Abstract

A photonic realization of the Dirac oscillator (DO), i.e., of the relativistic extension of the quantum harmonic oscillator, is proposed for light propagation in fiber Bragg gratings. Transmission spectra clearly show the existence of electron and positron bound states of the DO, corresponding to resonance modes above and below the Bragg frequency, as well as the asymmetry of the energy spectrum for electron and positron branches.

© 2010 Optical Society of America

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  1. M. Moshinsky and A. Szczepaniak, J. Phys. A  22, L817 (1989).
    [CrossRef]
  2. M. Moshinsky and Y. F. Smirnov, The Harmonic Oscillator in Modern Physics (Harwood, 1996).
  3. R. P. Martinez-y-Romero, H. N. Nunez-Yepez, and A. L. Salas-Brito, Eur. J. Phys.  16, 135 (1995).
    [CrossRef]
  4. D. Ito, K. Mori, and E. Carrieri, Nuovo Cimento  51A, 1119 (1967).
  5. J. Bentez, R. P. Martnez-y-Romero, H. N. Nuez-Yepez, and A. L. Salas-Brito, Phys. Rev. Lett.  64, 1643 (1990).
    [CrossRef] [PubMed]
  6. J. Bentez, R. P. Martnez-y-Romero, H. N. Nuez-Yepez, and A. L. Salas-Brito, Phys. Rev. Lett.  65, 2085 (1990).
    [CrossRef]
  7. R. P. Martriaanez-y-Romero, M. Moreno, and A. Zentella, Phys. Rev. D  43, 2036 (1991).
    [CrossRef]
  8. F. Dominguez-Adame and M. A. Gonzalez, Europhys. Lett.  13, 193 (1990).
    [CrossRef]
  9. W. Greiner, Relativistic Quantum Mechanics (Springer-Verlag, 1990).
  10. A. F. Young and P. Kim, Nat. Phys.  5, 222 (2009).
    [CrossRef]
  11. N. Stander, B. Huard, and D. Goldhaber-Gordon, Phys. Rev. Lett.  102, 026807 (2009).
    [CrossRef] [PubMed]
  12. R. Gerritsma, G. Kirchmair, F. Zähringer, E. Solano, R. Blatt, and C. F. Roos, Nature  463, 68 (2010).
    [CrossRef] [PubMed]
  13. R. A. Sepkhanov, Ya. B. Bazaliy, and C. W. J. Beenakker, Phys. Rev. A  75, 063813 (2007).
    [CrossRef]
  14. O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett.  98, 103901 (2007).
    [CrossRef] [PubMed]
  15. X. Zhang, Phys. Rev. Lett.  100, 113903 (2008).
    [CrossRef] [PubMed]
  16. L.-G. Wang, Z.-G. Wang, and S.-Y. Zhu, EPL  86, 47008 (2009).
    [CrossRef]
  17. P. Rozmej and R. Arvieu, J. Phys. A  32, 5367 (1999).
    [CrossRef]
  18. A. Bermudez, M. A. Martin-Delgado, and E. Solano, Phys. Rev. A  76, 041801(R) (2007).
    [CrossRef]
  19. A. Bermudez, M. A. Martin-Delgado, and A. Luis, Phys. Rev. A  77, 033832 (2008).
    [CrossRef]
  20. T. Erdogan, J. Lightwave Technol.  15, 1277 (1997).
    [CrossRef]
  21. L. Poladian, Phys. Rev. E  48, 4758 (1993).
    [CrossRef]
  22. A. Asseh, H. Storoy, B. Sahlgren, and R. Stubbe, J. Lightwave Technol.  15, 1419 (1997).
    [CrossRef]
  23. Y. Liu, L. Wei, and J. W. Y. Lit, Appl. Opt.  46, 6770 (2007).
    [CrossRef] [PubMed]

2010 (1)

R. Gerritsma, G. Kirchmair, F. Zähringer, E. Solano, R. Blatt, and C. F. Roos, Nature  463, 68 (2010).
[CrossRef] [PubMed]

2009 (3)

L.-G. Wang, Z.-G. Wang, and S.-Y. Zhu, EPL  86, 47008 (2009).
[CrossRef]

A. F. Young and P. Kim, Nat. Phys.  5, 222 (2009).
[CrossRef]

N. Stander, B. Huard, and D. Goldhaber-Gordon, Phys. Rev. Lett.  102, 026807 (2009).
[CrossRef] [PubMed]

2008 (2)

A. Bermudez, M. A. Martin-Delgado, and A. Luis, Phys. Rev. A  77, 033832 (2008).
[CrossRef]

X. Zhang, Phys. Rev. Lett.  100, 113903 (2008).
[CrossRef] [PubMed]

2007 (4)

R. A. Sepkhanov, Ya. B. Bazaliy, and C. W. J. Beenakker, Phys. Rev. A  75, 063813 (2007).
[CrossRef]

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett.  98, 103901 (2007).
[CrossRef] [PubMed]

A. Bermudez, M. A. Martin-Delgado, and E. Solano, Phys. Rev. A  76, 041801(R) (2007).
[CrossRef]

Y. Liu, L. Wei, and J. W. Y. Lit, Appl. Opt.  46, 6770 (2007).
[CrossRef] [PubMed]

1999 (1)

P. Rozmej and R. Arvieu, J. Phys. A  32, 5367 (1999).
[CrossRef]

1997 (2)

T. Erdogan, J. Lightwave Technol.  15, 1277 (1997).
[CrossRef]

A. Asseh, H. Storoy, B. Sahlgren, and R. Stubbe, J. Lightwave Technol.  15, 1419 (1997).
[CrossRef]

1995 (1)

R. P. Martinez-y-Romero, H. N. Nunez-Yepez, and A. L. Salas-Brito, Eur. J. Phys.  16, 135 (1995).
[CrossRef]

1993 (1)

L. Poladian, Phys. Rev. E  48, 4758 (1993).
[CrossRef]

1991 (1)

R. P. Martriaanez-y-Romero, M. Moreno, and A. Zentella, Phys. Rev. D  43, 2036 (1991).
[CrossRef]

1990 (3)

F. Dominguez-Adame and M. A. Gonzalez, Europhys. Lett.  13, 193 (1990).
[CrossRef]

J. Bentez, R. P. Martnez-y-Romero, H. N. Nuez-Yepez, and A. L. Salas-Brito, Phys. Rev. Lett.  64, 1643 (1990).
[CrossRef] [PubMed]

J. Bentez, R. P. Martnez-y-Romero, H. N. Nuez-Yepez, and A. L. Salas-Brito, Phys. Rev. Lett.  65, 2085 (1990).
[CrossRef]

1989 (1)

M. Moshinsky and A. Szczepaniak, J. Phys. A  22, L817 (1989).
[CrossRef]

1967 (1)

D. Ito, K. Mori, and E. Carrieri, Nuovo Cimento  51A, 1119 (1967).

Arvieu, R.

P. Rozmej and R. Arvieu, J. Phys. A  32, 5367 (1999).
[CrossRef]

Asseh, A.

A. Asseh, H. Storoy, B. Sahlgren, and R. Stubbe, J. Lightwave Technol.  15, 1419 (1997).
[CrossRef]

Bartal, G.

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett.  98, 103901 (2007).
[CrossRef] [PubMed]

Bazaliy, Ya. B.

R. A. Sepkhanov, Ya. B. Bazaliy, and C. W. J. Beenakker, Phys. Rev. A  75, 063813 (2007).
[CrossRef]

Beenakker, C. W. J.

R. A. Sepkhanov, Ya. B. Bazaliy, and C. W. J. Beenakker, Phys. Rev. A  75, 063813 (2007).
[CrossRef]

Bentez, J.

J. Bentez, R. P. Martnez-y-Romero, H. N. Nuez-Yepez, and A. L. Salas-Brito, Phys. Rev. Lett.  64, 1643 (1990).
[CrossRef] [PubMed]

J. Bentez, R. P. Martnez-y-Romero, H. N. Nuez-Yepez, and A. L. Salas-Brito, Phys. Rev. Lett.  65, 2085 (1990).
[CrossRef]

Bermudez, A.

A. Bermudez, M. A. Martin-Delgado, and A. Luis, Phys. Rev. A  77, 033832 (2008).
[CrossRef]

A. Bermudez, M. A. Martin-Delgado, and E. Solano, Phys. Rev. A  76, 041801(R) (2007).
[CrossRef]

Blatt, R.

R. Gerritsma, G. Kirchmair, F. Zähringer, E. Solano, R. Blatt, and C. F. Roos, Nature  463, 68 (2010).
[CrossRef] [PubMed]

Carrieri, E.

D. Ito, K. Mori, and E. Carrieri, Nuovo Cimento  51A, 1119 (1967).

Christodoulides, D. N.

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett.  98, 103901 (2007).
[CrossRef] [PubMed]

Dominguez-Adame, F.

F. Dominguez-Adame and M. A. Gonzalez, Europhys. Lett.  13, 193 (1990).
[CrossRef]

Erdogan, T.

T. Erdogan, J. Lightwave Technol.  15, 1277 (1997).
[CrossRef]

Freedman, B.

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett.  98, 103901 (2007).
[CrossRef] [PubMed]

Gerritsma, R.

R. Gerritsma, G. Kirchmair, F. Zähringer, E. Solano, R. Blatt, and C. F. Roos, Nature  463, 68 (2010).
[CrossRef] [PubMed]

Goldhaber-Gordon, D.

N. Stander, B. Huard, and D. Goldhaber-Gordon, Phys. Rev. Lett.  102, 026807 (2009).
[CrossRef] [PubMed]

Gonzalez, M. A.

F. Dominguez-Adame and M. A. Gonzalez, Europhys. Lett.  13, 193 (1990).
[CrossRef]

Greiner, W.

W. Greiner, Relativistic Quantum Mechanics (Springer-Verlag, 1990).

Huard, B.

N. Stander, B. Huard, and D. Goldhaber-Gordon, Phys. Rev. Lett.  102, 026807 (2009).
[CrossRef] [PubMed]

Ito, D.

D. Ito, K. Mori, and E. Carrieri, Nuovo Cimento  51A, 1119 (1967).

Kim, P.

A. F. Young and P. Kim, Nat. Phys.  5, 222 (2009).
[CrossRef]

Kirchmair, G.

R. Gerritsma, G. Kirchmair, F. Zähringer, E. Solano, R. Blatt, and C. F. Roos, Nature  463, 68 (2010).
[CrossRef] [PubMed]

Lit, J. W. Y.

Liu, Y.

Luis, A.

A. Bermudez, M. A. Martin-Delgado, and A. Luis, Phys. Rev. A  77, 033832 (2008).
[CrossRef]

Manela, O.

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett.  98, 103901 (2007).
[CrossRef] [PubMed]

Martin-Delgado, M. A.

A. Bermudez, M. A. Martin-Delgado, and A. Luis, Phys. Rev. A  77, 033832 (2008).
[CrossRef]

A. Bermudez, M. A. Martin-Delgado, and E. Solano, Phys. Rev. A  76, 041801(R) (2007).
[CrossRef]

Martinez-y-Romero, R. P.

R. P. Martinez-y-Romero, H. N. Nunez-Yepez, and A. L. Salas-Brito, Eur. J. Phys.  16, 135 (1995).
[CrossRef]

Martnez-y-Romero, R. P.

J. Bentez, R. P. Martnez-y-Romero, H. N. Nuez-Yepez, and A. L. Salas-Brito, Phys. Rev. Lett.  65, 2085 (1990).
[CrossRef]

J. Bentez, R. P. Martnez-y-Romero, H. N. Nuez-Yepez, and A. L. Salas-Brito, Phys. Rev. Lett.  64, 1643 (1990).
[CrossRef] [PubMed]

Martriaanez-y-Romero, R. P.

R. P. Martriaanez-y-Romero, M. Moreno, and A. Zentella, Phys. Rev. D  43, 2036 (1991).
[CrossRef]

Moreno, M.

R. P. Martriaanez-y-Romero, M. Moreno, and A. Zentella, Phys. Rev. D  43, 2036 (1991).
[CrossRef]

Mori, K.

D. Ito, K. Mori, and E. Carrieri, Nuovo Cimento  51A, 1119 (1967).

Moshinsky, M.

M. Moshinsky and A. Szczepaniak, J. Phys. A  22, L817 (1989).
[CrossRef]

M. Moshinsky and Y. F. Smirnov, The Harmonic Oscillator in Modern Physics (Harwood, 1996).

Nuez-Yepez, H. N.

J. Bentez, R. P. Martnez-y-Romero, H. N. Nuez-Yepez, and A. L. Salas-Brito, Phys. Rev. Lett.  64, 1643 (1990).
[CrossRef] [PubMed]

J. Bentez, R. P. Martnez-y-Romero, H. N. Nuez-Yepez, and A. L. Salas-Brito, Phys. Rev. Lett.  65, 2085 (1990).
[CrossRef]

Nunez-Yepez, H. N.

R. P. Martinez-y-Romero, H. N. Nunez-Yepez, and A. L. Salas-Brito, Eur. J. Phys.  16, 135 (1995).
[CrossRef]

Peleg, O.

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett.  98, 103901 (2007).
[CrossRef] [PubMed]

Poladian, L.

L. Poladian, Phys. Rev. E  48, 4758 (1993).
[CrossRef]

Roos, C. F.

R. Gerritsma, G. Kirchmair, F. Zähringer, E. Solano, R. Blatt, and C. F. Roos, Nature  463, 68 (2010).
[CrossRef] [PubMed]

Rozmej, P.

P. Rozmej and R. Arvieu, J. Phys. A  32, 5367 (1999).
[CrossRef]

Sahlgren, B.

A. Asseh, H. Storoy, B. Sahlgren, and R. Stubbe, J. Lightwave Technol.  15, 1419 (1997).
[CrossRef]

Salas-Brito, A. L.

R. P. Martinez-y-Romero, H. N. Nunez-Yepez, and A. L. Salas-Brito, Eur. J. Phys.  16, 135 (1995).
[CrossRef]

J. Bentez, R. P. Martnez-y-Romero, H. N. Nuez-Yepez, and A. L. Salas-Brito, Phys. Rev. Lett.  65, 2085 (1990).
[CrossRef]

J. Bentez, R. P. Martnez-y-Romero, H. N. Nuez-Yepez, and A. L. Salas-Brito, Phys. Rev. Lett.  64, 1643 (1990).
[CrossRef] [PubMed]

Segev, M.

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett.  98, 103901 (2007).
[CrossRef] [PubMed]

Sepkhanov, R. A.

R. A. Sepkhanov, Ya. B. Bazaliy, and C. W. J. Beenakker, Phys. Rev. A  75, 063813 (2007).
[CrossRef]

Smirnov, Y. F.

M. Moshinsky and Y. F. Smirnov, The Harmonic Oscillator in Modern Physics (Harwood, 1996).

Solano, E.

R. Gerritsma, G. Kirchmair, F. Zähringer, E. Solano, R. Blatt, and C. F. Roos, Nature  463, 68 (2010).
[CrossRef] [PubMed]

A. Bermudez, M. A. Martin-Delgado, and E. Solano, Phys. Rev. A  76, 041801(R) (2007).
[CrossRef]

Stander, N.

N. Stander, B. Huard, and D. Goldhaber-Gordon, Phys. Rev. Lett.  102, 026807 (2009).
[CrossRef] [PubMed]

Storoy, H.

A. Asseh, H. Storoy, B. Sahlgren, and R. Stubbe, J. Lightwave Technol.  15, 1419 (1997).
[CrossRef]

Stubbe, R.

A. Asseh, H. Storoy, B. Sahlgren, and R. Stubbe, J. Lightwave Technol.  15, 1419 (1997).
[CrossRef]

Szczepaniak, A.

M. Moshinsky and A. Szczepaniak, J. Phys. A  22, L817 (1989).
[CrossRef]

Wang, L.-G.

L.-G. Wang, Z.-G. Wang, and S.-Y. Zhu, EPL  86, 47008 (2009).
[CrossRef]

Wang, Z.-G.

L.-G. Wang, Z.-G. Wang, and S.-Y. Zhu, EPL  86, 47008 (2009).
[CrossRef]

Wei, L.

Young, A. F.

A. F. Young and P. Kim, Nat. Phys.  5, 222 (2009).
[CrossRef]

Zähringer, F.

R. Gerritsma, G. Kirchmair, F. Zähringer, E. Solano, R. Blatt, and C. F. Roos, Nature  463, 68 (2010).
[CrossRef] [PubMed]

Zentella, A.

R. P. Martriaanez-y-Romero, M. Moreno, and A. Zentella, Phys. Rev. D  43, 2036 (1991).
[CrossRef]

Zhang, X.

X. Zhang, Phys. Rev. Lett.  100, 113903 (2008).
[CrossRef] [PubMed]

Zhu, S.-Y.

L.-G. Wang, Z.-G. Wang, and S.-Y. Zhu, EPL  86, 47008 (2009).
[CrossRef]

Appl. Opt. (1)

EPL (1)

L.-G. Wang, Z.-G. Wang, and S.-Y. Zhu, EPL  86, 47008 (2009).
[CrossRef]

Eur. J. Phys. (1)

R. P. Martinez-y-Romero, H. N. Nunez-Yepez, and A. L. Salas-Brito, Eur. J. Phys.  16, 135 (1995).
[CrossRef]

Europhys. Lett. (1)

F. Dominguez-Adame and M. A. Gonzalez, Europhys. Lett.  13, 193 (1990).
[CrossRef]

J. Lightwave Technol. (2)

A. Asseh, H. Storoy, B. Sahlgren, and R. Stubbe, J. Lightwave Technol.  15, 1419 (1997).
[CrossRef]

T. Erdogan, J. Lightwave Technol.  15, 1277 (1997).
[CrossRef]

J. Phys. A (2)

M. Moshinsky and A. Szczepaniak, J. Phys. A  22, L817 (1989).
[CrossRef]

P. Rozmej and R. Arvieu, J. Phys. A  32, 5367 (1999).
[CrossRef]

Nat. Phys. (1)

A. F. Young and P. Kim, Nat. Phys.  5, 222 (2009).
[CrossRef]

Nature (1)

R. Gerritsma, G. Kirchmair, F. Zähringer, E. Solano, R. Blatt, and C. F. Roos, Nature  463, 68 (2010).
[CrossRef] [PubMed]

Nuovo Cimento (1)

D. Ito, K. Mori, and E. Carrieri, Nuovo Cimento  51A, 1119 (1967).

Phys. Rev. A (3)

R. A. Sepkhanov, Ya. B. Bazaliy, and C. W. J. Beenakker, Phys. Rev. A  75, 063813 (2007).
[CrossRef]

A. Bermudez, M. A. Martin-Delgado, and E. Solano, Phys. Rev. A  76, 041801(R) (2007).
[CrossRef]

A. Bermudez, M. A. Martin-Delgado, and A. Luis, Phys. Rev. A  77, 033832 (2008).
[CrossRef]

Phys. Rev. D (1)

R. P. Martriaanez-y-Romero, M. Moreno, and A. Zentella, Phys. Rev. D  43, 2036 (1991).
[CrossRef]

Phys. Rev. E (1)

L. Poladian, Phys. Rev. E  48, 4758 (1993).
[CrossRef]

Phys. Rev. Lett. (5)

N. Stander, B. Huard, and D. Goldhaber-Gordon, Phys. Rev. Lett.  102, 026807 (2009).
[CrossRef] [PubMed]

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett.  98, 103901 (2007).
[CrossRef] [PubMed]

X. Zhang, Phys. Rev. Lett.  100, 113903 (2008).
[CrossRef] [PubMed]

J. Bentez, R. P. Martnez-y-Romero, H. N. Nuez-Yepez, and A. L. Salas-Brito, Phys. Rev. Lett.  64, 1643 (1990).
[CrossRef] [PubMed]

J. Bentez, R. P. Martnez-y-Romero, H. N. Nuez-Yepez, and A. L. Salas-Brito, Phys. Rev. Lett.  65, 2085 (1990).
[CrossRef]

Other (2)

M. Moshinsky and Y. F. Smirnov, The Harmonic Oscillator in Modern Physics (Harwood, 1996).

W. Greiner, Relativistic Quantum Mechanics (Springer-Verlag, 1990).

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Figures (2)

Fig. 1
Fig. 1

(a) Schematic of an FBG. (b), (c) Example of FBG amplitude and phase profiles that realize the analog of the Dirac oscillator ( m 0 = 1 , ω s = 0.5 , L Z = 10 ).

Fig. 2
Fig. 2

(a) Numerically computed power transmission (in decibels) of a lossless FBG with amplitude and phase profiles shown in Fig. 1; (b) corresponding reflection band diagram (shaded area). In (b) a few low-order intensity profiles of trapped modes in the electron and positron branches are also depicted. (c) Same as (a) but for a lossy FBG (power loss rate α = 0.05 m 1 ).

Equations (9)

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

[ z + ( 1 v g ) t ] u = i q ( z ) v ,
[ z ( 1 v g ) t ] v = i q * ( z ) u ,
i τ ψ = σ x { p x i f ( x ) σ z } ψ + σ z m ( x ) ψ
σ x = ( 0 1 1 0 ) , σ z = ( 1 0 0 1 ) ,
m ( x ) = h ( x ) cos [ ϕ ( x ) ] , f ( x ) = h ( x ) sin [ ϕ ( x ) ] .
h ( x ) = m 0 1 + ( ω s x ) 2 , ϕ ( x ) = atan ( ω s x ) ,
1 2 m 0 d 2 ψ ± d x 2 + 1 2 m 0 ω s 2 x 2 ψ ± = δ 2 m 0 2 m 0 ω s 2 m 0 ψ ± .
δ n = m 0 2 + 2 m 0 ω s ( 1 + n ) , n = 0 , 1 , 2 , 3 , ,
δ n = m 0 2 + 2 m 0 ω s n , n = 0 , 1 , 2 , 3 , .

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