Abstract

We apply the notion of discrete supersymmetry based on matrix factorization to quantum systems consisting of coupled bosonic oscillators to construct isospectral bosonic quantum networks. By using the algebra that arises due to the indistinguishability of bosonic particles, we write down the Schrödinger equations for these oscillators in the different boson-number sectors. By doing so, we obtain, for every partner quantum network, a system of coupled differential equations that can be emulated by classical light propagation in optical waveguide arrays. This mathematical scheme allows us to build quasi-two-dimensional optical arrays that are either isospectral or share only a subset of their spectrum after deliberately omitting some chosen eigenstates from the spectrum. As an example, we use this technique (which we call bosonic discrete supersymmetry or BD-SUSY) to design two optical, silica-based waveguide arrays consisting of six and three elements, respectively, with overlapping eigenspectrum.

© 2019 Chinese Laser Press

Full Article  |  PDF Article
OSA Recommended Articles
Jacobi photonic lattices and their SUSY partners

A. Zúñiga-Segundo, B. M. Rodríguez-Lara, David J. Fernández C., and H. M. Moya-Cessa
Opt. Express 22(1) 987-994 (2014)

Supercharge optical arrays

Bikashkali Midya, Wiktor Walasik, Natalia M. Litchinitser, and Liang Feng
Opt. Lett. 43(20) 4927-4930 (2018)

SUSY-inspired one-dimensional transformation optics

Mohammad-Ali Miri, Matthias Heinrich, and Demetrios N. Christodoulides
Optica 1(2) 89-95 (2014)

References

  • View by:
  • |
  • |
  • |

  1. P. Ramond, “Dual theory for free fermions,” Phys. Rev. D 3, 2415–2418 (1971).
    [Crossref]
  2. A. Neveu and J. H. Schwarz, “Factorizable dual model of pions,” Nucl. Phys. B 31, 86–112 (1971).
    [Crossref]
  3. E. Witten, “Dynamical breaking of supersymmetry,” Nucl. Phys. B 188, 513–554 (1981).
    [Crossref]
  4. F. Cooper and B. Freedman, “Aspects of supersymmetric quantum mechanics,” Ann. Phys. 146, 262–288 (1983).
    [Crossref]
  5. A. Lahiri, P. K. Roy, and B. Bagchi, “Supersymmetry in quantum mechanics,” Int. J. Mod. Phys. A 05, 1383–1456 (1990).
    [Crossref]
  6. F. Cooper, A. Khare, and U. Sukhatme, “Supersymmetry and quantum mechanics,” Phys. Rep. 251, 267–385 (1995).
    [Crossref]
  7. R. El-Ganainy, K. G. Makris, and D. N. Christodoulides, “Local PT invariance and supersymmetric parametric oscillators,” Phys. Rev. A 86, 033813 (2012).
    [Crossref]
  8. M.-A. Miri, M. Heinrich, R. El-Ganainy, and D. N. Christodoulides, “Supersymmetric optical structures,” Phys. Rev. Lett. 110, 233902 (2013).
    [Crossref]
  9. M. Heinrich, M.-A. Miri, S. Stützer, R. El-Ganainy, S. Nolte, A. Szameit, and D. N. Christodoulides, “Supersymmetric mode converters,” Nat. Commun. 5, 3698 (2014).
    [Crossref]
  10. M.-A. Miri, M. Heinrich, and D. N. Christodoulides, “SUSY-inspired one-dimensional transformation optics,” Optica 1, 89–95 (2014).
    [Crossref]
  11. S. Longhi, “Supersymmetric transparent optical intersections,” Opt. Lett. 40, 463–466 (2015).
    [Crossref]
  12. S. Longhi, “Supersymmetric Bragg gratings,” J. Opt. 17, 045803 (2015).
    [Crossref]
  13. W. Walasik, B. Midya, L. Feng, and N. M. Litchinitser, “Supersymmetry-guided method for mode selection and optimization in coupled systems,” Opt. Lett. 43, 3758–3761 (2018).
    [Crossref]
  14. M. Heinrich, M.-A. Miri, S. Stützer, S. Nolte, D. N. Christodoulides, and A. Szameit, “Observation of supersymmetric scattering in photonic lattices,” Opt. Lett. 39, 6130–6133 (2014).
    [Crossref]
  15. R. El-Ganainy, L. Ge, M. Khajavikhan, and D. N. Christodoulides, “Supersymmetric laser arrays,” Phys. Rev. A 92, 033818 (2015).
    [Crossref]
  16. M. H. Teimourpour, L. Ge, D. N. Christodoulides, and R. El-Ganainy, “Non-Hermitian engineering of single mode two dimensional laser arrays,” Sci. Rep. 6, 33253 (2016).
    [Crossref]
  17. M. P. Hokmabadi, N. S. Nye, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Supersymmetric laser arrays,” Science 363, 623–626 (2019).
    [Crossref]
  18. B. Midya, H. Zhao, X. Qiao, P. Miao, W. Walasik, Z. Zhang, N. M. Litchinitser, and L. Feng, “Supersymmetric microring laser arrays,” Photon. Res. 7, 363–367 (2019).
    [Crossref]
  19. G. Strang, Introduction to Linear Algebra, 5th ed. (Wellesley-Cambridge, 2016).
  20. S. Yu, X. Piao, J. Hong, and N. Park, “Interdimensional optical isospectrality inspired by graph networks,” Optica 3, 836–839 (2016).
    [Crossref]
  21. M. H. Teimourpour, R. El-Ganainy, A. Eisfeld, A. Szameit, and D. N. Christodoulides, “Light transport in PT-invariant photonic structures with hidden symmetries,” Phys. Rev. A 90, 053817 (2014).
    [Crossref]
  22. M. H. Teimourpour, Q. Zhong, M. Khajavikhan, and R. El-Ganainy, Higher Order Exceptional Points in Discrete Photonics Platforms (Springer, 2018), pp. 261–275.
  23. W. C. Schieve and L. P. Horwitz, Quantum Statistical Mechanics (Cambridge University, 2009).
  24. L. Shah, A. Y. Arai, S. M. Eaton, and P. R. Herman, “Waveguide writing in fused silica with a femtosecond fiber laser at 522 nm and 1 MHz repetition rate,” Opt. Express 13, 1999–2006 (2005).
    [Crossref]
  25. A. Szameit, J. Burghoff, T. Pertsch, S. Nolte, A. Tünnermann, and F. Lederer, “Two-dimensional soliton in cubic fs laser written waveguide arrays in fused silica,” Opt. Express 14, 6055–6062 (2006).
    [Crossref]
  26. T. Meany, M. Gräfe, R. Heilmann, A. Perez-Leija, S. Gross, M. J. Steel, M. J. Withford, and A. Szameit, “Laser written circuits for quantum photonics,” Laser Photon. Rev. 9, 363–384 (2015).
    [Crossref]
  27. A. J. Kollár, M. Fitzpatrick, and A. A. Houck, “Hyperbolic lattices in circuit quantum electrodynamics,” Nature 571, 45–50 (2019).
    [Crossref]

2019 (3)

M. P. Hokmabadi, N. S. Nye, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Supersymmetric laser arrays,” Science 363, 623–626 (2019).
[Crossref]

A. J. Kollár, M. Fitzpatrick, and A. A. Houck, “Hyperbolic lattices in circuit quantum electrodynamics,” Nature 571, 45–50 (2019).
[Crossref]

B. Midya, H. Zhao, X. Qiao, P. Miao, W. Walasik, Z. Zhang, N. M. Litchinitser, and L. Feng, “Supersymmetric microring laser arrays,” Photon. Res. 7, 363–367 (2019).
[Crossref]

2018 (1)

2016 (2)

S. Yu, X. Piao, J. Hong, and N. Park, “Interdimensional optical isospectrality inspired by graph networks,” Optica 3, 836–839 (2016).
[Crossref]

M. H. Teimourpour, L. Ge, D. N. Christodoulides, and R. El-Ganainy, “Non-Hermitian engineering of single mode two dimensional laser arrays,” Sci. Rep. 6, 33253 (2016).
[Crossref]

2015 (4)

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

S. Longhi, “Supersymmetric Bragg gratings,” J. Opt. 17, 045803 (2015).
[Crossref]

R. El-Ganainy, L. Ge, M. Khajavikhan, and D. N. Christodoulides, “Supersymmetric laser arrays,” Phys. Rev. A 92, 033818 (2015).
[Crossref]

S. Longhi, “Supersymmetric transparent optical intersections,” Opt. Lett. 40, 463–466 (2015).
[Crossref]

2014 (4)

M.-A. Miri, M. Heinrich, and D. N. Christodoulides, “SUSY-inspired one-dimensional transformation optics,” Optica 1, 89–95 (2014).
[Crossref]

M. Heinrich, M.-A. Miri, S. Stützer, S. Nolte, D. N. Christodoulides, and A. Szameit, “Observation of supersymmetric scattering in photonic lattices,” Opt. Lett. 39, 6130–6133 (2014).
[Crossref]

M. Heinrich, M.-A. Miri, S. Stützer, R. El-Ganainy, S. Nolte, A. Szameit, and D. N. Christodoulides, “Supersymmetric mode converters,” Nat. Commun. 5, 3698 (2014).
[Crossref]

M. H. Teimourpour, R. El-Ganainy, A. Eisfeld, A. Szameit, and D. N. Christodoulides, “Light transport in PT-invariant photonic structures with hidden symmetries,” Phys. Rev. A 90, 053817 (2014).
[Crossref]

2013 (1)

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

2012 (1)

R. El-Ganainy, K. G. Makris, and D. N. Christodoulides, “Local PT invariance and supersymmetric parametric oscillators,” Phys. Rev. A 86, 033813 (2012).
[Crossref]

2006 (1)

2005 (1)

1995 (1)

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

1990 (1)

A. Lahiri, P. K. Roy, and B. Bagchi, “Supersymmetry in quantum mechanics,” Int. J. Mod. Phys. A 05, 1383–1456 (1990).
[Crossref]

1983 (1)

F. Cooper and B. Freedman, “Aspects of supersymmetric quantum mechanics,” Ann. Phys. 146, 262–288 (1983).
[Crossref]

1981 (1)

E. Witten, “Dynamical breaking of supersymmetry,” Nucl. Phys. B 188, 513–554 (1981).
[Crossref]

1971 (2)

P. Ramond, “Dual theory for free fermions,” Phys. Rev. D 3, 2415–2418 (1971).
[Crossref]

A. Neveu and J. H. Schwarz, “Factorizable dual model of pions,” Nucl. Phys. B 31, 86–112 (1971).
[Crossref]

Arai, A. Y.

Bagchi, B.

A. Lahiri, P. K. Roy, and B. Bagchi, “Supersymmetry in quantum mechanics,” Int. J. Mod. Phys. A 05, 1383–1456 (1990).
[Crossref]

Burghoff, J.

Christodoulides, D. N.

M. P. Hokmabadi, N. S. Nye, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Supersymmetric laser arrays,” Science 363, 623–626 (2019).
[Crossref]

M. H. Teimourpour, L. Ge, D. N. Christodoulides, and R. El-Ganainy, “Non-Hermitian engineering of single mode two dimensional laser arrays,” Sci. Rep. 6, 33253 (2016).
[Crossref]

R. El-Ganainy, L. Ge, M. Khajavikhan, and D. N. Christodoulides, “Supersymmetric laser arrays,” Phys. Rev. A 92, 033818 (2015).
[Crossref]

M. Heinrich, M.-A. Miri, S. Stützer, R. El-Ganainy, S. Nolte, A. Szameit, and D. N. Christodoulides, “Supersymmetric mode converters,” Nat. Commun. 5, 3698 (2014).
[Crossref]

M. H. Teimourpour, R. El-Ganainy, A. Eisfeld, A. Szameit, and D. N. Christodoulides, “Light transport in PT-invariant photonic structures with hidden symmetries,” Phys. Rev. A 90, 053817 (2014).
[Crossref]

M.-A. Miri, M. Heinrich, and D. N. Christodoulides, “SUSY-inspired one-dimensional transformation optics,” Optica 1, 89–95 (2014).
[Crossref]

M. Heinrich, M.-A. Miri, S. Stützer, S. Nolte, D. N. Christodoulides, and A. Szameit, “Observation of supersymmetric scattering in photonic lattices,” Opt. Lett. 39, 6130–6133 (2014).
[Crossref]

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, K. G. Makris, and D. N. Christodoulides, “Local PT invariance and supersymmetric parametric oscillators,” Phys. Rev. A 86, 033813 (2012).
[Crossref]

Cooper, F.

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

F. Cooper and B. Freedman, “Aspects of supersymmetric quantum mechanics,” Ann. Phys. 146, 262–288 (1983).
[Crossref]

Eaton, S. M.

Eisfeld, A.

M. H. Teimourpour, R. El-Ganainy, A. Eisfeld, A. Szameit, and D. N. Christodoulides, “Light transport in PT-invariant photonic structures with hidden symmetries,” Phys. Rev. A 90, 053817 (2014).
[Crossref]

El-Ganainy, R.

M. P. Hokmabadi, N. S. Nye, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Supersymmetric laser arrays,” Science 363, 623–626 (2019).
[Crossref]

M. H. Teimourpour, L. Ge, D. N. Christodoulides, and R. El-Ganainy, “Non-Hermitian engineering of single mode two dimensional laser arrays,” Sci. Rep. 6, 33253 (2016).
[Crossref]

R. El-Ganainy, L. Ge, M. Khajavikhan, and D. N. Christodoulides, “Supersymmetric laser arrays,” Phys. Rev. A 92, 033818 (2015).
[Crossref]

M. Heinrich, M.-A. Miri, S. Stützer, R. El-Ganainy, S. Nolte, A. Szameit, and D. N. Christodoulides, “Supersymmetric mode converters,” Nat. Commun. 5, 3698 (2014).
[Crossref]

M. H. Teimourpour, R. El-Ganainy, A. Eisfeld, A. Szameit, and D. N. Christodoulides, “Light transport in PT-invariant photonic structures with hidden symmetries,” Phys. Rev. A 90, 053817 (2014).
[Crossref]

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, K. G. Makris, and D. N. Christodoulides, “Local PT invariance and supersymmetric parametric oscillators,” Phys. Rev. A 86, 033813 (2012).
[Crossref]

M. H. Teimourpour, Q. Zhong, M. Khajavikhan, and R. El-Ganainy, Higher Order Exceptional Points in Discrete Photonics Platforms (Springer, 2018), pp. 261–275.

Feng, L.

Fitzpatrick, M.

A. J. Kollár, M. Fitzpatrick, and A. A. Houck, “Hyperbolic lattices in circuit quantum electrodynamics,” Nature 571, 45–50 (2019).
[Crossref]

Freedman, B.

F. Cooper and B. Freedman, “Aspects of supersymmetric quantum mechanics,” Ann. Phys. 146, 262–288 (1983).
[Crossref]

Ge, L.

M. H. Teimourpour, L. Ge, D. N. Christodoulides, and R. El-Ganainy, “Non-Hermitian engineering of single mode two dimensional laser arrays,” Sci. Rep. 6, 33253 (2016).
[Crossref]

R. El-Ganainy, L. Ge, M. Khajavikhan, and D. N. Christodoulides, “Supersymmetric laser arrays,” Phys. Rev. A 92, 033818 (2015).
[Crossref]

Gräfe, M.

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

Gross, S.

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

Heilmann, R.

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

Heinrich, M.

M. Heinrich, M.-A. Miri, S. Stützer, R. El-Ganainy, S. Nolte, A. Szameit, and D. N. Christodoulides, “Supersymmetric mode converters,” Nat. Commun. 5, 3698 (2014).
[Crossref]

M. Heinrich, M.-A. Miri, S. Stützer, S. Nolte, D. N. Christodoulides, and A. Szameit, “Observation of supersymmetric scattering in photonic lattices,” Opt. Lett. 39, 6130–6133 (2014).
[Crossref]

M.-A. Miri, M. Heinrich, and D. N. Christodoulides, “SUSY-inspired one-dimensional transformation optics,” Optica 1, 89–95 (2014).
[Crossref]

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

Herman, P. R.

Hokmabadi, M. P.

M. P. Hokmabadi, N. S. Nye, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Supersymmetric laser arrays,” Science 363, 623–626 (2019).
[Crossref]

Hong, J.

Horwitz, L. P.

W. C. Schieve and L. P. Horwitz, Quantum Statistical Mechanics (Cambridge University, 2009).

Houck, A. A.

A. J. Kollár, M. Fitzpatrick, and A. A. Houck, “Hyperbolic lattices in circuit quantum electrodynamics,” Nature 571, 45–50 (2019).
[Crossref]

Khajavikhan, M.

M. P. Hokmabadi, N. S. Nye, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Supersymmetric laser arrays,” Science 363, 623–626 (2019).
[Crossref]

R. El-Ganainy, L. Ge, M. Khajavikhan, and D. N. Christodoulides, “Supersymmetric laser arrays,” Phys. Rev. A 92, 033818 (2015).
[Crossref]

M. H. Teimourpour, Q. Zhong, M. Khajavikhan, and R. El-Ganainy, Higher Order Exceptional Points in Discrete Photonics Platforms (Springer, 2018), pp. 261–275.

Khare, A.

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

Kollár, A. J.

A. J. Kollár, M. Fitzpatrick, and A. A. Houck, “Hyperbolic lattices in circuit quantum electrodynamics,” Nature 571, 45–50 (2019).
[Crossref]

Lahiri, A.

A. Lahiri, P. K. Roy, and B. Bagchi, “Supersymmetry in quantum mechanics,” Int. J. Mod. Phys. A 05, 1383–1456 (1990).
[Crossref]

Lederer, F.

Litchinitser, N. M.

Longhi, S.

Makris, K. G.

R. El-Ganainy, K. G. Makris, and D. N. Christodoulides, “Local PT invariance and supersymmetric parametric oscillators,” Phys. Rev. A 86, 033813 (2012).
[Crossref]

Meany, T.

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

Miao, P.

Midya, B.

Miri, M.-A.

M.-A. Miri, M. Heinrich, and D. N. Christodoulides, “SUSY-inspired one-dimensional transformation optics,” Optica 1, 89–95 (2014).
[Crossref]

M. Heinrich, M.-A. Miri, S. Stützer, S. Nolte, D. N. Christodoulides, and A. Szameit, “Observation of supersymmetric scattering in photonic lattices,” Opt. Lett. 39, 6130–6133 (2014).
[Crossref]

M. Heinrich, M.-A. Miri, S. Stützer, R. El-Ganainy, S. Nolte, A. Szameit, and D. N. Christodoulides, “Supersymmetric mode converters,” Nat. Commun. 5, 3698 (2014).
[Crossref]

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

Neveu, A.

A. Neveu and J. H. Schwarz, “Factorizable dual model of pions,” Nucl. Phys. B 31, 86–112 (1971).
[Crossref]

Nolte, S.

Nye, N. S.

M. P. Hokmabadi, N. S. Nye, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Supersymmetric laser arrays,” Science 363, 623–626 (2019).
[Crossref]

Park, N.

Perez-Leija, A.

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

Pertsch, T.

Piao, X.

Qiao, X.

Ramond, P.

P. Ramond, “Dual theory for free fermions,” Phys. Rev. D 3, 2415–2418 (1971).
[Crossref]

Roy, P. K.

A. Lahiri, P. K. Roy, and B. Bagchi, “Supersymmetry in quantum mechanics,” Int. J. Mod. Phys. A 05, 1383–1456 (1990).
[Crossref]

Schieve, W. C.

W. C. Schieve and L. P. Horwitz, Quantum Statistical Mechanics (Cambridge University, 2009).

Schwarz, J. H.

A. Neveu and J. H. Schwarz, “Factorizable dual model of pions,” Nucl. Phys. B 31, 86–112 (1971).
[Crossref]

Shah, L.

Steel, M. J.

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

Strang, G.

G. Strang, Introduction to Linear Algebra, 5th ed. (Wellesley-Cambridge, 2016).

Stützer, S.

M. Heinrich, M.-A. Miri, S. Stützer, R. El-Ganainy, S. Nolte, A. Szameit, and D. N. Christodoulides, “Supersymmetric mode converters,” Nat. Commun. 5, 3698 (2014).
[Crossref]

M. Heinrich, M.-A. Miri, S. Stützer, S. Nolte, D. N. Christodoulides, and A. Szameit, “Observation of supersymmetric scattering in photonic lattices,” Opt. Lett. 39, 6130–6133 (2014).
[Crossref]

Sukhatme, U.

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

Szameit, A.

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

M. H. Teimourpour, R. El-Ganainy, A. Eisfeld, A. Szameit, and D. N. Christodoulides, “Light transport in PT-invariant photonic structures with hidden symmetries,” Phys. Rev. A 90, 053817 (2014).
[Crossref]

M. Heinrich, M.-A. Miri, S. Stützer, R. El-Ganainy, S. Nolte, A. Szameit, and D. N. Christodoulides, “Supersymmetric mode converters,” Nat. Commun. 5, 3698 (2014).
[Crossref]

M. Heinrich, M.-A. Miri, S. Stützer, S. Nolte, D. N. Christodoulides, and A. Szameit, “Observation of supersymmetric scattering in photonic lattices,” Opt. Lett. 39, 6130–6133 (2014).
[Crossref]

A. Szameit, J. Burghoff, T. Pertsch, S. Nolte, A. Tünnermann, and F. Lederer, “Two-dimensional soliton in cubic fs laser written waveguide arrays in fused silica,” Opt. Express 14, 6055–6062 (2006).
[Crossref]

Teimourpour, M. H.

M. H. Teimourpour, L. Ge, D. N. Christodoulides, and R. El-Ganainy, “Non-Hermitian engineering of single mode two dimensional laser arrays,” Sci. Rep. 6, 33253 (2016).
[Crossref]

M. H. Teimourpour, R. El-Ganainy, A. Eisfeld, A. Szameit, and D. N. Christodoulides, “Light transport in PT-invariant photonic structures with hidden symmetries,” Phys. Rev. A 90, 053817 (2014).
[Crossref]

M. H. Teimourpour, Q. Zhong, M. Khajavikhan, and R. El-Ganainy, Higher Order Exceptional Points in Discrete Photonics Platforms (Springer, 2018), pp. 261–275.

Tünnermann, A.

Walasik, W.

Withford, M. J.

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

Witten, E.

E. Witten, “Dynamical breaking of supersymmetry,” Nucl. Phys. B 188, 513–554 (1981).
[Crossref]

Yu, S.

Zhang, Z.

Zhao, H.

Zhong, Q.

M. H. Teimourpour, Q. Zhong, M. Khajavikhan, and R. El-Ganainy, Higher Order Exceptional Points in Discrete Photonics Platforms (Springer, 2018), pp. 261–275.

Ann. Phys. (1)

F. Cooper and B. Freedman, “Aspects of supersymmetric quantum mechanics,” Ann. Phys. 146, 262–288 (1983).
[Crossref]

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

A. Lahiri, P. K. Roy, and B. Bagchi, “Supersymmetry in quantum mechanics,” Int. J. Mod. Phys. A 05, 1383–1456 (1990).
[Crossref]

J. Opt. (1)

S. Longhi, “Supersymmetric Bragg gratings,” J. Opt. 17, 045803 (2015).
[Crossref]

Laser Photon. Rev. (1)

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

Nat. Commun. (1)

M. Heinrich, M.-A. Miri, S. Stützer, R. El-Ganainy, S. Nolte, A. Szameit, and D. N. Christodoulides, “Supersymmetric mode converters,” Nat. Commun. 5, 3698 (2014).
[Crossref]

Nature (1)

A. J. Kollár, M. Fitzpatrick, and A. A. Houck, “Hyperbolic lattices in circuit quantum electrodynamics,” Nature 571, 45–50 (2019).
[Crossref]

Nucl. Phys. B (2)

A. Neveu and J. H. Schwarz, “Factorizable dual model of pions,” Nucl. Phys. B 31, 86–112 (1971).
[Crossref]

E. Witten, “Dynamical breaking of supersymmetry,” Nucl. Phys. B 188, 513–554 (1981).
[Crossref]

Opt. Express (2)

Opt. Lett. (3)

Optica (2)

Photon. Res. (1)

Phys. Rep. (1)

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

Phys. Rev. A (3)

R. El-Ganainy, K. G. Makris, and D. N. Christodoulides, “Local PT invariance and supersymmetric parametric oscillators,” Phys. Rev. A 86, 033813 (2012).
[Crossref]

R. El-Ganainy, L. Ge, M. Khajavikhan, and D. N. Christodoulides, “Supersymmetric laser arrays,” Phys. Rev. A 92, 033818 (2015).
[Crossref]

M. H. Teimourpour, R. El-Ganainy, A. Eisfeld, A. Szameit, and D. N. Christodoulides, “Light transport in PT-invariant photonic structures with hidden symmetries,” Phys. Rev. A 90, 053817 (2014).
[Crossref]

Phys. Rev. D (1)

P. Ramond, “Dual theory for free fermions,” Phys. Rev. D 3, 2415–2418 (1971).
[Crossref]

Phys. Rev. Lett. (1)

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

Sci. Rep. (1)

M. H. Teimourpour, L. Ge, D. N. Christodoulides, and R. El-Ganainy, “Non-Hermitian engineering of single mode two dimensional laser arrays,” Sci. Rep. 6, 33253 (2016).
[Crossref]

Science (1)

M. P. Hokmabadi, N. S. Nye, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Supersymmetric laser arrays,” Science 363, 623–626 (2019).
[Crossref]

Other (3)

G. Strang, Introduction to Linear Algebra, 5th ed. (Wellesley-Cambridge, 2016).

M. H. Teimourpour, Q. Zhong, M. Khajavikhan, and R. El-Ganainy, Higher Order Exceptional Points in Discrete Photonics Platforms (Springer, 2018), pp. 261–275.

W. C. Schieve and L. P. Horwitz, Quantum Statistical Mechanics (Cambridge University, 2009).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1.
Fig. 1. Summary of our proposed approach. A discrete SUSY transformation is applied to a set of N coupled quantum oscillators (for demonstration, we take N=3). The resultant partner network made of N1 elements exhibits a subset of the spectrum of the original system. By populating both quantum networks with multiple bosons (2 bosons in the example shown here), we can construct classical arrays that exhibit partial spectral overlap.
Fig. 2.
Fig. 2. Optical implementation of the SUSY arrays example shown in Fig. 1 using a waveguide platform. The left panel shows the original array while right panel shows the BD-SUSY partner obtained as described in the text. The waveguides are all identical, having an elliptic geometry with main/minor diameters of 12 and 6 μm, respectively. The core and cladding refractive indices are taken to be ncore=1.461 and nclad=1.46, respectively [24,25]. Each waveguide supports only one optical mode for each polarization direction. Finally, the distances shown in panel (a) are: d1=23.765  μm, d2=18.275  μm, and d3=16.190  μm. These design parameters result in the following coupling coefficients: κ12=14.143  m1, κ23=13.141  m1, and κ24=10.000  m1. The second order nearest next neighbor coupling is found to be below 10% of the above values. Similarly, in panel (b) we have: d4=22.425  μm and κ12=20.012  m1. Note that we list the above values with high precision as per our numerical simulations; however, in practice the weakly guiding nature of the structure provides reasonable robustness against fabrication tolerance.
Fig. 3.
Fig. 3. Eigenmode structure of the waveguide arrays shown in Figs. 2(a) and 2(b) are depicted in the left and right panels, respectively (obtained by full-wave finite element simulations). The figures also indicate the values of the associated propagation constants as measured from the isolated waveguide value, i.e., Δβ=βmβo (in units of m1), where βm,o are the propagation constants of array mode m and the isolated waveguide mode, correspondingly. As anticipated from the coupled mode analysis, modes ② and ⑤ in the main array have no partner modes in the BD-SUSY partner array. Moreover, mode ② of the partner array corresponds to two degenerate states in the main array. These results confirm the feasibility of our approach for building quasi-2D supersymmetric optical systems.
Fig. 4.
Fig. 4. Light propagation dynamics in a waveguide array formed by introducing a weak coupling between the main structure and its partner, as shown in (a)–(e). When mode ① of the main array is excited, we observe an efficient optical power transfer to the partner array after a propagation distance corresponding to z=4π (in units of meter). On the other hand, if mode ② of the main array is excited, no appreciable power transfer to the partner array is observed (not shown here). The power transfer efficiency between the modes is illustrated in (f) where near perfect transfer is observed. The blue and red lines are the total power in the main and pattern arrays, respectively.

Equations (8)

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

H^=βon=13a^na^n+κn=12a^na^n+1+H.c.,
ida^dz=Ha^,H=[βoκ0κβoκ0κβo],
QR=HβSI,HS=RQ+βSI,
H^S=n=13Hn,nSa^na^n+n=12Hn,n+1Sa^na^n+1+H.c.
H^=n=13λnA^nA^n,H^S=n=13λnB^nB^n,
HS=[βo2κ02κβo000βo].
H^RS=βo(a^1a^1+a^2a^2)+2κa^1a^2+H.c.
H^RS=n=12μnB^nB^n,

Metrics