Abstract

We obtain stationary solution for optical solitons propagating in a Kerr-effect nonlinear cavity using elliptic functions and quantize them semiclassically. On invoking box boundary conditions, a constraint relating the number of particles, wavelength, and a parameter associated with the elliptic function emerges. This constraint fundamentally modifies the binding energy of the soliton and lends the system a rich plethora of solution types with diverse behavior as a function of excitation number. We also speculate on how the bright soliton can thermalize through a path of frequency conversion.

© 2006 Optical Society of America

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  1. R. Y. Chiao, T. H. Hansson, J. M. Leinaas, and S. Viefers, "Effective photon-photon interaction in a two-dimensional photon fluid," Phys. Rev. A 69, 063816 (2004).
    [CrossRef]
  2. M. W. Mitchel, C. I. Hancox, and R. Y. Chiao, "Dynamics of atom-mediated photon-photon scattering," Phys. Rev. A 62, 043819 (2000).
    [CrossRef]
  3. R. Y. Chiao and J. Boyce, "Bogoliubov dispersion relation and the possibility of superfluidity for weakly interacting photons in a two-dimensional photon fluid," Phys. Rev. A 60, 4114-4121 (1999).
    [CrossRef]
  4. J. Boyce and R. Chiao, "Transverse oscillation arising from spatial soliton formation in nonlinear optical cavities," Phys. Rev. A 59, 3953-3958 (1999).
    [CrossRef]
  5. P. Navez, "Frequency down-conversion through Bose condensation of light," Phys. Rev. A 68, 013811 (2003).
    [CrossRef]
  6. J. C. Martinez, "Dynamics of frequency conversion of an optical pulse in a microcavity," Phys. Rev. A 71, 015801 (2005).
    [CrossRef]
  7. F. Laussy, G. Malpuech, A. Kavokin, and P. Bigenwald, "Spontaneous coherence buildup in a polariton laser," Phys. Rev. Lett. 93, 016402 (2004).
    [CrossRef]
  8. P. G. Savvidis, J. J. Baumberg, R. M. Stevenson, M. Skolnick, D. M. Whittaker, and J. S. Roberts, "Angle-resonant stimulated polariton amplifier," Phys. Rev. Lett. 84, 1547-1550 (2000).
    [CrossRef] [PubMed]
  9. A. Gorlitz, J. M. Vogels, A. E. Leanhardt, C. Raman, T. L. Gustavson, J. R. Abo-Shaeer, A. P. Chikkatur, S. Gupta, S. Inouye, T. Rosenband, and W. Ketterle, "Realization of Bose-Einstein condensates in lower dimensions," Phys. Rev. Lett. 87, 130402 (2001).
    [CrossRef] [PubMed]
  10. D. S. Petrov, G. V. Shlyapnikov, and J. T. M. Walraven, "Regimes of quantum degeneracy in trapped 1D gases," Phys. Rev. Lett. 85, 3745-3749 (2000).
    [CrossRef] [PubMed]
  11. Y. S. Kivshar and G. P. Agrawal, Optical Solitons (Academic, 2003), pp. 31-46.
  12. A. Hasegawa, Optical Solitons in Fibers, Vol. 116 of Springer Tracts in Modern Physics (Springer, 1989), pp. 13-30.
    [CrossRef]
  13. Y. Lai and H. A. Haus, "Quantum theory of solitons in optical fibers. I. Time-dependent Hartree approximation," Phys. Rev. A 40, 844-853 (1989).
    [CrossRef] [PubMed]
  14. M. Wadati and M. Sakagami, "Classical soliton as the limit of the quantum field theory," J. Phys. Soc. Jpn. 53, 1933-1938 (1984).
    [CrossRef]
  15. C. Nohl, "Semiclassical quantization of the nonlinear Schrodinger equation," Ann. Phys. (N.Y.) 96, 234-260 (1976).
    [CrossRef]
  16. L. D. Carr, C. W. Clark, and W. P. Reinhardt, "Stationary solutions of the one-dimensional nonlinear Schrodinger equation. I. Case of repulsive nonlinearity," Phys. Rev. A 62, 063610 (2000).
    [CrossRef]
  17. L. D. Carr, C. W. Clark, and W. P. Reinhardt, "Stationary solutions of the one-dimensional nonlinear Schrodinger equation. II. Case of attractive nonlinearity," Phys. Rev. A 62, 063611 (2000).
    [CrossRef]
  18. R. Kanamoto, H. Saito, and M. Ueda, "Quantum phase transition in one-dimensional Bose-Einstein condensates with attractive interaction," Phys. Rev. A 67, 013608 (2003).
    [CrossRef]
  19. L. A. Lugiato and R. Lefever, "Spatial dissipative structures in passive optical systems," Phys. Rev. Lett. 58, 2209-2211 (1987).
    [CrossRef] [PubMed]
  20. L. M. Milne-Thomson, "Jacobian elliptic functions and theta functions," in Handbook of Mathematical Functions, M.Abramowitz and I.A.Stegun, eds. (National Bureau of Standards, 1964), pp. 569-588.
  21. R. F. Dashen, B. Hasslacher, and A. Neveu, "Nonperturbative methods and extended-hadron models in field theory. I. Semiclassical functional methods," Phys. Rev. D 10, 4114-4129 (1974).
    [CrossRef]
  22. R. Rajaraman, Solitons and Instantons (North-Holland, 1982), pp. 167-195.
  23. J. B. McGuire, "Study of exactly soluble one-dimensional N-body problems," J. Math. Phys. 5, 622-636 (1964).
    [CrossRef]
  24. J. W. Negele and H. Orland, Quantum Many-Particle Systems (Addison-Wesley, 1988), pp. 44-46.

2005

J. C. Martinez, "Dynamics of frequency conversion of an optical pulse in a microcavity," Phys. Rev. A 71, 015801 (2005).
[CrossRef]

2004

F. Laussy, G. Malpuech, A. Kavokin, and P. Bigenwald, "Spontaneous coherence buildup in a polariton laser," Phys. Rev. Lett. 93, 016402 (2004).
[CrossRef]

R. Y. Chiao, T. H. Hansson, J. M. Leinaas, and S. Viefers, "Effective photon-photon interaction in a two-dimensional photon fluid," Phys. Rev. A 69, 063816 (2004).
[CrossRef]

2003

R. Kanamoto, H. Saito, and M. Ueda, "Quantum phase transition in one-dimensional Bose-Einstein condensates with attractive interaction," Phys. Rev. A 67, 013608 (2003).
[CrossRef]

P. Navez, "Frequency down-conversion through Bose condensation of light," Phys. Rev. A 68, 013811 (2003).
[CrossRef]

2001

A. Gorlitz, J. M. Vogels, A. E. Leanhardt, C. Raman, T. L. Gustavson, J. R. Abo-Shaeer, A. P. Chikkatur, S. Gupta, S. Inouye, T. Rosenband, and W. Ketterle, "Realization of Bose-Einstein condensates in lower dimensions," Phys. Rev. Lett. 87, 130402 (2001).
[CrossRef] [PubMed]

2000

D. S. Petrov, G. V. Shlyapnikov, and J. T. M. Walraven, "Regimes of quantum degeneracy in trapped 1D gases," Phys. Rev. Lett. 85, 3745-3749 (2000).
[CrossRef] [PubMed]

M. W. Mitchel, C. I. Hancox, and R. Y. Chiao, "Dynamics of atom-mediated photon-photon scattering," Phys. Rev. A 62, 043819 (2000).
[CrossRef]

P. G. Savvidis, J. J. Baumberg, R. M. Stevenson, M. Skolnick, D. M. Whittaker, and J. S. Roberts, "Angle-resonant stimulated polariton amplifier," Phys. Rev. Lett. 84, 1547-1550 (2000).
[CrossRef] [PubMed]

L. D. Carr, C. W. Clark, and W. P. Reinhardt, "Stationary solutions of the one-dimensional nonlinear Schrodinger equation. I. Case of repulsive nonlinearity," Phys. Rev. A 62, 063610 (2000).
[CrossRef]

L. D. Carr, C. W. Clark, and W. P. Reinhardt, "Stationary solutions of the one-dimensional nonlinear Schrodinger equation. II. Case of attractive nonlinearity," Phys. Rev. A 62, 063611 (2000).
[CrossRef]

1999

R. Y. Chiao and J. Boyce, "Bogoliubov dispersion relation and the possibility of superfluidity for weakly interacting photons in a two-dimensional photon fluid," Phys. Rev. A 60, 4114-4121 (1999).
[CrossRef]

J. Boyce and R. Chiao, "Transverse oscillation arising from spatial soliton formation in nonlinear optical cavities," Phys. Rev. A 59, 3953-3958 (1999).
[CrossRef]

1989

Y. Lai and H. A. Haus, "Quantum theory of solitons in optical fibers. I. Time-dependent Hartree approximation," Phys. Rev. A 40, 844-853 (1989).
[CrossRef] [PubMed]

1987

L. A. Lugiato and R. Lefever, "Spatial dissipative structures in passive optical systems," Phys. Rev. Lett. 58, 2209-2211 (1987).
[CrossRef] [PubMed]

1984

M. Wadati and M. Sakagami, "Classical soliton as the limit of the quantum field theory," J. Phys. Soc. Jpn. 53, 1933-1938 (1984).
[CrossRef]

1976

C. Nohl, "Semiclassical quantization of the nonlinear Schrodinger equation," Ann. Phys. (N.Y.) 96, 234-260 (1976).
[CrossRef]

1974

R. F. Dashen, B. Hasslacher, and A. Neveu, "Nonperturbative methods and extended-hadron models in field theory. I. Semiclassical functional methods," Phys. Rev. D 10, 4114-4129 (1974).
[CrossRef]

1964

J. B. McGuire, "Study of exactly soluble one-dimensional N-body problems," J. Math. Phys. 5, 622-636 (1964).
[CrossRef]

Abo-Shaeer, J. R.

A. Gorlitz, J. M. Vogels, A. E. Leanhardt, C. Raman, T. L. Gustavson, J. R. Abo-Shaeer, A. P. Chikkatur, S. Gupta, S. Inouye, T. Rosenband, and W. Ketterle, "Realization of Bose-Einstein condensates in lower dimensions," Phys. Rev. Lett. 87, 130402 (2001).
[CrossRef] [PubMed]

Agrawal, G. P.

Y. S. Kivshar and G. P. Agrawal, Optical Solitons (Academic, 2003), pp. 31-46.

Baumberg, J. J.

P. G. Savvidis, J. J. Baumberg, R. M. Stevenson, M. Skolnick, D. M. Whittaker, and J. S. Roberts, "Angle-resonant stimulated polariton amplifier," Phys. Rev. Lett. 84, 1547-1550 (2000).
[CrossRef] [PubMed]

Bigenwald, P.

F. Laussy, G. Malpuech, A. Kavokin, and P. Bigenwald, "Spontaneous coherence buildup in a polariton laser," Phys. Rev. Lett. 93, 016402 (2004).
[CrossRef]

Boyce, J.

R. Y. Chiao and J. Boyce, "Bogoliubov dispersion relation and the possibility of superfluidity for weakly interacting photons in a two-dimensional photon fluid," Phys. Rev. A 60, 4114-4121 (1999).
[CrossRef]

J. Boyce and R. Chiao, "Transverse oscillation arising from spatial soliton formation in nonlinear optical cavities," Phys. Rev. A 59, 3953-3958 (1999).
[CrossRef]

Carr, L. D.

L. D. Carr, C. W. Clark, and W. P. Reinhardt, "Stationary solutions of the one-dimensional nonlinear Schrodinger equation. I. Case of repulsive nonlinearity," Phys. Rev. A 62, 063610 (2000).
[CrossRef]

L. D. Carr, C. W. Clark, and W. P. Reinhardt, "Stationary solutions of the one-dimensional nonlinear Schrodinger equation. II. Case of attractive nonlinearity," Phys. Rev. A 62, 063611 (2000).
[CrossRef]

Chiao, R.

J. Boyce and R. Chiao, "Transverse oscillation arising from spatial soliton formation in nonlinear optical cavities," Phys. Rev. A 59, 3953-3958 (1999).
[CrossRef]

Chiao, R. Y.

R. Y. Chiao, T. H. Hansson, J. M. Leinaas, and S. Viefers, "Effective photon-photon interaction in a two-dimensional photon fluid," Phys. Rev. A 69, 063816 (2004).
[CrossRef]

M. W. Mitchel, C. I. Hancox, and R. Y. Chiao, "Dynamics of atom-mediated photon-photon scattering," Phys. Rev. A 62, 043819 (2000).
[CrossRef]

R. Y. Chiao and J. Boyce, "Bogoliubov dispersion relation and the possibility of superfluidity for weakly interacting photons in a two-dimensional photon fluid," Phys. Rev. A 60, 4114-4121 (1999).
[CrossRef]

Chikkatur, A. P.

A. Gorlitz, J. M. Vogels, A. E. Leanhardt, C. Raman, T. L. Gustavson, J. R. Abo-Shaeer, A. P. Chikkatur, S. Gupta, S. Inouye, T. Rosenband, and W. Ketterle, "Realization of Bose-Einstein condensates in lower dimensions," Phys. Rev. Lett. 87, 130402 (2001).
[CrossRef] [PubMed]

Clark, C. W.

L. D. Carr, C. W. Clark, and W. P. Reinhardt, "Stationary solutions of the one-dimensional nonlinear Schrodinger equation. II. Case of attractive nonlinearity," Phys. Rev. A 62, 063611 (2000).
[CrossRef]

L. D. Carr, C. W. Clark, and W. P. Reinhardt, "Stationary solutions of the one-dimensional nonlinear Schrodinger equation. I. Case of repulsive nonlinearity," Phys. Rev. A 62, 063610 (2000).
[CrossRef]

Dashen, R. F.

R. F. Dashen, B. Hasslacher, and A. Neveu, "Nonperturbative methods and extended-hadron models in field theory. I. Semiclassical functional methods," Phys. Rev. D 10, 4114-4129 (1974).
[CrossRef]

Gorlitz, A.

A. Gorlitz, J. M. Vogels, A. E. Leanhardt, C. Raman, T. L. Gustavson, J. R. Abo-Shaeer, A. P. Chikkatur, S. Gupta, S. Inouye, T. Rosenband, and W. Ketterle, "Realization of Bose-Einstein condensates in lower dimensions," Phys. Rev. Lett. 87, 130402 (2001).
[CrossRef] [PubMed]

Gupta, S.

A. Gorlitz, J. M. Vogels, A. E. Leanhardt, C. Raman, T. L. Gustavson, J. R. Abo-Shaeer, A. P. Chikkatur, S. Gupta, S. Inouye, T. Rosenband, and W. Ketterle, "Realization of Bose-Einstein condensates in lower dimensions," Phys. Rev. Lett. 87, 130402 (2001).
[CrossRef] [PubMed]

Gustavson, T. L.

A. Gorlitz, J. M. Vogels, A. E. Leanhardt, C. Raman, T. L. Gustavson, J. R. Abo-Shaeer, A. P. Chikkatur, S. Gupta, S. Inouye, T. Rosenband, and W. Ketterle, "Realization of Bose-Einstein condensates in lower dimensions," Phys. Rev. Lett. 87, 130402 (2001).
[CrossRef] [PubMed]

Hancox, C. I.

M. W. Mitchel, C. I. Hancox, and R. Y. Chiao, "Dynamics of atom-mediated photon-photon scattering," Phys. Rev. A 62, 043819 (2000).
[CrossRef]

Hansson, T. H.

R. Y. Chiao, T. H. Hansson, J. M. Leinaas, and S. Viefers, "Effective photon-photon interaction in a two-dimensional photon fluid," Phys. Rev. A 69, 063816 (2004).
[CrossRef]

Hasegawa, A.

A. Hasegawa, Optical Solitons in Fibers, Vol. 116 of Springer Tracts in Modern Physics (Springer, 1989), pp. 13-30.
[CrossRef]

Hasslacher, B.

R. F. Dashen, B. Hasslacher, and A. Neveu, "Nonperturbative methods and extended-hadron models in field theory. I. Semiclassical functional methods," Phys. Rev. D 10, 4114-4129 (1974).
[CrossRef]

Haus, H. A.

Y. Lai and H. A. Haus, "Quantum theory of solitons in optical fibers. I. Time-dependent Hartree approximation," Phys. Rev. A 40, 844-853 (1989).
[CrossRef] [PubMed]

Inouye, S.

A. Gorlitz, J. M. Vogels, A. E. Leanhardt, C. Raman, T. L. Gustavson, J. R. Abo-Shaeer, A. P. Chikkatur, S. Gupta, S. Inouye, T. Rosenband, and W. Ketterle, "Realization of Bose-Einstein condensates in lower dimensions," Phys. Rev. Lett. 87, 130402 (2001).
[CrossRef] [PubMed]

Kanamoto, R.

R. Kanamoto, H. Saito, and M. Ueda, "Quantum phase transition in one-dimensional Bose-Einstein condensates with attractive interaction," Phys. Rev. A 67, 013608 (2003).
[CrossRef]

Kavokin, A.

F. Laussy, G. Malpuech, A. Kavokin, and P. Bigenwald, "Spontaneous coherence buildup in a polariton laser," Phys. Rev. Lett. 93, 016402 (2004).
[CrossRef]

Ketterle, W.

A. Gorlitz, J. M. Vogels, A. E. Leanhardt, C. Raman, T. L. Gustavson, J. R. Abo-Shaeer, A. P. Chikkatur, S. Gupta, S. Inouye, T. Rosenband, and W. Ketterle, "Realization of Bose-Einstein condensates in lower dimensions," Phys. Rev. Lett. 87, 130402 (2001).
[CrossRef] [PubMed]

Kivshar, Y. S.

Y. S. Kivshar and G. P. Agrawal, Optical Solitons (Academic, 2003), pp. 31-46.

Lai, Y.

Y. Lai and H. A. Haus, "Quantum theory of solitons in optical fibers. I. Time-dependent Hartree approximation," Phys. Rev. A 40, 844-853 (1989).
[CrossRef] [PubMed]

Laussy, F.

F. Laussy, G. Malpuech, A. Kavokin, and P. Bigenwald, "Spontaneous coherence buildup in a polariton laser," Phys. Rev. Lett. 93, 016402 (2004).
[CrossRef]

Leanhardt, A. E.

A. Gorlitz, J. M. Vogels, A. E. Leanhardt, C. Raman, T. L. Gustavson, J. R. Abo-Shaeer, A. P. Chikkatur, S. Gupta, S. Inouye, T. Rosenband, and W. Ketterle, "Realization of Bose-Einstein condensates in lower dimensions," Phys. Rev. Lett. 87, 130402 (2001).
[CrossRef] [PubMed]

Lefever, R.

L. A. Lugiato and R. Lefever, "Spatial dissipative structures in passive optical systems," Phys. Rev. Lett. 58, 2209-2211 (1987).
[CrossRef] [PubMed]

Leinaas, J. M.

R. Y. Chiao, T. H. Hansson, J. M. Leinaas, and S. Viefers, "Effective photon-photon interaction in a two-dimensional photon fluid," Phys. Rev. A 69, 063816 (2004).
[CrossRef]

Lugiato, L. A.

L. A. Lugiato and R. Lefever, "Spatial dissipative structures in passive optical systems," Phys. Rev. Lett. 58, 2209-2211 (1987).
[CrossRef] [PubMed]

Malpuech, G.

F. Laussy, G. Malpuech, A. Kavokin, and P. Bigenwald, "Spontaneous coherence buildup in a polariton laser," Phys. Rev. Lett. 93, 016402 (2004).
[CrossRef]

Martinez, J. C.

J. C. Martinez, "Dynamics of frequency conversion of an optical pulse in a microcavity," Phys. Rev. A 71, 015801 (2005).
[CrossRef]

McGuire, J. B.

J. B. McGuire, "Study of exactly soluble one-dimensional N-body problems," J. Math. Phys. 5, 622-636 (1964).
[CrossRef]

Milne-Thomson, L. M.

L. M. Milne-Thomson, "Jacobian elliptic functions and theta functions," in Handbook of Mathematical Functions, M.Abramowitz and I.A.Stegun, eds. (National Bureau of Standards, 1964), pp. 569-588.

Mitchel, M. W.

M. W. Mitchel, C. I. Hancox, and R. Y. Chiao, "Dynamics of atom-mediated photon-photon scattering," Phys. Rev. A 62, 043819 (2000).
[CrossRef]

Navez, P.

P. Navez, "Frequency down-conversion through Bose condensation of light," Phys. Rev. A 68, 013811 (2003).
[CrossRef]

Negele, J. W.

J. W. Negele and H. Orland, Quantum Many-Particle Systems (Addison-Wesley, 1988), pp. 44-46.

Neveu, A.

R. F. Dashen, B. Hasslacher, and A. Neveu, "Nonperturbative methods and extended-hadron models in field theory. I. Semiclassical functional methods," Phys. Rev. D 10, 4114-4129 (1974).
[CrossRef]

Nohl, C.

C. Nohl, "Semiclassical quantization of the nonlinear Schrodinger equation," Ann. Phys. (N.Y.) 96, 234-260 (1976).
[CrossRef]

Orland, H.

J. W. Negele and H. Orland, Quantum Many-Particle Systems (Addison-Wesley, 1988), pp. 44-46.

Petrov, D. S.

D. S. Petrov, G. V. Shlyapnikov, and J. T. M. Walraven, "Regimes of quantum degeneracy in trapped 1D gases," Phys. Rev. Lett. 85, 3745-3749 (2000).
[CrossRef] [PubMed]

Rajaraman, R.

R. Rajaraman, Solitons and Instantons (North-Holland, 1982), pp. 167-195.

Raman, C.

A. Gorlitz, J. M. Vogels, A. E. Leanhardt, C. Raman, T. L. Gustavson, J. R. Abo-Shaeer, A. P. Chikkatur, S. Gupta, S. Inouye, T. Rosenband, and W. Ketterle, "Realization of Bose-Einstein condensates in lower dimensions," Phys. Rev. Lett. 87, 130402 (2001).
[CrossRef] [PubMed]

Reinhardt, W. P.

L. D. Carr, C. W. Clark, and W. P. Reinhardt, "Stationary solutions of the one-dimensional nonlinear Schrodinger equation. II. Case of attractive nonlinearity," Phys. Rev. A 62, 063611 (2000).
[CrossRef]

L. D. Carr, C. W. Clark, and W. P. Reinhardt, "Stationary solutions of the one-dimensional nonlinear Schrodinger equation. I. Case of repulsive nonlinearity," Phys. Rev. A 62, 063610 (2000).
[CrossRef]

Roberts, J. S.

P. G. Savvidis, J. J. Baumberg, R. M. Stevenson, M. Skolnick, D. M. Whittaker, and J. S. Roberts, "Angle-resonant stimulated polariton amplifier," Phys. Rev. Lett. 84, 1547-1550 (2000).
[CrossRef] [PubMed]

Rosenband, T.

A. Gorlitz, J. M. Vogels, A. E. Leanhardt, C. Raman, T. L. Gustavson, J. R. Abo-Shaeer, A. P. Chikkatur, S. Gupta, S. Inouye, T. Rosenband, and W. Ketterle, "Realization of Bose-Einstein condensates in lower dimensions," Phys. Rev. Lett. 87, 130402 (2001).
[CrossRef] [PubMed]

Saito, H.

R. Kanamoto, H. Saito, and M. Ueda, "Quantum phase transition in one-dimensional Bose-Einstein condensates with attractive interaction," Phys. Rev. A 67, 013608 (2003).
[CrossRef]

Sakagami, M.

M. Wadati and M. Sakagami, "Classical soliton as the limit of the quantum field theory," J. Phys. Soc. Jpn. 53, 1933-1938 (1984).
[CrossRef]

Savvidis, P. G.

P. G. Savvidis, J. J. Baumberg, R. M. Stevenson, M. Skolnick, D. M. Whittaker, and J. S. Roberts, "Angle-resonant stimulated polariton amplifier," Phys. Rev. Lett. 84, 1547-1550 (2000).
[CrossRef] [PubMed]

Shlyapnikov, G. V.

D. S. Petrov, G. V. Shlyapnikov, and J. T. M. Walraven, "Regimes of quantum degeneracy in trapped 1D gases," Phys. Rev. Lett. 85, 3745-3749 (2000).
[CrossRef] [PubMed]

Skolnick, M.

P. G. Savvidis, J. J. Baumberg, R. M. Stevenson, M. Skolnick, D. M. Whittaker, and J. S. Roberts, "Angle-resonant stimulated polariton amplifier," Phys. Rev. Lett. 84, 1547-1550 (2000).
[CrossRef] [PubMed]

Stevenson, R. M.

P. G. Savvidis, J. J. Baumberg, R. M. Stevenson, M. Skolnick, D. M. Whittaker, and J. S. Roberts, "Angle-resonant stimulated polariton amplifier," Phys. Rev. Lett. 84, 1547-1550 (2000).
[CrossRef] [PubMed]

Ueda, M.

R. Kanamoto, H. Saito, and M. Ueda, "Quantum phase transition in one-dimensional Bose-Einstein condensates with attractive interaction," Phys. Rev. A 67, 013608 (2003).
[CrossRef]

Viefers, S.

R. Y. Chiao, T. H. Hansson, J. M. Leinaas, and S. Viefers, "Effective photon-photon interaction in a two-dimensional photon fluid," Phys. Rev. A 69, 063816 (2004).
[CrossRef]

Vogels, J. M.

A. Gorlitz, J. M. Vogels, A. E. Leanhardt, C. Raman, T. L. Gustavson, J. R. Abo-Shaeer, A. P. Chikkatur, S. Gupta, S. Inouye, T. Rosenband, and W. Ketterle, "Realization of Bose-Einstein condensates in lower dimensions," Phys. Rev. Lett. 87, 130402 (2001).
[CrossRef] [PubMed]

Wadati, M.

M. Wadati and M. Sakagami, "Classical soliton as the limit of the quantum field theory," J. Phys. Soc. Jpn. 53, 1933-1938 (1984).
[CrossRef]

Walraven, J. T. M.

D. S. Petrov, G. V. Shlyapnikov, and J. T. M. Walraven, "Regimes of quantum degeneracy in trapped 1D gases," Phys. Rev. Lett. 85, 3745-3749 (2000).
[CrossRef] [PubMed]

Whittaker, D. M.

P. G. Savvidis, J. J. Baumberg, R. M. Stevenson, M. Skolnick, D. M. Whittaker, and J. S. Roberts, "Angle-resonant stimulated polariton amplifier," Phys. Rev. Lett. 84, 1547-1550 (2000).
[CrossRef] [PubMed]

Ann. Phys. (N.Y.)

C. Nohl, "Semiclassical quantization of the nonlinear Schrodinger equation," Ann. Phys. (N.Y.) 96, 234-260 (1976).
[CrossRef]

J. Math. Phys.

J. B. McGuire, "Study of exactly soluble one-dimensional N-body problems," J. Math. Phys. 5, 622-636 (1964).
[CrossRef]

J. Phys. Soc. Jpn.

M. Wadati and M. Sakagami, "Classical soliton as the limit of the quantum field theory," J. Phys. Soc. Jpn. 53, 1933-1938 (1984).
[CrossRef]

Phys. Rev. A

Y. Lai and H. A. Haus, "Quantum theory of solitons in optical fibers. I. Time-dependent Hartree approximation," Phys. Rev. A 40, 844-853 (1989).
[CrossRef] [PubMed]

L. D. Carr, C. W. Clark, and W. P. Reinhardt, "Stationary solutions of the one-dimensional nonlinear Schrodinger equation. I. Case of repulsive nonlinearity," Phys. Rev. A 62, 063610 (2000).
[CrossRef]

L. D. Carr, C. W. Clark, and W. P. Reinhardt, "Stationary solutions of the one-dimensional nonlinear Schrodinger equation. II. Case of attractive nonlinearity," Phys. Rev. A 62, 063611 (2000).
[CrossRef]

R. Kanamoto, H. Saito, and M. Ueda, "Quantum phase transition in one-dimensional Bose-Einstein condensates with attractive interaction," Phys. Rev. A 67, 013608 (2003).
[CrossRef]

R. Y. Chiao, T. H. Hansson, J. M. Leinaas, and S. Viefers, "Effective photon-photon interaction in a two-dimensional photon fluid," Phys. Rev. A 69, 063816 (2004).
[CrossRef]

M. W. Mitchel, C. I. Hancox, and R. Y. Chiao, "Dynamics of atom-mediated photon-photon scattering," Phys. Rev. A 62, 043819 (2000).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Logarithm of the absolute value of the detuning plotted against m. Note that Δ ω is negative when m < 1 2 and positive for m > 1 2 .

Fig. 2
Fig. 2

Plot of m K ( m ) S cl j 2 m 1 versus m. The minimum at the right occurs at m * = 0.774915 .

Fig. 3
Fig. 3

Plots of ϵ ( x ) for pairs of m values grouped side by side: (a) m = 0.1 , j = 5 and (b) m = 0.05 , j = 10 (hence m j = 1 2 for these two cases); (c) m = 0.48 , j = 5 and (d) m = 0.52 , j = 5 ; (e) m = 0.99 , j = 1 and (f) m = 0.99999 , j = 1 ; (g) m = 0.99 , j = 2 and (h) m = 0.99999 , j = 2 . In all cases, ϵ ( x ) is normalized to unity for the right-hand graphs and the corresponding left-hand ones are measured relative to this.

Equations (36)

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ϵ t ̃ = i c 2 k 2 ϵ t ̃ 2 + i ω n 2 ϵ 2 ϵ + i ( Δ ω ) ϵ Γ ( ϵ ϵ d ) ,
i E t + 2 E x 2 + 1 2 E 2 E ( Δ ω ) E = 0 .
E ( x , t ) = ϵ ( x ) exp [ i ( α + Δ ω ) t + i ν 2 t 4 i ν ( x + ν t ) 2 ] ,
ϵ ( x ) = A cn [ K ̃ ( x + ν t ) + δ m ] ,
L ( x , t ) = i 2 ( E * E ̇ E ̇ * E ) E x 2 Δ ω E 2 + 1 2 E 4 ,
S cl = f 1 4 m A 3 S cl red T ,
S cl red = d u [ 2 ( 1 2 m ) cn 2 ( u m ) + 2 m cn 4 ( u m ) ( 1 m ) ]
τ = 2 π ( α + Δ ω + 1 4 ν 2 ) .
G ( E 0 e ) i A t 0 x 0 0 d t exp ( i u e t ) exp ( i S cl ) d 2 q 2 S ( q , q ) q q q = q = q 1 2 ,
0 x 0 d q 0 0 2 π d θ 2 S q q q 1 2 = if 1 4 m S cl red g 3 2 3 π L x 0 2 l τ ,
d t l l d τ d A δ [ 2 π ( α + Δ ω 2 + ν 2 4 τ ) ] ( τ A ) 1 .
G ( E 0 e ) n , l 0 l d τ d A d 2 q 2 π i 2 S q q 1 2 exp ( i u e l τ ) exp [ i S cl n , l ( τ ) ] × δ [ 2 π g A 2 + Δ ω τ ( 1 + ν 2 g A 2 + Δ ω ) ] 2 g A g A 2 + Δ ω ( ν 2 τ 4 2 π ) τ g A 2 + Δ ω 1 2 .
l d M exp [ i M l 2 π g A 2 + Δ ω i τ M l ( 1 + ν 2 4 g A 2 + Δ ω ) ] ,
exp [ i l ( u e τ 2 π W + ν 2 τ 4 W τ M ) ] ,
W SP = 3 γ 2 g A .
G ( E ) n , l l d τ d M g A τ l g A 2 + Δ ω 2 2 π i l τ g A 2 + Δ ω W SP exp [ i τ l ( u e M ) ] exp [ i W SP ( n 2 L 2 4 τ l 2 π l ) ] .
G ( E ) n , l d M g A g A 2 + Δ ω 3 2 2 π 2 W SP ( u e M ) exp ( 2 π i l W SP ) exp [ i n L W SP ( u e M ) ] .
2 π W SP = 2 π N ,
L W SP ( u e M ) = 2 π r ,
3 2 f 1 4 m S cl red g A = N .
E N , r ( m ) = t 0 [ 1 N ( 2 π L r ) 2 + M ] .
M = Δ ω N 2 ( N 1 ) ,
E N , r ( m ) = t 0 [ 1 N ( 2 π L r ) 2 Δ ω N 2 ( N 1 ) ] .
( 1 2 ) ( 1 N ) ( c k ) ( h x 0 L r ) 2 ,
Δ ω = m ( g 3 ) 3 ( 4 f S cl red ) 2 ,
ϵ ( 0 ) = ϵ ( 1 ) = 0 .
j = 2 r .
S cl red = 2 j 3 { 2 ( 1 2 m ) Cn [ K ( m ) ] ( 1 m ) K ( m ) } ,
S cl red = π j m ( 3 8 + m 32 + 9 m 2 1024 + 15 m 3 4096 + ) ,
S cl j 2 m 1 m K ( m ) = 1 12 j 2 f N ,
0 1 ϵ ( x ) 2 d x = ( 4 j ) 2 m K ( m ) Cn [ K ( m ) ] ,
N f ( 3 π 2 m j ) 2 ,
A 2 π j m , Δ ω 1 3 ( π j N ) 2 ,
J K ( 1 2 ) = ( x f N ) 1 2 ,
A ( 8 f N x ) 1 2 = 2 3 2 j K ( 1 2 ) , Δ ω 8 3 N f x 2 ,
N f ( 4 j ) 2 log ( 4 z ) , A 4 j log ( 4 z ) .

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