Abstract

We present a class of waveguide arrays that is the classical analog of a quantum harmonic oscillator, where the mass and frequency depend on the propagation distance. In these photonic lattices, refractive indices and second-neighbor couplings define the mass and frequency of the analog quantum oscillator, while first-neighbor couplings are a free parameter to adjust the model. The quantum model conserves the Ermakov–Lewis invariant, thus the photonic crystal also possesses this symmetry.

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  1. C. P. Boyer, Helv. Phys. Acta 47, 589 (1974).
  2. C. P. Boyer, E. G. Kalnins, and W. Miller, Nagoya Math. J. 60, 35 (1976).
  3. A. Leviathan, Prog. Part. Nucl. Phys. 66, 93 (2011).
    [CrossRef]
  4. E. Noether, Nachr. v. d. Ges. d. Wiss. zu Göttingen 1918, 235 (1918).
  5. D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
    [CrossRef]
  6. V. P. Ermakov, Izv. Kievskoyo Univ. 9, 1 (1880).
  7. H. R. Lewis, Phys. Rev. Lett. 18, 510 (1967).
    [CrossRef]
  8. H. R. Lewis, J. Math. Phys. 9, 1976 (1968).
    [CrossRef]
  9. H. R. Lewis and W. B. Riesenfeld, J. Math. Phys. 10, 1458 (1969).
    [CrossRef]
  10. C. J. Eliezer and A. Gray, SIAM J. Appl. Math. 30, 463 (1976).
    [CrossRef]
  11. M. Lutzky, Phys. Lett. A 68, 3 (1978).
    [CrossRef]
  12. I. A. Pedrosa, Phys. Rev. A 55, 3219 (1997).
    [CrossRef]
  13. M. Fernández-Guasti and H. Moya-Cessa, J. Phys. A Math. Theor. 36, 2069 (2003).
    [CrossRef]
  14. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions. (Dover, 1970).
  15. I. A. Pedrosa, J. Math. Phys. 28, 2662 (1987).
    [CrossRef]
  16. P. Caldirola, Nuovo Cimento 18, 393 (1941).
    [CrossRef]
  17. E. Kanai, Prog. Theor. Phys. 3, 440 (1948).
    [CrossRef]
  18. R. Keil, A. Perez-Leija, P. Aleahmad, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, Opt. Lett. 37, 3801 (2012).
    [CrossRef]
  19. H. Moya-Cessa and M. Fernández-Guasti, Phys. Lett. A 311, 1 (2003).
    [CrossRef]
  20. M. Fernández-Guasti, Physica D 189, 188 (2004).
    [CrossRef]

2012 (1)

2011 (1)

A. Leviathan, Prog. Part. Nucl. Phys. 66, 93 (2011).
[CrossRef]

2004 (1)

M. Fernández-Guasti, Physica D 189, 188 (2004).
[CrossRef]

2003 (3)

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[CrossRef]

M. Fernández-Guasti and H. Moya-Cessa, J. Phys. A Math. Theor. 36, 2069 (2003).
[CrossRef]

H. Moya-Cessa and M. Fernández-Guasti, Phys. Lett. A 311, 1 (2003).
[CrossRef]

1997 (1)

I. A. Pedrosa, Phys. Rev. A 55, 3219 (1997).
[CrossRef]

1987 (1)

I. A. Pedrosa, J. Math. Phys. 28, 2662 (1987).
[CrossRef]

1978 (1)

M. Lutzky, Phys. Lett. A 68, 3 (1978).
[CrossRef]

1976 (2)

C. J. Eliezer and A. Gray, SIAM J. Appl. Math. 30, 463 (1976).
[CrossRef]

C. P. Boyer, E. G. Kalnins, and W. Miller, Nagoya Math. J. 60, 35 (1976).

1974 (1)

C. P. Boyer, Helv. Phys. Acta 47, 589 (1974).

1969 (1)

H. R. Lewis and W. B. Riesenfeld, J. Math. Phys. 10, 1458 (1969).
[CrossRef]

1968 (1)

H. R. Lewis, J. Math. Phys. 9, 1976 (1968).
[CrossRef]

1967 (1)

H. R. Lewis, Phys. Rev. Lett. 18, 510 (1967).
[CrossRef]

1948 (1)

E. Kanai, Prog. Theor. Phys. 3, 440 (1948).
[CrossRef]

1941 (1)

P. Caldirola, Nuovo Cimento 18, 393 (1941).
[CrossRef]

1918 (1)

E. Noether, Nachr. v. d. Ges. d. Wiss. zu Göttingen 1918, 235 (1918).

1880 (1)

V. P. Ermakov, Izv. Kievskoyo Univ. 9, 1 (1880).

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions. (Dover, 1970).

Aleahmad, P.

Boyer, C. P.

C. P. Boyer, E. G. Kalnins, and W. Miller, Nagoya Math. J. 60, 35 (1976).

C. P. Boyer, Helv. Phys. Acta 47, 589 (1974).

Caldirola, P.

P. Caldirola, Nuovo Cimento 18, 393 (1941).
[CrossRef]

Christodoulides, D. N.

Eliezer, C. J.

C. J. Eliezer and A. Gray, SIAM J. Appl. Math. 30, 463 (1976).
[CrossRef]

Ermakov, V. P.

V. P. Ermakov, Izv. Kievskoyo Univ. 9, 1 (1880).

Fernández-Guasti, M.

M. Fernández-Guasti, Physica D 189, 188 (2004).
[CrossRef]

M. Fernández-Guasti and H. Moya-Cessa, J. Phys. A Math. Theor. 36, 2069 (2003).
[CrossRef]

H. Moya-Cessa and M. Fernández-Guasti, Phys. Lett. A 311, 1 (2003).
[CrossRef]

Gray, A.

C. J. Eliezer and A. Gray, SIAM J. Appl. Math. 30, 463 (1976).
[CrossRef]

Kalnins, E. G.

C. P. Boyer, E. G. Kalnins, and W. Miller, Nagoya Math. J. 60, 35 (1976).

Kanai, E.

E. Kanai, Prog. Theor. Phys. 3, 440 (1948).
[CrossRef]

Keil, R.

Lederer, F.

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[CrossRef]

Leviathan, A.

A. Leviathan, Prog. Part. Nucl. Phys. 66, 93 (2011).
[CrossRef]

Lewis, H. R.

H. R. Lewis and W. B. Riesenfeld, J. Math. Phys. 10, 1458 (1969).
[CrossRef]

H. R. Lewis, J. Math. Phys. 9, 1976 (1968).
[CrossRef]

H. R. Lewis, Phys. Rev. Lett. 18, 510 (1967).
[CrossRef]

Lutzky, M.

M. Lutzky, Phys. Lett. A 68, 3 (1978).
[CrossRef]

Miller, W.

C. P. Boyer, E. G. Kalnins, and W. Miller, Nagoya Math. J. 60, 35 (1976).

Moya-Cessa, H.

R. Keil, A. Perez-Leija, P. Aleahmad, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, Opt. Lett. 37, 3801 (2012).
[CrossRef]

H. Moya-Cessa and M. Fernández-Guasti, Phys. Lett. A 311, 1 (2003).
[CrossRef]

M. Fernández-Guasti and H. Moya-Cessa, J. Phys. A Math. Theor. 36, 2069 (2003).
[CrossRef]

Noether, E.

E. Noether, Nachr. v. d. Ges. d. Wiss. zu Göttingen 1918, 235 (1918).

Nolte, S.

Pedrosa, I. A.

I. A. Pedrosa, Phys. Rev. A 55, 3219 (1997).
[CrossRef]

I. A. Pedrosa, J. Math. Phys. 28, 2662 (1987).
[CrossRef]

Perez-Leija, A.

Riesenfeld, W. B.

H. R. Lewis and W. B. Riesenfeld, J. Math. Phys. 10, 1458 (1969).
[CrossRef]

Silberberg, Y.

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[CrossRef]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions. (Dover, 1970).

Szameit, A.

Helv. Phys. Acta (1)

C. P. Boyer, Helv. Phys. Acta 47, 589 (1974).

Izv. Kievskoyo Univ. (1)

V. P. Ermakov, Izv. Kievskoyo Univ. 9, 1 (1880).

J. Math. Phys. (3)

H. R. Lewis, J. Math. Phys. 9, 1976 (1968).
[CrossRef]

H. R. Lewis and W. B. Riesenfeld, J. Math. Phys. 10, 1458 (1969).
[CrossRef]

I. A. Pedrosa, J. Math. Phys. 28, 2662 (1987).
[CrossRef]

J. Phys. A Math. Theor. (1)

M. Fernández-Guasti and H. Moya-Cessa, J. Phys. A Math. Theor. 36, 2069 (2003).
[CrossRef]

Nachr. v. d. Ges. d. Wiss. zu Göttingen (1)

E. Noether, Nachr. v. d. Ges. d. Wiss. zu Göttingen 1918, 235 (1918).

Nagoya Math. J. (1)

C. P. Boyer, E. G. Kalnins, and W. Miller, Nagoya Math. J. 60, 35 (1976).

Nature (1)

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[CrossRef]

Nuovo Cimento (1)

P. Caldirola, Nuovo Cimento 18, 393 (1941).
[CrossRef]

Opt. Lett. (1)

Phys. Lett. A (2)

H. Moya-Cessa and M. Fernández-Guasti, Phys. Lett. A 311, 1 (2003).
[CrossRef]

M. Lutzky, Phys. Lett. A 68, 3 (1978).
[CrossRef]

Phys. Rev. A (1)

I. A. Pedrosa, Phys. Rev. A 55, 3219 (1997).
[CrossRef]

Phys. Rev. Lett. (1)

H. R. Lewis, Phys. Rev. Lett. 18, 510 (1967).
[CrossRef]

Physica D (1)

M. Fernández-Guasti, Physica D 189, 188 (2004).
[CrossRef]

Prog. Part. Nucl. Phys. (1)

A. Leviathan, Prog. Part. Nucl. Phys. 66, 93 (2011).
[CrossRef]

Prog. Theor. Phys. (1)

E. Kanai, Prog. Theor. Phys. 3, 440 (1948).
[CrossRef]

SIAM J. Appl. Math. (1)

C. J. Eliezer and A. Gray, SIAM J. Appl. Math. 30, 463 (1976).
[CrossRef]

Other (1)

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions. (Dover, 1970).

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Figures (3)

Fig. 1.
Fig. 1.

Light intensity propagation for a beam impinging on the j=10 waveguide of a lattice with parameters, α0(z)=α1(z)=1, and α2(z)=0; this is equivalent to an oscillator with constant mass and frequency, M(z)=Ω(z)=1.

Fig. 2.
Fig. 2.

Light intensity propagation for a beam impinging on the j=10 waveguide of a lattice with parameters α0(z)=5/2, α1(z)=1, and α2(z)=3/4; this is equivalent to an oscillator with constant mass and frequency, M(z)=1 and Ω(z)=2.

Fig. 3.
Fig. 3.

(a) Profile of the driving frequency, Ω(z). (b) Light intensity propagation for a beam impinging on the j=10 waveguide of a lattice with parameters α0(z)=[M2(z)Ω(z)2+1]/[2M(z)], α1(z)=1, and α2(z)=[M2(z)Ω(z)21]/[4M(z)]; with harmonic oscillator parameters M(z)=1 and Ω(z)=[3+tanh20(z12.5)]/2.

Equations (14)

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

izEn(z)+α0(z)nEn(z)+α1(z)[fn+1En+1(z)+fnEn1(z)]+α2(z)[gn+2En+2(z)+gnEn2(z)]=0,
iz|ψ(z)=H^|ψ(z),
H^=[α0(z)n^+α1(z)(a^+a^)+α2(z)(a^2+a^2)].
H^=[12M(z)p^2+12M(z)Ω2(z)q^2+2α1(z)q^α0(z)2],
M(z)=1α0(z)2α2(z),
Ω2(z)=α02(z)4α22(z).
|ψ(z)=eiφ(z)dzei[u(z)p^+M(z)u˙(z)q^]|ξ(z),
it|ξ(t)=[12m(t)p^2+12m(t)ω2(t)q^2]|ξ(t),
I^=12{[q^ρ(t)]2+[ρ(t)p^m(t)ρ˙(t)q^]2},
ρ¨(t)+m˙(t)m(t)ρ˙(t)+ω2(t)ρ(t)=1m2(t)ρ3(t).
|ξ(t)=eim(t)ρ˙(t)2ρ(t)q^2eilnρ(t)2(p^q^+q^p^)|ζ(t),
it|ζ(t)=12m(t)ρ2(t)(p^2+q^2)|ζ(t),
=1m(t)ρ2(t)(a^a^+12)|ζ(t).
|ζ(t)=ei1m(t)ρ2(t)dt(a^a^+12)|ζ(0),

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