0 e 0 d e 0 0 e c 0 e 0 0 d ] ;
c = 1 2 { 1 + 2 e 2 γt sinh 2 ( r ) + e 2 γt sinh ( 2 r ) cos ( 2 J t ) }
d = 1 2 { 1 + 2 e 2 γt sinh 2 ( r ) e 2 γt sinh ( 2 r ) cos ( 2 J t ) } ; e = 1 2 e 2 γt sinh ( 2 r ) sin ( 2 J t ) .
σ = [ f g h 0 g f 0 h h 0 f g 0 h g f ]
f = 1 2 + e 2 γt sinh 2 ( r ) ; g = 1 2 e 2 γt sinh ( 2 r ) sin ( 2 J t ) ; h = 1 2 e 2 γt sinh ( 2 r ) cos ( 2 J t ) .
ν ̃ ± = m + m ± h , m ± ( t ) = 1 e 2 γt [ 1 { cosh ( 2 r ) ± sinh ( 2 r ) sin ( 2 J t ) } ]

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