Abstract

We propose a scheme for generating and manipulating vector (or two-component) optical rogue waves using Akhmediev and Kuznetsov-Ma breathers in a coherent atomic system with an M-type five-level configuration via electromagnetically induced transparency (EIT). We show that the propagation velocity of these nonlinear excitations can be reduced to 10−4 c and their generation power can be lowered to microwatts. We also show that the motion trajectories of the two polarization components in these excitations can be deflected significantly by using a transversal gradient magnetic field, similar to the Stern-Gerlach effect of an atomic beam. We find that the deflection angle can reach to 10−4 radian within the propagation distance of only several centimeters; at variance with the atomic Stern-Gerlach effect, the deflection angle can be made different for different polarization components and may be actively adjusted in a controllable way. The results obtained may have promising applications, including the precise measurement of gradient magnetic fields.

© 2017 Optical Society of America

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References

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  7. A. Chabchoub, N. P. Hoffmann, and N. Akhmediev, “Rogue Wave Observation in a Water Wave Tank,” Phys. Rev. Lett. 106, 204502 (2011).
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  8. A. Chabchoub, N. Hoffmann, M. Onorato, and N. Akhmediev, “Super Rogue Waves: Observation of a Higher-Order Breather in Water Waves,” Phys. Rev. X 2, 011015 (2012).
  9. A. Chabchoub, N. Hoffmann, M. Onorato, A. Slunyaev, A. Sergeeva, E. Pelinovsky, and N. Akhmediev, “Observation of a hierarchy of up to fifth-order rogue waves in a water tank,” Phys. Rev. E 86, 056601 (2012).
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  12. B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, “The Peregrine soliton in nonlinear fibre optics,” Nat. Phys. 6, 790 (2010).
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    [Crossref]
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    [Crossref]
  30. S. Chen and L.-Y. Song, “Rogue waves in coupled Hirota systems,” Phys. Rev. E 87, 032910 (2013).
    [Crossref]
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    [Crossref]
  32. F. Baronio, M. Conforti, A. Degasperis, and S. Lombardo, “Rogue waves emerging from the resonant interaction of three waves,” Phys. Rev. Lett. 111, 114101 (2013).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  38. Z. Chen and G. Huang, “Stern-Gerlach effect of multi-component ultraslow optical solitons via electromagnetically induced transparency,” J. Opt. Soc. Am. 30, 2248 (2013).
    [Crossref]
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2017 (1)

D. Mihalache, “Multidimensional localized structures in optical and matter-wave media: A topical survey of recent literature”, Rom. Rep. Phys. 69, 403 (2017).

2016 (1)

J. Liu, C. Hang, and G. Huang, “Weak-light rogue waves, breathers, and their active control in a cold atomic gas via electromagnetically induced transparency,” Phys. Rev. A 93, 063836 (2016).
[Crossref]

2015 (2)

D. Mihalache, “Localized structures in nonlinear optical media: A selection of recent studies”, Rom. Rep. Phys. 67, 1383 (2015).

S. Chen and D. Mihalache, “Vector rogue waves in the Manakov system: diversity and compossibility,” J. Phys. A: Math. Theor. 48, 215202 (2015).
[Crossref]

2014 (2)

S. Chen, Ph. Grelu, and J. M. Soto-Crespo, “Dark- and bright-rogue-wave solutions for media with long-wave–short-wave resonance,” Phys. Rev. E 89, 011201(R) (2014).
[Crossref]

J. M. Dudley, F. Dias, M. Erkintalo, and G. Genty, “Instabilities, breathers and rogue waves in optics,” Nat. Photon. 8, 755 (2014).
[Crossref]

2013 (7)

M. Onorato, S. Residori, U. Bortolozzo, A. Montinad, and F. T. Arecchi, “Rogue waves and their generating mechanisms in different physical contexts,” Phys. Reports 528, 47 (2013).
[Crossref]

N. Akhmediev, J. M. Dudley, D. R. Solli, and S. K. Turitsyn, “Recent progress in investigating optical rogue waves,” J. Opt. 15, 060201 (2013).
[Crossref]

S. Chen and L.-Y. Song, “Rogue waves in coupled Hirota systems,” Phys. Rev. E 87, 032910 (2013).
[Crossref]

K. W. Chow, H. N. Chan, D. J. Kedziora, and R. H. J. Grimshaw, “Rogue wave modes for the long wave-short wave resonance model,” J. Phys. Soc. Jpn. 82, 074001 (2013).
[Crossref]

F. Baronio, M. Conforti, A. Degasperis, and S. Lombardo, “Rogue waves emerging from the resonant interaction of three waves,” Phys. Rev. Lett. 111, 114101 (2013).
[Crossref] [PubMed]

Z. Chen and G. Huang, “Stern-Gerlach effect of multi-component ultraslow optical solitons via electromagnetically induced transparency,” J. Opt. Soc. Am. 30, 2248 (2013).
[Crossref]

S. Chen, “Twisted rogue-wave pairs in the Sasa-Satsuma equation,” Phys. Rev. E 88, 023202 (2013).
[Crossref]

2012 (9)

Y. Ohta and J. Yang, “Rogue waves in the Davey-Stewartson I equation,” Phys. Rev. E 86, 036604 (2012).
[Crossref]

F. Baronio, A. Degasperis, M. Conforti, and S. Wabnitz, “Solutions of the vector nonlinear Schrödinger equations: Evidence for deterministic Rogue waves,” Phys. Rev. Lett. 109, 044102 (2012).
[Crossref]

U. Bandelow and N. Akhmediev, “Sasa-Satsuma equation: Soliton on a background and its limiting cases,” Phys. Rev. E 86, 026606 (2012).
[Crossref]

C. Lecaplain, Ph. Grelu, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative Rogue waves generated by chaotic pulse bunching in a mode-locked laser,” Phys. Rev. Lett. 108, 233901 (2012).
[Crossref] [PubMed]

Ph. Grelu and N. Akhmediev, “Dissipative solitons for mode-locked lasers,” Nat. Photon. 6, 84 (2012).
[Crossref]

C. Hang and G. Huang, “Stern-Gerlach effect of weak-light ultraslow vector solitons,” Phys. Rev. A 86, 043809 (2012).
[Crossref]

A. Chabchoub, N. Hoffmann, M. Onorato, and N. Akhmediev, “Super Rogue Waves: Observation of a Higher-Order Breather in Water Waves,” Phys. Rev. X 2, 011015 (2012).

A. Chabchoub, N. Hoffmann, M. Onorato, A. Slunyaev, A. Sergeeva, E. Pelinovsky, and N. Akhmediev, “Observation of a hierarchy of up to fifth-order rogue waves in a water tank,” Phys. Rev. E 86, 056601 (2012).
[Crossref]

A. Zaviyalov, O. Egorov, R. Iliew, and F. Lederer, “Rogue waves in mode-locked fiber lasers,” Phys. Rev. A 85, 013828 (2012).
[Crossref]

2011 (4)

H. Bailung, S. K. Sharma, and Y. Nakamura, “Observation of Peregrine solitons in a multicomponent plasma with negative ions,” Phys. Rev. Lett. 107, 255005 (2011).
[Crossref]

Z. Yan, “Vector financial rogue waves,” Phys. Lett. A 375, 4274 (2011).
[Crossref]

A. Chabchoub, N. P. Hoffmann, and N. Akhmediev, “Rogue Wave Observation in a Water Wave Tank,” Phys. Rev. Lett. 106, 204502 (2011).
[Crossref] [PubMed]

J. M. Soto-Crespo, Ph. Grelu, and N. Akhmediev, “Dissipative rogue waves: Extreme pulses generated by passively mode-locked lasers,” Phys. Rev. E 84, 016604 (2011).
[Crossref]

2010 (4)

A. Ankiewicz, J. M. Soto-Crespo, and N. Akhmediev, “Rogue waves and rational solutions of the Hirota equation,” Phys. Rev. E 81, 046602 (2010).
[Crossref]

Yu. V. Bludov, V. V. Konotop, and N. Akhmediev, “Vector rogue waves in binary mixtures of Bose-Einstein condensates,” Eur. Phys. J. Special Topics. 185, 169 (2010).
[Crossref]

M. Shatz, H. Punzmann, and H. Xia, “Capillary rogue waves,” Phys. Rev. Lett 104, 104503 (2010).
[Crossref]

B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, “The Peregrine soliton in nonlinear fibre optics,” Nat. Phys. 6, 790 (2010).
[Crossref]

2009 (3)

Yu. V. Bludov, V. V. Konotop, and N. Akhmediev, “Matter rogue waves,” Phys. Rev. A 80, 033610 (2009).
[Crossref]

N. Akhmediev, J. M. Soto-Crespo, and A. Ankiewicz, “Extreme waves that appear from nowhere: On the nature of rogue waves,” Phys. Lett. A 373, 2137–2145 (2009).
[Crossref]

A. Montina, U. Bortolozzo, S. Residori, and F. T. Arecchi, “Non-Gaussian Statistics and Extreme Waves in a Nonlinear Optical Cavity,” Phys. Rev. Lett. 103, 173901 (2009).
[Crossref] [PubMed]

2008 (2)

A. N. Ganshin, V. B. Efimov, G. V. Kolmakov, L. P. Mezhov-Deglin, and P. V. E. McClintock, “Observation of an inverse energy cascade in developed acoustic turbulence in superfluid helium,” Phys. Rev. Lett. 101, 065303 (2008).
[Crossref] [PubMed]

Y. Guo, L. Zhou, L-M. Kuang, and C. P. Sun, “Magneto-optical Stern-Gerlach effect in an atomic ensemble,” Phys. Rev. A 78, 013833 (2008).
[Crossref]

2007 (1)

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature (London) 450, 1054 (2007).
[Crossref]

2006 (1)

L. Karpa and M. Weitz, “A Stern-Gerlach experiment for slow light,” Nat. Phys. 2, 332 (2006).
[Crossref]

1973 (1)

S. V. Manakov, “On the theory of two-dimensional stationary selffocusing of electromagnetic waves,” Zh. Eksp. Teor. Fiz. 65, 505 (1973) [Sov. Phys. JETP 38, 248 (1974)].

Akhmediev, N.

N. Akhmediev, J. M. Dudley, D. R. Solli, and S. K. Turitsyn, “Recent progress in investigating optical rogue waves,” J. Opt. 15, 060201 (2013).
[Crossref]

C. Lecaplain, Ph. Grelu, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative Rogue waves generated by chaotic pulse bunching in a mode-locked laser,” Phys. Rev. Lett. 108, 233901 (2012).
[Crossref] [PubMed]

A. Chabchoub, N. Hoffmann, M. Onorato, and N. Akhmediev, “Super Rogue Waves: Observation of a Higher-Order Breather in Water Waves,” Phys. Rev. X 2, 011015 (2012).

U. Bandelow and N. Akhmediev, “Sasa-Satsuma equation: Soliton on a background and its limiting cases,” Phys. Rev. E 86, 026606 (2012).
[Crossref]

Ph. Grelu and N. Akhmediev, “Dissipative solitons for mode-locked lasers,” Nat. Photon. 6, 84 (2012).
[Crossref]

A. Chabchoub, N. Hoffmann, M. Onorato, A. Slunyaev, A. Sergeeva, E. Pelinovsky, and N. Akhmediev, “Observation of a hierarchy of up to fifth-order rogue waves in a water tank,” Phys. Rev. E 86, 056601 (2012).
[Crossref]

A. Chabchoub, N. P. Hoffmann, and N. Akhmediev, “Rogue Wave Observation in a Water Wave Tank,” Phys. Rev. Lett. 106, 204502 (2011).
[Crossref] [PubMed]

J. M. Soto-Crespo, Ph. Grelu, and N. Akhmediev, “Dissipative rogue waves: Extreme pulses generated by passively mode-locked lasers,” Phys. Rev. E 84, 016604 (2011).
[Crossref]

A. Ankiewicz, J. M. Soto-Crespo, and N. Akhmediev, “Rogue waves and rational solutions of the Hirota equation,” Phys. Rev. E 81, 046602 (2010).
[Crossref]

Yu. V. Bludov, V. V. Konotop, and N. Akhmediev, “Vector rogue waves in binary mixtures of Bose-Einstein condensates,” Eur. Phys. J. Special Topics. 185, 169 (2010).
[Crossref]

B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, “The Peregrine soliton in nonlinear fibre optics,” Nat. Phys. 6, 790 (2010).
[Crossref]

Yu. V. Bludov, V. V. Konotop, and N. Akhmediev, “Matter rogue waves,” Phys. Rev. A 80, 033610 (2009).
[Crossref]

N. Akhmediev, J. M. Soto-Crespo, and A. Ankiewicz, “Extreme waves that appear from nowhere: On the nature of rogue waves,” Phys. Lett. A 373, 2137–2145 (2009).
[Crossref]

Ankiewicz, A.

A. Ankiewicz, J. M. Soto-Crespo, and N. Akhmediev, “Rogue waves and rational solutions of the Hirota equation,” Phys. Rev. E 81, 046602 (2010).
[Crossref]

N. Akhmediev, J. M. Soto-Crespo, and A. Ankiewicz, “Extreme waves that appear from nowhere: On the nature of rogue waves,” Phys. Lett. A 373, 2137–2145 (2009).
[Crossref]

Arecchi, F. T.

M. Onorato, S. Residori, U. Bortolozzo, A. Montinad, and F. T. Arecchi, “Rogue waves and their generating mechanisms in different physical contexts,” Phys. Reports 528, 47 (2013).
[Crossref]

A. Montina, U. Bortolozzo, S. Residori, and F. T. Arecchi, “Non-Gaussian Statistics and Extreme Waves in a Nonlinear Optical Cavity,” Phys. Rev. Lett. 103, 173901 (2009).
[Crossref] [PubMed]

Bailung, H.

H. Bailung, S. K. Sharma, and Y. Nakamura, “Observation of Peregrine solitons in a multicomponent plasma with negative ions,” Phys. Rev. Lett. 107, 255005 (2011).
[Crossref]

Bandelow, U.

U. Bandelow and N. Akhmediev, “Sasa-Satsuma equation: Soliton on a background and its limiting cases,” Phys. Rev. E 86, 026606 (2012).
[Crossref]

Baronio, F.

F. Baronio, M. Conforti, A. Degasperis, and S. Lombardo, “Rogue waves emerging from the resonant interaction of three waves,” Phys. Rev. Lett. 111, 114101 (2013).
[Crossref] [PubMed]

F. Baronio, A. Degasperis, M. Conforti, and S. Wabnitz, “Solutions of the vector nonlinear Schrödinger equations: Evidence for deterministic Rogue waves,” Phys. Rev. Lett. 109, 044102 (2012).
[Crossref]

Bludov, Yu. V.

Yu. V. Bludov, V. V. Konotop, and N. Akhmediev, “Vector rogue waves in binary mixtures of Bose-Einstein condensates,” Eur. Phys. J. Special Topics. 185, 169 (2010).
[Crossref]

Yu. V. Bludov, V. V. Konotop, and N. Akhmediev, “Matter rogue waves,” Phys. Rev. A 80, 033610 (2009).
[Crossref]

Bortolozzo, U.

M. Onorato, S. Residori, U. Bortolozzo, A. Montinad, and F. T. Arecchi, “Rogue waves and their generating mechanisms in different physical contexts,” Phys. Reports 528, 47 (2013).
[Crossref]

A. Montina, U. Bortolozzo, S. Residori, and F. T. Arecchi, “Non-Gaussian Statistics and Extreme Waves in a Nonlinear Optical Cavity,” Phys. Rev. Lett. 103, 173901 (2009).
[Crossref] [PubMed]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic, Elsevier, 2008).

Chabchoub, A.

A. Chabchoub, N. Hoffmann, M. Onorato, A. Slunyaev, A. Sergeeva, E. Pelinovsky, and N. Akhmediev, “Observation of a hierarchy of up to fifth-order rogue waves in a water tank,” Phys. Rev. E 86, 056601 (2012).
[Crossref]

A. Chabchoub, N. Hoffmann, M. Onorato, and N. Akhmediev, “Super Rogue Waves: Observation of a Higher-Order Breather in Water Waves,” Phys. Rev. X 2, 011015 (2012).

A. Chabchoub, N. P. Hoffmann, and N. Akhmediev, “Rogue Wave Observation in a Water Wave Tank,” Phys. Rev. Lett. 106, 204502 (2011).
[Crossref] [PubMed]

Chan, H. N.

K. W. Chow, H. N. Chan, D. J. Kedziora, and R. H. J. Grimshaw, “Rogue wave modes for the long wave-short wave resonance model,” J. Phys. Soc. Jpn. 82, 074001 (2013).
[Crossref]

Chen, S.

S. Chen and D. Mihalache, “Vector rogue waves in the Manakov system: diversity and compossibility,” J. Phys. A: Math. Theor. 48, 215202 (2015).
[Crossref]

S. Chen, Ph. Grelu, and J. M. Soto-Crespo, “Dark- and bright-rogue-wave solutions for media with long-wave–short-wave resonance,” Phys. Rev. E 89, 011201(R) (2014).
[Crossref]

S. Chen and L.-Y. Song, “Rogue waves in coupled Hirota systems,” Phys. Rev. E 87, 032910 (2013).
[Crossref]

S. Chen, “Twisted rogue-wave pairs in the Sasa-Satsuma equation,” Phys. Rev. E 88, 023202 (2013).
[Crossref]

Chen, Z.

Z. Chen and G. Huang, “Stern-Gerlach effect of multi-component ultraslow optical solitons via electromagnetically induced transparency,” J. Opt. Soc. Am. 30, 2248 (2013).
[Crossref]

Chow, K. W.

K. W. Chow, H. N. Chan, D. J. Kedziora, and R. H. J. Grimshaw, “Rogue wave modes for the long wave-short wave resonance model,” J. Phys. Soc. Jpn. 82, 074001 (2013).
[Crossref]

Conforti, M.

F. Baronio, M. Conforti, A. Degasperis, and S. Lombardo, “Rogue waves emerging from the resonant interaction of three waves,” Phys. Rev. Lett. 111, 114101 (2013).
[Crossref] [PubMed]

F. Baronio, A. Degasperis, M. Conforti, and S. Wabnitz, “Solutions of the vector nonlinear Schrödinger equations: Evidence for deterministic Rogue waves,” Phys. Rev. Lett. 109, 044102 (2012).
[Crossref]

Degasperis, A.

F. Baronio, M. Conforti, A. Degasperis, and S. Lombardo, “Rogue waves emerging from the resonant interaction of three waves,” Phys. Rev. Lett. 111, 114101 (2013).
[Crossref] [PubMed]

F. Baronio, A. Degasperis, M. Conforti, and S. Wabnitz, “Solutions of the vector nonlinear Schrödinger equations: Evidence for deterministic Rogue waves,” Phys. Rev. Lett. 109, 044102 (2012).
[Crossref]

Dias, F.

J. M. Dudley, F. Dias, M. Erkintalo, and G. Genty, “Instabilities, breathers and rogue waves in optics,” Nat. Photon. 8, 755 (2014).
[Crossref]

B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, “The Peregrine soliton in nonlinear fibre optics,” Nat. Phys. 6, 790 (2010).
[Crossref]

Dudley, J. M.

J. M. Dudley, F. Dias, M. Erkintalo, and G. Genty, “Instabilities, breathers and rogue waves in optics,” Nat. Photon. 8, 755 (2014).
[Crossref]

N. Akhmediev, J. M. Dudley, D. R. Solli, and S. K. Turitsyn, “Recent progress in investigating optical rogue waves,” J. Opt. 15, 060201 (2013).
[Crossref]

B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, “The Peregrine soliton in nonlinear fibre optics,” Nat. Phys. 6, 790 (2010).
[Crossref]

Efimov, V. B.

A. N. Ganshin, V. B. Efimov, G. V. Kolmakov, L. P. Mezhov-Deglin, and P. V. E. McClintock, “Observation of an inverse energy cascade in developed acoustic turbulence in superfluid helium,” Phys. Rev. Lett. 101, 065303 (2008).
[Crossref] [PubMed]

Egorov, O.

A. Zaviyalov, O. Egorov, R. Iliew, and F. Lederer, “Rogue waves in mode-locked fiber lasers,” Phys. Rev. A 85, 013828 (2012).
[Crossref]

Erkintalo, M.

J. M. Dudley, F. Dias, M. Erkintalo, and G. Genty, “Instabilities, breathers and rogue waves in optics,” Nat. Photon. 8, 755 (2014).
[Crossref]

Fatome, J.

B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, “The Peregrine soliton in nonlinear fibre optics,” Nat. Phys. 6, 790 (2010).
[Crossref]

Finot, C.

B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, “The Peregrine soliton in nonlinear fibre optics,” Nat. Phys. 6, 790 (2010).
[Crossref]

Foot, C. J.

C. J. Foot, Atomic Physics (Oxford University, 2005).

Ganshin, A. N.

A. N. Ganshin, V. B. Efimov, G. V. Kolmakov, L. P. Mezhov-Deglin, and P. V. E. McClintock, “Observation of an inverse energy cascade in developed acoustic turbulence in superfluid helium,” Phys. Rev. Lett. 101, 065303 (2008).
[Crossref] [PubMed]

Genty, G.

J. M. Dudley, F. Dias, M. Erkintalo, and G. Genty, “Instabilities, breathers and rogue waves in optics,” Nat. Photon. 8, 755 (2014).
[Crossref]

B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, “The Peregrine soliton in nonlinear fibre optics,” Nat. Phys. 6, 790 (2010).
[Crossref]

Grelu, Ph.

S. Chen, Ph. Grelu, and J. M. Soto-Crespo, “Dark- and bright-rogue-wave solutions for media with long-wave–short-wave resonance,” Phys. Rev. E 89, 011201(R) (2014).
[Crossref]

C. Lecaplain, Ph. Grelu, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative Rogue waves generated by chaotic pulse bunching in a mode-locked laser,” Phys. Rev. Lett. 108, 233901 (2012).
[Crossref] [PubMed]

Ph. Grelu and N. Akhmediev, “Dissipative solitons for mode-locked lasers,” Nat. Photon. 6, 84 (2012).
[Crossref]

J. M. Soto-Crespo, Ph. Grelu, and N. Akhmediev, “Dissipative rogue waves: Extreme pulses generated by passively mode-locked lasers,” Phys. Rev. E 84, 016604 (2011).
[Crossref]

Grimshaw, R. H. J.

K. W. Chow, H. N. Chan, D. J. Kedziora, and R. H. J. Grimshaw, “Rogue wave modes for the long wave-short wave resonance model,” J. Phys. Soc. Jpn. 82, 074001 (2013).
[Crossref]

Guo, Y.

Y. Guo, L. Zhou, L-M. Kuang, and C. P. Sun, “Magneto-optical Stern-Gerlach effect in an atomic ensemble,” Phys. Rev. A 78, 013833 (2008).
[Crossref]

Hang, C.

J. Liu, C. Hang, and G. Huang, “Weak-light rogue waves, breathers, and their active control in a cold atomic gas via electromagnetically induced transparency,” Phys. Rev. A 93, 063836 (2016).
[Crossref]

C. Hang and G. Huang, “Stern-Gerlach effect of weak-light ultraslow vector solitons,” Phys. Rev. A 86, 043809 (2012).
[Crossref]

Hoffmann, N.

A. Chabchoub, N. Hoffmann, M. Onorato, A. Slunyaev, A. Sergeeva, E. Pelinovsky, and N. Akhmediev, “Observation of a hierarchy of up to fifth-order rogue waves in a water tank,” Phys. Rev. E 86, 056601 (2012).
[Crossref]

A. Chabchoub, N. Hoffmann, M. Onorato, and N. Akhmediev, “Super Rogue Waves: Observation of a Higher-Order Breather in Water Waves,” Phys. Rev. X 2, 011015 (2012).

Hoffmann, N. P.

A. Chabchoub, N. P. Hoffmann, and N. Akhmediev, “Rogue Wave Observation in a Water Wave Tank,” Phys. Rev. Lett. 106, 204502 (2011).
[Crossref] [PubMed]

Huang, G.

J. Liu, C. Hang, and G. Huang, “Weak-light rogue waves, breathers, and their active control in a cold atomic gas via electromagnetically induced transparency,” Phys. Rev. A 93, 063836 (2016).
[Crossref]

Z. Chen and G. Huang, “Stern-Gerlach effect of multi-component ultraslow optical solitons via electromagnetically induced transparency,” J. Opt. Soc. Am. 30, 2248 (2013).
[Crossref]

C. Hang and G. Huang, “Stern-Gerlach effect of weak-light ultraslow vector solitons,” Phys. Rev. A 86, 043809 (2012).
[Crossref]

Iliew, R.

A. Zaviyalov, O. Egorov, R. Iliew, and F. Lederer, “Rogue waves in mode-locked fiber lasers,” Phys. Rev. A 85, 013828 (2012).
[Crossref]

Jalali, B.

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature (London) 450, 1054 (2007).
[Crossref]

Karpa, L.

L. Karpa and M. Weitz, “A Stern-Gerlach experiment for slow light,” Nat. Phys. 2, 332 (2006).
[Crossref]

Kedziora, D. J.

K. W. Chow, H. N. Chan, D. J. Kedziora, and R. H. J. Grimshaw, “Rogue wave modes for the long wave-short wave resonance model,” J. Phys. Soc. Jpn. 82, 074001 (2013).
[Crossref]

Kharif, C.

C. Kharif, E. Pelinovsky, and A. Slunyaev, Rogue waves in the ocean (Springer, Heidelberg, 2009).

Kibler, B.

B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, “The Peregrine soliton in nonlinear fibre optics,” Nat. Phys. 6, 790 (2010).
[Crossref]

Kolmakov, G. V.

A. N. Ganshin, V. B. Efimov, G. V. Kolmakov, L. P. Mezhov-Deglin, and P. V. E. McClintock, “Observation of an inverse energy cascade in developed acoustic turbulence in superfluid helium,” Phys. Rev. Lett. 101, 065303 (2008).
[Crossref] [PubMed]

Konotop, V. V.

Yu. V. Bludov, V. V. Konotop, and N. Akhmediev, “Vector rogue waves in binary mixtures of Bose-Einstein condensates,” Eur. Phys. J. Special Topics. 185, 169 (2010).
[Crossref]

Yu. V. Bludov, V. V. Konotop, and N. Akhmediev, “Matter rogue waves,” Phys. Rev. A 80, 033610 (2009).
[Crossref]

Koonath, P.

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature (London) 450, 1054 (2007).
[Crossref]

Kuang, L-M.

Y. Guo, L. Zhou, L-M. Kuang, and C. P. Sun, “Magneto-optical Stern-Gerlach effect in an atomic ensemble,” Phys. Rev. A 78, 013833 (2008).
[Crossref]

Lecaplain, C.

C. Lecaplain, Ph. Grelu, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative Rogue waves generated by chaotic pulse bunching in a mode-locked laser,” Phys. Rev. Lett. 108, 233901 (2012).
[Crossref] [PubMed]

Lederer, F.

A. Zaviyalov, O. Egorov, R. Iliew, and F. Lederer, “Rogue waves in mode-locked fiber lasers,” Phys. Rev. A 85, 013828 (2012).
[Crossref]

Liu, J.

J. Liu, C. Hang, and G. Huang, “Weak-light rogue waves, breathers, and their active control in a cold atomic gas via electromagnetically induced transparency,” Phys. Rev. A 93, 063836 (2016).
[Crossref]

Lombardo, S.

F. Baronio, M. Conforti, A. Degasperis, and S. Lombardo, “Rogue waves emerging from the resonant interaction of three waves,” Phys. Rev. Lett. 111, 114101 (2013).
[Crossref] [PubMed]

Manakov, S. V.

S. V. Manakov, “On the theory of two-dimensional stationary selffocusing of electromagnetic waves,” Zh. Eksp. Teor. Fiz. 65, 505 (1973) [Sov. Phys. JETP 38, 248 (1974)].

McClintock, P. V. E.

A. N. Ganshin, V. B. Efimov, G. V. Kolmakov, L. P. Mezhov-Deglin, and P. V. E. McClintock, “Observation of an inverse energy cascade in developed acoustic turbulence in superfluid helium,” Phys. Rev. Lett. 101, 065303 (2008).
[Crossref] [PubMed]

Mezhov-Deglin, L. P.

A. N. Ganshin, V. B. Efimov, G. V. Kolmakov, L. P. Mezhov-Deglin, and P. V. E. McClintock, “Observation of an inverse energy cascade in developed acoustic turbulence in superfluid helium,” Phys. Rev. Lett. 101, 065303 (2008).
[Crossref] [PubMed]

Mihalache, D.

D. Mihalache, “Multidimensional localized structures in optical and matter-wave media: A topical survey of recent literature”, Rom. Rep. Phys. 69, 403 (2017).

D. Mihalache, “Localized structures in nonlinear optical media: A selection of recent studies”, Rom. Rep. Phys. 67, 1383 (2015).

S. Chen and D. Mihalache, “Vector rogue waves in the Manakov system: diversity and compossibility,” J. Phys. A: Math. Theor. 48, 215202 (2015).
[Crossref]

Millot, G.

B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, “The Peregrine soliton in nonlinear fibre optics,” Nat. Phys. 6, 790 (2010).
[Crossref]

Montina, A.

A. Montina, U. Bortolozzo, S. Residori, and F. T. Arecchi, “Non-Gaussian Statistics and Extreme Waves in a Nonlinear Optical Cavity,” Phys. Rev. Lett. 103, 173901 (2009).
[Crossref] [PubMed]

Montinad, A.

M. Onorato, S. Residori, U. Bortolozzo, A. Montinad, and F. T. Arecchi, “Rogue waves and their generating mechanisms in different physical contexts,” Phys. Reports 528, 47 (2013).
[Crossref]

Nakamura, Y.

H. Bailung, S. K. Sharma, and Y. Nakamura, “Observation of Peregrine solitons in a multicomponent plasma with negative ions,” Phys. Rev. Lett. 107, 255005 (2011).
[Crossref]

Ohta, Y.

Y. Ohta and J. Yang, “Rogue waves in the Davey-Stewartson I equation,” Phys. Rev. E 86, 036604 (2012).
[Crossref]

Onorato, M.

M. Onorato, S. Residori, U. Bortolozzo, A. Montinad, and F. T. Arecchi, “Rogue waves and their generating mechanisms in different physical contexts,” Phys. Reports 528, 47 (2013).
[Crossref]

A. Chabchoub, N. Hoffmann, M. Onorato, A. Slunyaev, A. Sergeeva, E. Pelinovsky, and N. Akhmediev, “Observation of a hierarchy of up to fifth-order rogue waves in a water tank,” Phys. Rev. E 86, 056601 (2012).
[Crossref]

A. Chabchoub, N. Hoffmann, M. Onorato, and N. Akhmediev, “Super Rogue Waves: Observation of a Higher-Order Breather in Water Waves,” Phys. Rev. X 2, 011015 (2012).

Osborne, A. R.

A. R. Osborne, Nonlinear Ocean Waves (Academic Press, 2009).

Pelinovsky, E.

A. Chabchoub, N. Hoffmann, M. Onorato, A. Slunyaev, A. Sergeeva, E. Pelinovsky, and N. Akhmediev, “Observation of a hierarchy of up to fifth-order rogue waves in a water tank,” Phys. Rev. E 86, 056601 (2012).
[Crossref]

C. Kharif, E. Pelinovsky, and A. Slunyaev, Rogue waves in the ocean (Springer, Heidelberg, 2009).

Punzmann, H.

M. Shatz, H. Punzmann, and H. Xia, “Capillary rogue waves,” Phys. Rev. Lett 104, 104503 (2010).
[Crossref]

Residori, S.

M. Onorato, S. Residori, U. Bortolozzo, A. Montinad, and F. T. Arecchi, “Rogue waves and their generating mechanisms in different physical contexts,” Phys. Reports 528, 47 (2013).
[Crossref]

A. Montina, U. Bortolozzo, S. Residori, and F. T. Arecchi, “Non-Gaussian Statistics and Extreme Waves in a Nonlinear Optical Cavity,” Phys. Rev. Lett. 103, 173901 (2009).
[Crossref] [PubMed]

Ropers, C.

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature (London) 450, 1054 (2007).
[Crossref]

Sergeeva, A.

A. Chabchoub, N. Hoffmann, M. Onorato, A. Slunyaev, A. Sergeeva, E. Pelinovsky, and N. Akhmediev, “Observation of a hierarchy of up to fifth-order rogue waves in a water tank,” Phys. Rev. E 86, 056601 (2012).
[Crossref]

Sharma, S. K.

H. Bailung, S. K. Sharma, and Y. Nakamura, “Observation of Peregrine solitons in a multicomponent plasma with negative ions,” Phys. Rev. Lett. 107, 255005 (2011).
[Crossref]

Shatz, M.

M. Shatz, H. Punzmann, and H. Xia, “Capillary rogue waves,” Phys. Rev. Lett 104, 104503 (2010).
[Crossref]

Slunyaev, A.

A. Chabchoub, N. Hoffmann, M. Onorato, A. Slunyaev, A. Sergeeva, E. Pelinovsky, and N. Akhmediev, “Observation of a hierarchy of up to fifth-order rogue waves in a water tank,” Phys. Rev. E 86, 056601 (2012).
[Crossref]

C. Kharif, E. Pelinovsky, and A. Slunyaev, Rogue waves in the ocean (Springer, Heidelberg, 2009).

Solli, D. R.

N. Akhmediev, J. M. Dudley, D. R. Solli, and S. K. Turitsyn, “Recent progress in investigating optical rogue waves,” J. Opt. 15, 060201 (2013).
[Crossref]

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature (London) 450, 1054 (2007).
[Crossref]

Song, L.-Y.

S. Chen and L.-Y. Song, “Rogue waves in coupled Hirota systems,” Phys. Rev. E 87, 032910 (2013).
[Crossref]

Soto-Crespo, J. M.

S. Chen, Ph. Grelu, and J. M. Soto-Crespo, “Dark- and bright-rogue-wave solutions for media with long-wave–short-wave resonance,” Phys. Rev. E 89, 011201(R) (2014).
[Crossref]

C. Lecaplain, Ph. Grelu, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative Rogue waves generated by chaotic pulse bunching in a mode-locked laser,” Phys. Rev. Lett. 108, 233901 (2012).
[Crossref] [PubMed]

J. M. Soto-Crespo, Ph. Grelu, and N. Akhmediev, “Dissipative rogue waves: Extreme pulses generated by passively mode-locked lasers,” Phys. Rev. E 84, 016604 (2011).
[Crossref]

A. Ankiewicz, J. M. Soto-Crespo, and N. Akhmediev, “Rogue waves and rational solutions of the Hirota equation,” Phys. Rev. E 81, 046602 (2010).
[Crossref]

N. Akhmediev, J. M. Soto-Crespo, and A. Ankiewicz, “Extreme waves that appear from nowhere: On the nature of rogue waves,” Phys. Lett. A 373, 2137–2145 (2009).
[Crossref]

Sun, C. P.

Y. Guo, L. Zhou, L-M. Kuang, and C. P. Sun, “Magneto-optical Stern-Gerlach effect in an atomic ensemble,” Phys. Rev. A 78, 013833 (2008).
[Crossref]

Turitsyn, S. K.

N. Akhmediev, J. M. Dudley, D. R. Solli, and S. K. Turitsyn, “Recent progress in investigating optical rogue waves,” J. Opt. 15, 060201 (2013).
[Crossref]

Wabnitz, S.

F. Baronio, A. Degasperis, M. Conforti, and S. Wabnitz, “Solutions of the vector nonlinear Schrödinger equations: Evidence for deterministic Rogue waves,” Phys. Rev. Lett. 109, 044102 (2012).
[Crossref]

Weitz, M.

L. Karpa and M. Weitz, “A Stern-Gerlach experiment for slow light,” Nat. Phys. 2, 332 (2006).
[Crossref]

Xia, H.

M. Shatz, H. Punzmann, and H. Xia, “Capillary rogue waves,” Phys. Rev. Lett 104, 104503 (2010).
[Crossref]

Yan, Z.

Z. Yan, “Vector financial rogue waves,” Phys. Lett. A 375, 4274 (2011).
[Crossref]

Yang, J.

Y. Ohta and J. Yang, “Rogue waves in the Davey-Stewartson I equation,” Phys. Rev. E 86, 036604 (2012).
[Crossref]

Zaviyalov, A.

A. Zaviyalov, O. Egorov, R. Iliew, and F. Lederer, “Rogue waves in mode-locked fiber lasers,” Phys. Rev. A 85, 013828 (2012).
[Crossref]

Zhou, L.

Y. Guo, L. Zhou, L-M. Kuang, and C. P. Sun, “Magneto-optical Stern-Gerlach effect in an atomic ensemble,” Phys. Rev. A 78, 013833 (2008).
[Crossref]

Eur. Phys. J. Special Topics. (1)

Yu. V. Bludov, V. V. Konotop, and N. Akhmediev, “Vector rogue waves in binary mixtures of Bose-Einstein condensates,” Eur. Phys. J. Special Topics. 185, 169 (2010).
[Crossref]

J. Opt. (1)

N. Akhmediev, J. M. Dudley, D. R. Solli, and S. K. Turitsyn, “Recent progress in investigating optical rogue waves,” J. Opt. 15, 060201 (2013).
[Crossref]

J. Opt. Soc. Am. (1)

Z. Chen and G. Huang, “Stern-Gerlach effect of multi-component ultraslow optical solitons via electromagnetically induced transparency,” J. Opt. Soc. Am. 30, 2248 (2013).
[Crossref]

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

S. Chen and D. Mihalache, “Vector rogue waves in the Manakov system: diversity and compossibility,” J. Phys. A: Math. Theor. 48, 215202 (2015).
[Crossref]

J. Phys. Soc. Jpn. (1)

K. W. Chow, H. N. Chan, D. J. Kedziora, and R. H. J. Grimshaw, “Rogue wave modes for the long wave-short wave resonance model,” J. Phys. Soc. Jpn. 82, 074001 (2013).
[Crossref]

Nat. Photon. (2)

Ph. Grelu and N. Akhmediev, “Dissipative solitons for mode-locked lasers,” Nat. Photon. 6, 84 (2012).
[Crossref]

J. M. Dudley, F. Dias, M. Erkintalo, and G. Genty, “Instabilities, breathers and rogue waves in optics,” Nat. Photon. 8, 755 (2014).
[Crossref]

Nat. Phys. (2)

B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, “The Peregrine soliton in nonlinear fibre optics,” Nat. Phys. 6, 790 (2010).
[Crossref]

L. Karpa and M. Weitz, “A Stern-Gerlach experiment for slow light,” Nat. Phys. 2, 332 (2006).
[Crossref]

Nature (London) (1)

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature (London) 450, 1054 (2007).
[Crossref]

Phys. Lett. A (2)

Z. Yan, “Vector financial rogue waves,” Phys. Lett. A 375, 4274 (2011).
[Crossref]

N. Akhmediev, J. M. Soto-Crespo, and A. Ankiewicz, “Extreme waves that appear from nowhere: On the nature of rogue waves,” Phys. Lett. A 373, 2137–2145 (2009).
[Crossref]

Phys. Reports (1)

M. Onorato, S. Residori, U. Bortolozzo, A. Montinad, and F. T. Arecchi, “Rogue waves and their generating mechanisms in different physical contexts,” Phys. Reports 528, 47 (2013).
[Crossref]

Phys. Rev. A (5)

A. Zaviyalov, O. Egorov, R. Iliew, and F. Lederer, “Rogue waves in mode-locked fiber lasers,” Phys. Rev. A 85, 013828 (2012).
[Crossref]

J. Liu, C. Hang, and G. Huang, “Weak-light rogue waves, breathers, and their active control in a cold atomic gas via electromagnetically induced transparency,” Phys. Rev. A 93, 063836 (2016).
[Crossref]

Yu. V. Bludov, V. V. Konotop, and N. Akhmediev, “Matter rogue waves,” Phys. Rev. A 80, 033610 (2009).
[Crossref]

Y. Guo, L. Zhou, L-M. Kuang, and C. P. Sun, “Magneto-optical Stern-Gerlach effect in an atomic ensemble,” Phys. Rev. A 78, 013833 (2008).
[Crossref]

C. Hang and G. Huang, “Stern-Gerlach effect of weak-light ultraslow vector solitons,” Phys. Rev. A 86, 043809 (2012).
[Crossref]

Phys. Rev. E (8)

S. Chen and L.-Y. Song, “Rogue waves in coupled Hirota systems,” Phys. Rev. E 87, 032910 (2013).
[Crossref]

S. Chen, Ph. Grelu, and J. M. Soto-Crespo, “Dark- and bright-rogue-wave solutions for media with long-wave–short-wave resonance,” Phys. Rev. E 89, 011201(R) (2014).
[Crossref]

J. M. Soto-Crespo, Ph. Grelu, and N. Akhmediev, “Dissipative rogue waves: Extreme pulses generated by passively mode-locked lasers,” Phys. Rev. E 84, 016604 (2011).
[Crossref]

A. Ankiewicz, J. M. Soto-Crespo, and N. Akhmediev, “Rogue waves and rational solutions of the Hirota equation,” Phys. Rev. E 81, 046602 (2010).
[Crossref]

U. Bandelow and N. Akhmediev, “Sasa-Satsuma equation: Soliton on a background and its limiting cases,” Phys. Rev. E 86, 026606 (2012).
[Crossref]

S. Chen, “Twisted rogue-wave pairs in the Sasa-Satsuma equation,” Phys. Rev. E 88, 023202 (2013).
[Crossref]

Y. Ohta and J. Yang, “Rogue waves in the Davey-Stewartson I equation,” Phys. Rev. E 86, 036604 (2012).
[Crossref]

A. Chabchoub, N. Hoffmann, M. Onorato, A. Slunyaev, A. Sergeeva, E. Pelinovsky, and N. Akhmediev, “Observation of a hierarchy of up to fifth-order rogue waves in a water tank,” Phys. Rev. E 86, 056601 (2012).
[Crossref]

Phys. Rev. Lett (1)

M. Shatz, H. Punzmann, and H. Xia, “Capillary rogue waves,” Phys. Rev. Lett 104, 104503 (2010).
[Crossref]

Phys. Rev. Lett. (7)

A. Chabchoub, N. P. Hoffmann, and N. Akhmediev, “Rogue Wave Observation in a Water Wave Tank,” Phys. Rev. Lett. 106, 204502 (2011).
[Crossref] [PubMed]

H. Bailung, S. K. Sharma, and Y. Nakamura, “Observation of Peregrine solitons in a multicomponent plasma with negative ions,” Phys. Rev. Lett. 107, 255005 (2011).
[Crossref]

A. N. Ganshin, V. B. Efimov, G. V. Kolmakov, L. P. Mezhov-Deglin, and P. V. E. McClintock, “Observation of an inverse energy cascade in developed acoustic turbulence in superfluid helium,” Phys. Rev. Lett. 101, 065303 (2008).
[Crossref] [PubMed]

A. Montina, U. Bortolozzo, S. Residori, and F. T. Arecchi, “Non-Gaussian Statistics and Extreme Waves in a Nonlinear Optical Cavity,” Phys. Rev. Lett. 103, 173901 (2009).
[Crossref] [PubMed]

F. Baronio, A. Degasperis, M. Conforti, and S. Wabnitz, “Solutions of the vector nonlinear Schrödinger equations: Evidence for deterministic Rogue waves,” Phys. Rev. Lett. 109, 044102 (2012).
[Crossref]

C. Lecaplain, Ph. Grelu, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative Rogue waves generated by chaotic pulse bunching in a mode-locked laser,” Phys. Rev. Lett. 108, 233901 (2012).
[Crossref] [PubMed]

F. Baronio, M. Conforti, A. Degasperis, and S. Lombardo, “Rogue waves emerging from the resonant interaction of three waves,” Phys. Rev. Lett. 111, 114101 (2013).
[Crossref] [PubMed]

Phys. Rev. X (1)

A. Chabchoub, N. Hoffmann, M. Onorato, and N. Akhmediev, “Super Rogue Waves: Observation of a Higher-Order Breather in Water Waves,” Phys. Rev. X 2, 011015 (2012).

Rom. Rep. Phys. (2)

D. Mihalache, “Localized structures in nonlinear optical media: A selection of recent studies”, Rom. Rep. Phys. 67, 1383 (2015).

D. Mihalache, “Multidimensional localized structures in optical and matter-wave media: A topical survey of recent literature”, Rom. Rep. Phys. 69, 403 (2017).

Zh. Eksp. Teor. Fiz. (1)

S. V. Manakov, “On the theory of two-dimensional stationary selffocusing of electromagnetic waves,” Zh. Eksp. Teor. Fiz. 65, 505 (1973) [Sov. Phys. JETP 38, 248 (1974)].

Other (4)

C. J. Foot, Atomic Physics (Oxford University, 2005).

R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic, Elsevier, 2008).

C. Kharif, E. Pelinovsky, and A. Slunyaev, Rogue waves in the ocean (Springer, Heidelberg, 2009).

A. R. Osborne, Nonlinear Ocean Waves (Academic Press, 2009).

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Figures (7)

Fig. 1
Fig. 1 Possible geometrical arrangement and the coordinate frame chosen for the system. A cold atomic gas with a M-type five-level configuration (right side) are loaded in a gas cell (left side). A pulsed probe field E p drives respectively the transitions |3〉 ↔ |2〉 and |3〉 ↔ |4〉 by its σ polarization component and σ+ polarization component; Two strong CW control field Ec1 and Ec2 drive respectively the transitions |1〉 ↔ |2〉 and |5〉 ↔ |4〉. δcj, δp, and δp + Δ are detunings. Atoms are assumed to be prepared in the ground-state |3〉, indicated by black dots. B is a magnetic field applied to the atomic gas.
Fig. 2
Fig. 2 Linear dispersion relation of the system. (a) Im(K1) and Re(K1) as functions of ω. The red dashed (solid) line is Im(K1) for the case Ωc1 = 0 (Ωc1 = 1.0 × 107 s−1); the black dashed-dotted line is Re(K1) for Ωc1 = 1.0 × 107 s−1. (b) The same as (a) but for Im(K2) and Re(K2).
Fig. 3
Fig. 3 Vector optical rogue waves and breathers as functions of 2 τ / τ 0 and z/LD. The left and right panels are the intensity distributions for the first (i.e. σ) and the second (i.e. σ+) polarization components of the probe field, respectively. (a) [(b)] |ΩpKMB1/U0|2 [|ΩpKMB2/U0|2] obtained by using the vector Kuznetsov-Ma breather (13) as an initial condition; (c) [(d)] |Ωp1/U0|2 (left) [|Ωp2/U0|2] obtained by using the vector Peregrine soliton (11) as an initial condition; (e) [(f)] |ΩpAB1/U0|2 [|ΩpAB2/U0|2] obtained by using the vector Ahkmediev breathers (14) as an initial condition.
Fig. 4
Fig. 4 Collisions between two vector optical rogue waves and breathers. (a) ((b)) Motion trajectories of two Peregrine solitons with attractive (repulsive) interaction during collision; (c) ((d)) Motion trajectories of two Kuznetsov-Ma breathers with attractive (repulsive) interaction during collision. In panels (a) and (c), the initial phase between the two solitons φ = 0; in panels (b) and (d), φ = π/4.
Fig. 5
Fig. 5 Stern-Gerlach deflection of the vector Kuznetsov-Ma breather in the presence of a gradient magnetic field. (a) Intensity distribution of the first polarization component of the probe field, i.e. |ΩpKMB1/U0|2, as a function of t/τ0 and z/LDiff, with (b) the corresponding contour map (motion trajectory in the xz plane); (c) Intensity distribution of the second polarization component of the probe field, i.e. |ΩpKMB2/U0|2, with (d) the corresponding contour map (motion trajectory in the xz plane). The initial condition is chosen to be the Kuznetsov-Ma breather (24), with α = π/3 and Ω = 2 .
Fig. 6
Fig. 6 Stern-Gerlach deflection of the Kuznetsov-Ma breather in the presence of a gradient magnetic field. (a) Intensity distribution of the first polarization component of probe field, i.e. |ΩpKMB1/U0|2 versus 2 x / R and z/LDiff, with (b) the corresponding contour map; (c)Intensity distribution of the second polarization component of probe field, i.e. |ΩpKMB2/U0|2, with (d) the corresponding contour map. Due to the different velocities, the deflections of the two polarization components are asymmetric.
Fig. 7
Fig. 7 Symmetrical and asymmetrical SG deflections of the Kuznetsov-Ma breather. (a) SG deflection distance x/R as a function of z/LDiff for B1 = 2.2 × 10−3 G cm−1 (blue lines) and 4.4 × 10−3 G cm−1 (red lines). The upbent lines are for the first (i.e. σ) polarization component, and downbent lines are for the second (i.e. σ+) polarization component. The solid lines (dashed lines) are for the symmetrical (asymmetrical) SG deflections, obtained by taking Ωc2 to be 1.6 × 108 s−1 (1.54 × 108 s−1). (b) SG deflection angles θ as a function of the magnetic field gradient B1 when the breather propagating to z = 4LDiff ≈ 10.4 cm. The blue (red) line is for σ (σ+) polarization component.

Equations (57)

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( t + Γ ) ρ = i [ H int , ρ ] ,
[ i ( z + 1 c t ) + c 2 ω p 2 ] Ω p 1 , p 2 κ 32 , 34 ρ 23 , 43 = 0 ,
K 1 , 2 ( ω ) = ω c + κ 32 , 34 ω d 1 , 5 D 1 , 2 ,
i ( F j z 1 + 1 V g j F j t 1 ) = 0 , ( j = 1 , 2 )
V g 1 , g 2 = { 1 c + κ 32 , 34 | Ω c 1 , c 2 | 2 + ( ω + d 1 , 5 ) 2 D 1 , 2 2 } 1 ,
i F 1 z 2 K 12 2 2 F 1 t 1 2 ( W 11 | F 1 | 2 + W 12 | F 2 | 2 ) e 2 α ¯ 1 z 2 F 1 = 0 ,
i F 2 z 2 K 22 2 2 F 2 t 1 2 ( W 21 | F 1 | 2 + W 22 | F 2 | 2 ) e 2 α ¯ 2 z 2 F 2 = 0 ,
K 12 , 22 = 2 κ 32 , 34 d 2 , 4 | Ω c 1 , c 2 | 2 + 2 d 1 , 5 | Ω c 1 , c 2 | 2 + d 1 , 5 3 ( | Ω c 1 , c 2 | 2 d 2 , 4 d 1 , 5 ) 3 ,
W 11 , 22 = κ 32 , 34 d 1 , 5 | d 1 , 5 | 2 + | Ω c 1 , c 2 | 2 D 1 , 2 | D 1 , 2 | 2 ,
W 12 , 21 = κ 32 , 34 d 1 , 5 | d 5 , 1 | 2 + | Ω c 2 , c 1 | 2 D 1 , 2 | D 2 , 1 | 2 ,
i u 1 s + i g δ u 1 σ g D 1 2 u 1 σ 2 ( g 11 | u 1 | 2 + g 12 | u 2 | 2 ) u 1 = i a 1 u 1 ,
i u 2 s i g δ u 2 σ g D 2 2 u 2 σ 2 ( g 22 | u 2 | 2 + g 21 | u 1 | 2 ) u 2 = i a 2 u 2 ,
V g 1 2.28 × 10 4 c ,
V g 2 2.26 × 10 4 c ,
i u 1 s + 2 u 1 σ 2 + ( | u 1 | 2 + | u 2 | 2 ) u 1 = 0 ,
i u 2 s + 2 u 2 σ 2 + ( | u 2 | 2 + | u 1 | 2 ) u 2 = 0 .
u 1 ( s , σ ) = 1 2 ( cos α + sin α ) ( 1 4 1 + 2 i s 1 + 4 s 2 + 2 σ 2 ) e i s ,
u 2 ( s , σ ) = 1 2 ( cos α sin α ) ( 1 4 1 + 2 i s 1 + 4 s 2 + 2 σ 2 ) e i s ,
Ω p 1 ( z , τ ) = U 0 2 ( cos α + sin α ) [ 1 4 1 + 2 i ( z / L D ) 1 + 4 ( z / L D ) 2 + 2 ( 2 τ / τ 0 ) 2 ] e i z / L D ,
Ω p 2 ( z , τ ) = U 0 2 ( cos α sin α ) [ 1 4 1 + 2 i ( z / L D ) 1 + 4 ( z / L D ) 2 + 2 ( 2 τ / τ 0 ) 2 ] e i z / L D ,
Ω p K B M 1 ( z , τ ) ( z , τ ) = U 0 2 ( 1 4 q ) cos ( a z / L D ) + 2 q cosh ( Ω τ / τ 0 ) i a sin ( a z / L D ) 2 q cosh ( Ω τ / τ 0 ) cos ( a z / L D ) × ( cos α + sin α ) e i z / L D ,
Ω p K B M 2 ( z , τ ) ( z , τ ) = U 0 2 ( 1 4 q ) cos ( a z / L D ) + 2 q cosh ( Ω τ / τ 0 ) i a sin ( a z / L D ) 2 q cosh ( Ω τ / τ 0 ) cos ( a z / L D ) × ( cos α sin α ) e i z / L D .
Ω p A B 1 ( z , τ ) = U 0 2 ( 1 4 q ) cosh ( a z / L D ) + 2 q cos ( Ω τ / τ 0 ) + i a sinh ( a z / L D ) 2 q cos ( Ω τ / τ 0 ) cosh ( a z / L D )
Ω p A B 2 ( z , τ ) = U 0 2 ( 1 4 q ) cosh ( a z / L D ) + 2 q cos ( Ω τ / τ 0 ) + i a sinh ( a z / L D ) 2 q cos ( Ω τ / τ 0 ) cosh ( a z / L D )
V g 2.27 × 10 4 c .
P th 0.97 μ W .
B = z ^ B ( x ) = z ^ ( B 0 + B 1 x ) ,
i ( z 2 + 1 V g 1 t 2 ) F 1 + c 2 ω p 2 F 1 x 1 2 ( W 11 | F 1 | 2 + W 12 | F 2 | 2 ) e 2 a ¯ 1 z 2 F 1 + M 1 B 1 x 1 F 1 = 0 ,
i ( z 2 + 1 V g 2 t 2 ) F 2 + c 2 ω p 2 F 2 x 1 2 ( W 22 | F 2 | 2 + W 21 | F 1 | 2 ) e 2 a ¯ 2 z 2 F 2 + M 2 B 2 x 1 F 2 = 0 ,
i ( s + 1 λ 1 τ ) u 1 + 2 u 1 ξ 2 ( g 11 | u 1 | 2 + g 12 | u 2 | 2 ) u 1 + 1 ξ u 1 = i b 1 u 1 ,
i ( s + 1 λ 2 τ ) u 2 + 2 u 2 ξ 2 ( g 22 | u 1 | 2 + g 21 | u 2 | 2 ) u 2 + 2 ξ u 2 = i b 2 u 2 ,
i ( s + 1 λ 1 τ ) u 1 + 2 u 1 ξ 2 g 11 | u 1 | 2 u 1 + 1 ξ u 1 = i b 1 u 1 ,
i ( s + 1 λ 2 τ ) u 2 + 2 u 2 ξ 2 g 22 | u 1 | 2 u 1 + 2 ξ u 2 = i b 2 u 2 .
G 1 , 2 ( s , τ ) = 2 1 / 4 g 11 , 22 e ( s λ 1 , 2 τ ) 2 / 4 = 2 1 / 4 g 11 , 22 e ( z V g 1 , 2 t ) 2 / ( 4 L Diff 2 ) .
i λ 1 v 1 τ + 2 v 1 ξ 2 | v 1 | 2 v 1 + 1 ξ v 1 = 0 ,
i λ 2 v 2 τ + 2 v 2 ξ 2 | v 2 | 2 v 2 + 2 ξ v 2 = 0 .
u 1 ( s , τ , ξ ) = ( 1 4 q ) cos ( a λ 1 τ ) + 2 q cosh [ 2 Ω ( ξ 1 λ 1 2 τ 2 / 2 ) ] i a sin ( a λ 1 τ ) 2 q cosh [ 2 Ω ( ξ 1 λ 1 2 τ 2 / 2 ) ] cos ( a λ 1 τ ) × 2 1 / 4 g 11 ( cos α + sin α ) e i λ 1 τ + i 1 λ ( ξ 1 λ 1 2 τ 2 / 6 ) τ ( s λ 1 τ ) 2 / 4 ,
u 2 ( s , τ , ξ ) = ( 1 4 q ) cos ( a λ 2 τ ) + 2 q cosh [ 2 Ω ( ξ 2 λ 2 2 τ 2 / 2 ) ] i a sin ( a λ 2 τ ) 2 q cosh [ 2 Ω ( ξ 2 λ 2 2 τ 2 / 2 ) ] cos ( a λ 2 τ ) × 2 1 / 4 g 22 ( cos α + sin α ) e i λ 2 τ + i 2 λ ( ξ 2 λ 2 2 τ 2 / 6 ) τ ( s λ 2 τ ) 2 / 4 ,
Ω p K B M 1 ( z , t ) = U 0 ( 1 4 q ) cos ( a λ 1 t / τ 0 ) + 2 q cosh [ 2 Ω ( x / R 1 λ 1 2 t 2 / ( 2 τ 0 2 ) ) ] i a sin ( a λ 1 t / τ 0 ) 2 q cosh [ 2 Ω ( x / R 1 λ 1 2 t 2 / ( 2 τ 0 2 ) ) ] cos ( a λ 1 t / τ 0 ) × 2 1 / 4 g 11 ( cos α + sin α ) e i λ 1 t / τ 0 + i 1 λ 1 [ x / R 1 λ 1 2 t 2 / ( 6 τ 0 2 ) ] t / τ 0 ( z V g 1 t ) 2 / ( 4 L Diff 2 ) ,
Ω p K B M 2 ( z , t ) = U 0 ( 1 4 q ) cos ( a λ 2 t / τ 0 ) + 2 q cosh [ 2 Ω ( x / R 2 λ 2 2 t 2 / ( 2 τ 0 2 ) ) ] i a sin ( a λ 2 t / τ 0 ) 2 q cosh [ 2 Ω ( x / R 2 λ 2 2 t 2 / ( 2 τ 0 2 ) ) ] cos ( a λ 2 t / τ 0 ) × 2 1 / 4 g 22 ( cos α sin α ) e i λ 2 t / τ 0 + i 2 λ 2 [ x / R 2 λ 2 2 t 2 / ( 6 τ 0 2 ) ] t / τ 0 ( z V g 2 t ) 2 / ( 4 L Diff 2 ) .
( x , y , z ) = ( R j λ j 2 2 τ 0 2 t 2 , 0 , V g j t ) .
θ j = V j V g j = M j R 2 2 L L Diff B 1
ρ 11 t = i Ω c 1 * ρ 21 + i Ω c 1 ρ 12 + Γ 41 ρ 44 + Γ 21 ρ 22 ,
ρ 22 t = i Ω c 1 ρ 12 + i Ω c 1 * ρ 21 + i Ω p 1 * ρ 23 i Ω p 1 ρ 32 Γ 2 ρ 22 ,
ρ 33 t = i Ω p 1 * ρ 23 + i Ω p 1 ρ 32 + i Ω p 2 ρ 34 i Ω p 2 * ρ 43 + Γ 43 ρ 44 + Γ 23 ρ 22 ,
ρ 44 t = i Ω p 2 ρ 34 + i Ω p 2 * ρ 43 + i Ω c 2 * ρ 45 i Ω c 2 ρ 54 Γ 4 ρ 44 ,
ρ 55 t = i Ω c 2 * ρ 45 + i Ω c 2 ρ 54 + Γ 45 ρ 44 + Γ 25 ρ 22 ,
ρ 12 t = i δ c 1 ρ 12 i Ω c 1 * ( ρ 22 ρ 11 ) + i Ω p 1 * σ 13 Γ 2 + γ 12 2 ρ 12 ,
ρ 13 t = i ( δ c 1 δ p ) ρ 13 + i Ω p 1 ρ 12 + i Ω p 2 ρ 14 i Ω c 1 * ρ 23 γ 13 2 ρ 13 ,
ρ 14 t = i ( Δ + δ c 1 ) ρ 14 + i Ω p 2 * ρ 13 + i Ω c 2 * ρ 15 i Ω c 1 * ρ 24 Γ 4 + γ 14 2 ρ 14 ,
ρ 15 t = i ( Δ + δ c 1 δ c 2 ) ρ 15 + i Ω c 2 ρ 14 i Ω c 1 * ρ 25 γ 15 2 ρ 15 ,
ρ 23 t = i δ p ρ 23 i Ω c 1 ρ 13 + i Ω p 2 ρ 24 i Ω p 1 ( ρ 33 ρ 22 ) Γ 2 + γ 23 2 ρ 23 ,
ρ 24 t = i Δ ρ 24 i Ω c 1 ρ 14 + i Ω c 2 * ρ 25 + i Ω p 2 * ρ 23 i Ω p 1 ρ 34 Γ 2 + Γ 4 + γ 24 2 ρ 24 ,
ρ 25 t = i ( Δ δ c 2 ) ρ 25 i Ω c 1 ρ 15 i Ω p 1 ρ 35 + i Ω c 2 ρ 24 Γ 2 + γ 25 2 ρ 25 ,
ρ 34 t = i ( δ p + Δ ) ρ 34 i Ω p 1 * ρ 24 + i Ω c 2 * ρ 35 i Ω p 2 * ( ρ 44 ρ 33 ) Γ 4 + γ 34 2 ρ 34 ,
ρ 35 t = i ( δ p + Δ δ c 2 ) ρ 35 i Ω p 1 * ρ 25 i Ω p 2 * ρ 45 + i Ω c 2 ρ 34 γ 35 2 ρ 35 ,
ρ 45 t = i δ c 2 ρ 45 i Ω c 2 ( ρ 55 ρ 44 ) i Ω p 2 ρ 35 Γ 4 + γ 45 2 ρ 45 ,

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