Abstract

We provide a quantum description of X waves by considering two different kinds of quantum states of the electromagnetic field whose field amplitude expectation value coincides with a propagation-invariant solution of the Maxwell equation. The difference between the two states is of pure quantum mechanical origin since they are internally entangled and disentangled and can be generated by different linear or nonlinear processes.

© 2007 Optical Society of America

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References

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  1. E. Recami, M. Zamboni-Rached, and H. E. Hernandez-Figueroa, Localized Waves (Wiley-Interscience, 2007).
  2. J. Lu and J. F. Greenleaf, "Nondiffracting X waves--exact solutions to free-space scalar wave equation and their finite aperture realizations," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19-31 (1992).
    [CrossRef] [PubMed]
  3. R. W. Ziolkowski, D. K. Lewis, and B. D. Cook, "Evidence of localized wave transmission," Phys. Rev. Lett. 62, 147-150 (1989).
    [CrossRef] [PubMed]
  4. A. Ciattoni, C. Conti, and P. Di Porto, "Vector electromagnetic X waves," Phys. Rev. E 69, 036608 (2004).
    [CrossRef]
  5. H. Sonajalg and P. Saari, "Suppression of temporal spread of ultrashort pulses in dispersive media by Bessel beam generators," Opt. Lett. 21, 1162-1164 (1996).
    [CrossRef] [PubMed]
  6. A. Ciattoni and P. Di Porto, "Electromagnetic nondiffracting pulses in lossless isotropic plasmalike media," Phys. Rev. E 70, 035601(R) (2004).
    [CrossRef]
  7. L. Bergé, K. Germaschewski, R. Grauer, and J. J. Rasmussen, "Hyperbolic shock waves of the optical self-focusing with normal group-velocity dispersion," Phys. Rev. Lett. 89, 153902 (2002).
    [CrossRef] [PubMed]
  8. S. Orlov, A. Piskarskas, and A. Stabinis, "Focus wave modes in optical parametric generators," Opt. Lett. 27, 2103-2105 (2002).
    [CrossRef]
  9. C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, "Nonlinear electromagnetic X waves," Phys. Rev. Lett. 90, 170406 (2003).
    [CrossRef] [PubMed]
  10. K. Staliunas and M. Tlidi, "Hyperbolic transverse patterns in nonlinear optical resonators," Phys. Rev. Lett. 94, 133902 (2005).
    [CrossRef] [PubMed]
  11. C. Conti and S. Trillo, "Nonspreading wave packets in three dimensions formed by an ultracold Bose gas in an optical lattice," Phys. Rev. Lett. 92, 120404 (2004).
    [CrossRef] [PubMed]
  12. S. Longhi, "Spatial-temporal Gauss-Laguerre waves in dispersive media," Phys. Rev. E 68, 066612 (2004).
    [CrossRef]
  13. O. Manela, M. Segev, and D. N. Christodoulides, "Nondiffracting beams in periodic media," Opt. Lett. 30, 2611-2613 (2005).
    [CrossRef] [PubMed]
  14. Y. Lahini, E. Frumker, Y. Silberberg, S. Droulias, K. Hizanidis, R. Morandotti, and D. N. Christodoulides, "Discrete X-wave formation in nonlinear waveguide arrays," Phys. Rev. Lett. 98, 023901 (2007).
    [CrossRef] [PubMed]
  15. P. Saari and K. Reivelt, "Evidence of X-shaped propagation-invariant localized light waves," Phys. Rev. Lett. 79, 4135-4138 (1997).
    [CrossRef]
  16. O. Jedrkiewicz, A. Picozzi, M. Clerici, D. Faccio, and P. D. Trapani, "Emergence of X-shaped spatiotemporal coherence in optical waves," Phys. Rev. Lett. 97, 243903 (2006).
    [CrossRef]
  17. A. Picozzi and M. Haelterman, "Hidden coherence along space-time trajectories in parametric wave mixing," Phys. Rev. Lett. 88, 083901 (2002).
    [CrossRef] [PubMed]
  18. P. Saari, M. Menert, and H. Valtna, "Photon localization barrier can be overcome," Opt. Commun. 246, 445-450 (2005).
    [CrossRef]
  19. C. Conti, arXiv:quant-ph/0309069v3 (2003).
  20. C. Conti, arXiv.org:quant-ph/0409130 (2004).
  21. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
  22. R. Loudon, The Quantum Theory of Light, 3rd ed. (Oxford U. Press, 2000).
  23. R. Butkus, S. Orlov, A. Piskarskas, V. Smilgevicius, and A. Stabinis, "Phase matching of optical X-waves in nonlinear crystals," Opt. Commun. 244, 411-421 (2005).
    [CrossRef]
  24. S. Trillo, C. Conti, P. D. Trapani, O. Jedrkiewicz, J. Trull, G. Valiulis, and G. Bellanca, "Colored conical emission by means of second-harmonic generation," Opt. Lett. 27, 1451-1453 (2002).
    [CrossRef]
  25. C. Conti and S. Trillo, "X waves generated at the second harmonic," Opt. Lett. 28, 1251-1253 (2003).
    [CrossRef] [PubMed]
  26. S. Longhi, "Parametric amplification of spatiotemporal localized envelope waves," Phys. Rev. E 69, 016606 (2004).
    [CrossRef]
  27. B. Yurke, "Wideband photon counting and homodyne detection," Phys. Rev. A 32, 311-323 (1985).
    [CrossRef] [PubMed]
  28. B. Yurke and D. Stoler, "Generating quantum mechanical superpositions of macroscopically distinguishable states via amplitude dispersion," Phys. Rev. Lett. 57, 13-16 (1986).
    [CrossRef] [PubMed]
  29. B. C. Sanders, "Entangled coherent states," Phys. Rev. A 45, 6811-6815 (1992).
    [CrossRef] [PubMed]

2007 (1)

Y. Lahini, E. Frumker, Y. Silberberg, S. Droulias, K. Hizanidis, R. Morandotti, and D. N. Christodoulides, "Discrete X-wave formation in nonlinear waveguide arrays," Phys. Rev. Lett. 98, 023901 (2007).
[CrossRef] [PubMed]

2006 (1)

O. Jedrkiewicz, A. Picozzi, M. Clerici, D. Faccio, and P. D. Trapani, "Emergence of X-shaped spatiotemporal coherence in optical waves," Phys. Rev. Lett. 97, 243903 (2006).
[CrossRef]

2005 (4)

P. Saari, M. Menert, and H. Valtna, "Photon localization barrier can be overcome," Opt. Commun. 246, 445-450 (2005).
[CrossRef]

R. Butkus, S. Orlov, A. Piskarskas, V. Smilgevicius, and A. Stabinis, "Phase matching of optical X-waves in nonlinear crystals," Opt. Commun. 244, 411-421 (2005).
[CrossRef]

K. Staliunas and M. Tlidi, "Hyperbolic transverse patterns in nonlinear optical resonators," Phys. Rev. Lett. 94, 133902 (2005).
[CrossRef] [PubMed]

O. Manela, M. Segev, and D. N. Christodoulides, "Nondiffracting beams in periodic media," Opt. Lett. 30, 2611-2613 (2005).
[CrossRef] [PubMed]

2004 (5)

A. Ciattoni and P. Di Porto, "Electromagnetic nondiffracting pulses in lossless isotropic plasmalike media," Phys. Rev. E 70, 035601(R) (2004).
[CrossRef]

C. Conti and S. Trillo, "Nonspreading wave packets in three dimensions formed by an ultracold Bose gas in an optical lattice," Phys. Rev. Lett. 92, 120404 (2004).
[CrossRef] [PubMed]

S. Longhi, "Spatial-temporal Gauss-Laguerre waves in dispersive media," Phys. Rev. E 68, 066612 (2004).
[CrossRef]

A. Ciattoni, C. Conti, and P. Di Porto, "Vector electromagnetic X waves," Phys. Rev. E 69, 036608 (2004).
[CrossRef]

S. Longhi, "Parametric amplification of spatiotemporal localized envelope waves," Phys. Rev. E 69, 016606 (2004).
[CrossRef]

2003 (2)

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, "Nonlinear electromagnetic X waves," Phys. Rev. Lett. 90, 170406 (2003).
[CrossRef] [PubMed]

C. Conti and S. Trillo, "X waves generated at the second harmonic," Opt. Lett. 28, 1251-1253 (2003).
[CrossRef] [PubMed]

2002 (4)

L. Bergé, K. Germaschewski, R. Grauer, and J. J. Rasmussen, "Hyperbolic shock waves of the optical self-focusing with normal group-velocity dispersion," Phys. Rev. Lett. 89, 153902 (2002).
[CrossRef] [PubMed]

S. Trillo, C. Conti, P. D. Trapani, O. Jedrkiewicz, J. Trull, G. Valiulis, and G. Bellanca, "Colored conical emission by means of second-harmonic generation," Opt. Lett. 27, 1451-1453 (2002).
[CrossRef]

S. Orlov, A. Piskarskas, and A. Stabinis, "Focus wave modes in optical parametric generators," Opt. Lett. 27, 2103-2105 (2002).
[CrossRef]

A. Picozzi and M. Haelterman, "Hidden coherence along space-time trajectories in parametric wave mixing," Phys. Rev. Lett. 88, 083901 (2002).
[CrossRef] [PubMed]

1997 (1)

P. Saari and K. Reivelt, "Evidence of X-shaped propagation-invariant localized light waves," Phys. Rev. Lett. 79, 4135-4138 (1997).
[CrossRef]

1996 (1)

1992 (2)

J. Lu and J. F. Greenleaf, "Nondiffracting X waves--exact solutions to free-space scalar wave equation and their finite aperture realizations," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19-31 (1992).
[CrossRef] [PubMed]

B. C. Sanders, "Entangled coherent states," Phys. Rev. A 45, 6811-6815 (1992).
[CrossRef] [PubMed]

1989 (1)

R. W. Ziolkowski, D. K. Lewis, and B. D. Cook, "Evidence of localized wave transmission," Phys. Rev. Lett. 62, 147-150 (1989).
[CrossRef] [PubMed]

1986 (1)

B. Yurke and D. Stoler, "Generating quantum mechanical superpositions of macroscopically distinguishable states via amplitude dispersion," Phys. Rev. Lett. 57, 13-16 (1986).
[CrossRef] [PubMed]

1985 (1)

B. Yurke, "Wideband photon counting and homodyne detection," Phys. Rev. A 32, 311-323 (1985).
[CrossRef] [PubMed]

IEEE Trans. Ultrason. Ferroelectr. Freq. Control (1)

J. Lu and J. F. Greenleaf, "Nondiffracting X waves--exact solutions to free-space scalar wave equation and their finite aperture realizations," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19-31 (1992).
[CrossRef] [PubMed]

Opt. Commun. (2)

R. Butkus, S. Orlov, A. Piskarskas, V. Smilgevicius, and A. Stabinis, "Phase matching of optical X-waves in nonlinear crystals," Opt. Commun. 244, 411-421 (2005).
[CrossRef]

P. Saari, M. Menert, and H. Valtna, "Photon localization barrier can be overcome," Opt. Commun. 246, 445-450 (2005).
[CrossRef]

Opt. Lett. (5)

Phys. Rev. A (2)

B. Yurke, "Wideband photon counting and homodyne detection," Phys. Rev. A 32, 311-323 (1985).
[CrossRef] [PubMed]

B. C. Sanders, "Entangled coherent states," Phys. Rev. A 45, 6811-6815 (1992).
[CrossRef] [PubMed]

Phys. Rev. E (4)

A. Ciattoni and P. Di Porto, "Electromagnetic nondiffracting pulses in lossless isotropic plasmalike media," Phys. Rev. E 70, 035601(R) (2004).
[CrossRef]

S. Longhi, "Parametric amplification of spatiotemporal localized envelope waves," Phys. Rev. E 69, 016606 (2004).
[CrossRef]

A. Ciattoni, C. Conti, and P. Di Porto, "Vector electromagnetic X waves," Phys. Rev. E 69, 036608 (2004).
[CrossRef]

S. Longhi, "Spatial-temporal Gauss-Laguerre waves in dispersive media," Phys. Rev. E 68, 066612 (2004).
[CrossRef]

Phys. Rev. Lett. (10)

Y. Lahini, E. Frumker, Y. Silberberg, S. Droulias, K. Hizanidis, R. Morandotti, and D. N. Christodoulides, "Discrete X-wave formation in nonlinear waveguide arrays," Phys. Rev. Lett. 98, 023901 (2007).
[CrossRef] [PubMed]

P. Saari and K. Reivelt, "Evidence of X-shaped propagation-invariant localized light waves," Phys. Rev. Lett. 79, 4135-4138 (1997).
[CrossRef]

O. Jedrkiewicz, A. Picozzi, M. Clerici, D. Faccio, and P. D. Trapani, "Emergence of X-shaped spatiotemporal coherence in optical waves," Phys. Rev. Lett. 97, 243903 (2006).
[CrossRef]

A. Picozzi and M. Haelterman, "Hidden coherence along space-time trajectories in parametric wave mixing," Phys. Rev. Lett. 88, 083901 (2002).
[CrossRef] [PubMed]

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, "Nonlinear electromagnetic X waves," Phys. Rev. Lett. 90, 170406 (2003).
[CrossRef] [PubMed]

K. Staliunas and M. Tlidi, "Hyperbolic transverse patterns in nonlinear optical resonators," Phys. Rev. Lett. 94, 133902 (2005).
[CrossRef] [PubMed]

C. Conti and S. Trillo, "Nonspreading wave packets in three dimensions formed by an ultracold Bose gas in an optical lattice," Phys. Rev. Lett. 92, 120404 (2004).
[CrossRef] [PubMed]

R. W. Ziolkowski, D. K. Lewis, and B. D. Cook, "Evidence of localized wave transmission," Phys. Rev. Lett. 62, 147-150 (1989).
[CrossRef] [PubMed]

L. Bergé, K. Germaschewski, R. Grauer, and J. J. Rasmussen, "Hyperbolic shock waves of the optical self-focusing with normal group-velocity dispersion," Phys. Rev. Lett. 89, 153902 (2002).
[CrossRef] [PubMed]

B. Yurke and D. Stoler, "Generating quantum mechanical superpositions of macroscopically distinguishable states via amplitude dispersion," Phys. Rev. Lett. 57, 13-16 (1986).
[CrossRef] [PubMed]

Other (5)

E. Recami, M. Zamboni-Rached, and H. E. Hernandez-Figueroa, Localized Waves (Wiley-Interscience, 2007).

C. Conti, arXiv:quant-ph/0309069v3 (2003).

C. Conti, arXiv.org:quant-ph/0409130 (2004).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

R. Loudon, The Quantum Theory of Light, 3rd ed. (Oxford U. Press, 2000).

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Equations (15)

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E x ( r , t ) = R 0 d k f ( k ) J 0 ( sin θ 0 k r ) e i k cos θ 0 [ z ( c cos θ 0 ) ] ,
E ̂ ( r , t ) = k , s c k 2 ε 0 L 3 i a ̂ k , s e i ( k r c k t ) e k , s + h.c. ,
X = k , s D ̂ k , s [ Δ θ 0 ( k ) β ( k , s ) ] 0 ,
X E ̂ X = R [ k , s Δ θ 0 ( k ) 2 c k ε 0 L 3 i β ( k , s ) e i ( k r c k t ) e k , s ] ,
β ( k , s ) = ( 2 π ) 3 i δ θ ε 0 2 c k L 3 k k f s ( k ) ,
X E ̂ X = R { 0 d k k 2 0 2 π d ϕ [ f 1 ( k ) e k 1 + f 2 ( k ) e k 2 ] e i ( k r c k t ) } ,
X E ̂ X = R 0 d k f ( k ) [ J 0 ( sin θ 0 k r ) e x x tan θ 0 r J 1 ( sin θ 0 k r ) e z ] e i k cos θ 0 ( z V t ) ,
X E = k , s Δ θ 0 ( k ) Φ ( k , s ) D ̂ k , s [ α ( k , s ) ] 0 ,
X E a ̂ k s X E = Δ θ 0 ( k ) α ( k , s ) Q ( k , s ) ,
Q ( k , s ) = Φ ( k , s ) 2 + Φ ( k , s ) e ( 1 2 ) α ( k , s ) 2 k k , s s Δ θ 0 ( k ) Φ * ( k , s ) e ( 1 2 ) α ( k , s ) 2 ,
X E E ̂ X E = R [ k , s Δ θ 0 ( k ) G ( k , s ) e i ( k r c k t ) e k , s ] ,
X E projected = α 1 + Φ E 0 ,
x α 2 = e x 2 π [ 1 + Φ E 2 + 2 Φ E cos ( 2 α 1 x arg Φ E ) ] .
X out = e ( 1 2 ) r = 0 M α r 2 n 1 , n 2 , , n M = 0 e ( i 2 ) χ t m , p = 0 M n m n p q = 1 , 2 , , M α q n q n q ! n q .
X out = 1 2 ( e i π 4 X + e i π 4 X ) ,

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