Abstract

Spatial adiabatic passage represents a new way to design integrated photonic devices. In conventional adiabatic passage, designs require smoothly varying waveguide separations. Here we show modelling of adiabatic passage devices where the waveguide separation is varied digitally. Despite digitisation, our designs show robustness against variations in the input wavelength and refractive index contrast of the waveguides relative to the cladding. This approach to spatial adiabatic passage opens new design strategies and hence the potential for new photonics devices.

© 2017 Optical Society of America

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References

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    [Crossref]
  4. A. D. Greentree, J. H. Cole, A. R. Hamilton, and L. C. L. Hollenberg, “Coherent electronic transfer in quantum dot systems using adiabatic passage,” Phys. Rev. B 70, 235317 (2004).
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  5. E. Paspalakis, “Adiabatic three-waveguide directional coupler,” Optics Communications 258, 30–34 (2006).
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  6. S. Longhi, G. Della Valle, M. Ornigotti, and P. Laporta, “Coherent tunneling by adiabatic passage in an optical waveguide system,” Phys. Rev. B 76, 20110(R) (2007).
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  8. P. Král, I. Thanopulos, and M. Shapiro, “Colloquium: Coherently controlled adiabatic passage,” Rev. Mod. Phys. 79, 53–77 (2007).
    [Crossref]
  9. E. A. Shapiro, V. Milner, C. Menzel-Jones, and M. Shapiro, “Piecewise adiabatic passage with a series of femtosecond pulses,” Phys. Rev. Lett. 99, 033002 (2007).
    [Crossref] [PubMed]
  10. A. A. Rangelov and N. V. Vitanov, “Complete population transfer in a three-state quantum system by a train of pairs of coincident pulses,” Phys. Rev. A 85, 043407 (2012).
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    [Crossref]
  13. E. Dimova, A. Rangelov, and E. Kyoseva, “Broadband and ultra-broadband polarization rotators with adiabatic modular design,” J. Opt. 17, 075605 (2015).
    [Crossref]
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  15. T. Meany, M. Gräfe, R. Heilmann, A. Perez-Leija, S. Gross, M. J. Steel, M. J. Withford, and A. Szameit, “Laser written circuits for quantum photonics,” Laser and Photonics Reviews 9, 363–384 (2015).
    [Crossref]
  16. M. Delanty and M. J. Steel, “Discretely observable continuous-time quantum walks on Möbius strips and other exotic structures in three-dimensional integrated photonics,” Phys. Rev. A 86, 043821 (2012).
    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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  20. E. A. Shapiro, V. Milner, and M. Shapiro, “Complete transfer of populations from a single state to a preselected superposition of states using piecewise adiabatic passage: Theory,” Phys. Rev. A 79, 023422 (2009).
    [Crossref]
  21. R. Menchon-Enrich, J. Mompart, and V. Ahufinger, “Spatial adiabatic passage processes in sonic crystals with linear defects,” Phys. Rev. B 89, 094304 (2014).
    [Crossref]
  22. I. H. Malitson, “Interspecimen Comparison of the Refractive Index of Fused Silica,” J. Opt. Soc. Am. 55, 1205 (1965).
    [Crossref]
  23. A. Snyder and J. Love, Optical Waveguide Theory (Springer Science & Business Media, 1983).
  24. L. M. Jong, A. D. Greentree, V. I. Conrad, L. C. L. Hollenberg, and D. N. Jamieson, “Coherent tunneling adiabatic passage with the alternating coupling scheme,” Nanotechnology 20, 405402 (2009).
    [Crossref] [PubMed]
  25. K. Chung, T. J. Karle, M. Rab, A. D. Greentree, and S. Tomljenovic-Hanic, “Broadband and robust optical waveguide devices using coherent tunnelling adiabatic passage,” Opt. Express 20, 23108 (2012).
    [Crossref] [PubMed]
  26. S. J. Devitt, A. D. Greentree, and L. C. L. Hollenberg, “Information free quantum bus for generating stabiliser states,” Quantum Information Processing 6, 229–242 (2007).
    [Crossref]
  27. C. D. Hill, A. D. Greentree, and L. C. L. Hollenberg, “Parallel interaction-free measurement using spatial adiabatic passage,” New J. Phys. 13, 125002 (2011).
    [Crossref]
  28. A. P. Hope, T. G. Nguyen, A. Mitchell, and A. D. Greentree, “Adiabatic two-photon quantum gate operations using a long-range photonic bus,” J. Phys. B 48, 055503 (2015).
    [Crossref]
  29. R. Unanyan, B. Shore, and K. Bergmann, “Laser-driven population transfer in four-level atoms: Consequences of non-Abelian geometrical adiabatic phase factors,” Phys. Rev. A 59, 2910–2919 (1999).
    [Crossref]

2016 (1)

R. Menchon-Enrich, A. Benseny, V. Ahufinger, A. D. Greentree, T. Busch, and J. Mompart, “Spatial adiabatic passage: a review of recent progress,” Rep. Prog. Phys. 79, 74401 (2016).
[Crossref]

2015 (4)

E. Dimova, A. Rangelov, and E. Kyoseva, “Broadband and ultra-broadband polarization rotators with adiabatic modular design,” J. Opt. 17, 075605 (2015).
[Crossref]

Z. Chaboyer, T. Meany, L. G. Helt, M. J. Withford, and M. J. Steel, “Tunable quantum interference in a 3D integrated circuit,” Sci. Rep. 5, 9601 (2015).
[Crossref] [PubMed]

T. Meany, M. Gräfe, R. Heilmann, A. Perez-Leija, S. Gross, M. J. Steel, M. J. Withford, and A. Szameit, “Laser written circuits for quantum photonics,” Laser and Photonics Reviews 9, 363–384 (2015).
[Crossref]

A. P. Hope, T. G. Nguyen, A. Mitchell, and A. D. Greentree, “Adiabatic two-photon quantum gate operations using a long-range photonic bus,” J. Phys. B 48, 055503 (2015).
[Crossref]

2014 (1)

R. Menchon-Enrich, J. Mompart, and V. Ahufinger, “Spatial adiabatic passage processes in sonic crystals with linear defects,” Phys. Rev. B 89, 094304 (2014).
[Crossref]

2013 (1)

J. A. Vaitkus and A. D. Greentree, “Digital three-state adiabatic passage,” Phys. Rev. A 87, 063820 (2013).
[Crossref]

2012 (4)

A. A. Rangelov and N. V. Vitanov, “Complete population transfer in a three-state quantum system by a train of pairs of coincident pulses,” Phys. Rev. A 85, 043407 (2012).
[Crossref]

M. Delanty and M. J. Steel, “Discretely observable continuous-time quantum walks on Möbius strips and other exotic structures in three-dimensional integrated photonics,” Phys. Rev. A 86, 043821 (2012).
[Crossref]

G. Porat and A. Arie, “Efficient two-process frequency conversion through a dark intermediate state,” J. Opt. Soc. Am. B 29, 2901 (2012).
[Crossref]

K. Chung, T. J. Karle, M. Rab, A. D. Greentree, and S. Tomljenovic-Hanic, “Broadband and robust optical waveguide devices using coherent tunnelling adiabatic passage,” Opt. Express 20, 23108 (2012).
[Crossref] [PubMed]

2011 (1)

C. D. Hill, A. D. Greentree, and L. C. L. Hollenberg, “Parallel interaction-free measurement using spatial adiabatic passage,” New J. Phys. 13, 125002 (2011).
[Crossref]

2009 (5)

F. Dreisow, A. Szameit, M. Heinrich, R. Keil, S. Nolte, A. Tünnermann, and S. Longhi, “Adiabatic transfer of light via a continuum in optical waveguides,” Opt. Lett. 34, 2405–2407 (2009).
[Crossref] [PubMed]

L. M. Jong, A. D. Greentree, V. I. Conrad, L. C. L. Hollenberg, and D. N. Jamieson, “Coherent tunneling adiabatic passage with the alternating coupling scheme,” Nanotechnology 20, 405402 (2009).
[Crossref] [PubMed]

T. G. Nguyen, R. S. Tummidi, T. L. Koch, and A. Mitchell, “Rigorous modeling of lateral leakage loss in SOI thin-ridge waveguides and couplers,” IEEE Photon. Technol. Lett. 21, 486–488 (2009).
[Crossref]

E. A. Shapiro, V. Milner, and M. Shapiro, “Complete transfer of populations from a single state to a preselected superposition of states using piecewise adiabatic passage: Theory,” Phys. Rev. A 79, 023422 (2009).
[Crossref]

J. L. O’Brien, A. Furusawa, and J. Vučković, “Photonic quantum technologies,” Nature Photonics 3, 687–695 (2009).
[Crossref]

2007 (4)

S. Longhi, G. Della Valle, M. Ornigotti, and P. Laporta, “Coherent tunneling by adiabatic passage in an optical waveguide system,” Phys. Rev. B 76, 20110(R) (2007).
[Crossref]

P. Král, I. Thanopulos, and M. Shapiro, “Colloquium: Coherently controlled adiabatic passage,” Rev. Mod. Phys. 79, 53–77 (2007).
[Crossref]

E. A. Shapiro, V. Milner, C. Menzel-Jones, and M. Shapiro, “Piecewise adiabatic passage with a series of femtosecond pulses,” Phys. Rev. Lett. 99, 033002 (2007).
[Crossref] [PubMed]

S. J. Devitt, A. D. Greentree, and L. C. L. Hollenberg, “Information free quantum bus for generating stabiliser states,” Quantum Information Processing 6, 229–242 (2007).
[Crossref]

2006 (1)

E. Paspalakis, “Adiabatic three-waveguide directional coupler,” Optics Communications 258, 30–34 (2006).
[Crossref]

2004 (2)

K. Eckert, M. Lewenstein, R. Corbalán, G. Birkl, W. Ertmer, and J. Mompart, “Three-level atom optics via the tunneling interaction,” Phys. Rev. A 70, 023606 (2004).
[Crossref]

A. D. Greentree, J. H. Cole, A. R. Hamilton, and L. C. L. Hollenberg, “Coherent electronic transfer in quantum dot systems using adiabatic passage,” Phys. Rev. B 70, 235317 (2004).
[Crossref]

1999 (1)

R. Unanyan, B. Shore, and K. Bergmann, “Laser-driven population transfer in four-level atoms: Consequences of non-Abelian geometrical adiabatic phase factors,” Phys. Rev. A 59, 2910–2919 (1999).
[Crossref]

1965 (1)

Ahufinger, V.

R. Menchon-Enrich, A. Benseny, V. Ahufinger, A. D. Greentree, T. Busch, and J. Mompart, “Spatial adiabatic passage: a review of recent progress,” Rep. Prog. Phys. 79, 74401 (2016).
[Crossref]

R. Menchon-Enrich, J. Mompart, and V. Ahufinger, “Spatial adiabatic passage processes in sonic crystals with linear defects,” Phys. Rev. B 89, 094304 (2014).
[Crossref]

Arie, A.

Benseny, A.

R. Menchon-Enrich, A. Benseny, V. Ahufinger, A. D. Greentree, T. Busch, and J. Mompart, “Spatial adiabatic passage: a review of recent progress,” Rep. Prog. Phys. 79, 74401 (2016).
[Crossref]

Bergmann, K.

R. Unanyan, B. Shore, and K. Bergmann, “Laser-driven population transfer in four-level atoms: Consequences of non-Abelian geometrical adiabatic phase factors,” Phys. Rev. A 59, 2910–2919 (1999).
[Crossref]

Birkl, G.

K. Eckert, M. Lewenstein, R. Corbalán, G. Birkl, W. Ertmer, and J. Mompart, “Three-level atom optics via the tunneling interaction,” Phys. Rev. A 70, 023606 (2004).
[Crossref]

Busch, T.

R. Menchon-Enrich, A. Benseny, V. Ahufinger, A. D. Greentree, T. Busch, and J. Mompart, “Spatial adiabatic passage: a review of recent progress,” Rep. Prog. Phys. 79, 74401 (2016).
[Crossref]

Chaboyer, Z.

Z. Chaboyer, T. Meany, L. G. Helt, M. J. Withford, and M. J. Steel, “Tunable quantum interference in a 3D integrated circuit,” Sci. Rep. 5, 9601 (2015).
[Crossref] [PubMed]

Chaboyer, Z. J.

V. Ng, J. A. Vaitkus, Z. J. Chaboyer, T. Nguyen, J. M. Dawes, M. J. Withford, A. D. Greentree, and M. J. Steel, “Digital Waveguide Adiabatic Passage Part 2: Experiment,” arXiv:1608.07385 (2016).

Chung, K.

Cole, J. H.

A. D. Greentree, J. H. Cole, A. R. Hamilton, and L. C. L. Hollenberg, “Coherent electronic transfer in quantum dot systems using adiabatic passage,” Phys. Rev. B 70, 235317 (2004).
[Crossref]

Conrad, V. I.

L. M. Jong, A. D. Greentree, V. I. Conrad, L. C. L. Hollenberg, and D. N. Jamieson, “Coherent tunneling adiabatic passage with the alternating coupling scheme,” Nanotechnology 20, 405402 (2009).
[Crossref] [PubMed]

Corbalán, R.

K. Eckert, M. Lewenstein, R. Corbalán, G. Birkl, W. Ertmer, and J. Mompart, “Three-level atom optics via the tunneling interaction,” Phys. Rev. A 70, 023606 (2004).
[Crossref]

Dawes, J. M.

V. Ng, J. A. Vaitkus, Z. J. Chaboyer, T. Nguyen, J. M. Dawes, M. J. Withford, A. D. Greentree, and M. J. Steel, “Digital Waveguide Adiabatic Passage Part 2: Experiment,” arXiv:1608.07385 (2016).

Delanty, M.

M. Delanty and M. J. Steel, “Discretely observable continuous-time quantum walks on Möbius strips and other exotic structures in three-dimensional integrated photonics,” Phys. Rev. A 86, 043821 (2012).
[Crossref]

Della Valle, G.

S. Longhi, G. Della Valle, M. Ornigotti, and P. Laporta, “Coherent tunneling by adiabatic passage in an optical waveguide system,” Phys. Rev. B 76, 20110(R) (2007).
[Crossref]

Devitt, S. J.

S. J. Devitt, A. D. Greentree, and L. C. L. Hollenberg, “Information free quantum bus for generating stabiliser states,” Quantum Information Processing 6, 229–242 (2007).
[Crossref]

Dimova, E.

E. Dimova, A. Rangelov, and E. Kyoseva, “Broadband and ultra-broadband polarization rotators with adiabatic modular design,” J. Opt. 17, 075605 (2015).
[Crossref]

Dreisow, F.

Eckert, K.

K. Eckert, M. Lewenstein, R. Corbalán, G. Birkl, W. Ertmer, and J. Mompart, “Three-level atom optics via the tunneling interaction,” Phys. Rev. A 70, 023606 (2004).
[Crossref]

Ertmer, W.

K. Eckert, M. Lewenstein, R. Corbalán, G. Birkl, W. Ertmer, and J. Mompart, “Three-level atom optics via the tunneling interaction,” Phys. Rev. A 70, 023606 (2004).
[Crossref]

Furusawa, A.

J. L. O’Brien, A. Furusawa, and J. Vučković, “Photonic quantum technologies,” Nature Photonics 3, 687–695 (2009).
[Crossref]

Gräfe, M.

T. Meany, M. Gräfe, R. Heilmann, A. Perez-Leija, S. Gross, M. J. Steel, M. J. Withford, and A. Szameit, “Laser written circuits for quantum photonics,” Laser and Photonics Reviews 9, 363–384 (2015).
[Crossref]

Greentree, A. D.

R. Menchon-Enrich, A. Benseny, V. Ahufinger, A. D. Greentree, T. Busch, and J. Mompart, “Spatial adiabatic passage: a review of recent progress,” Rep. Prog. Phys. 79, 74401 (2016).
[Crossref]

A. P. Hope, T. G. Nguyen, A. Mitchell, and A. D. Greentree, “Adiabatic two-photon quantum gate operations using a long-range photonic bus,” J. Phys. B 48, 055503 (2015).
[Crossref]

J. A. Vaitkus and A. D. Greentree, “Digital three-state adiabatic passage,” Phys. Rev. A 87, 063820 (2013).
[Crossref]

K. Chung, T. J. Karle, M. Rab, A. D. Greentree, and S. Tomljenovic-Hanic, “Broadband and robust optical waveguide devices using coherent tunnelling adiabatic passage,” Opt. Express 20, 23108 (2012).
[Crossref] [PubMed]

C. D. Hill, A. D. Greentree, and L. C. L. Hollenberg, “Parallel interaction-free measurement using spatial adiabatic passage,” New J. Phys. 13, 125002 (2011).
[Crossref]

L. M. Jong, A. D. Greentree, V. I. Conrad, L. C. L. Hollenberg, and D. N. Jamieson, “Coherent tunneling adiabatic passage with the alternating coupling scheme,” Nanotechnology 20, 405402 (2009).
[Crossref] [PubMed]

S. J. Devitt, A. D. Greentree, and L. C. L. Hollenberg, “Information free quantum bus for generating stabiliser states,” Quantum Information Processing 6, 229–242 (2007).
[Crossref]

A. D. Greentree, J. H. Cole, A. R. Hamilton, and L. C. L. Hollenberg, “Coherent electronic transfer in quantum dot systems using adiabatic passage,” Phys. Rev. B 70, 235317 (2004).
[Crossref]

V. Ng, J. A. Vaitkus, Z. J. Chaboyer, T. Nguyen, J. M. Dawes, M. J. Withford, A. D. Greentree, and M. J. Steel, “Digital Waveguide Adiabatic Passage Part 2: Experiment,” arXiv:1608.07385 (2016).

Gross, S.

T. Meany, M. Gräfe, R. Heilmann, A. Perez-Leija, S. Gross, M. J. Steel, M. J. Withford, and A. Szameit, “Laser written circuits for quantum photonics,” Laser and Photonics Reviews 9, 363–384 (2015).
[Crossref]

Hamilton, A. R.

A. D. Greentree, J. H. Cole, A. R. Hamilton, and L. C. L. Hollenberg, “Coherent electronic transfer in quantum dot systems using adiabatic passage,” Phys. Rev. B 70, 235317 (2004).
[Crossref]

Heilmann, R.

T. Meany, M. Gräfe, R. Heilmann, A. Perez-Leija, S. Gross, M. J. Steel, M. J. Withford, and A. Szameit, “Laser written circuits for quantum photonics,” Laser and Photonics Reviews 9, 363–384 (2015).
[Crossref]

Heinrich, M.

Helt, L. G.

Z. Chaboyer, T. Meany, L. G. Helt, M. J. Withford, and M. J. Steel, “Tunable quantum interference in a 3D integrated circuit,” Sci. Rep. 5, 9601 (2015).
[Crossref] [PubMed]

Hill, C. D.

C. D. Hill, A. D. Greentree, and L. C. L. Hollenberg, “Parallel interaction-free measurement using spatial adiabatic passage,” New J. Phys. 13, 125002 (2011).
[Crossref]

Hollenberg, L. C. L.

C. D. Hill, A. D. Greentree, and L. C. L. Hollenberg, “Parallel interaction-free measurement using spatial adiabatic passage,” New J. Phys. 13, 125002 (2011).
[Crossref]

L. M. Jong, A. D. Greentree, V. I. Conrad, L. C. L. Hollenberg, and D. N. Jamieson, “Coherent tunneling adiabatic passage with the alternating coupling scheme,” Nanotechnology 20, 405402 (2009).
[Crossref] [PubMed]

S. J. Devitt, A. D. Greentree, and L. C. L. Hollenberg, “Information free quantum bus for generating stabiliser states,” Quantum Information Processing 6, 229–242 (2007).
[Crossref]

A. D. Greentree, J. H. Cole, A. R. Hamilton, and L. C. L. Hollenberg, “Coherent electronic transfer in quantum dot systems using adiabatic passage,” Phys. Rev. B 70, 235317 (2004).
[Crossref]

Hope, A. P.

A. P. Hope, T. G. Nguyen, A. Mitchell, and A. D. Greentree, “Adiabatic two-photon quantum gate operations using a long-range photonic bus,” J. Phys. B 48, 055503 (2015).
[Crossref]

Jamieson, D. N.

L. M. Jong, A. D. Greentree, V. I. Conrad, L. C. L. Hollenberg, and D. N. Jamieson, “Coherent tunneling adiabatic passage with the alternating coupling scheme,” Nanotechnology 20, 405402 (2009).
[Crossref] [PubMed]

Jong, L. M.

L. M. Jong, A. D. Greentree, V. I. Conrad, L. C. L. Hollenberg, and D. N. Jamieson, “Coherent tunneling adiabatic passage with the alternating coupling scheme,” Nanotechnology 20, 405402 (2009).
[Crossref] [PubMed]

Karle, T. J.

Keil, R.

Koch, T. L.

T. G. Nguyen, R. S. Tummidi, T. L. Koch, and A. Mitchell, “Rigorous modeling of lateral leakage loss in SOI thin-ridge waveguides and couplers,” IEEE Photon. Technol. Lett. 21, 486–488 (2009).
[Crossref]

Král, P.

P. Král, I. Thanopulos, and M. Shapiro, “Colloquium: Coherently controlled adiabatic passage,” Rev. Mod. Phys. 79, 53–77 (2007).
[Crossref]

Kyoseva, E.

E. Dimova, A. Rangelov, and E. Kyoseva, “Broadband and ultra-broadband polarization rotators with adiabatic modular design,” J. Opt. 17, 075605 (2015).
[Crossref]

Laporta, P.

S. Longhi, G. Della Valle, M. Ornigotti, and P. Laporta, “Coherent tunneling by adiabatic passage in an optical waveguide system,” Phys. Rev. B 76, 20110(R) (2007).
[Crossref]

Lewenstein, M.

K. Eckert, M. Lewenstein, R. Corbalán, G. Birkl, W. Ertmer, and J. Mompart, “Three-level atom optics via the tunneling interaction,” Phys. Rev. A 70, 023606 (2004).
[Crossref]

Lipson, M.

C. R. Pollock and M. Lipson, Integrated Photonics, vol. 20 (Springer, 2003).
[Crossref]

Longhi, S.

F. Dreisow, A. Szameit, M. Heinrich, R. Keil, S. Nolte, A. Tünnermann, and S. Longhi, “Adiabatic transfer of light via a continuum in optical waveguides,” Opt. Lett. 34, 2405–2407 (2009).
[Crossref] [PubMed]

S. Longhi, G. Della Valle, M. Ornigotti, and P. Laporta, “Coherent tunneling by adiabatic passage in an optical waveguide system,” Phys. Rev. B 76, 20110(R) (2007).
[Crossref]

Love, J.

A. Snyder and J. Love, Optical Waveguide Theory (Springer Science & Business Media, 1983).

Malitson, I. H.

Meany, T.

Z. Chaboyer, T. Meany, L. G. Helt, M. J. Withford, and M. J. Steel, “Tunable quantum interference in a 3D integrated circuit,” Sci. Rep. 5, 9601 (2015).
[Crossref] [PubMed]

T. Meany, M. Gräfe, R. Heilmann, A. Perez-Leija, S. Gross, M. J. Steel, M. J. Withford, and A. Szameit, “Laser written circuits for quantum photonics,” Laser and Photonics Reviews 9, 363–384 (2015).
[Crossref]

Menchon-Enrich, R.

R. Menchon-Enrich, A. Benseny, V. Ahufinger, A. D. Greentree, T. Busch, and J. Mompart, “Spatial adiabatic passage: a review of recent progress,” Rep. Prog. Phys. 79, 74401 (2016).
[Crossref]

R. Menchon-Enrich, J. Mompart, and V. Ahufinger, “Spatial adiabatic passage processes in sonic crystals with linear defects,” Phys. Rev. B 89, 094304 (2014).
[Crossref]

Menzel-Jones, C.

E. A. Shapiro, V. Milner, C. Menzel-Jones, and M. Shapiro, “Piecewise adiabatic passage with a series of femtosecond pulses,” Phys. Rev. Lett. 99, 033002 (2007).
[Crossref] [PubMed]

Milner, V.

E. A. Shapiro, V. Milner, and M. Shapiro, “Complete transfer of populations from a single state to a preselected superposition of states using piecewise adiabatic passage: Theory,” Phys. Rev. A 79, 023422 (2009).
[Crossref]

E. A. Shapiro, V. Milner, C. Menzel-Jones, and M. Shapiro, “Piecewise adiabatic passage with a series of femtosecond pulses,” Phys. Rev. Lett. 99, 033002 (2007).
[Crossref] [PubMed]

Mitchell, A.

A. P. Hope, T. G. Nguyen, A. Mitchell, and A. D. Greentree, “Adiabatic two-photon quantum gate operations using a long-range photonic bus,” J. Phys. B 48, 055503 (2015).
[Crossref]

T. G. Nguyen, R. S. Tummidi, T. L. Koch, and A. Mitchell, “Rigorous modeling of lateral leakage loss in SOI thin-ridge waveguides and couplers,” IEEE Photon. Technol. Lett. 21, 486–488 (2009).
[Crossref]

Mompart, J.

R. Menchon-Enrich, A. Benseny, V. Ahufinger, A. D. Greentree, T. Busch, and J. Mompart, “Spatial adiabatic passage: a review of recent progress,” Rep. Prog. Phys. 79, 74401 (2016).
[Crossref]

R. Menchon-Enrich, J. Mompart, and V. Ahufinger, “Spatial adiabatic passage processes in sonic crystals with linear defects,” Phys. Rev. B 89, 094304 (2014).
[Crossref]

K. Eckert, M. Lewenstein, R. Corbalán, G. Birkl, W. Ertmer, and J. Mompart, “Three-level atom optics via the tunneling interaction,” Phys. Rev. A 70, 023606 (2004).
[Crossref]

Ng, V.

V. Ng, J. A. Vaitkus, Z. J. Chaboyer, T. Nguyen, J. M. Dawes, M. J. Withford, A. D. Greentree, and M. J. Steel, “Digital Waveguide Adiabatic Passage Part 2: Experiment,” arXiv:1608.07385 (2016).

Nguyen, T.

V. Ng, J. A. Vaitkus, Z. J. Chaboyer, T. Nguyen, J. M. Dawes, M. J. Withford, A. D. Greentree, and M. J. Steel, “Digital Waveguide Adiabatic Passage Part 2: Experiment,” arXiv:1608.07385 (2016).

Nguyen, T. G.

A. P. Hope, T. G. Nguyen, A. Mitchell, and A. D. Greentree, “Adiabatic two-photon quantum gate operations using a long-range photonic bus,” J. Phys. B 48, 055503 (2015).
[Crossref]

T. G. Nguyen, R. S. Tummidi, T. L. Koch, and A. Mitchell, “Rigorous modeling of lateral leakage loss in SOI thin-ridge waveguides and couplers,” IEEE Photon. Technol. Lett. 21, 486–488 (2009).
[Crossref]

Nolte, S.

O’Brien, J. L.

J. L. O’Brien, A. Furusawa, and J. Vučković, “Photonic quantum technologies,” Nature Photonics 3, 687–695 (2009).
[Crossref]

Ornigotti, M.

S. Longhi, G. Della Valle, M. Ornigotti, and P. Laporta, “Coherent tunneling by adiabatic passage in an optical waveguide system,” Phys. Rev. B 76, 20110(R) (2007).
[Crossref]

Paspalakis, E.

E. Paspalakis, “Adiabatic three-waveguide directional coupler,” Optics Communications 258, 30–34 (2006).
[Crossref]

Perez-Leija, A.

T. Meany, M. Gräfe, R. Heilmann, A. Perez-Leija, S. Gross, M. J. Steel, M. J. Withford, and A. Szameit, “Laser written circuits for quantum photonics,” Laser and Photonics Reviews 9, 363–384 (2015).
[Crossref]

Pollock, C. R.

C. R. Pollock and M. Lipson, Integrated Photonics, vol. 20 (Springer, 2003).
[Crossref]

Porat, G.

Rab, M.

Rangelov, A.

E. Dimova, A. Rangelov, and E. Kyoseva, “Broadband and ultra-broadband polarization rotators with adiabatic modular design,” J. Opt. 17, 075605 (2015).
[Crossref]

Rangelov, A. A.

A. A. Rangelov and N. V. Vitanov, “Complete population transfer in a three-state quantum system by a train of pairs of coincident pulses,” Phys. Rev. A 85, 043407 (2012).
[Crossref]

Shapiro, E. A.

E. A. Shapiro, V. Milner, and M. Shapiro, “Complete transfer of populations from a single state to a preselected superposition of states using piecewise adiabatic passage: Theory,” Phys. Rev. A 79, 023422 (2009).
[Crossref]

E. A. Shapiro, V. Milner, C. Menzel-Jones, and M. Shapiro, “Piecewise adiabatic passage with a series of femtosecond pulses,” Phys. Rev. Lett. 99, 033002 (2007).
[Crossref] [PubMed]

Shapiro, M.

E. A. Shapiro, V. Milner, and M. Shapiro, “Complete transfer of populations from a single state to a preselected superposition of states using piecewise adiabatic passage: Theory,” Phys. Rev. A 79, 023422 (2009).
[Crossref]

E. A. Shapiro, V. Milner, C. Menzel-Jones, and M. Shapiro, “Piecewise adiabatic passage with a series of femtosecond pulses,” Phys. Rev. Lett. 99, 033002 (2007).
[Crossref] [PubMed]

P. Král, I. Thanopulos, and M. Shapiro, “Colloquium: Coherently controlled adiabatic passage,” Rev. Mod. Phys. 79, 53–77 (2007).
[Crossref]

Shore, B.

R. Unanyan, B. Shore, and K. Bergmann, “Laser-driven population transfer in four-level atoms: Consequences of non-Abelian geometrical adiabatic phase factors,” Phys. Rev. A 59, 2910–2919 (1999).
[Crossref]

Snyder, A.

A. Snyder and J. Love, Optical Waveguide Theory (Springer Science & Business Media, 1983).

Steel, M. J.

Z. Chaboyer, T. Meany, L. G. Helt, M. J. Withford, and M. J. Steel, “Tunable quantum interference in a 3D integrated circuit,” Sci. Rep. 5, 9601 (2015).
[Crossref] [PubMed]

T. Meany, M. Gräfe, R. Heilmann, A. Perez-Leija, S. Gross, M. J. Steel, M. J. Withford, and A. Szameit, “Laser written circuits for quantum photonics,” Laser and Photonics Reviews 9, 363–384 (2015).
[Crossref]

M. Delanty and M. J. Steel, “Discretely observable continuous-time quantum walks on Möbius strips and other exotic structures in three-dimensional integrated photonics,” Phys. Rev. A 86, 043821 (2012).
[Crossref]

V. Ng, J. A. Vaitkus, Z. J. Chaboyer, T. Nguyen, J. M. Dawes, M. J. Withford, A. D. Greentree, and M. J. Steel, “Digital Waveguide Adiabatic Passage Part 2: Experiment,” arXiv:1608.07385 (2016).

Szameit, A.

T. Meany, M. Gräfe, R. Heilmann, A. Perez-Leija, S. Gross, M. J. Steel, M. J. Withford, and A. Szameit, “Laser written circuits for quantum photonics,” Laser and Photonics Reviews 9, 363–384 (2015).
[Crossref]

F. Dreisow, A. Szameit, M. Heinrich, R. Keil, S. Nolte, A. Tünnermann, and S. Longhi, “Adiabatic transfer of light via a continuum in optical waveguides,” Opt. Lett. 34, 2405–2407 (2009).
[Crossref] [PubMed]

Thanopulos, I.

P. Král, I. Thanopulos, and M. Shapiro, “Colloquium: Coherently controlled adiabatic passage,” Rev. Mod. Phys. 79, 53–77 (2007).
[Crossref]

Tomljenovic-Hanic, S.

Tummidi, R. S.

T. G. Nguyen, R. S. Tummidi, T. L. Koch, and A. Mitchell, “Rigorous modeling of lateral leakage loss in SOI thin-ridge waveguides and couplers,” IEEE Photon. Technol. Lett. 21, 486–488 (2009).
[Crossref]

Tünnermann, A.

Unanyan, R.

R. Unanyan, B. Shore, and K. Bergmann, “Laser-driven population transfer in four-level atoms: Consequences of non-Abelian geometrical adiabatic phase factors,” Phys. Rev. A 59, 2910–2919 (1999).
[Crossref]

Vaitkus, J. A.

J. A. Vaitkus and A. D. Greentree, “Digital three-state adiabatic passage,” Phys. Rev. A 87, 063820 (2013).
[Crossref]

V. Ng, J. A. Vaitkus, Z. J. Chaboyer, T. Nguyen, J. M. Dawes, M. J. Withford, A. D. Greentree, and M. J. Steel, “Digital Waveguide Adiabatic Passage Part 2: Experiment,” arXiv:1608.07385 (2016).

Vitanov, N. V.

A. A. Rangelov and N. V. Vitanov, “Complete population transfer in a three-state quantum system by a train of pairs of coincident pulses,” Phys. Rev. A 85, 043407 (2012).
[Crossref]

Vuckovic, J.

J. L. O’Brien, A. Furusawa, and J. Vučković, “Photonic quantum technologies,” Nature Photonics 3, 687–695 (2009).
[Crossref]

Withford, M. J.

Z. Chaboyer, T. Meany, L. G. Helt, M. J. Withford, and M. J. Steel, “Tunable quantum interference in a 3D integrated circuit,” Sci. Rep. 5, 9601 (2015).
[Crossref] [PubMed]

T. Meany, M. Gräfe, R. Heilmann, A. Perez-Leija, S. Gross, M. J. Steel, M. J. Withford, and A. Szameit, “Laser written circuits for quantum photonics,” Laser and Photonics Reviews 9, 363–384 (2015).
[Crossref]

V. Ng, J. A. Vaitkus, Z. J. Chaboyer, T. Nguyen, J. M. Dawes, M. J. Withford, A. D. Greentree, and M. J. Steel, “Digital Waveguide Adiabatic Passage Part 2: Experiment,” arXiv:1608.07385 (2016).

IEEE Photon. Technol. Lett. (1)

T. G. Nguyen, R. S. Tummidi, T. L. Koch, and A. Mitchell, “Rigorous modeling of lateral leakage loss in SOI thin-ridge waveguides and couplers,” IEEE Photon. Technol. Lett. 21, 486–488 (2009).
[Crossref]

J. Opt. (1)

E. Dimova, A. Rangelov, and E. Kyoseva, “Broadband and ultra-broadband polarization rotators with adiabatic modular design,” J. Opt. 17, 075605 (2015).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (1)

J. Phys. B (1)

A. P. Hope, T. G. Nguyen, A. Mitchell, and A. D. Greentree, “Adiabatic two-photon quantum gate operations using a long-range photonic bus,” J. Phys. B 48, 055503 (2015).
[Crossref]

Laser and Photonics Reviews (1)

T. Meany, M. Gräfe, R. Heilmann, A. Perez-Leija, S. Gross, M. J. Steel, M. J. Withford, and A. Szameit, “Laser written circuits for quantum photonics,” Laser and Photonics Reviews 9, 363–384 (2015).
[Crossref]

Nanotechnology (1)

L. M. Jong, A. D. Greentree, V. I. Conrad, L. C. L. Hollenberg, and D. N. Jamieson, “Coherent tunneling adiabatic passage with the alternating coupling scheme,” Nanotechnology 20, 405402 (2009).
[Crossref] [PubMed]

Nature Photonics (1)

J. L. O’Brien, A. Furusawa, and J. Vučković, “Photonic quantum technologies,” Nature Photonics 3, 687–695 (2009).
[Crossref]

New J. Phys. (1)

C. D. Hill, A. D. Greentree, and L. C. L. Hollenberg, “Parallel interaction-free measurement using spatial adiabatic passage,” New J. Phys. 13, 125002 (2011).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Optics Communications (1)

E. Paspalakis, “Adiabatic three-waveguide directional coupler,” Optics Communications 258, 30–34 (2006).
[Crossref]

Phys. Rev. A (6)

K. Eckert, M. Lewenstein, R. Corbalán, G. Birkl, W. Ertmer, and J. Mompart, “Three-level atom optics via the tunneling interaction,” Phys. Rev. A 70, 023606 (2004).
[Crossref]

A. A. Rangelov and N. V. Vitanov, “Complete population transfer in a three-state quantum system by a train of pairs of coincident pulses,” Phys. Rev. A 85, 043407 (2012).
[Crossref]

M. Delanty and M. J. Steel, “Discretely observable continuous-time quantum walks on Möbius strips and other exotic structures in three-dimensional integrated photonics,” Phys. Rev. A 86, 043821 (2012).
[Crossref]

J. A. Vaitkus and A. D. Greentree, “Digital three-state adiabatic passage,” Phys. Rev. A 87, 063820 (2013).
[Crossref]

E. A. Shapiro, V. Milner, and M. Shapiro, “Complete transfer of populations from a single state to a preselected superposition of states using piecewise adiabatic passage: Theory,” Phys. Rev. A 79, 023422 (2009).
[Crossref]

R. Unanyan, B. Shore, and K. Bergmann, “Laser-driven population transfer in four-level atoms: Consequences of non-Abelian geometrical adiabatic phase factors,” Phys. Rev. A 59, 2910–2919 (1999).
[Crossref]

Phys. Rev. B (3)

R. Menchon-Enrich, J. Mompart, and V. Ahufinger, “Spatial adiabatic passage processes in sonic crystals with linear defects,” Phys. Rev. B 89, 094304 (2014).
[Crossref]

A. D. Greentree, J. H. Cole, A. R. Hamilton, and L. C. L. Hollenberg, “Coherent electronic transfer in quantum dot systems using adiabatic passage,” Phys. Rev. B 70, 235317 (2004).
[Crossref]

S. Longhi, G. Della Valle, M. Ornigotti, and P. Laporta, “Coherent tunneling by adiabatic passage in an optical waveguide system,” Phys. Rev. B 76, 20110(R) (2007).
[Crossref]

Phys. Rev. Lett. (1)

E. A. Shapiro, V. Milner, C. Menzel-Jones, and M. Shapiro, “Piecewise adiabatic passage with a series of femtosecond pulses,” Phys. Rev. Lett. 99, 033002 (2007).
[Crossref] [PubMed]

Quantum Information Processing (1)

S. J. Devitt, A. D. Greentree, and L. C. L. Hollenberg, “Information free quantum bus for generating stabiliser states,” Quantum Information Processing 6, 229–242 (2007).
[Crossref]

Rep. Prog. Phys. (1)

R. Menchon-Enrich, A. Benseny, V. Ahufinger, A. D. Greentree, T. Busch, and J. Mompart, “Spatial adiabatic passage: a review of recent progress,” Rep. Prog. Phys. 79, 74401 (2016).
[Crossref]

Rev. Mod. Phys. (1)

P. Král, I. Thanopulos, and M. Shapiro, “Colloquium: Coherently controlled adiabatic passage,” Rev. Mod. Phys. 79, 53–77 (2007).
[Crossref]

Sci. Rep. (1)

Z. Chaboyer, T. Meany, L. G. Helt, M. J. Withford, and M. J. Steel, “Tunable quantum interference in a 3D integrated circuit,” Sci. Rep. 5, 9601 (2015).
[Crossref] [PubMed]

Other (3)

C. R. Pollock and M. Lipson, Integrated Photonics, vol. 20 (Springer, 2003).
[Crossref]

A. Snyder and J. Love, Optical Waveguide Theory (Springer Science & Business Media, 1983).

V. Ng, J. A. Vaitkus, Z. J. Chaboyer, T. Nguyen, J. M. Dawes, M. J. Withford, A. D. Greentree, and M. J. Steel, “Digital Waveguide Adiabatic Passage Part 2: Experiment,” arXiv:1608.07385 (2016).

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Figures (8)

Fig. 1
Fig. 1

(a) Structure for digital waveguide adiabatic passage showing the segmented waveguide with circular geometry. The counter-intuitive coupling sequence is achieved by light propagating in the z-direction entering at the bottom left waveguide, and exiting via the top right, with the coupling mediated in the x-direction by the central waveguidelets (shown colored). Figures (b), (c) and (d) show the refractive index profiles for the red, green and blue cases from (a), demonstrating the additive nature of a continuous refractive index profile. The red lines are the refractive index of each element independently, and the blue lines show the sum of the refractive indices. When the central waveguidelet is closest to one of the outer waveguides, the independent waveguide approximation breaks down. The last two waveguidelet images are mirror images of the first two and are not shown for brevity. Note that these images are purely for illustrative purposes only; the particular device parameters can be found in Table 1.

Fig. 2
Fig. 2

(a) Effective change to propagation constant due to the presence of another guide. (b) Numerically (solid) and semi-analytically (dashed) obtained coupling of Gaussian index fibers. The minimum separation is 2ρ so that the waveguides are clearly distinguishable. Device parameters are given in Table 1.

Fig. 3
Fig. 3

(a) Pseudo-colour plot showing the final state population (colour axis) as a function of δ and λ.(b) Pseudo-colour plot showing final state population as a function of λ and device length, L. In both cases note the wide wavelength range over which devices provide high-fidelity transport. The fidelity is periodic, and we have highlighted only one period here. The dark patch in the top right of the length subfigure is actually a pessimal resonance [12] with 90% in the initial state despite being designed for a completely different length and wavelength. All other parameters not being varied are the same as they are in Table 1.

Fig. 4
Fig. 4

Pb = |〈E2|b〉|2 as a function of scaled perturbative parameters using analytical (solid) and approximate (dashed) forms for Ωac (left) and βdiff (right). Values are symmetric with respect to Ωab ↔ Ωbc, to represent data in reduced units we divide through all parameters by Ωab. We also divide the perturbative parameters by Ωbc, as any value exceeding Ωbc/2 would no longer be a perturbation. The functions each have a local maximum at Ω b c = Ω a b / 5 and Ω b c = Ω a b / 2 (see (17) and (19) respectively) and so values are linearly spaced up to those points. These data show that the approximations are good over a wide range of possible values with deviations strongest at the turning point.

Fig. 5
Fig. 5

Illustrations of the mirrored five-state device considered in this subsection. (left) schematic demonstrating the couplings of the device and (right) illustration demonstrating the mirror geometry, each color corresponds to a set of waveguides described in Table 2. Light is injected in |c〉 and coupled into a superposition of |a〉 and |e〉. For simplicity, we will only consider when Ωbc = Ωcd and Ωab = Ωde resulting in a equal division of power. The x-coordinate describes the centre of the waveguide. Segments are varied in the x-direction to vary the couplings. All waveguides are placed such that no next-nearest neighbour coupling is possible.

Fig. 6
Fig. 6

Populations Pi = |〈ψ|i〉|2 during the five-state digital adiabatic 50:50 power division protocol on a regular (left) and log (right) scale. Faint dashed lines show the end/beginning of waveguidelets. It can be seen that the population traces Pa, Pe and Pb, Pd are directly on top of each other and that population in the intermediate waveguides goes to zero at the end of each waveguidelet. Coupling values were chosen to transfer equal population per step and device parameters can be found in Table 2.

Fig. 7
Fig. 7

(left) Illustration demonstrating couplings in the triangular four-state system considered in this subsection. (right) Schematic to-scale side-view of the parameters found in Table 3, waveguide sizes are decreased for distinguishability and the inset provided demonstrates the relatively small vertical movement. Light is injected into |a〉 and is brought into a controlled superposition of |c〉 and |d〉 by controlling the ratio α = Ω b c / Ω b d = P c / P d. The x-coordinate describes the centre of the waveguide. The x and y-coordinates describe the centre of the waveguides. Segments are varied in the x and y-directions to vary the couplings while maintaining a constant ratio of couplings α = Ω b c / Ω b d = P c / P d. All waveguides are placed such that no next-nearest neighbour coupling is possible.

Fig. 8
Fig. 8

Populations Pi = |〈ψ|i〉|2 during the four-state adiabatic 1/3:2/3 power division protocol on a regular (left) and log (right) scale. Faint dashed lines show the end/beginning of waveguidelets. It can be seen that the population in the intermediate waveguides goes to zero at the end of each waveguidelet. Coupling values were chosen to transfer equal population per step and device parameters can be found in Table 3.

Tables (3)

Tables Icon

Table 1 Device geometry and parameters used in all calculations regarding the three-state coupler. DAP is from |a〉 to |c〉, and the central waveguide, |b〉, is split into 5 waveguidelets, |b1 to |b5. Propagation occurs in the z-direction and all segments are aligned at y = 0. |a〉, |b1, |c〉 all begin at z = 0. Segment |bi+1 is connected to the end of segment |bi. The x-coordinate describes the centre of the waveguide. Segments are varied in the x-direction to vary the couplings.

Tables Icon

Table 2 Device geometry and parameters used in transport calculations for five-state based 50:50 splitter. DAP is from |c〉 to |a〉 and |e〉, and the intermediate waveguides, |b〉, |d〉, are split into 5 waveguidelets each, |b1, |d1 to |b5, |d5. Propagation occurs in the z-direction and all segments are aligned at y = 0. |a〉, |b1, |c〉, |d1, |e〉 all begin at z = 0. Segment |bi+1 (|di+1) is connected to the end of segment |bi (|di). The x-coordinate describes the centre of the waveguide. Segments are varied in the x-direction to vary the couplings. Recall that the device is symmetric about |c〉, and the positions of |a〉, |e〉 and |b〉, |d〉 are related by x → −x. Properties ρ, δ, λopt are the same as Table 1.

Tables Icon

Table 3 Device geometry and parameters used in transport calculations for four-state based 1/3:2/3 splitter. DAP is from |a〉 to |c〉 and |d〉, and the intermediate waveguide, |b〉, is split into 5 waveguidelets, |b1 to |b5. Propagation occurs in the z-direction. |a〉, |b1, |c〉, |d〉 all begin at z = 0. Segment |bi+1 is connected to the end of segment |bi. The x and y-coordinates describe the centre of the waveguides. Segments are varied in the x and y-directions to vary the couplings while maintaining a constant ratio of couplings Ω b c / Ω b d = P c / P d. Properties ρ, δ, λopt are the same as Table 1. Distances between |a〉, |c〉, and |d〉 are all 21μm to minimize cross-talk.

Equations (22)

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

H = [ β a Ω a b Ω a c Ω a b * β b Ω b c Ω a c * Ω b c * β c ] ,
ε 0 + ε 1 E k + ε 2 E k 2 + E k 3 = 0 ,
Q ( ε 2 2 3 ε 1 9 ) 1 / 2 , R 9 ε 1 ε 2 27 ε 0 2 ε 2 3 54 ,
E k = 2 Q cos [ 1 3 arccos ( R Q 3 ) + 2 π k 3 ] ε 2 / 3 .
| E k = a k | a + b k | b + c k | c | a k | 2 + | b k | 2 + | c k | 2 ,
a k = Ω b c [ 1 Ω a c ( β b E k ) Ω a b Ω b c ] , b k = 1 Ω a b [ Ω a b * Ω a c ( β a E k ) Ω b c ] , c k = Ω a b * [ 1 ( β a E k ) ( β b E k ) | Ω a b | 2 ] .
| E 2 = Ω b c | a Ω a b * | c | Ω a b | 2 + | Ω b c | 2 ,
P i = | ψ | i | 2 = | i | ψ | 2 ,
n cl ( λ ) = ( 0.897479 λ 2 λ 2 97.934 + 0.696166 λ 2 λ 2 4.67915 × 10 3 + 0.407943 λ 2 λ 2 1.35121 × 10 2 + 1 ) 1 / 2 ,
δ = n co n cl , Δ = n co 2 n cl 2 2 n co 2 .
δ ( λ ) = δ ref n cl ( λ ref ) n cl ( λ ) n co ( λ ) = ( 1 + δ ref n cl ( λ ref ) ) n cl ( λ ) .
n = n cl + δ [ exp ( r a 2 ρ 2 ) + exp ( r b 2 ρ 2 ) + exp ( r c 2 ρ 2 ) ] ,
r a 2 = ( x + D / 2 ) 2 + y 2 , r b 2 = [ x ( D 2 d ) ( 2 z z max 1 ) 2 ] + y 2 , r c 2 = ( x D / 2 ) 2 + y 2 ,
n 2 = n co 2 { 1 2 Δ [ 1 exp ( r a 2 ρ 2 ) exp ( r b 2 ρ 2 ) exp ( r c 2 ρ 2 ) ] } .
Ω i j = 2 Δ ρ V 3 ( V 1 ) ( V + 1 ) 2 exp [ ( V 1 ) 2 V + 1 ] K 0 [ ( V 1 ) R i j / ρ ] ,
( π Δ R i j ρ ) 1 / 2 V 3 ( V 1 ) 1 / 2 ( V + 1 ) 2 exp [ ( V 1 ) ( V 1 V + 1 R i j ρ ) ] ,
| E 2 | b | 2 Ω a c 2 ( Ω a b Ω b c ) 2 ( Ω a b + Ω b c ) 2 ( Ω a b 2 + Ω b c 2 ) 3 .
1 P Loss = [ 1 ( 10 27 Ω a c Ω ) 2 ] N exp ( 10 N Ω a c 2 27 Ω 2 ) ,
| E 2 | b | 2 β diff 2 Ω a b 2 Ω b c 2 ( Ω a b 2 + Ω b c 2 ) 3 .
1 P Loss = [ 1 ( 2 β diff 3 3 Ω ) 2 ] N exp ( 4 N β diff 2 27 Ω 2 ) ,
H = [ 0 Ω a b 0 0 0 Ω a b 0 Ω b c 0 0 0 Ω b c 0 Ω c d 0 0 0 Ω c d 0 Ω d e 0 0 0 Ω d e 0 ] .
H = [ 0 Ω a b 0 0 Ω a b 0 Ω b c Ω b d 0 Ω b c 0 0 0 Ω b d 0 0 ] .

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