Abstract

We propose short and robust directional couplers designed by shortcuts to adiabaticity, based on Lewis-Riesenfeld invariant theory. The design of directional couplers is discussed by combining invariant-based inverse engineering and perturbation theory. The error sensitivity of the coupler is minimized by optimizing the evolution of dynamical invariant with respect to coupling coefficient/input wavelength variations. The proposed robust coupler devices are verified with beam propagation simulations.

© 2014 Optical Society of America

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  1. A. Syahriar, V. M. Schneider, and S. Al-Bader, “The design of mode evolution couplers,” J. Lightwave Technol. 16, 1907–1914 (1998).
    [CrossRef]
  2. T. A. Ramadan, R. Scarmozzino, and R. M. Osgood, “Adiabatic couplers: design rules and optimization,” J. Lightwave Technol. 16, 277–283 (1998).
    [CrossRef]
  3. X. Sun, H.-C. Liu, and A. Yariv, “Adiabaticity criterion and the shortest adiabatic mode transformer in a coupled-waveguide system,” Opt. Lett. 34, 280–282 (2009).
    [CrossRef] [PubMed]
  4. R. R. A. Syms, “The digital directional coupler: improved design,” IEEE Photon. Technol. Lett. 4, 1135–1138 (1992).
    [CrossRef]
  5. G. T. Paloczi, A. Eyal, and A. Yariv, “Wavelength-insensitive nonadiabatic mode evolution couplers,” IEEE Photon. Technol. Lett. 16, 515–517 (2004).
    [CrossRef]
  6. S.-Y. Tseng, “Counterdiabatic mode-evolution based coupled-waveguide devices,” Opt. Express 21, 21224–21235 (2013).
    [CrossRef] [PubMed]
  7. S. Longhi, “Quantum-optical analogies using photonic structures,” Laser and Photon. Rev. 3, 243–261 (2009).
    [CrossRef]
  8. E. Torrontegui, S. Ibáñez, S. Martínez-Garaot, M. Modugno, A. del Campo, D. Guéry-Odelin, A. Ruschhaupt, X. Chen, and J. G. Muga, “Shortcuts to adiabaticity,” Adv. At. Mol. Opt. Phys. 62, 117–169 (2013).
    [CrossRef]
  9. T.-Y. Lin, F.-C. Hsiao, Y.-W. Jhang, C. Hu, and S.-Y. Tseng, “Mode conversion using optical analogy of shortcut to adiabatic passage in engineered multimode waveguides,” Opt. Express 20, 24085–24092 (2012).
    [CrossRef] [PubMed]
  10. S.-Y. Tseng and X. Chen, “Engineering of fast mode conversion in multimode waveguides,” Opt. Lett. 37, 5118–5120 (2012).
    [CrossRef] [PubMed]
  11. S. Martínez-Garaot, S.-Y. Tseng, and J. G. Muga, “Compact and high conversion efficiency mode-sorting asymmetric Y junction using shortcuts to adiabaticity,” Opt. Lett. 39, 2306 (2014).
    [CrossRef] [PubMed]
  12. X. Chen and J. G. Muga, “Engineering of fast population transfer in three-level systems,” Phys. Rev. A 86, 033405 (2012).
    [CrossRef]
  13. S.-Y. Tseng and Y.-W. Jhang, “Fast and robust beam coupling in a three waveguide directional coupler,” IEEE Photon. Technol. Lett. 25, 2478–2481 (2013).
    [CrossRef]
  14. A. Ruschhaupt, X. Chen, D. Alonso, and J. G. Muga, “Optimally robust shortcuts to population inversion in two-level quantum systems,” New J. Phys. 14, 093040 (2012).
    [CrossRef]
  15. D. Daems, A. Ruschhaupt, D. Sugny, and S. Guérin, “Robust quantum control by a single-shot shaped pulse,” Phys. Rev. Lett. 111, 050404 (2013).
    [CrossRef] [PubMed]
  16. X.-J. Lu, X. Chen, A. Ruschhaupt, D. Alonso, S. Guérin, and J. G. Muga, “Fast and robust population transfer in two-level quantum systems with dephasing noise and/or systematic frequency errors,” Phys. Rev. A 88, 033406 (2013).
    [CrossRef]
  17. X. Chen, E. Torrontegui, and J. G. Muga, “Lewis-Riesenfeld invariants and transitionless quantum driving,” Phys. Rev. A 83, 062116 (2011).
    [CrossRef]
  18. H. R. Lewis and W. B. Riesenfeld, “An exact quantum theory of the time-dependent harmonic oscillator and of a charged particle in a time-dependent electromagnetic field,” J. Math. Phys. 10, 1458–1473 (1969).
    [CrossRef]
  19. C.-L. Chen, Foundations for Guided-Wave Optics (Wiley, 2007).
  20. K. Kawano and T. Kitoh, Introduction to Optical Waveguide Analysis: Solving Maxwell’s Equations (Wiley, 2001).
    [CrossRef]
  21. K. Okamoto, Fundamentals of Optical Waveguides (Academic, 2006).
  22. D. R. Rowland, Y. Chen, and A. W. Snyder, “Tapered mismatched couplers,” J. Lightwave Technol. 9, 567–570 (1991).
    [CrossRef]

2014

2013

S.-Y. Tseng, “Counterdiabatic mode-evolution based coupled-waveguide devices,” Opt. Express 21, 21224–21235 (2013).
[CrossRef] [PubMed]

E. Torrontegui, S. Ibáñez, S. Martínez-Garaot, M. Modugno, A. del Campo, D. Guéry-Odelin, A. Ruschhaupt, X. Chen, and J. G. Muga, “Shortcuts to adiabaticity,” Adv. At. Mol. Opt. Phys. 62, 117–169 (2013).
[CrossRef]

D. Daems, A. Ruschhaupt, D. Sugny, and S. Guérin, “Robust quantum control by a single-shot shaped pulse,” Phys. Rev. Lett. 111, 050404 (2013).
[CrossRef] [PubMed]

X.-J. Lu, X. Chen, A. Ruschhaupt, D. Alonso, S. Guérin, and J. G. Muga, “Fast and robust population transfer in two-level quantum systems with dephasing noise and/or systematic frequency errors,” Phys. Rev. A 88, 033406 (2013).
[CrossRef]

S.-Y. Tseng and Y.-W. Jhang, “Fast and robust beam coupling in a three waveguide directional coupler,” IEEE Photon. Technol. Lett. 25, 2478–2481 (2013).
[CrossRef]

2012

A. Ruschhaupt, X. Chen, D. Alonso, and J. G. Muga, “Optimally robust shortcuts to population inversion in two-level quantum systems,” New J. Phys. 14, 093040 (2012).
[CrossRef]

T.-Y. Lin, F.-C. Hsiao, Y.-W. Jhang, C. Hu, and S.-Y. Tseng, “Mode conversion using optical analogy of shortcut to adiabatic passage in engineered multimode waveguides,” Opt. Express 20, 24085–24092 (2012).
[CrossRef] [PubMed]

S.-Y. Tseng and X. Chen, “Engineering of fast mode conversion in multimode waveguides,” Opt. Lett. 37, 5118–5120 (2012).
[CrossRef] [PubMed]

X. Chen and J. G. Muga, “Engineering of fast population transfer in three-level systems,” Phys. Rev. A 86, 033405 (2012).
[CrossRef]

2011

X. Chen, E. Torrontegui, and J. G. Muga, “Lewis-Riesenfeld invariants and transitionless quantum driving,” Phys. Rev. A 83, 062116 (2011).
[CrossRef]

2009

2004

G. T. Paloczi, A. Eyal, and A. Yariv, “Wavelength-insensitive nonadiabatic mode evolution couplers,” IEEE Photon. Technol. Lett. 16, 515–517 (2004).
[CrossRef]

1998

1992

R. R. A. Syms, “The digital directional coupler: improved design,” IEEE Photon. Technol. Lett. 4, 1135–1138 (1992).
[CrossRef]

1991

D. R. Rowland, Y. Chen, and A. W. Snyder, “Tapered mismatched couplers,” J. Lightwave Technol. 9, 567–570 (1991).
[CrossRef]

1969

H. R. Lewis and W. B. Riesenfeld, “An exact quantum theory of the time-dependent harmonic oscillator and of a charged particle in a time-dependent electromagnetic field,” J. Math. Phys. 10, 1458–1473 (1969).
[CrossRef]

Al-Bader, S.

Alonso, D.

X.-J. Lu, X. Chen, A. Ruschhaupt, D. Alonso, S. Guérin, and J. G. Muga, “Fast and robust population transfer in two-level quantum systems with dephasing noise and/or systematic frequency errors,” Phys. Rev. A 88, 033406 (2013).
[CrossRef]

A. Ruschhaupt, X. Chen, D. Alonso, and J. G. Muga, “Optimally robust shortcuts to population inversion in two-level quantum systems,” New J. Phys. 14, 093040 (2012).
[CrossRef]

Chen, C.-L.

C.-L. Chen, Foundations for Guided-Wave Optics (Wiley, 2007).

Chen, X.

X.-J. Lu, X. Chen, A. Ruschhaupt, D. Alonso, S. Guérin, and J. G. Muga, “Fast and robust population transfer in two-level quantum systems with dephasing noise and/or systematic frequency errors,” Phys. Rev. A 88, 033406 (2013).
[CrossRef]

E. Torrontegui, S. Ibáñez, S. Martínez-Garaot, M. Modugno, A. del Campo, D. Guéry-Odelin, A. Ruschhaupt, X. Chen, and J. G. Muga, “Shortcuts to adiabaticity,” Adv. At. Mol. Opt. Phys. 62, 117–169 (2013).
[CrossRef]

S.-Y. Tseng and X. Chen, “Engineering of fast mode conversion in multimode waveguides,” Opt. Lett. 37, 5118–5120 (2012).
[CrossRef] [PubMed]

X. Chen and J. G. Muga, “Engineering of fast population transfer in three-level systems,” Phys. Rev. A 86, 033405 (2012).
[CrossRef]

A. Ruschhaupt, X. Chen, D. Alonso, and J. G. Muga, “Optimally robust shortcuts to population inversion in two-level quantum systems,” New J. Phys. 14, 093040 (2012).
[CrossRef]

X. Chen, E. Torrontegui, and J. G. Muga, “Lewis-Riesenfeld invariants and transitionless quantum driving,” Phys. Rev. A 83, 062116 (2011).
[CrossRef]

Chen, Y.

D. R. Rowland, Y. Chen, and A. W. Snyder, “Tapered mismatched couplers,” J. Lightwave Technol. 9, 567–570 (1991).
[CrossRef]

Daems, D.

D. Daems, A. Ruschhaupt, D. Sugny, and S. Guérin, “Robust quantum control by a single-shot shaped pulse,” Phys. Rev. Lett. 111, 050404 (2013).
[CrossRef] [PubMed]

del Campo, A.

E. Torrontegui, S. Ibáñez, S. Martínez-Garaot, M. Modugno, A. del Campo, D. Guéry-Odelin, A. Ruschhaupt, X. Chen, and J. G. Muga, “Shortcuts to adiabaticity,” Adv. At. Mol. Opt. Phys. 62, 117–169 (2013).
[CrossRef]

Eyal, A.

G. T. Paloczi, A. Eyal, and A. Yariv, “Wavelength-insensitive nonadiabatic mode evolution couplers,” IEEE Photon. Technol. Lett. 16, 515–517 (2004).
[CrossRef]

Guérin, S.

D. Daems, A. Ruschhaupt, D. Sugny, and S. Guérin, “Robust quantum control by a single-shot shaped pulse,” Phys. Rev. Lett. 111, 050404 (2013).
[CrossRef] [PubMed]

X.-J. Lu, X. Chen, A. Ruschhaupt, D. Alonso, S. Guérin, and J. G. Muga, “Fast and robust population transfer in two-level quantum systems with dephasing noise and/or systematic frequency errors,” Phys. Rev. A 88, 033406 (2013).
[CrossRef]

Guéry-Odelin, D.

E. Torrontegui, S. Ibáñez, S. Martínez-Garaot, M. Modugno, A. del Campo, D. Guéry-Odelin, A. Ruschhaupt, X. Chen, and J. G. Muga, “Shortcuts to adiabaticity,” Adv. At. Mol. Opt. Phys. 62, 117–169 (2013).
[CrossRef]

Hsiao, F.-C.

Hu, C.

Ibáñez, S.

E. Torrontegui, S. Ibáñez, S. Martínez-Garaot, M. Modugno, A. del Campo, D. Guéry-Odelin, A. Ruschhaupt, X. Chen, and J. G. Muga, “Shortcuts to adiabaticity,” Adv. At. Mol. Opt. Phys. 62, 117–169 (2013).
[CrossRef]

Jhang, Y.-W.

Kawano, K.

K. Kawano and T. Kitoh, Introduction to Optical Waveguide Analysis: Solving Maxwell’s Equations (Wiley, 2001).
[CrossRef]

Kitoh, T.

K. Kawano and T. Kitoh, Introduction to Optical Waveguide Analysis: Solving Maxwell’s Equations (Wiley, 2001).
[CrossRef]

Lewis, H. R.

H. R. Lewis and W. B. Riesenfeld, “An exact quantum theory of the time-dependent harmonic oscillator and of a charged particle in a time-dependent electromagnetic field,” J. Math. Phys. 10, 1458–1473 (1969).
[CrossRef]

Lin, T.-Y.

Liu, H.-C.

Longhi, S.

S. Longhi, “Quantum-optical analogies using photonic structures,” Laser and Photon. Rev. 3, 243–261 (2009).
[CrossRef]

Lu, X.-J.

X.-J. Lu, X. Chen, A. Ruschhaupt, D. Alonso, S. Guérin, and J. G. Muga, “Fast and robust population transfer in two-level quantum systems with dephasing noise and/or systematic frequency errors,” Phys. Rev. A 88, 033406 (2013).
[CrossRef]

Martínez-Garaot, S.

S. Martínez-Garaot, S.-Y. Tseng, and J. G. Muga, “Compact and high conversion efficiency mode-sorting asymmetric Y junction using shortcuts to adiabaticity,” Opt. Lett. 39, 2306 (2014).
[CrossRef] [PubMed]

E. Torrontegui, S. Ibáñez, S. Martínez-Garaot, M. Modugno, A. del Campo, D. Guéry-Odelin, A. Ruschhaupt, X. Chen, and J. G. Muga, “Shortcuts to adiabaticity,” Adv. At. Mol. Opt. Phys. 62, 117–169 (2013).
[CrossRef]

Modugno, M.

E. Torrontegui, S. Ibáñez, S. Martínez-Garaot, M. Modugno, A. del Campo, D. Guéry-Odelin, A. Ruschhaupt, X. Chen, and J. G. Muga, “Shortcuts to adiabaticity,” Adv. At. Mol. Opt. Phys. 62, 117–169 (2013).
[CrossRef]

Muga, J. G.

S. Martínez-Garaot, S.-Y. Tseng, and J. G. Muga, “Compact and high conversion efficiency mode-sorting asymmetric Y junction using shortcuts to adiabaticity,” Opt. Lett. 39, 2306 (2014).
[CrossRef] [PubMed]

E. Torrontegui, S. Ibáñez, S. Martínez-Garaot, M. Modugno, A. del Campo, D. Guéry-Odelin, A. Ruschhaupt, X. Chen, and J. G. Muga, “Shortcuts to adiabaticity,” Adv. At. Mol. Opt. Phys. 62, 117–169 (2013).
[CrossRef]

X.-J. Lu, X. Chen, A. Ruschhaupt, D. Alonso, S. Guérin, and J. G. Muga, “Fast and robust population transfer in two-level quantum systems with dephasing noise and/or systematic frequency errors,” Phys. Rev. A 88, 033406 (2013).
[CrossRef]

X. Chen and J. G. Muga, “Engineering of fast population transfer in three-level systems,” Phys. Rev. A 86, 033405 (2012).
[CrossRef]

A. Ruschhaupt, X. Chen, D. Alonso, and J. G. Muga, “Optimally robust shortcuts to population inversion in two-level quantum systems,” New J. Phys. 14, 093040 (2012).
[CrossRef]

X. Chen, E. Torrontegui, and J. G. Muga, “Lewis-Riesenfeld invariants and transitionless quantum driving,” Phys. Rev. A 83, 062116 (2011).
[CrossRef]

Okamoto, K.

K. Okamoto, Fundamentals of Optical Waveguides (Academic, 2006).

Osgood, R. M.

Paloczi, G. T.

G. T. Paloczi, A. Eyal, and A. Yariv, “Wavelength-insensitive nonadiabatic mode evolution couplers,” IEEE Photon. Technol. Lett. 16, 515–517 (2004).
[CrossRef]

Ramadan, T. A.

Riesenfeld, W. B.

H. R. Lewis and W. B. Riesenfeld, “An exact quantum theory of the time-dependent harmonic oscillator and of a charged particle in a time-dependent electromagnetic field,” J. Math. Phys. 10, 1458–1473 (1969).
[CrossRef]

Rowland, D. R.

D. R. Rowland, Y. Chen, and A. W. Snyder, “Tapered mismatched couplers,” J. Lightwave Technol. 9, 567–570 (1991).
[CrossRef]

Ruschhaupt, A.

E. Torrontegui, S. Ibáñez, S. Martínez-Garaot, M. Modugno, A. del Campo, D. Guéry-Odelin, A. Ruschhaupt, X. Chen, and J. G. Muga, “Shortcuts to adiabaticity,” Adv. At. Mol. Opt. Phys. 62, 117–169 (2013).
[CrossRef]

X.-J. Lu, X. Chen, A. Ruschhaupt, D. Alonso, S. Guérin, and J. G. Muga, “Fast and robust population transfer in two-level quantum systems with dephasing noise and/or systematic frequency errors,” Phys. Rev. A 88, 033406 (2013).
[CrossRef]

D. Daems, A. Ruschhaupt, D. Sugny, and S. Guérin, “Robust quantum control by a single-shot shaped pulse,” Phys. Rev. Lett. 111, 050404 (2013).
[CrossRef] [PubMed]

A. Ruschhaupt, X. Chen, D. Alonso, and J. G. Muga, “Optimally robust shortcuts to population inversion in two-level quantum systems,” New J. Phys. 14, 093040 (2012).
[CrossRef]

Scarmozzino, R.

Schneider, V. M.

Snyder, A. W.

D. R. Rowland, Y. Chen, and A. W. Snyder, “Tapered mismatched couplers,” J. Lightwave Technol. 9, 567–570 (1991).
[CrossRef]

Sugny, D.

D. Daems, A. Ruschhaupt, D. Sugny, and S. Guérin, “Robust quantum control by a single-shot shaped pulse,” Phys. Rev. Lett. 111, 050404 (2013).
[CrossRef] [PubMed]

Sun, X.

Syahriar, A.

Syms, R. R. A.

R. R. A. Syms, “The digital directional coupler: improved design,” IEEE Photon. Technol. Lett. 4, 1135–1138 (1992).
[CrossRef]

Torrontegui, E.

E. Torrontegui, S. Ibáñez, S. Martínez-Garaot, M. Modugno, A. del Campo, D. Guéry-Odelin, A. Ruschhaupt, X. Chen, and J. G. Muga, “Shortcuts to adiabaticity,” Adv. At. Mol. Opt. Phys. 62, 117–169 (2013).
[CrossRef]

X. Chen, E. Torrontegui, and J. G. Muga, “Lewis-Riesenfeld invariants and transitionless quantum driving,” Phys. Rev. A 83, 062116 (2011).
[CrossRef]

Tseng, S.-Y.

Yariv, A.

X. Sun, H.-C. Liu, and A. Yariv, “Adiabaticity criterion and the shortest adiabatic mode transformer in a coupled-waveguide system,” Opt. Lett. 34, 280–282 (2009).
[CrossRef] [PubMed]

G. T. Paloczi, A. Eyal, and A. Yariv, “Wavelength-insensitive nonadiabatic mode evolution couplers,” IEEE Photon. Technol. Lett. 16, 515–517 (2004).
[CrossRef]

Adv. At. Mol. Opt. Phys.

E. Torrontegui, S. Ibáñez, S. Martínez-Garaot, M. Modugno, A. del Campo, D. Guéry-Odelin, A. Ruschhaupt, X. Chen, and J. G. Muga, “Shortcuts to adiabaticity,” Adv. At. Mol. Opt. Phys. 62, 117–169 (2013).
[CrossRef]

IEEE Photon. Technol. Lett.

R. R. A. Syms, “The digital directional coupler: improved design,” IEEE Photon. Technol. Lett. 4, 1135–1138 (1992).
[CrossRef]

G. T. Paloczi, A. Eyal, and A. Yariv, “Wavelength-insensitive nonadiabatic mode evolution couplers,” IEEE Photon. Technol. Lett. 16, 515–517 (2004).
[CrossRef]

S.-Y. Tseng and Y.-W. Jhang, “Fast and robust beam coupling in a three waveguide directional coupler,” IEEE Photon. Technol. Lett. 25, 2478–2481 (2013).
[CrossRef]

J. Lightwave Technol.

J. Math. Phys.

H. R. Lewis and W. B. Riesenfeld, “An exact quantum theory of the time-dependent harmonic oscillator and of a charged particle in a time-dependent electromagnetic field,” J. Math. Phys. 10, 1458–1473 (1969).
[CrossRef]

Laser and Photon. Rev.

S. Longhi, “Quantum-optical analogies using photonic structures,” Laser and Photon. Rev. 3, 243–261 (2009).
[CrossRef]

New J. Phys.

A. Ruschhaupt, X. Chen, D. Alonso, and J. G. Muga, “Optimally robust shortcuts to population inversion in two-level quantum systems,” New J. Phys. 14, 093040 (2012).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. A

X.-J. Lu, X. Chen, A. Ruschhaupt, D. Alonso, S. Guérin, and J. G. Muga, “Fast and robust population transfer in two-level quantum systems with dephasing noise and/or systematic frequency errors,” Phys. Rev. A 88, 033406 (2013).
[CrossRef]

X. Chen, E. Torrontegui, and J. G. Muga, “Lewis-Riesenfeld invariants and transitionless quantum driving,” Phys. Rev. A 83, 062116 (2011).
[CrossRef]

X. Chen and J. G. Muga, “Engineering of fast population transfer in three-level systems,” Phys. Rev. A 86, 033405 (2012).
[CrossRef]

Phys. Rev. Lett.

D. Daems, A. Ruschhaupt, D. Sugny, and S. Guérin, “Robust quantum control by a single-shot shaped pulse,” Phys. Rev. Lett. 111, 050404 (2013).
[CrossRef] [PubMed]

Other

C.-L. Chen, Foundations for Guided-Wave Optics (Wiley, 2007).

K. Kawano and T. Kitoh, Introduction to Optical Waveguide Analysis: Solving Maxwell’s Equations (Wiley, 2001).
[CrossRef]

K. Okamoto, Fundamentals of Optical Waveguides (Academic, 2006).

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Figures (9)

Fig. 1
Fig. 1

Cross-sectional schematic of the polymer channel waveguide structure considered for device design.

Fig. 2
Fig. 2

Coupling coefficient Ω and detuning Δ variation along z for the coupler designs. (a) Ω-coupler. (c) adiabatic coupler. The corresponding waveguide parameters. (b) Ω-coupler. (d) adiabatic coupler.

Fig. 3
Fig. 3

Waveguide geometries and the corresponding beam propagation simulations for (a) Ω-coupler, (b) adiabatic coupler, and (c) resonant coupler. White lines indicate the waveguide cores.

Fig. 4
Fig. 4

Coupling efficiencies as a function of device length L.

Fig. 5
Fig. 5

Coupling efficiencies as a function of waveguide spacing error δD

Fig. 6
Fig. 6

Coupling efficiencies as a function of refractive index variations.

Fig. 7
Fig. 7

Coupling efficiency Ω and detuning Δ variation along z for the coupler designs. (a) Δ-coupler. (c) adiabatic coupler. The corresponding waveguide parameters. (b) Δ-coupler. (d) adiabatic coupler.

Fig. 8
Fig. 8

Waveguide geometries and the corresponding beam propagation simulations for (a) Δ-coupler and (b) adiabatic coupler. White lines indicate the waveguide cores.

Fig. 9
Fig. 9

Coupling efficiencies as a function of input wavelength.

Equations (20)

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

d d z [ A + A ] = i [ Δ Ω Ω Δ ] [ A + A ] ,
I ( z ) = κ 2 ( cos θ sin θ e i β sin θ e i β cos θ ) ,
θ ˙ = Ω sin β ,
β ˙ = Ω cot θ cos β Δ .
| ϕ + ( z ) = ( cos ( θ / 2 ) e i β sin ( θ / 2 ) ) ,
| ϕ ( z ) = ( sin ( θ / 2 ) cos ( θ / 2 ) e i β ) ,
| Ψ ( z ) = Σ ± c ± e i γ ± ( z ) | ϕ ± ( z ) ,
γ ˙ ± ( z ) = ± 1 2 ( β ˙ + θ ˙ cot β sin θ ) .
θ ( 0 ) = π , θ ( L ) = 0 ,
θ ˙ ( 0 ) = 0 , θ ˙ ( L ) = 0 ,
q Ω = 1 2 2 P 1 ( L ) δ 2 | δ = 0 ,
q Ω = 1 4 | 0 L d z θ ˙ sin 2 θ e i m ( z ) | 2 ,
β = arctan ( 4 sin 3 θ ) ,
θ = π 2 [ 1 sin π ( 2 z L ) 2 L ] ,
Ω = θ ˙ / sin β ,
Δ = 4 θ ˙ cos θ sin 2 θ β ˙ ,
q Δ = 1 4 | 0 L d z sin θ e i m ( z ) | 2 ,
β = cos 1 ( 2 M sin θ 1 + 4 M 2 sin 2 θ ) .
Ω = θ ˙ 1 + 4 M 2 sin 2 θ ,
Δ = 2 θ ˙ cos θ [ M + 1 4 α + 6 α cos ( 2 θ ) 1 + 4 M 2 sin 2 θ ] ,

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