Abstract

A shortcut to adiabatic mode conversion in multimode waveguides using optical analogy of stimulated Raman adiabatic passage is investigated. The design of mode converters using the shortcut scheme is discussed. Computer-generated planar holograms are used to mimic the shaped pulses used to speed up adiabatic passage in quantum systems based on the transitionless quantum driving algorithm. The mode coupling properties are analyzed using the coupled mode theory and beam propagation simulations. We show reduced device length using the shortcut scheme as compared to the common adiabatic scheme. Modal evolution in the shortened device indeed follows the adiabatic eigenmode exactly amid the violation of adiabatic criterion.

© 2012 OSA

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  14. R. G. Unanyan, B. W. Shore, and K. Bergmann, “Preparation of an N-component maximal coherent superposition state using the stimulated Raman adiabatic passage method,” Phys. Rev. A63, 043401 (2001).
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    [CrossRef]
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2012 (1)

A. A. Rangelov and N. V. Vitanov, “Achromatic multiple beam splitting by adiabatic passage in optical waveguides,” Phys. Rev. A85, 055803 (2012).
[CrossRef]

2010 (3)

S.-Y. Tseng and M.-C. Wu, “Adiabatic mode conversion in multimode waveguides using computer-generated planar holograms,” IEEE Photon. Technol. Lett.22, 1211–1213 (2010).
[CrossRef]

S.-Y. Tseng and M.-C. Wu, “Mode conversion/splitting by optical analogy of multistate stimulated Raman adiabatic passage in multimode waveguides,” J. Lightwave Technol.28, 3529–3534 (2010).

X. Chen, I. Lizuain, A. Ruschhaupt, D. Guéry-Odelin, and J. G. Muga, “Shortcut to adiabatic passage in two- and three-level atoms,” Phys. Rev. Lett.105, 123003 (2010).
[CrossRef] [PubMed]

2009 (8)

M. V. Berry, “Transitionless quantum driving,” J. Phys. A: Math. Theor.42, 365303 (2009).
[CrossRef]

M. H. S. Amin, “Consistency of the adiabatic theorem,” Phys. Rev. Lett.102, 220401 (2009).
[CrossRef] [PubMed]

S.-Y. Tseng, S. K. Choi, and B. Kippelen, “Variable-ratio power splitters using computer-generated planar holograms on multimode interference couplers,” Opt. Lett.34, 512–514 (2009).
[CrossRef] [PubMed]

G. S. Vasilev, A. Kuhn, and N. V. Vitanov, “Optimum pulse shapes for stimulated Raman adiabatic passage,” Phys. Rev. A80, 013417 (2009).
[CrossRef]

G. Dridi, S. Guérin, V. Hakobyan, H. R. Jauslin, and E. Eleuch, “Ultrafast stimulated Raman parallel adiabatic passage by shaped pulses,” Phys. Rev. A80, 043408 (2009).
[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]

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

F. Dreisow, M. Ornigotti, A. Szameit, M. Heinrich, R. Keil, S. Nolte, A. Tünnermann, and S. Longhi, “Polychromatic beam splitting by fractional stimulated Raman adiabatic passage,” Appl. Phys. Lett.95, 261102 (2009).
[CrossRef]

2008 (1)

G. Della Valle, M. Ornigotti, T. Toney Fernandez, P. Laporta, S. Longhi, A. Coppa, and V. Foglietti, “Adiabatic light transfer via dressed states in optical waveguide arrays,” Appl. Phys. Lett.92, 011106 (2008).
[CrossRef]

2007 (1)

S. Longhi, G. Della Valle, M. Ornigotti, and P. Laporta, “Coherent tunneling by adiabatic passage in an optical waveguide system,” Phys. Rev. B76, 201101(R) (2007).
[CrossRef]

2006 (3)

S. Longhi, “Adiabatic passage of light in coupled optical waveguides,” Phys. Rev. E73, 026607 (2006).
[CrossRef]

E. Paspalakis, “Adiabatic three-waveguide directional coupler,” Opt. Commun.258, 30–34 (2006).
[CrossRef]

J. B. Park, D.-M. Yeo, and S.-Y. Shin, “Variable optical mode generator in a multimode waveguide,” IEEE Photon. Technol. Lett.18, 2084–2086 (2006).
[CrossRef]

2005 (1)

2003 (1)

F. Vewinger, M. Heinz, R. Garcia Fernandez, N. V. Vitanov, and K. Bergmann, “Creation and measurement of a coherent superposition of quantum states,” Phys. Rev. Lett.91, 213001 (2003).
[CrossRef] [PubMed]

2002 (1)

T. Ando, T. Murata, H. Nakayama, J. Yamauchi, and H. Nakano, “Analysis and measurement of polarization conversion in a periodically loaded dielectric waveguide,” IEEE Photon. Technol. Lett.14, 1288–1290 (2002).
[CrossRef]

2001 (1)

R. G. Unanyan, B. W. Shore, and K. Bergmann, “Preparation of an N-component maximal coherent superposition state using the stimulated Raman adiabatic passage method,” Phys. Rev. A63, 043401 (2001).
[CrossRef]

2000 (1)

R. G. Unanyan, N. V. Vitanov, B. W. Shore, and K. Bergmann, “Coherent properties of a tripod system coupled via a continuum,” Phys. Rev. A61, 043408 (2000).
[CrossRef]

1999 (1)

1998 (2)

E. Paspalakis, M. Protopapas, and P. L. Knight, “Time-dependent pulse and frequency effects in population trapping via the continuum,” J Phys. B31, 775–794 (1998).
[CrossRef]

K. Bergmann, H. Theuer, and B. W. Shore, “Coherent population transfer among quantum states of atoms and molecules,” Rev. Mod. Phys.70, 1003–1025 (1998).
[CrossRef]

1982 (1)

Amin, M. H. S.

M. H. S. Amin, “Consistency of the adiabatic theorem,” Phys. Rev. Lett.102, 220401 (2009).
[CrossRef] [PubMed]

Ando, T.

T. Ando, T. Murata, H. Nakayama, J. Yamauchi, and H. Nakano, “Analysis and measurement of polarization conversion in a periodically loaded dielectric waveguide,” IEEE Photon. Technol. Lett.14, 1288–1290 (2002).
[CrossRef]

Berdague, S.

Bergmann, K.

F. Vewinger, M. Heinz, R. Garcia Fernandez, N. V. Vitanov, and K. Bergmann, “Creation and measurement of a coherent superposition of quantum states,” Phys. Rev. Lett.91, 213001 (2003).
[CrossRef] [PubMed]

R. G. Unanyan, B. W. Shore, and K. Bergmann, “Preparation of an N-component maximal coherent superposition state using the stimulated Raman adiabatic passage method,” Phys. Rev. A63, 043401 (2001).
[CrossRef]

R. G. Unanyan, N. V. Vitanov, B. W. Shore, and K. Bergmann, “Coherent properties of a tripod system coupled via a continuum,” Phys. Rev. A61, 043408 (2000).
[CrossRef]

H. Theuer, R. G. Unanyan, C. Habscheid, K. Klein, and K. Bergmann, “Novel laser controlled variable matter wave beamsplitter,” Opt. Express4, 77–83 (1999).
[CrossRef] [PubMed]

K. Bergmann, H. Theuer, and B. W. Shore, “Coherent population transfer among quantum states of atoms and molecules,” Rev. Mod. Phys.70, 1003–1025 (1998).
[CrossRef]

Berry, M. V.

M. V. Berry, “Transitionless quantum driving,” J. Phys. A: Math. Theor.42, 365303 (2009).
[CrossRef]

Chen, X.

X. Chen, I. Lizuain, A. Ruschhaupt, D. Guéry-Odelin, and J. G. Muga, “Shortcut to adiabatic passage in two- and three-level atoms,” Phys. Rev. Lett.105, 123003 (2010).
[CrossRef] [PubMed]

Choi, S. K.

Coppa, A.

G. Della Valle, M. Ornigotti, T. Toney Fernandez, P. Laporta, S. Longhi, A. Coppa, and V. Foglietti, “Adiabatic light transfer via dressed states in optical waveguide arrays,” Appl. Phys. Lett.92, 011106 (2008).
[CrossRef]

Della Valle, G.

G. Della Valle, M. Ornigotti, T. Toney Fernandez, P. Laporta, S. Longhi, A. Coppa, and V. Foglietti, “Adiabatic light transfer via dressed states in optical waveguide arrays,” Appl. Phys. Lett.92, 011106 (2008).
[CrossRef]

S. Longhi, G. Della Valle, M. Ornigotti, and P. Laporta, “Coherent tunneling by adiabatic passage in an optical waveguide system,” Phys. Rev. B76, 201101(R) (2007).
[CrossRef]

Dreisow, F.

F. Dreisow, M. Ornigotti, A. Szameit, M. Heinrich, R. Keil, S. Nolte, A. Tünnermann, and S. Longhi, “Polychromatic beam splitting by fractional stimulated Raman adiabatic passage,” Appl. Phys. Lett.95, 261102 (2009).
[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]

Dridi, G.

G. Dridi, S. Guérin, V. Hakobyan, H. R. Jauslin, and E. Eleuch, “Ultrafast stimulated Raman parallel adiabatic passage by shaped pulses,” Phys. Rev. A80, 043408 (2009).
[CrossRef]

Eleuch, E.

G. Dridi, S. Guérin, V. Hakobyan, H. R. Jauslin, and E. Eleuch, “Ultrafast stimulated Raman parallel adiabatic passage by shaped pulses,” Phys. Rev. A80, 043408 (2009).
[CrossRef]

Facq, P.

Foglietti, V.

G. Della Valle, M. Ornigotti, T. Toney Fernandez, P. Laporta, S. Longhi, A. Coppa, and V. Foglietti, “Adiabatic light transfer via dressed states in optical waveguide arrays,” Appl. Phys. Lett.92, 011106 (2008).
[CrossRef]

Garcia Fernandez, R.

F. Vewinger, M. Heinz, R. Garcia Fernandez, N. V. Vitanov, and K. Bergmann, “Creation and measurement of a coherent superposition of quantum states,” Phys. Rev. Lett.91, 213001 (2003).
[CrossRef] [PubMed]

Greenberg, M.

Guérin, S.

G. Dridi, S. Guérin, V. Hakobyan, H. R. Jauslin, and E. Eleuch, “Ultrafast stimulated Raman parallel adiabatic passage by shaped pulses,” Phys. Rev. A80, 043408 (2009).
[CrossRef]

Guéry-Odelin, D.

X. Chen, I. Lizuain, A. Ruschhaupt, D. Guéry-Odelin, and J. G. Muga, “Shortcut to adiabatic passage in two- and three-level atoms,” Phys. Rev. Lett.105, 123003 (2010).
[CrossRef] [PubMed]

Habscheid, C.

Hakobyan, V.

G. Dridi, S. Guérin, V. Hakobyan, H. R. Jauslin, and E. Eleuch, “Ultrafast stimulated Raman parallel adiabatic passage by shaped pulses,” Phys. Rev. A80, 043408 (2009).
[CrossRef]

Heinrich, M.

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]

F. Dreisow, M. Ornigotti, A. Szameit, M. Heinrich, R. Keil, S. Nolte, A. Tünnermann, and S. Longhi, “Polychromatic beam splitting by fractional stimulated Raman adiabatic passage,” Appl. Phys. Lett.95, 261102 (2009).
[CrossRef]

Heinz, M.

F. Vewinger, M. Heinz, R. Garcia Fernandez, N. V. Vitanov, and K. Bergmann, “Creation and measurement of a coherent superposition of quantum states,” Phys. Rev. Lett.91, 213001 (2003).
[CrossRef] [PubMed]

Jauslin, H. R.

G. Dridi, S. Guérin, V. Hakobyan, H. R. Jauslin, and E. Eleuch, “Ultrafast stimulated Raman parallel adiabatic passage by shaped pulses,” Phys. Rev. A80, 043408 (2009).
[CrossRef]

Keil, R.

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]

F. Dreisow, M. Ornigotti, A. Szameit, M. Heinrich, R. Keil, S. Nolte, A. Tünnermann, and S. Longhi, “Polychromatic beam splitting by fractional stimulated Raman adiabatic passage,” Appl. Phys. Lett.95, 261102 (2009).
[CrossRef]

Kippelen, B.

Klein, K.

Knight, P. L.

E. Paspalakis, M. Protopapas, and P. L. Knight, “Time-dependent pulse and frequency effects in population trapping via the continuum,” J Phys. B31, 775–794 (1998).
[CrossRef]

Kuhn, A.

G. S. Vasilev, A. Kuhn, and N. V. Vitanov, “Optimum pulse shapes for stimulated Raman adiabatic passage,” Phys. Rev. A80, 013417 (2009).
[CrossRef]

Laporta, P.

G. Della Valle, M. Ornigotti, T. Toney Fernandez, P. Laporta, S. Longhi, A. Coppa, and V. Foglietti, “Adiabatic light transfer via dressed states in optical waveguide arrays,” Appl. Phys. Lett.92, 011106 (2008).
[CrossRef]

S. Longhi, G. Della Valle, M. Ornigotti, and P. Laporta, “Coherent tunneling by adiabatic passage in an optical waveguide system,” Phys. Rev. B76, 201101(R) (2007).
[CrossRef]

Lizuain, I.

X. Chen, I. Lizuain, A. Ruschhaupt, D. Guéry-Odelin, and J. G. Muga, “Shortcut to adiabatic passage in two- and three-level atoms,” Phys. Rev. Lett.105, 123003 (2010).
[CrossRef] [PubMed]

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

F. Dreisow, M. Ornigotti, A. Szameit, M. Heinrich, R. Keil, S. Nolte, A. Tünnermann, and S. Longhi, “Polychromatic beam splitting by fractional stimulated Raman adiabatic passage,” Appl. Phys. Lett.95, 261102 (2009).
[CrossRef]

G. Della Valle, M. Ornigotti, T. Toney Fernandez, P. Laporta, S. Longhi, A. Coppa, and V. Foglietti, “Adiabatic light transfer via dressed states in optical waveguide arrays,” Appl. Phys. Lett.92, 011106 (2008).
[CrossRef]

S. Longhi, G. Della Valle, M. Ornigotti, and P. Laporta, “Coherent tunneling by adiabatic passage in an optical waveguide system,” Phys. Rev. B76, 201101(R) (2007).
[CrossRef]

S. Longhi, “Adiabatic passage of light in coupled optical waveguides,” Phys. Rev. E73, 026607 (2006).
[CrossRef]

Muga, J. G.

X. Chen, I. Lizuain, A. Ruschhaupt, D. Guéry-Odelin, and J. G. Muga, “Shortcut to adiabatic passage in two- and three-level atoms,” Phys. Rev. Lett.105, 123003 (2010).
[CrossRef] [PubMed]

Murata, T.

T. Ando, T. Murata, H. Nakayama, J. Yamauchi, and H. Nakano, “Analysis and measurement of polarization conversion in a periodically loaded dielectric waveguide,” IEEE Photon. Technol. Lett.14, 1288–1290 (2002).
[CrossRef]

Nakano, H.

T. Ando, T. Murata, H. Nakayama, J. Yamauchi, and H. Nakano, “Analysis and measurement of polarization conversion in a periodically loaded dielectric waveguide,” IEEE Photon. Technol. Lett.14, 1288–1290 (2002).
[CrossRef]

Nakayama, H.

T. Ando, T. Murata, H. Nakayama, J. Yamauchi, and H. Nakano, “Analysis and measurement of polarization conversion in a periodically loaded dielectric waveguide,” IEEE Photon. Technol. Lett.14, 1288–1290 (2002).
[CrossRef]

Nolte, S.

F. Dreisow, M. Ornigotti, A. Szameit, M. Heinrich, R. Keil, S. Nolte, A. Tünnermann, and S. Longhi, “Polychromatic beam splitting by fractional stimulated Raman adiabatic passage,” Appl. Phys. Lett.95, 261102 (2009).
[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]

Orenstein, M.

Ornigotti, M.

F. Dreisow, M. Ornigotti, A. Szameit, M. Heinrich, R. Keil, S. Nolte, A. Tünnermann, and S. Longhi, “Polychromatic beam splitting by fractional stimulated Raman adiabatic passage,” Appl. Phys. Lett.95, 261102 (2009).
[CrossRef]

G. Della Valle, M. Ornigotti, T. Toney Fernandez, P. Laporta, S. Longhi, A. Coppa, and V. Foglietti, “Adiabatic light transfer via dressed states in optical waveguide arrays,” Appl. Phys. Lett.92, 011106 (2008).
[CrossRef]

S. Longhi, G. Della Valle, M. Ornigotti, and P. Laporta, “Coherent tunneling by adiabatic passage in an optical waveguide system,” Phys. Rev. B76, 201101(R) (2007).
[CrossRef]

Park, J. B.

J. B. Park, D.-M. Yeo, and S.-Y. Shin, “Variable optical mode generator in a multimode waveguide,” IEEE Photon. Technol. Lett.18, 2084–2086 (2006).
[CrossRef]

Paspalakis, E.

E. Paspalakis, “Adiabatic three-waveguide directional coupler,” Opt. Commun.258, 30–34 (2006).
[CrossRef]

E. Paspalakis, M. Protopapas, and P. L. Knight, “Time-dependent pulse and frequency effects in population trapping via the continuum,” J Phys. B31, 775–794 (1998).
[CrossRef]

Protopapas, M.

E. Paspalakis, M. Protopapas, and P. L. Knight, “Time-dependent pulse and frequency effects in population trapping via the continuum,” J Phys. B31, 775–794 (1998).
[CrossRef]

Rangelov, A. A.

A. A. Rangelov and N. V. Vitanov, “Achromatic multiple beam splitting by adiabatic passage in optical waveguides,” Phys. Rev. A85, 055803 (2012).
[CrossRef]

Ruschhaupt, A.

X. Chen, I. Lizuain, A. Ruschhaupt, D. Guéry-Odelin, and J. G. Muga, “Shortcut to adiabatic passage in two- and three-level atoms,” Phys. Rev. Lett.105, 123003 (2010).
[CrossRef] [PubMed]

Shin, S.-Y.

J. B. Park, D.-M. Yeo, and S.-Y. Shin, “Variable optical mode generator in a multimode waveguide,” IEEE Photon. Technol. Lett.18, 2084–2086 (2006).
[CrossRef]

Shore, B. W.

R. G. Unanyan, B. W. Shore, and K. Bergmann, “Preparation of an N-component maximal coherent superposition state using the stimulated Raman adiabatic passage method,” Phys. Rev. A63, 043401 (2001).
[CrossRef]

R. G. Unanyan, N. V. Vitanov, B. W. Shore, and K. Bergmann, “Coherent properties of a tripod system coupled via a continuum,” Phys. Rev. A61, 043408 (2000).
[CrossRef]

K. Bergmann, H. Theuer, and B. W. Shore, “Coherent population transfer among quantum states of atoms and molecules,” Rev. Mod. Phys.70, 1003–1025 (1998).
[CrossRef]

Szameit, A.

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]

F. Dreisow, M. Ornigotti, A. Szameit, M. Heinrich, R. Keil, S. Nolte, A. Tünnermann, and S. Longhi, “Polychromatic beam splitting by fractional stimulated Raman adiabatic passage,” Appl. Phys. Lett.95, 261102 (2009).
[CrossRef]

Theuer, H.

H. Theuer, R. G. Unanyan, C. Habscheid, K. Klein, and K. Bergmann, “Novel laser controlled variable matter wave beamsplitter,” Opt. Express4, 77–83 (1999).
[CrossRef] [PubMed]

K. Bergmann, H. Theuer, and B. W. Shore, “Coherent population transfer among quantum states of atoms and molecules,” Rev. Mod. Phys.70, 1003–1025 (1998).
[CrossRef]

Toney Fernandez, T.

G. Della Valle, M. Ornigotti, T. Toney Fernandez, P. Laporta, S. Longhi, A. Coppa, and V. Foglietti, “Adiabatic light transfer via dressed states in optical waveguide arrays,” Appl. Phys. Lett.92, 011106 (2008).
[CrossRef]

Tseng, S.-Y.

Tünnermann, A.

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]

F. Dreisow, M. Ornigotti, A. Szameit, M. Heinrich, R. Keil, S. Nolte, A. Tünnermann, and S. Longhi, “Polychromatic beam splitting by fractional stimulated Raman adiabatic passage,” Appl. Phys. Lett.95, 261102 (2009).
[CrossRef]

Unanyan, R. G.

R. G. Unanyan, B. W. Shore, and K. Bergmann, “Preparation of an N-component maximal coherent superposition state using the stimulated Raman adiabatic passage method,” Phys. Rev. A63, 043401 (2001).
[CrossRef]

R. G. Unanyan, N. V. Vitanov, B. W. Shore, and K. Bergmann, “Coherent properties of a tripod system coupled via a continuum,” Phys. Rev. A61, 043408 (2000).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

(a) Level scheme of STIRAP (b) Schematic of a CGPH loaded multimode waveguide composed of three multiplexed gratings (c) Amplitude profiles of the three guided modes which correspond to levels 1, 2, and 3 in the level scheme (a).

Fig. 2
Fig. 2

Evolution of the coupling coefficients for STIRAP (κ12, κ23) and SHAPE (κ13) schemes.

Fig. 3
Fig. 3

Calculated CGPH pattern to implement the SHAPE scheme for mode conversion. Dashed lines indicate the waveguide core.

Fig. 4
Fig. 4

BPM simulation of light propagation in a 5 mm SHAPE mode converter using CGPH. Dashed lines indicate the waveguide core.

Fig. 5
Fig. 5

BPM simulation of light propagation in a 5 mm STIRAP mode converter using CGPH. Dashed lines indicate the waveguide core.

Fig. 6
Fig. 6

Modal power evolution in a 5 mm mode converter using (a) SHAPE and (b) STIRAP scheme.

Fig. 7
Fig. 7

Normalized modal power at the mode converter output for different device lengths. |Ψ1〉 is used as the input. (a) STIRAP (b) SHAPE.

Equations (6)

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H ( t ) = H 0 ( t ) + H 1 ( t ) ,
H 1 ( t ) = i h ¯ n | t Ψ n ( t ) Ψ n ( t ) | .
i d d z [ A 1 A 2 A 3 ] = [ 0 κ 12 ( z ) 0 κ 21 ( z ) 0 κ 23 ( z ) 0 κ 32 ( z ) 0 ] [ A 1 A 2 A 3 ] = H 0 ( z ) [ A 1 A 2 A 3 ] .
| Ψ D = 1 κ 12 2 + κ 23 2 ( κ 23 | Ψ 1 κ 12 | Ψ 3 ) .
H 1 ( z ) = [ 0 0 i κ 13 ( z ) 0 0 0 i κ 13 ( z ) 0 0 ] ,
κ 13 ( z ) = κ ˙ 12 ( z ) κ 23 ( z ) κ ˙ 23 ( z ) κ 12 ( z ) κ 12 2 ( z ) + κ 23 2 ( z ) .

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