Abstract

We revisit electromagnetic field propagation through tight-binding arrays of coupled photonic waveguides, with properties independent of the propagation distance, and recast it as a symmetry problem. We focus our analysis on photonic lattices with underlying symmetries given by three well-known groups, SU(2), SU(1, 1) and Heisenberg-Weyl, to show that disperssion relations, normal states and impulse functions can be constructed following a Gilmore-Perelomov coherent state approach. Furthermore, this symmetry based approach can be followed for each an every lattice with an underlying symmetry given by a dynamical group.

© 2015 Optical Society of America

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References

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    [Crossref] [PubMed]
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2015 (1)

B. M. Rodríguez-Lara, F. Soto-Eguibar, and D. N. Christodoulides, “Quantum optics as a tool for photonic lattice design,” Phys. Scr. 90, 068014 (2015).
[Crossref]

2014 (3)

2013 (2)

A. Perez-Leija, R. Keil, A. Kay, H. Moya-Cessa, S. Nolte, L.-C. Kwek, B. Rodríguez-Lara, A. Szameit, and D. Christodoulides, “Coherent quantum transport in photonic lattices,” Phys. Rev. A 87, 012309 (2013).
[Crossref]

A. Perez-Leija, R. Keil, H. Moya-Cessa, A. Szameit, and D. N. Christodoulides, “Perfect transfer of path-entangled photons in Jx photonic lattices,” Phys. Rev. A 87, 022303 (2013).
[Crossref]

2012 (2)

R. Keil, A. Perez-Leija, P. Aleahmad, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Observation of Bloch-like revivals in semi-infinite Glauber-Fock photonic lattices,” Opt. Lett. 37, 3801– 3803 (2012).
[Crossref] [PubMed]

A. Perez-Leija, R. Keil, A. Szameit, A. F. Abouraddy, H. Moya-Cessa, and D. N. Christodoulides, “Tailoring the correlation and anticorrelation behavior of path-entangled photons in Glauber-Fock lattices,” Phys. Rev. A 85, 013848 (2012).
[Crossref]

2011 (4)

R. Keil, A. Perez-Leija, F. Dreisow, M. Heinrich, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Classical analogue of displaced Fock states and quantum correlations in Glauber-Fock photonic lattices,” Phys. Rev. Lett. 107, 103601 (2011).
[Crossref] [PubMed]

B. M. Rodríguez-Lara, “Exact dynamics of finite Glauber-Fock photonic lattices,” Phys. Rev. A 84, 053845 (2011).
[Crossref]

L. E. Vicent and K. B. Wolf, “Analysis of digital images into energy-angular momentum modes,” J. Opt. Soc. Am. A 28, 808–814 (2011).
[Crossref]

S. Longhi, “Classical simulation of relativistic quantum mechanics in periodic optical structures,” Appl. Phys. B 104, 453–468 (2011).
[Crossref]

2010 (1)

2009 (2)

K. B. Wolf, “Mode analysis and signal restoration with Kravchuk functions,” J. Opt. Soc. Am. A 26, 509–516 (2009).
[Crossref]

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

2008 (1)

2007 (1)

1994 (1)

1993 (1)

1991 (1)

A. Wünsche, “Displaced Fock states and their connection to quasiprobabilities,” Quantum Opt. 3, 359–383 (1991).
[Crossref]

1990 (1)

W. M. Zhang, D. H. Feng, and R. Gilmore, “Coherent states: Theory and some applications,” Rev. Mod. Phys. 62, 867 (1990).
[Crossref]

1987 (2)

W. Streifer, M. Osiński, and A. Hardy, “Reformulation of the coupled-mode theory of mutiwaveguide systems,” J. Lightwave Technol. 5, 1–4 (1987).
[Crossref]

H. A. Haus, W. P. Huang, S. Kawakami, and N. A. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol. 5, 16–23 (1987).
[Crossref]

1985 (2)

1973 (1)

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. 9, 919–933 (1973).
[Crossref]

1972 (4)

R. Gilmore, “Geometry of symmetrized states,” Ann. Phys. 74, 391–463 (1972).
[Crossref]

A. M. Perelomov, “Coherent states for arbitrary Lie group,” Comm. Math. Phys. 26, 222–236 (1972).
[Crossref]

A. W. Snyder, “Coupled-mode theory for optical fibers,” J. Opt. Soc. Am. 62, 1267–1277 (1972).
[Crossref]

H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[Crossref]

1967 (1)

R. M. Wilcox, “Exponential operators and parameter differentiation in quantum physics,” J. Math. Phys. 8, 962–982 (1967).
[Crossref]

1965 (1)

1963 (2)

R. J. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev. 131, 2766–2788 (1963).
[Crossref]

E. C. G. Sudarshan, “Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams,” Phys. Rev. Lett. 10, 277–279 (1963).
[Crossref]

Abouraddy, A. F.

A. Perez-Leija, R. Keil, A. Szameit, A. F. Abouraddy, H. Moya-Cessa, and D. N. Christodoulides, “Tailoring the correlation and anticorrelation behavior of path-entangled photons in Glauber-Fock lattices,” Phys. Rev. A 85, 013848 (2012).
[Crossref]

Aleahmad, P.

Askey, R.

R. Askey, Orthogonal Polynomials and Special Functions, in CBMS-NSF Regional Conference Series in Applied Mathematics (SIAM, 1975).
[Crossref]

Ban, M.

Barut, A. O.

A. O. Barut and R. Raczka, Theory of group representations and applications (PWN Polish Scientific Publishers, 1980).

Christodoulides, D.

A. Perez-Leija, R. Keil, A. Kay, H. Moya-Cessa, S. Nolte, L.-C. Kwek, B. Rodríguez-Lara, A. Szameit, and D. Christodoulides, “Coherent quantum transport in photonic lattices,” Phys. Rev. A 87, 012309 (2013).
[Crossref]

Christodoulides, D. N.

B. M. Rodríguez-Lara, F. Soto-Eguibar, and D. N. Christodoulides, “Quantum optics as a tool for photonic lattice design,” Phys. Scr. 90, 068014 (2015).
[Crossref]

B. M. Rodríguez-Lara, H. M. Moya-Cessa, and D. N. Christodoulides, “Propagation and perfect transmission in three-waveguide axially varying couplers,” Phys. Rev. A 89, 013802 (2014).
[Crossref]

B. M. Rodríguez-Lara, P. Aleahmad, H. M. Moya-Cessa, and D. N. Christodoulides, “Ermakov-lewis symmetry in photonic lattices,” Opt. Lett. 39, 2083–2085 (2014).
[Crossref] [PubMed]

A. Perez-Leija, R. Keil, H. Moya-Cessa, A. Szameit, and D. N. Christodoulides, “Perfect transfer of path-entangled photons in Jx photonic lattices,” Phys. Rev. A 87, 022303 (2013).
[Crossref]

A. Perez-Leija, R. Keil, A. Szameit, A. F. Abouraddy, H. Moya-Cessa, and D. N. Christodoulides, “Tailoring the correlation and anticorrelation behavior of path-entangled photons in Glauber-Fock lattices,” Phys. Rev. A 85, 013848 (2012).
[Crossref]

R. Keil, A. Perez-Leija, P. Aleahmad, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Observation of Bloch-like revivals in semi-infinite Glauber-Fock photonic lattices,” Opt. Lett. 37, 3801– 3803 (2012).
[Crossref] [PubMed]

R. Keil, A. Perez-Leija, F. Dreisow, M. Heinrich, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Classical analogue of displaced Fock states and quantum correlations in Glauber-Fock photonic lattices,” Phys. Rev. Lett. 107, 103601 (2011).
[Crossref] [PubMed]

A. Perez-Leija, H. Moya-Cessa, A. Szameit, and D. N. Christodoulides, “Glauber-Fock photonic lattices,” Opt. Lett. 35, 2409–2411 (2010).
[Crossref] [PubMed]

Dreisow, F.

R. Keil, A. Perez-Leija, F. Dreisow, M. Heinrich, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Classical analogue of displaced Fock states and quantum correlations in Glauber-Fock photonic lattices,” Phys. Rev. Lett. 107, 103601 (2011).
[Crossref] [PubMed]

Eberly, J. H.

Feng, D. H.

W. M. Zhang, D. H. Feng, and R. Gilmore, “Coherent states: Theory and some applications,” Rev. Mod. Phys. 62, 867 (1990).
[Crossref]

Fernández C., D. J.

Gilmore, R.

W. M. Zhang, D. H. Feng, and R. Gilmore, “Coherent states: Theory and some applications,” Rev. Mod. Phys. 62, 867 (1990).
[Crossref]

R. Gilmore, “Geometry of symmetrized states,” Ann. Phys. 74, 391–463 (1972).
[Crossref]

Glauber, R. J.

R. J. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev. 131, 2766–2788 (1963).
[Crossref]

Hardy, A.

W. Streifer, M. Osiński, and A. Hardy, “Reformulation of the coupled-mode theory of mutiwaveguide systems,” J. Lightwave Technol. 5, 1–4 (1987).
[Crossref]

Haus, H. A.

H. A. Haus, W. P. Huang, S. Kawakami, and N. A. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol. 5, 16–23 (1987).
[Crossref]

Heinrich, M.

R. Keil, A. Perez-Leija, F. Dreisow, M. Heinrich, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Classical analogue of displaced Fock states and quantum correlations in Glauber-Fock photonic lattices,” Phys. Rev. Lett. 107, 103601 (2011).
[Crossref] [PubMed]

Huang, W. P.

H. A. Haus, W. P. Huang, S. Kawakami, and N. A. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol. 5, 16–23 (1987).
[Crossref]

Huang, W.-P.

Jones, A. L.

Kawakami, S.

H. A. Haus, W. P. Huang, S. Kawakami, and N. A. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol. 5, 16–23 (1987).
[Crossref]

Kay, A.

A. Perez-Leija, R. Keil, A. Kay, H. Moya-Cessa, S. Nolte, L.-C. Kwek, B. Rodríguez-Lara, A. Szameit, and D. Christodoulides, “Coherent quantum transport in photonic lattices,” Phys. Rev. A 87, 012309 (2013).
[Crossref]

Keil, R.

A. Perez-Leija, R. Keil, H. Moya-Cessa, A. Szameit, and D. N. Christodoulides, “Perfect transfer of path-entangled photons in Jx photonic lattices,” Phys. Rev. A 87, 022303 (2013).
[Crossref]

A. Perez-Leija, R. Keil, A. Kay, H. Moya-Cessa, S. Nolte, L.-C. Kwek, B. Rodríguez-Lara, A. Szameit, and D. Christodoulides, “Coherent quantum transport in photonic lattices,” Phys. Rev. A 87, 012309 (2013).
[Crossref]

A. Perez-Leija, R. Keil, A. Szameit, A. F. Abouraddy, H. Moya-Cessa, and D. N. Christodoulides, “Tailoring the correlation and anticorrelation behavior of path-entangled photons in Glauber-Fock lattices,” Phys. Rev. A 85, 013848 (2012).
[Crossref]

R. Keil, A. Perez-Leija, P. Aleahmad, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Observation of Bloch-like revivals in semi-infinite Glauber-Fock photonic lattices,” Opt. Lett. 37, 3801– 3803 (2012).
[Crossref] [PubMed]

R. Keil, A. Perez-Leija, F. Dreisow, M. Heinrich, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Classical analogue of displaced Fock states and quantum correlations in Glauber-Fock photonic lattices,” Phys. Rev. Lett. 107, 103601 (2011).
[Crossref] [PubMed]

Klauder, J. R.

J. R. Klauder and B.-S. Skagerstam, Coherent States. Applications in Physics and Mathematical Physics (World Scientific, 1985).
[Crossref]

Kogelnik, H.

H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[Crossref]

Krötzsch, G.

Kwek, L.-C.

A. Perez-Leija, R. Keil, A. Kay, H. Moya-Cessa, S. Nolte, L.-C. Kwek, B. Rodríguez-Lara, A. Szameit, and D. Christodoulides, “Coherent quantum transport in photonic lattices,” Phys. Rev. A 87, 012309 (2013).
[Crossref]

Lebedev, N. N.

N. N. Lebedev, Special Functions and their Applications (Prentice-Hall, 1965).

Longhi, S.

S. Longhi, “Classical simulation of relativistic quantum mechanics in periodic optical structures,” Appl. Phys. B 104, 453–468 (2011).
[Crossref]

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

Moya-Cessa, H.

A. Perez-Leija, R. Keil, A. Kay, H. Moya-Cessa, S. Nolte, L.-C. Kwek, B. Rodríguez-Lara, A. Szameit, and D. Christodoulides, “Coherent quantum transport in photonic lattices,” Phys. Rev. A 87, 012309 (2013).
[Crossref]

A. Perez-Leija, R. Keil, H. Moya-Cessa, A. Szameit, and D. N. Christodoulides, “Perfect transfer of path-entangled photons in Jx photonic lattices,” Phys. Rev. A 87, 022303 (2013).
[Crossref]

A. Perez-Leija, R. Keil, A. Szameit, A. F. Abouraddy, H. Moya-Cessa, and D. N. Christodoulides, “Tailoring the correlation and anticorrelation behavior of path-entangled photons in Glauber-Fock lattices,” Phys. Rev. A 85, 013848 (2012).
[Crossref]

R. Keil, A. Perez-Leija, P. Aleahmad, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Observation of Bloch-like revivals in semi-infinite Glauber-Fock photonic lattices,” Opt. Lett. 37, 3801– 3803 (2012).
[Crossref] [PubMed]

R. Keil, A. Perez-Leija, F. Dreisow, M. Heinrich, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Classical analogue of displaced Fock states and quantum correlations in Glauber-Fock photonic lattices,” Phys. Rev. Lett. 107, 103601 (2011).
[Crossref] [PubMed]

A. Perez-Leija, H. Moya-Cessa, A. Szameit, and D. N. Christodoulides, “Glauber-Fock photonic lattices,” Opt. Lett. 35, 2409–2411 (2010).
[Crossref] [PubMed]

Moya-Cessa, H. M.

Nolte, S.

A. Perez-Leija, R. Keil, A. Kay, H. Moya-Cessa, S. Nolte, L.-C. Kwek, B. Rodríguez-Lara, A. Szameit, and D. Christodoulides, “Coherent quantum transport in photonic lattices,” Phys. Rev. A 87, 012309 (2013).
[Crossref]

R. Keil, A. Perez-Leija, P. Aleahmad, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Observation of Bloch-like revivals in semi-infinite Glauber-Fock photonic lattices,” Opt. Lett. 37, 3801– 3803 (2012).
[Crossref] [PubMed]

R. Keil, A. Perez-Leija, F. Dreisow, M. Heinrich, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Classical analogue of displaced Fock states and quantum correlations in Glauber-Fock photonic lattices,” Phys. Rev. Lett. 107, 103601 (2011).
[Crossref] [PubMed]

Osinski, M.

W. Streifer, M. Osiński, and A. Hardy, “Reformulation of the coupled-mode theory of mutiwaveguide systems,” J. Lightwave Technol. 5, 1–4 (1987).
[Crossref]

Perelomov, A. M.

A. M. Perelomov, “Coherent states for arbitrary Lie group,” Comm. Math. Phys. 26, 222–236 (1972).
[Crossref]

Perez-Leija, A.

A. Perez-Leija, R. Keil, A. Kay, H. Moya-Cessa, S. Nolte, L.-C. Kwek, B. Rodríguez-Lara, A. Szameit, and D. Christodoulides, “Coherent quantum transport in photonic lattices,” Phys. Rev. A 87, 012309 (2013).
[Crossref]

A. Perez-Leija, R. Keil, H. Moya-Cessa, A. Szameit, and D. N. Christodoulides, “Perfect transfer of path-entangled photons in Jx photonic lattices,” Phys. Rev. A 87, 022303 (2013).
[Crossref]

A. Perez-Leija, R. Keil, A. Szameit, A. F. Abouraddy, H. Moya-Cessa, and D. N. Christodoulides, “Tailoring the correlation and anticorrelation behavior of path-entangled photons in Glauber-Fock lattices,” Phys. Rev. A 85, 013848 (2012).
[Crossref]

R. Keil, A. Perez-Leija, P. Aleahmad, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Observation of Bloch-like revivals in semi-infinite Glauber-Fock photonic lattices,” Opt. Lett. 37, 3801– 3803 (2012).
[Crossref] [PubMed]

R. Keil, A. Perez-Leija, F. Dreisow, M. Heinrich, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Classical analogue of displaced Fock states and quantum correlations in Glauber-Fock photonic lattices,” Phys. Rev. Lett. 107, 103601 (2011).
[Crossref] [PubMed]

A. Perez-Leija, H. Moya-Cessa, A. Szameit, and D. N. Christodoulides, “Glauber-Fock photonic lattices,” Opt. Lett. 35, 2409–2411 (2010).
[Crossref] [PubMed]

Puri, R. R.

R. R. Puri, Mathematical Methods of Quantum Optics (Springer, 2001).
[Crossref]

Raczka, R.

A. O. Barut and R. Raczka, Theory of group representations and applications (PWN Polish Scientific Publishers, 1980).

Rodríguez-Lara, B.

A. Perez-Leija, R. Keil, A. Kay, H. Moya-Cessa, S. Nolte, L.-C. Kwek, B. Rodríguez-Lara, A. Szameit, and D. Christodoulides, “Coherent quantum transport in photonic lattices,” Phys. Rev. A 87, 012309 (2013).
[Crossref]

Rodríguez-Lara, B. M.

B. M. Rodríguez-Lara, F. Soto-Eguibar, and D. N. Christodoulides, “Quantum optics as a tool for photonic lattice design,” Phys. Scr. 90, 068014 (2015).
[Crossref]

B. M. Rodríguez-Lara, H. M. Moya-Cessa, and D. N. Christodoulides, “Propagation and perfect transmission in three-waveguide axially varying couplers,” Phys. Rev. A 89, 013802 (2014).
[Crossref]

B. M. Rodríguez-Lara, P. Aleahmad, H. M. Moya-Cessa, and D. N. Christodoulides, “Ermakov-lewis symmetry in photonic lattices,” Opt. Lett. 39, 2083–2085 (2014).
[Crossref] [PubMed]

A. Zuñiga-Segundo, B. M. Rodríguez-Lara, D. J. Fernández C., and H. M. Moya-Cessa, “Jacobi photonic lattices and their SUSY partners,” Opt. Express 22, 987–994 (2014).
[Crossref] [PubMed]

B. M. Rodríguez-Lara, “Exact dynamics of finite Glauber-Fock photonic lattices,” Phys. Rev. A 84, 053845 (2011).
[Crossref]

Satyanarayana, M. V.

M. V. Satyanarayana, “Generalized coherent states and generalized squeezed coherent states,” Phys. Rev. D 32, 400–404 (1985).
[Crossref]

Shank, C. V.

H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[Crossref]

Skagerstam, B.-S.

J. R. Klauder and B.-S. Skagerstam, Coherent States. Applications in Physics and Mathematical Physics (World Scientific, 1985).
[Crossref]

Snyder, A. W.

Soto-Eguibar, F.

B. M. Rodríguez-Lara, F. Soto-Eguibar, and D. N. Christodoulides, “Quantum optics as a tool for photonic lattice design,” Phys. Scr. 90, 068014 (2015).
[Crossref]

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W. Streifer, M. Osiński, and A. Hardy, “Reformulation of the coupled-mode theory of mutiwaveguide systems,” J. Lightwave Technol. 5, 1–4 (1987).
[Crossref]

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E. C. G. Sudarshan, “Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams,” Phys. Rev. Lett. 10, 277–279 (1963).
[Crossref]

Szameit, A.

A. Perez-Leija, R. Keil, A. Kay, H. Moya-Cessa, S. Nolte, L.-C. Kwek, B. Rodríguez-Lara, A. Szameit, and D. Christodoulides, “Coherent quantum transport in photonic lattices,” Phys. Rev. A 87, 012309 (2013).
[Crossref]

A. Perez-Leija, R. Keil, H. Moya-Cessa, A. Szameit, and D. N. Christodoulides, “Perfect transfer of path-entangled photons in Jx photonic lattices,” Phys. Rev. A 87, 022303 (2013).
[Crossref]

A. Perez-Leija, R. Keil, A. Szameit, A. F. Abouraddy, H. Moya-Cessa, and D. N. Christodoulides, “Tailoring the correlation and anticorrelation behavior of path-entangled photons in Glauber-Fock lattices,” Phys. Rev. A 85, 013848 (2012).
[Crossref]

R. Keil, A. Perez-Leija, P. Aleahmad, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Observation of Bloch-like revivals in semi-infinite Glauber-Fock photonic lattices,” Opt. Lett. 37, 3801– 3803 (2012).
[Crossref] [PubMed]

R. Keil, A. Perez-Leija, F. Dreisow, M. Heinrich, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Classical analogue of displaced Fock states and quantum correlations in Glauber-Fock photonic lattices,” Phys. Rev. Lett. 107, 103601 (2011).
[Crossref] [PubMed]

A. Perez-Leija, H. Moya-Cessa, A. Szameit, and D. N. Christodoulides, “Glauber-Fock photonic lattices,” Opt. Lett. 35, 2409–2411 (2010).
[Crossref] [PubMed]

Vicent, L. E.

Whitaker, N. A.

H. A. Haus, W. P. Huang, S. Kawakami, and N. A. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol. 5, 16–23 (1987).
[Crossref]

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R. M. Wilcox, “Exponential operators and parameter differentiation in quantum physics,” J. Math. Phys. 8, 962–982 (1967).
[Crossref]

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A. Wünsche, “Displaced Fock states and their connection to quasiprobabilities,” Quantum Opt. 3, 359–383 (1991).
[Crossref]

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A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. 9, 919–933 (1973).
[Crossref]

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W. M. Zhang, D. H. Feng, and R. Gilmore, “Coherent states: Theory and some applications,” Rev. Mod. Phys. 62, 867 (1990).
[Crossref]

Zuñiga-Segundo, A.

Ann. Phys. (1)

R. Gilmore, “Geometry of symmetrized states,” Ann. Phys. 74, 391–463 (1972).
[Crossref]

Appl. Phys. B (1)

S. Longhi, “Classical simulation of relativistic quantum mechanics in periodic optical structures,” Appl. Phys. B 104, 453–468 (2011).
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Comm. Math. Phys. (1)

A. M. Perelomov, “Coherent states for arbitrary Lie group,” Comm. Math. Phys. 26, 222–236 (1972).
[Crossref]

IEEE J. Quantum Electron. (1)

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. 9, 919–933 (1973).
[Crossref]

J. Appl. Phys. (1)

H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[Crossref]

J. Lightwave Technol. (2)

W. Streifer, M. Osiński, and A. Hardy, “Reformulation of the coupled-mode theory of mutiwaveguide systems,” J. Lightwave Technol. 5, 1–4 (1987).
[Crossref]

H. A. Haus, W. P. Huang, S. Kawakami, and N. A. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol. 5, 16–23 (1987).
[Crossref]

J. Math. Phys. (1)

R. M. Wilcox, “Exponential operators and parameter differentiation in quantum physics,” J. Math. Phys. 8, 962–982 (1967).
[Crossref]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (5)

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

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S. Longhi, “Quantum-optical analogies using photonic structures,” Laser Photon. Rev. 3, 243–261 (2009).
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Opt. Express (1)

Opt. Lett. (3)

Phys. Rev. (1)

R. J. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev. 131, 2766–2788 (1963).
[Crossref]

Phys. Rev. A (5)

A. Perez-Leija, R. Keil, A. Szameit, A. F. Abouraddy, H. Moya-Cessa, and D. N. Christodoulides, “Tailoring the correlation and anticorrelation behavior of path-entangled photons in Glauber-Fock lattices,” Phys. Rev. A 85, 013848 (2012).
[Crossref]

B. M. Rodríguez-Lara, H. M. Moya-Cessa, and D. N. Christodoulides, “Propagation and perfect transmission in three-waveguide axially varying couplers,” Phys. Rev. A 89, 013802 (2014).
[Crossref]

A. Perez-Leija, R. Keil, A. Kay, H. Moya-Cessa, S. Nolte, L.-C. Kwek, B. Rodríguez-Lara, A. Szameit, and D. Christodoulides, “Coherent quantum transport in photonic lattices,” Phys. Rev. A 87, 012309 (2013).
[Crossref]

A. Perez-Leija, R. Keil, H. Moya-Cessa, A. Szameit, and D. N. Christodoulides, “Perfect transfer of path-entangled photons in Jx photonic lattices,” Phys. Rev. A 87, 022303 (2013).
[Crossref]

B. M. Rodríguez-Lara, “Exact dynamics of finite Glauber-Fock photonic lattices,” Phys. Rev. A 84, 053845 (2011).
[Crossref]

Phys. Rev. D (1)

M. V. Satyanarayana, “Generalized coherent states and generalized squeezed coherent states,” Phys. Rev. D 32, 400–404 (1985).
[Crossref]

Phys. Rev. Lett. (2)

E. C. G. Sudarshan, “Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams,” Phys. Rev. Lett. 10, 277–279 (1963).
[Crossref]

R. Keil, A. Perez-Leija, F. Dreisow, M. Heinrich, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Classical analogue of displaced Fock states and quantum correlations in Glauber-Fock photonic lattices,” Phys. Rev. Lett. 107, 103601 (2011).
[Crossref] [PubMed]

Phys. Scr. (1)

B. M. Rodríguez-Lara, F. Soto-Eguibar, and D. N. Christodoulides, “Quantum optics as a tool for photonic lattice design,” Phys. Scr. 90, 068014 (2015).
[Crossref]

Quantum Opt. (1)

A. Wünsche, “Displaced Fock states and their connection to quasiprobabilities,” Quantum Opt. 3, 359–383 (1991).
[Crossref]

Rev. Mod. Phys. (1)

W. M. Zhang, D. H. Feng, and R. Gilmore, “Coherent states: Theory and some applications,” Rev. Mod. Phys. 62, 867 (1990).
[Crossref]

Other (5)

N. N. Lebedev, Special Functions and their Applications (Prentice-Hall, 1965).

J. R. Klauder and B.-S. Skagerstam, Coherent States. Applications in Physics and Mathematical Physics (World Scientific, 1985).
[Crossref]

A. O. Barut and R. Raczka, Theory of group representations and applications (PWN Polish Scientific Publishers, 1980).

R. Askey, Orthogonal Polynomials and Special Functions, in CBMS-NSF Regional Conference Series in Applied Mathematics (SIAM, 1975).
[Crossref]

R. R. Puri, Mathematical Methods of Quantum Optics (Springer, 2001).
[Crossref]

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Figures (7)

Fig. 1
Fig. 1

Intensity at the n-th waveguide, | m , n ( SU ( 2 ) ) ( ω , λ , j , z ) | 2, for light initially impinging the middle waveguide, m = 5, of an SU(2) array with parameters λ = 0.4ω and j = 5.

Fig. 2
Fig. 2

Intensity at the n-th waveguide, | m , n ( SU ( 2 ) ) ( ω , λ , j , z ) | 2, for light initially impinging the m = 2 waveguide of an SU(2) array with parameters ω = 0 and j = 5.

Fig. 3
Fig. 3

Intensity at the n-th waveguide, | m , n ( SU ( 1 , 1 ) ) ( ω , λ , k , z ) | 2, for light initially impinging the m = 15 waveguide of a SU(1, 1) array with parameters λ = 0.4ω and k = 1/4.

Fig. 4
Fig. 4

Intensity at the n-th waveguide, | m , n ( SU ( 1 , 1 ) ) ( ω , λ , k , z ) | 2, for light initially impinging the m = 15 waveguide of a SU(1, 1) array with parameters ω = 2λ and k = 1/4.

Fig. 5
Fig. 5

Intensity at the n-th waveguide, | m , n ( SU ( 1 , 1 ) ) ( ω , λ , k , z ) | 2, for light initially impinging the m = 15 waveguide of a SU(1, 1) array with parameters ω = 0 and k = 1/4.

Fig. 6
Fig. 6

Intensity at the n-th waveguide, | m , n ( HW ) ( ω , λ , z ) | 2, for light initially impinging the m = 24 waveguide of a Heisenberg-Weyl array with parameter λ = 0.4 ω.

Fig. 7
Fig. 7

Intensity at the n-th waveguide, | m , n ( HW ) ( ω , λ , z ) | 2, for light initially impinging the m = 24 waveguide of a Heisenberg-Weyl array with parameter ω = 0.

Equations (52)

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i d d z l ( z ) = ω f ( l , z ) l ( z ) + λ [ g ( l , z ) l 1 ( z ) + g ( l + 1 , z ) l + 1 ( z ) ] ,
i d d z | ( z ) = H ^ ( z ) | ( z ) .
| ( z ) = ( 0 ( z ) 1 ( z ) 2 ( z ) ) ,
l = 0 l ( z ) | j ,
H ^ ( z ) = ( ω f ( 0 , z ) λ g ( 0 , z ) 0 λ g ( 0 , z ) ω f ( 1 , z ) λ g ( 1 , z ) 0 λ g ( 1 , z ) ω f ( 2 , z ) λ g ( 2 , z ) ) ,
ω A ^ 0 ( z ) + λ [ A ^ + ( z ) + A ^ ( z ) ] .
| ( z ) = e i 0 z H ^ ( x ) d x | ( 0 ) ,
m , n ( ω , λ , z ) = n | e i 0 z H ^ ( x ) d x | m .
m , n ( ω , λ , z ) = n | e i z H ^ | m ,
= n | e i z [ ω A ^ 0 + λ ( A ^ + + A ^ ) ] | m ,
= n | e a + ( z ) A ^ + e a 0 ( z ) A ^ 0 e a ( z ) A ^ | m ,
D ^ [ a 0 ( z ) , a ± ( z ) ] | m e a + ( z ) A ^ + e a 0 ( z ) A ^ 0 e a ( z ) A ^ | m .
H ^ SU ( 2 ) ( ω , λ , j ) = ω ( n ^ j ) + λ [ V ^ n ^ ( 2 j + 1 n ^ ) + n ^ ( 2 j + 1 n ^ ) V ^ ] ,
ω J ^ 0 + λ ( J ^ + J ^ + ) ,
D ^ SU ( 2 ) ( θ ) = e θ J ^ + θ * J ^ ,
= e θ | θ | tan | θ | J ^ + e ln sec 2 | θ | J ^ 0 e θ * | θ | tan | θ | J ^ ,
D ^ SU ( 2 ) ( θ ) H ^ SU ( 2 ) ( ω , λ ) D ^ SU ( 2 ) ( θ ) = ω 2 + 4 λ 2 ( n ^ j ) ,
tan 2 θ = 2 λ ω .
Ω SU ( 2 ) ( m , ω , λ , j ) = ω 2 + 4 λ 2 ( m j ) ,
n | Ω SU ( 2 ) ( m , ω , λ , j ) n | D ^ SU ( 2 ) ( θ ) | m ,
= ( 1 ) n ( 2 j m ) ( 2 j n ) ( cos | θ | ) 2 j m n ( sin | θ | ) m + n × × K m ( n , sin 2 | θ | , 2 j ) ,
m , n ( SU ( 2 ) ) ( ω , λ , j , z ) = n | e i z H ^ SU ( 2 ) | m ,
= ( 2 j m ) ( 2 j n ) ( 2 i λ sin z 2 ω 2 + 4 λ 2 ) m + n ( ω 2 + 4 λ 2 ) j × × ( ω 2 + 4 λ 2 cos z 2 ω 2 + 4 λ 2 i ω sin z 2 ω 2 + 4 λ 2 ) 2 j m n × × K m ( n , 4 λ 2 ω 2 + 4 λ 2 sin 2 z 2 ω 2 + 4 λ 2 , 2 j ) .
H ^ SU ( 2 ) ( 0 , λ , j ) = 2 λ J ^ x , J ^ x = 1 2 ( J ^ + + J ^ ) .
lim ω 0 1 2 arctan 2 λ ω = π 4
n | Ω ( m , 0 , λ , j ) SU ( 2 ) = 1 2 j K m ( n , 1 2 , 2 j ) ,
m , n ( SU ( 2 ) ) ( 0 , λ , j , z ) = ( 2 j m ) ( 2 j n ) ( i sin λ z ) m + n ( cos λ z ) 2 j m n × × K m ( n , sin 2 λ z , 2 j ) .
m , n ( SU ( 2 ) ) ( 0 , λ , j π 2 λ ) = δ n , 2 j m ,
H ^ SU ( 1 , 1 ) ( ω , λ , k ) = ω ( n ^ + k ) + λ [ V ^ n ^ ( 2 k 1 + n ^ ) + n ^ ( 2 k 1 + n ^ ) V ^ ] ,
ω K ^ 0 + λ ( K ^ + K ^ + ) .
D ^ SU ( 1 , 1 ) ( ξ ) H ^ SU ( 1 , 1 ) ( ω , λ , k ) D ^ SU ( 1 , 1 ) ( ξ ) = ω 2 4 λ 2 ( n ^ + k ) ,
tanh 2 ξ = 2 λ ω , ω > 2 λ .
D ^ SU ( 1 , 1 ) ( ξ ) = e ξ K ^ + ξ * K ^ ,
= e ξ | ξ | tanh | ξ | K ^ + e ln sech 2 | ξ | K ^ 0 e ξ * | ξ | tanh | ξ | K ^ ,
Ω SU ( 1 , 1 ) ( m , ω , λ , k ) = ω 2 4 λ 2 ( m + k ) ,
n | Ω SU ( 1 , 1 ) ( m , ω , λ , k ) = n | D ^ SU ( 1 , 1 ) ( ξ ) | m ,
= ( 1 ) n Γ ( 2 k ) Γ ( m + 2 k ) Γ ( n + 2 k ) m ! n ! ( sinh | ξ | ) m + n ( cosh | ξ | ) 2 k m n × × F 2 1 ( m , n , 2 k , csch 2 | ξ | ) ,
m , n ( SU ( 1 , 1 ) ) ( ω , λ , k , z ) = n | e i z H ^ SU ( 2 ) | m ,
= ( 4 λ 2 ω 2 ) Γ ( 2 k ) Γ ( m + 2 k ) Γ ( n + 2 k ) m ! n ! ( 2 i λ sinh z 2 4 λ 2 w 2 ) m + n × × ( 4 λ 2 ω 2 cosh z 2 4 λ 2 w 2 i ω sinh z 2 4 λ 2 w 2 ) 2 k m n × × F 2 1 ( n , m ; 2 k ; ( ω 2 4 λ 2 4 λ 2 ) csch 2 z 2 4 λ 2 w 2 ) .
m , n ( SU ( 1 , 1 ) ) ( 2 λ , λ , k , z ) = Γ ( m + 2 k ) Γ ( n + 2 k ) m ! n ! ( 1 ) k + m + n ( λ z ) m + n ( λ z + i ) 2 k m n Γ ( 2 k ) × × F 2 1 ( n , m ; 2 k ; 1 λ 2 z 2 ) .
m , n ( SU ( 1 , 1 ) ) ( 0 , λ , k , z ) = Γ ( m + 2 k ) Γ ( n + 2 k ) m ! n ! ( i sinh λ z ) m + n ( cosh λ z ) 2 k m n Γ ( 2 k ) × × F 2 1 ( n , m ; 2 k ; csch 2 λ z ) .
H ^ HW ( ω , λ , z ) = ω n ^ + λ ( V ^ n ^ + n ^ V ^ ) ,
= ω n ^ + λ ( a ^ + a ^ ) ,
D ^ HW ( α ) H ^ HW D ^ HW ( α ) = ω n ^ λ 2 ω , α = λ ω .
D ^ HW ( α ) = e α a ^ α * a ^ .
Ω HW ( m , ω , λ ) = ω m λ 2 ω ,
n | Ω HW ( m , ω , λ ) = n | D ^ HW ( α ) | m ,
= ( 1 ) m m ! n ! e 1 2 | λ ω | 2 ( λ ω ) n m U ( m , n m + 1 , | λ ω | 2 ) ,
m , n ( HW ) ( ω , λ , z ) = n | e i z H ^ HW | m ,
= ( 1 ) m m ! n ! e i z ( m ω λ 2 ω ) e | λ ω | 2 ( 1 e i w z ) [ λ ω ( e i w z 1 ) ] n m × × U ( m , n m + 1 , 4 | λ ω | 2 sin 2 ω z 2 ) ,
m , n ( HW ) ( 0 , λ , z ) = lim ω 0 m , n ( HW ) ( ω , λ , z ) ,
= ( 1 ) m m ! n ! e 1 2 | λ z | 2 ( i λ z ) n m U ( m , n m + 1 , | λ z | 2 ) ,

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