Abstract

We present a classical analog of quantum optical deformed oscillators in arrays of waveguides. The normal modes of these one-dimensional photonic crystals are given in terms of Jacobi polynomials. We show that it is possible to attack the problem via factorization by exploiting the corresponding quantum optical model. This allows us to provide an unbroken supersymmetric partner of the proposed Jacobi lattices. Thanks to the underlying SU(1, 1) group symmetry of the lattices, we present the analytic propagators and impulse functions for these one-dimensional photonic crystals.

© 2014 Optical Society of America

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  1. F. Cooper, A. Khare, and U. Sukhatme, “Supersymmetry and quantum mechanics,” Phys. Rep. 251, 267–385 (1995).
    [Crossref]
  2. D. J. Fernández C. and N. Fernández-García, “Higher-order supersymmetric quantum mechanics,” AIP Conf. Proc. 744, 236–273 (2005).
    [Crossref]
  3. D. J. Fernández C., “Supersymmetric quantum mechanics,” AIP Conf. Proc. 1287, 3–36 (2010).
  4. S. M. Chumakov and K. B. Wolf, “Supersymmetry in Helmholtz optics,” Phys. Lett. A 193, 51–53 (1994).
    [Crossref]
  5. S. Tomić, V. Milanović, and Z. Ikonić, “Optimization of intersubband resonant second-order susceptibility in assymetric graded AlxGa1−xAs quantum wells using supersymmetric quantum mechanics,” Phys. Rev. B 56, 1033–1036 (1997).
    [Crossref]
  6. J. Bai and D. S. Citrin, “Supersymmetric optimization of second-harmonic generation in mid-infrared quantum cascade lasers,” Opt. Express 14, 4043–4048 (2006).
    [Crossref] [PubMed]
  7. R. G. Unanyan and M. Fleischhauer, “Decoherence-free generation of many-particle entanglement by adiabatic ground-state transitions,” Phys. Rev. Lett. 90, 133601 (2003).
    [Crossref] [PubMed]
  8. Y. Yu and K. Yang, “Simulating the Wess-Zumino supersymmetry model in optical lattices,” Phys. Rev. Lett. 105, 150605 (2010).
    [Crossref]
  9. T. G. Tenev, P. A. Ivanov, and N. V. Vitanov, “Proposal for trapped-ion emulation of the electric dipole moment of neutral relativistic particles,” Phys. Rev. A 87, 022103 (2013).
    [Crossref]
  10. R. El-Ganainy, A. Eisfeld, M. Levy, and D. N. Christodoulides, “On-chip non-reciprocal optical devices based on quantum inspired photonic lattices,” Appl. Phys. Lett. 103, 161105 (2013).
    [Crossref]
  11. M.-A. Miri, M. Heinrich, R. El-Ganainy, and D. N. Christodoulides, “Supersymmetric optical structures,” Phys. Rev. Lett. 110, 233902 (2013).
    [Crossref]
  12. S. Longhi and G. Della Valle, “Transparency at the interface between two isospectral crystals,” Europhys. Lett. 102, 40008 (2013).
    [Crossref]
  13. C. Daskaloyannis, “Generalized deformed oscillator and nonlinear algebras,” J. Phys. A: Math. Gen. 24, 789–794 (1991).
    [Crossref]
  14. V. V. Dodonov, M. A. Marchiollo, Y. A. Korennoy, V. I. Man’ko, and Y. A. Moukhin, “Parametric excitation of photon-added coherent states,” Phys. Scr. 58, 469–480 (1998).
    [Crossref]
  15. A. A. Sukhorukov, A. S. Solntsev, and J. E. Sipe, “Classical simulation of squeezed light in optical waveguide arrays,” Phys. Rev. A 87, 053823 (2013).
    [Crossref]
  16. L. Infeld and T. E. Hull, “The factorization method,” Rev. Mod. Phys. 23, 21–68 (1951).
    [Crossref]
  17. A. L. Jones, “Coupling of optical fibers and scattering in fibers,” J. Opt. Soc. Am. 55, 261–271 (1965).
    [Crossref]
  18. D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behavior in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003).
    [Crossref] [PubMed]
  19. B. M. Rodríguez-Lara, “Exact dynamics of finite Glauber-Fock photonic lattices,” Phys. Rev. A 84, 053845 (2011).
    [Crossref]
  20. 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]
  21. B. M. Rodríguez-Lara, F. Soto-Eguibar, A. Z. Cárdenas, and H. M. Moya-Cessa, “A classical simulation of nonlinear Jaynes-Cummings and Rabi models in photonic lattices,” Opt. Express 21, 12888–128981 (2013).
    [Crossref] [PubMed]
  22. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions(Dover1970).
  23. B. M. Rodríguez-Lara, H. M. Moya-Cessa, and D. N. Christodoulides, “Propagation, perfect transmission and trapping in three-waveguide axially varying couplers,” arXiv:1310.4754 [physics.optics] (2013).
  24. R. R. Puri and G. S. Agarwal, “Unitarily inequivalent classes of minimum uncertainty states of SU(1,1),” Int. J. Mod. Phys. B 10, 1563–1572 (1996).
    [Crossref]
  25. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 2007).

2013 (6)

T. G. Tenev, P. A. Ivanov, and N. V. Vitanov, “Proposal for trapped-ion emulation of the electric dipole moment of neutral relativistic particles,” Phys. Rev. A 87, 022103 (2013).
[Crossref]

R. El-Ganainy, A. Eisfeld, M. Levy, and D. N. Christodoulides, “On-chip non-reciprocal optical devices based on quantum inspired photonic lattices,” Appl. Phys. Lett. 103, 161105 (2013).
[Crossref]

M.-A. Miri, M. Heinrich, R. El-Ganainy, and D. N. Christodoulides, “Supersymmetric optical structures,” Phys. Rev. Lett. 110, 233902 (2013).
[Crossref]

S. Longhi and G. Della Valle, “Transparency at the interface between two isospectral crystals,” Europhys. Lett. 102, 40008 (2013).
[Crossref]

A. A. Sukhorukov, A. S. Solntsev, and J. E. Sipe, “Classical simulation of squeezed light in optical waveguide arrays,” Phys. Rev. A 87, 053823 (2013).
[Crossref]

B. M. Rodríguez-Lara, F. Soto-Eguibar, A. Z. Cárdenas, and H. M. Moya-Cessa, “A classical simulation of nonlinear Jaynes-Cummings and Rabi models in photonic lattices,” Opt. Express 21, 12888–128981 (2013).
[Crossref] [PubMed]

2012 (1)

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 (1)

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

2010 (2)

D. J. Fernández C., “Supersymmetric quantum mechanics,” AIP Conf. Proc. 1287, 3–36 (2010).

Y. Yu and K. Yang, “Simulating the Wess-Zumino supersymmetry model in optical lattices,” Phys. Rev. Lett. 105, 150605 (2010).
[Crossref]

2006 (1)

2005 (1)

D. J. Fernández C. and N. Fernández-García, “Higher-order supersymmetric quantum mechanics,” AIP Conf. Proc. 744, 236–273 (2005).
[Crossref]

2003 (2)

R. G. Unanyan and M. Fleischhauer, “Decoherence-free generation of many-particle entanglement by adiabatic ground-state transitions,” Phys. Rev. Lett. 90, 133601 (2003).
[Crossref] [PubMed]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behavior in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003).
[Crossref] [PubMed]

1998 (1)

V. V. Dodonov, M. A. Marchiollo, Y. A. Korennoy, V. I. Man’ko, and Y. A. Moukhin, “Parametric excitation of photon-added coherent states,” Phys. Scr. 58, 469–480 (1998).
[Crossref]

1997 (1)

S. Tomić, V. Milanović, and Z. Ikonić, “Optimization of intersubband resonant second-order susceptibility in assymetric graded AlxGa1−xAs quantum wells using supersymmetric quantum mechanics,” Phys. Rev. B 56, 1033–1036 (1997).
[Crossref]

1996 (1)

R. R. Puri and G. S. Agarwal, “Unitarily inequivalent classes of minimum uncertainty states of SU(1,1),” Int. J. Mod. Phys. B 10, 1563–1572 (1996).
[Crossref]

1995 (1)

F. Cooper, A. Khare, and U. Sukhatme, “Supersymmetry and quantum mechanics,” Phys. Rep. 251, 267–385 (1995).
[Crossref]

1994 (1)

S. M. Chumakov and K. B. Wolf, “Supersymmetry in Helmholtz optics,” Phys. Lett. A 193, 51–53 (1994).
[Crossref]

1991 (1)

C. Daskaloyannis, “Generalized deformed oscillator and nonlinear algebras,” J. Phys. A: Math. Gen. 24, 789–794 (1991).
[Crossref]

1965 (1)

1951 (1)

L. Infeld and T. E. Hull, “The factorization method,” Rev. Mod. Phys. 23, 21–68 (1951).
[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]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions(Dover1970).

Agarwal, G. S.

R. R. Puri and G. S. Agarwal, “Unitarily inequivalent classes of minimum uncertainty states of SU(1,1),” Int. J. Mod. Phys. B 10, 1563–1572 (1996).
[Crossref]

Bai, J.

Cárdenas, A. Z.

Christodoulides, D. N.

R. El-Ganainy, A. Eisfeld, M. Levy, and D. N. Christodoulides, “On-chip non-reciprocal optical devices based on quantum inspired photonic lattices,” Appl. Phys. Lett. 103, 161105 (2013).
[Crossref]

M.-A. Miri, M. Heinrich, R. El-Ganainy, and D. N. Christodoulides, “Supersymmetric optical structures,” Phys. Rev. Lett. 110, 233902 (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]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behavior in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003).
[Crossref] [PubMed]

B. M. Rodríguez-Lara, H. M. Moya-Cessa, and D. N. Christodoulides, “Propagation, perfect transmission and trapping in three-waveguide axially varying couplers,” arXiv:1310.4754 [physics.optics] (2013).

Chumakov, S. M.

S. M. Chumakov and K. B. Wolf, “Supersymmetry in Helmholtz optics,” Phys. Lett. A 193, 51–53 (1994).
[Crossref]

Citrin, D. S.

Cooper, F.

F. Cooper, A. Khare, and U. Sukhatme, “Supersymmetry and quantum mechanics,” Phys. Rep. 251, 267–385 (1995).
[Crossref]

Daskaloyannis, C.

C. Daskaloyannis, “Generalized deformed oscillator and nonlinear algebras,” J. Phys. A: Math. Gen. 24, 789–794 (1991).
[Crossref]

Della Valle, G.

S. Longhi and G. Della Valle, “Transparency at the interface between two isospectral crystals,” Europhys. Lett. 102, 40008 (2013).
[Crossref]

Dodonov, V. V.

V. V. Dodonov, M. A. Marchiollo, Y. A. Korennoy, V. I. Man’ko, and Y. A. Moukhin, “Parametric excitation of photon-added coherent states,” Phys. Scr. 58, 469–480 (1998).
[Crossref]

Eisfeld, A.

R. El-Ganainy, A. Eisfeld, M. Levy, and D. N. Christodoulides, “On-chip non-reciprocal optical devices based on quantum inspired photonic lattices,” Appl. Phys. Lett. 103, 161105 (2013).
[Crossref]

El-Ganainy, R.

M.-A. Miri, M. Heinrich, R. El-Ganainy, and D. N. Christodoulides, “Supersymmetric optical structures,” Phys. Rev. Lett. 110, 233902 (2013).
[Crossref]

R. El-Ganainy, A. Eisfeld, M. Levy, and D. N. Christodoulides, “On-chip non-reciprocal optical devices based on quantum inspired photonic lattices,” Appl. Phys. Lett. 103, 161105 (2013).
[Crossref]

Fernández C., D. J.

D. J. Fernández C., “Supersymmetric quantum mechanics,” AIP Conf. Proc. 1287, 3–36 (2010).

D. J. Fernández C. and N. Fernández-García, “Higher-order supersymmetric quantum mechanics,” AIP Conf. Proc. 744, 236–273 (2005).
[Crossref]

Fernández-García, N.

D. J. Fernández C. and N. Fernández-García, “Higher-order supersymmetric quantum mechanics,” AIP Conf. Proc. 744, 236–273 (2005).
[Crossref]

Fleischhauer, M.

R. G. Unanyan and M. Fleischhauer, “Decoherence-free generation of many-particle entanglement by adiabatic ground-state transitions,” Phys. Rev. Lett. 90, 133601 (2003).
[Crossref] [PubMed]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 2007).

Heinrich, M.

M.-A. Miri, M. Heinrich, R. El-Ganainy, and D. N. Christodoulides, “Supersymmetric optical structures,” Phys. Rev. Lett. 110, 233902 (2013).
[Crossref]

Hull, T. E.

L. Infeld and T. E. Hull, “The factorization method,” Rev. Mod. Phys. 23, 21–68 (1951).
[Crossref]

Ikonic, Z.

S. Tomić, V. Milanović, and Z. Ikonić, “Optimization of intersubband resonant second-order susceptibility in assymetric graded AlxGa1−xAs quantum wells using supersymmetric quantum mechanics,” Phys. Rev. B 56, 1033–1036 (1997).
[Crossref]

Infeld, L.

L. Infeld and T. E. Hull, “The factorization method,” Rev. Mod. Phys. 23, 21–68 (1951).
[Crossref]

Ivanov, P. A.

T. G. Tenev, P. A. Ivanov, and N. V. Vitanov, “Proposal for trapped-ion emulation of the electric dipole moment of neutral relativistic particles,” Phys. Rev. A 87, 022103 (2013).
[Crossref]

Jones, A. L.

Keil, R.

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]

Khare, A.

F. Cooper, A. Khare, and U. Sukhatme, “Supersymmetry and quantum mechanics,” Phys. Rep. 251, 267–385 (1995).
[Crossref]

Korennoy, Y. A.

V. V. Dodonov, M. A. Marchiollo, Y. A. Korennoy, V. I. Man’ko, and Y. A. Moukhin, “Parametric excitation of photon-added coherent states,” Phys. Scr. 58, 469–480 (1998).
[Crossref]

Lederer, F.

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behavior in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003).
[Crossref] [PubMed]

Levy, M.

R. El-Ganainy, A. Eisfeld, M. Levy, and D. N. Christodoulides, “On-chip non-reciprocal optical devices based on quantum inspired photonic lattices,” Appl. Phys. Lett. 103, 161105 (2013).
[Crossref]

Longhi, S.

S. Longhi and G. Della Valle, “Transparency at the interface between two isospectral crystals,” Europhys. Lett. 102, 40008 (2013).
[Crossref]

Man’ko, V. I.

V. V. Dodonov, M. A. Marchiollo, Y. A. Korennoy, V. I. Man’ko, and Y. A. Moukhin, “Parametric excitation of photon-added coherent states,” Phys. Scr. 58, 469–480 (1998).
[Crossref]

Marchiollo, M. A.

V. V. Dodonov, M. A. Marchiollo, Y. A. Korennoy, V. I. Man’ko, and Y. A. Moukhin, “Parametric excitation of photon-added coherent states,” Phys. Scr. 58, 469–480 (1998).
[Crossref]

Milanovic, V.

S. Tomić, V. Milanović, and Z. Ikonić, “Optimization of intersubband resonant second-order susceptibility in assymetric graded AlxGa1−xAs quantum wells using supersymmetric quantum mechanics,” Phys. Rev. B 56, 1033–1036 (1997).
[Crossref]

Miri, M.-A.

M.-A. Miri, M. Heinrich, R. El-Ganainy, and D. N. Christodoulides, “Supersymmetric optical structures,” Phys. Rev. Lett. 110, 233902 (2013).
[Crossref]

Moukhin, Y. A.

V. V. Dodonov, M. A. Marchiollo, Y. A. Korennoy, V. I. Man’ko, and Y. A. Moukhin, “Parametric excitation of photon-added coherent states,” Phys. Scr. 58, 469–480 (1998).
[Crossref]

Moya-Cessa, H.

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]

Moya-Cessa, H. M.

B. M. Rodríguez-Lara, F. Soto-Eguibar, A. Z. Cárdenas, and H. M. Moya-Cessa, “A classical simulation of nonlinear Jaynes-Cummings and Rabi models in photonic lattices,” Opt. Express 21, 12888–128981 (2013).
[Crossref] [PubMed]

B. M. Rodríguez-Lara, H. M. Moya-Cessa, and D. N. Christodoulides, “Propagation, perfect transmission and trapping in three-waveguide axially varying couplers,” arXiv:1310.4754 [physics.optics] (2013).

Perez-Leija, A.

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]

Puri, R. R.

R. R. Puri and G. S. Agarwal, “Unitarily inequivalent classes of minimum uncertainty states of SU(1,1),” Int. J. Mod. Phys. B 10, 1563–1572 (1996).
[Crossref]

Rodríguez-Lara, B. M.

B. M. Rodríguez-Lara, F. Soto-Eguibar, A. Z. Cárdenas, and H. M. Moya-Cessa, “A classical simulation of nonlinear Jaynes-Cummings and Rabi models in photonic lattices,” Opt. Express 21, 12888–128981 (2013).
[Crossref] [PubMed]

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

B. M. Rodríguez-Lara, H. M. Moya-Cessa, and D. N. Christodoulides, “Propagation, perfect transmission and trapping in three-waveguide axially varying couplers,” arXiv:1310.4754 [physics.optics] (2013).

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 2007).

Silberberg, Y.

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behavior in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003).
[Crossref] [PubMed]

Sipe, J. E.

A. A. Sukhorukov, A. S. Solntsev, and J. E. Sipe, “Classical simulation of squeezed light in optical waveguide arrays,” Phys. Rev. A 87, 053823 (2013).
[Crossref]

Solntsev, A. S.

A. A. Sukhorukov, A. S. Solntsev, and J. E. Sipe, “Classical simulation of squeezed light in optical waveguide arrays,” Phys. Rev. A 87, 053823 (2013).
[Crossref]

Soto-Eguibar, F.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions(Dover1970).

Sukhatme, U.

F. Cooper, A. Khare, and U. Sukhatme, “Supersymmetry and quantum mechanics,” Phys. Rep. 251, 267–385 (1995).
[Crossref]

Sukhorukov, A. A.

A. A. Sukhorukov, A. S. Solntsev, and J. E. Sipe, “Classical simulation of squeezed light in optical waveguide arrays,” Phys. Rev. A 87, 053823 (2013).
[Crossref]

Szameit, A.

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]

Tenev, T. G.

T. G. Tenev, P. A. Ivanov, and N. V. Vitanov, “Proposal for trapped-ion emulation of the electric dipole moment of neutral relativistic particles,” Phys. Rev. A 87, 022103 (2013).
[Crossref]

Tomic, S.

S. Tomić, V. Milanović, and Z. Ikonić, “Optimization of intersubband resonant second-order susceptibility in assymetric graded AlxGa1−xAs quantum wells using supersymmetric quantum mechanics,” Phys. Rev. B 56, 1033–1036 (1997).
[Crossref]

Unanyan, R. G.

R. G. Unanyan and M. Fleischhauer, “Decoherence-free generation of many-particle entanglement by adiabatic ground-state transitions,” Phys. Rev. Lett. 90, 133601 (2003).
[Crossref] [PubMed]

Vitanov, N. V.

T. G. Tenev, P. A. Ivanov, and N. V. Vitanov, “Proposal for trapped-ion emulation of the electric dipole moment of neutral relativistic particles,” Phys. Rev. A 87, 022103 (2013).
[Crossref]

Wolf, K. B.

S. M. Chumakov and K. B. Wolf, “Supersymmetry in Helmholtz optics,” Phys. Lett. A 193, 51–53 (1994).
[Crossref]

Yang, K.

Y. Yu and K. Yang, “Simulating the Wess-Zumino supersymmetry model in optical lattices,” Phys. Rev. Lett. 105, 150605 (2010).
[Crossref]

Yu, Y.

Y. Yu and K. Yang, “Simulating the Wess-Zumino supersymmetry model in optical lattices,” Phys. Rev. Lett. 105, 150605 (2010).
[Crossref]

AIP Conf. Proc. (2)

D. J. Fernández C. and N. Fernández-García, “Higher-order supersymmetric quantum mechanics,” AIP Conf. Proc. 744, 236–273 (2005).
[Crossref]

D. J. Fernández C., “Supersymmetric quantum mechanics,” AIP Conf. Proc. 1287, 3–36 (2010).

Appl. Phys. Lett. (1)

R. El-Ganainy, A. Eisfeld, M. Levy, and D. N. Christodoulides, “On-chip non-reciprocal optical devices based on quantum inspired photonic lattices,” Appl. Phys. Lett. 103, 161105 (2013).
[Crossref]

Europhys. Lett. (1)

S. Longhi and G. Della Valle, “Transparency at the interface between two isospectral crystals,” Europhys. Lett. 102, 40008 (2013).
[Crossref]

Int. J. Mod. Phys. B (1)

R. R. Puri and G. S. Agarwal, “Unitarily inequivalent classes of minimum uncertainty states of SU(1,1),” Int. J. Mod. Phys. B 10, 1563–1572 (1996).
[Crossref]

J. Opt. Soc. Am. (1)

J. Phys. A: Math. Gen. (1)

C. Daskaloyannis, “Generalized deformed oscillator and nonlinear algebras,” J. Phys. A: Math. Gen. 24, 789–794 (1991).
[Crossref]

Nature (1)

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behavior in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003).
[Crossref] [PubMed]

Opt. Express (2)

Phys. Lett. A (1)

S. M. Chumakov and K. B. Wolf, “Supersymmetry in Helmholtz optics,” Phys. Lett. A 193, 51–53 (1994).
[Crossref]

Phys. Rep. (1)

F. Cooper, A. Khare, and U. Sukhatme, “Supersymmetry and quantum mechanics,” Phys. Rep. 251, 267–385 (1995).
[Crossref]

Phys. Rev. A (4)

T. G. Tenev, P. A. Ivanov, and N. V. Vitanov, “Proposal for trapped-ion emulation of the electric dipole moment of neutral relativistic particles,” Phys. Rev. A 87, 022103 (2013).
[Crossref]

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

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Figures (6)

Fig. 1
Fig. 1 The effective refractive index, ωj(α), and coupling, gj(α), functions for the Jacobi lattices.

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Fig. 2
Fig. 2 The squared amplitudes corresponding to the (a) ground, k = 0, and (b) tenth, k = 10, normal mode, | α k 〉 = ∑ j = 0 ∞ E k , j ( α ) | j 〉 for different values of the deformation parameter α. The lines are a fit with a polynomial of order 20.

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Fig. 3
Fig. 3 (a) Numerical propagation of light intensity for an initial field impinging at the j = 9 waveguide of a Jacobi lattice with a deformation parameter α = 0.5. (b) Comparison of the numerical (solid black) and theoretical (dotted red) intensities at the j = 9 waveguide.

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Fig. 4
Fig. 4 The effective refractive index, ωj(α), and coupling, gj(α), functions for the SUSY partner of our Jacobi lattices.

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Fig. 5
Fig. 5 Relation between the normal modes of the Jacobi lattices, |αk〉 with spectrum (1 − α2)(k + 1), and the normal modes of their susy partner, | α k ( p ) 〉 with spectrum (1 − α2)k.

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Fig. 6
Fig. 6 (a) Numerical propagation of light intensity for an initial field impinging at the j = 9 waveguide of a Jacobi SUSY partner lattice with a deformation parameter α = 0.5. (b) Comparison of the numerical (solid black) and theoretical (dotted red) intensities at the j = 19 waveguide.

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Equations (27)

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[ i ∂ z + ( 1 + α 2 ) ( n ^ + 1 ) − α ( n ^ + 1     a ^ † + a ^ n ^ + 1 ) ] | ψ 〉 = 0 ,
i ∂ z ℰ j + ( 1 + α 2 ) ( j + 1 ) ℰ j − α ( j + 1 ) ( j + 2 ) ℰ j + 1 − α j ( j + 1 ) ℰ j − 1 = 0 ,         ℰ − 1 = 0 .
H ^ = ( 1 + α 2 ) K ^ 0 − α ( K ^ + + K ^ − ) ,
H ^ R = R ^ ( ξ ) H ^ R ^ ( − ξ ) ,
= [ ( 1 + α 2 ) cosh ξ − 2 α sinh ξ ] K ^ 0 ,
= ( 1 − α 2 ) K ^ 0 ,
Ω k ( α ) = ( 1 − α 2 ) ( k + 1 ) ,             k = 0 , 1 , 2 , …
| α k 〉 = ∑ j = 0 ∞ k + 1 j + 1 ( − 1 ) k ( 1 − α 2 ) α k − j P j ( 1 , k − j ) ( 2 α 2 − 1 ) | j 〉 ,
| ψ ( z ) 〉 = e f ( z ) K ^ + e − 2 ln g ( z )             K ^ 0 e f ( z ) K ^ − | ψ ( 0 ) 〉
f ( z ) = α ( 1 − e − i ( 1 − α 2 ) z 1 − α 2 e − i ( 1 − α 2 ) z ) ,
g ( z ) = cos   [ 1 2 ( 1 − α 2 ) z ] − i ( α 2 + 1 α 2 − 1 ) sin   [ 1 2 ( 1 − α 2 ) z ] ,
I j , k ( z ) = k + 1 j + 1 f k − j ( z ) g 2 ( j + 1 ) ( z ) [ 2 h ( z ) − 1 ] j P j ( 1 , k − j ) [ h ( z ) ] ,
h ( z ) = 1 − 2 ( α 2 − 1 ) 2 α 4 − 2 α 2 cos [ ( 1 − α 2 ) z ] + 1 .
A ^ α † = a ^ † − α n ^ + 1 ,             A ^ α = a ^ − α n ^ + 1 ,
[ i ∂ z + A ^ α A ^ α † ] | ψ 〉 = 0 .
[ i ∂ z + A ^ α † A ^ α ] | ψ 〉 = 0 .
i ∂ z ℰ j + [ ( 1 + α 2 ) j + α 2 ] ℰ j − α ( j + 1 ) ℰ j + 1 − α j ℰ j − 1 = 0 .             ℰ − 1 = 0 ,
H ^ p = ( 1 + α 2 ) K ˜ 0 − α ( K ˜ + + K ˜ − ) − 1 2 ( 1 − α 2 ) .
H ^ p R = R ˜ ( ξ ) H ^ p R ˜ ( − ξ ) ,
= [ ( 1 + α 2 ) cosh ξ − 2 α sinh ξ ] K ˜ 0 − 1 2 ( 1 − α 2 ) ,
= ( 1 − α 2 ) ( K ˜ 0 − 1 2 ) ,
Ω k ( p ) ( α ) = ( 1 − α 2 ) k ,             k = 0 , 1 , 2 , … .
| α k ( p ) 〉 = ∑ j = 0 ∞ 1 − α 2 ( − 1 ) k α k − j P j ( 0 , k − j ) ( 2 α 2 − 1 ) | j 〉
A ^ α | α k ( p ) 〉 = k ( 1 − α 2 ) | α k − 1 〉 ,                   k ≥ 1 ,
A ^ α † | α k 〉 = ( k + 1 ) ( 1 − α 2 ) | α k + 1 ( p ) 〉 ,                 k ≥ 0 .
| ψ ( z ) 〉 = e − i z 2 ( 1 − α 2 ) e f ( z ) K ˜ + e − 2 ln g ( z )           K ˜ 0 e f ( z ) K ˜ − | ψ ( 0 ) 〉 ,
I j , k ( z ) = e − i z 2 ( 1 − α 2 ) f k − j ( z ) g 2 ( j + 1 ) ( z ) [ 2 h ( z ) − 1 ] j P j ( 0 , k − j ) [ h ( z ) ] .

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