Abstract

We propose a scheme to exhibit a Stern–Gerlach effect of n-component (n>2) high-dimensional ultraslow optical solitons in a coherent atomic system with (n+1)-pod level configuration via electromagnetically induced transparency (EIT). Based on Maxwell–Bloch equations, we derive coupled (3+1)-dimensional nonlinear Schrödinger equations governing the spatial-temporal evolution of n probe-field envelopes. We show that under EIT conditions significant deflections of the n components of coupled ultraslow optical solitons can be achieved by using a Stern–Gerlach gradient magnetic field. The stability of the ultraslow optical solitons can be realized by an optical lattice potential contributed from a far-detuned laser field.

© 2013 Optical Society of America

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  1. Y. Silberberg, “Collapse of optical pulses,” Opt. Lett. 15, 1282–1284 (1990).
    [CrossRef]
  2. Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).
  3. M. Fleischhauer, A. Imamoğlu, and J. P. Marangos, “Electromagnetically induced transparency: optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
    [CrossRef]
  4. H.-j. Li, Y.-p. Wu, and G. Huang, “Stable weak light ultraslow spatiotemporal solitons via atomic coherence,” Phys. Rev. A 84, 033816 (2011).
    [CrossRef]
  5. R. Harré, Great Scientific Experiments: 20 Experiments that Changed our View of the World (Phaidon, Oxford, 1981).
  6. J. J. Sakurai, Modern Quantum Mechanics, rev. ed. (Addison-Wesley, 1994).
  7. M. A. Nielsen and I. L. Chang, Quantum Computation and Quantum Information (Cambridge, 2000).
  8. N. A. Kuebler, M. B. Robin, J. J. Yang, and A. Gedanken, “Fully resolved Zeeman pattern in the Stern–Gerlach deflection spectrum of O2 (Σg−3, K=1),” Phys. Rev. A 38, 737–749 (1988).
    [CrossRef]
  9. A. Gedanken, N. A. Kuebler, M. B. Robin, and D. R. Herrick, “Stern–Gerlach deflection spectra of nitrogen oxide radicals,” J. Chem. Phys. 90, 3981–3993 (1989).
    [CrossRef]
  10. Y. Li, C. Bruder, and C. P. Sun, “Generalized Stern–Gerlach effect for chiral molecules,” Phys. Rev. Lett. 99, 130403 (2007).
    [CrossRef]
  11. W. D. Knight, R. Monot, E. R. Dietz, and A. R. George, “Stern–Gerlach deflection of metallic-cluster beams,” Phys. Rev. Lett. 40, 1324–1326 (1978).
    [CrossRef]
  12. S. Pokrant, “Evidence for adiabatic magnetization of cold DyN clusters,” Phys. Rev. A 62, 051201(R) (2000).
    [CrossRef]
  13. X. Xu, S. Yin, R. Moro, and W. A. de Heer, “Magnetic moments and adiabatic magnetization of free cobalt clusters,” Phys. Rev. Lett. 95, 237209 (2005).
    [CrossRef]
  14. F. W. Payne, W. Jiang, J. W. Emmert, J. Deng, and L. A. Bloomfield, “Magnetic structure of free cobalt clusters studied with Stern–Gerlach deflection experiments,” Phys. Rev. B 75, 094431 (2007).
    [CrossRef]
  15. S. Peredkov, M. Neeb, W. Eberhardt, J. Meyer, M. Tombers, H. Kampschulte, and G. Niedner-Schatteburg, “Spin and orbital magnetic moments of free nanoparticles,” Phys. Rev. Lett. 107, 233401 (2011).
    [CrossRef]
  16. L. Karpa and M. Weitz, “A Stern–Gerlach experiment for slow light,” Nat. Phys. 2, 332–335 (2006).
    [CrossRef]
  17. C. Hang and G. Huang, “Stern–Gerlach effect of weak-light ultraslow vector solitons,” Phys. Rev. A 86, 043809 (2012).
    [CrossRef]
  18. G. Herzberg, Atomic Spectra and Atomic Structure (Dover, 1944).
  19. The far-detuned optical field (2) will generates a magnetic field Bm(x,t)=z^(E0/c)sin(x/R⊥)sin(ωLt). In our model, we choose ωL∼1013  Hz, which is far from any resonance in atoms. It is easy to show that the magnitude of the effective magnetic field resulted from Bm(x,t) is proportional to E02/(4c2B0), which is around 6.94×10−4 Gauss when taking E0∼5×104  V/m and B0=1.0×103 Gauss. Because the second term of the SG magnetic field |B1y|≤|B1Ly|∼4.32×10−3 Gauss for Ly∼100R⊥ (Ly is the deflection distance in the y direction used below), the magnetic field Bm generated by the far-detuned optical field (2) can be safely neglected for the SG deflection considered here.
  20. B. B. Baizakov, B. A. Malomed, and M. Salerno, “Multidimensional solitons in a low-dimensional periodic potential,” Phys. Rev. A 70, 053613 (2004).
    [CrossRef]
  21. R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic, 2008).
  22. G. Huang, L. Deng, and M. G. Payne, “Dynamics of ultraslow optical solitons in a cold three-state atomic system,” Phys. Rev. E 72, 016617 (2005).
    [CrossRef]
  23. In the atomic medium, the frequency and wave number of the jth probe field are given by ωpj+ω and kpj+Kj(ω), respectively. Thus ω=0 corresponds to the center frequency of all probe fields.
  24. D. A. Steck, “Rubidium 87 D line data,” http://steck.us/alkalidata/ .
  25. 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]
  26. J. Yan, G. Zhou, and J. You, “Nonpropagating soliton and kink soliton in a mildly sloping channel,” Phys. Fluids A 4, 690–694 (1992).
    [CrossRef]
  27. Photons in vacuum have no magnetic moment, and hence experience no force and have no trajectory deflection when passing through an inhomogeneous magnetic field. However, photons may acquire effective magnetic moments in atomic media, thus experience a magnetic force and display the SG effect.

2012

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

2011

S. Peredkov, M. Neeb, W. Eberhardt, J. Meyer, M. Tombers, H. Kampschulte, and G. Niedner-Schatteburg, “Spin and orbital magnetic moments of free nanoparticles,” Phys. Rev. Lett. 107, 233401 (2011).
[CrossRef]

H.-j. Li, Y.-p. Wu, and G. Huang, “Stable weak light ultraslow spatiotemporal solitons via atomic coherence,” Phys. Rev. A 84, 033816 (2011).
[CrossRef]

2008

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

F. W. Payne, W. Jiang, J. W. Emmert, J. Deng, and L. A. Bloomfield, “Magnetic structure of free cobalt clusters studied with Stern–Gerlach deflection experiments,” Phys. Rev. B 75, 094431 (2007).
[CrossRef]

Y. Li, C. Bruder, and C. P. Sun, “Generalized Stern–Gerlach effect for chiral molecules,” Phys. Rev. Lett. 99, 130403 (2007).
[CrossRef]

2006

L. Karpa and M. Weitz, “A Stern–Gerlach experiment for slow light,” Nat. Phys. 2, 332–335 (2006).
[CrossRef]

2005

G. Huang, L. Deng, and M. G. Payne, “Dynamics of ultraslow optical solitons in a cold three-state atomic system,” Phys. Rev. E 72, 016617 (2005).
[CrossRef]

X. Xu, S. Yin, R. Moro, and W. A. de Heer, “Magnetic moments and adiabatic magnetization of free cobalt clusters,” Phys. Rev. Lett. 95, 237209 (2005).
[CrossRef]

M. Fleischhauer, A. Imamoğlu, and J. P. Marangos, “Electromagnetically induced transparency: optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
[CrossRef]

2004

B. B. Baizakov, B. A. Malomed, and M. Salerno, “Multidimensional solitons in a low-dimensional periodic potential,” Phys. Rev. A 70, 053613 (2004).
[CrossRef]

2000

S. Pokrant, “Evidence for adiabatic magnetization of cold DyN clusters,” Phys. Rev. A 62, 051201(R) (2000).
[CrossRef]

1992

J. Yan, G. Zhou, and J. You, “Nonpropagating soliton and kink soliton in a mildly sloping channel,” Phys. Fluids A 4, 690–694 (1992).
[CrossRef]

1990

1989

A. Gedanken, N. A. Kuebler, M. B. Robin, and D. R. Herrick, “Stern–Gerlach deflection spectra of nitrogen oxide radicals,” J. Chem. Phys. 90, 3981–3993 (1989).
[CrossRef]

1988

N. A. Kuebler, M. B. Robin, J. J. Yang, and A. Gedanken, “Fully resolved Zeeman pattern in the Stern–Gerlach deflection spectrum of O2 (Σg−3, K=1),” Phys. Rev. A 38, 737–749 (1988).
[CrossRef]

1978

W. D. Knight, R. Monot, E. R. Dietz, and A. R. George, “Stern–Gerlach deflection of metallic-cluster beams,” Phys. Rev. Lett. 40, 1324–1326 (1978).
[CrossRef]

Agrawal, G. P.

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).

Baizakov, B. B.

B. B. Baizakov, B. A. Malomed, and M. Salerno, “Multidimensional solitons in a low-dimensional periodic potential,” Phys. Rev. A 70, 053613 (2004).
[CrossRef]

Bloomfield, L. A.

F. W. Payne, W. Jiang, J. W. Emmert, J. Deng, and L. A. Bloomfield, “Magnetic structure of free cobalt clusters studied with Stern–Gerlach deflection experiments,” Phys. Rev. B 75, 094431 (2007).
[CrossRef]

Boyd, R. W.

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

Bruder, C.

Y. Li, C. Bruder, and C. P. Sun, “Generalized Stern–Gerlach effect for chiral molecules,” Phys. Rev. Lett. 99, 130403 (2007).
[CrossRef]

Chang, I. L.

M. A. Nielsen and I. L. Chang, Quantum Computation and Quantum Information (Cambridge, 2000).

de Heer, W. A.

X. Xu, S. Yin, R. Moro, and W. A. de Heer, “Magnetic moments and adiabatic magnetization of free cobalt clusters,” Phys. Rev. Lett. 95, 237209 (2005).
[CrossRef]

Deng, J.

F. W. Payne, W. Jiang, J. W. Emmert, J. Deng, and L. A. Bloomfield, “Magnetic structure of free cobalt clusters studied with Stern–Gerlach deflection experiments,” Phys. Rev. B 75, 094431 (2007).
[CrossRef]

Deng, L.

G. Huang, L. Deng, and M. G. Payne, “Dynamics of ultraslow optical solitons in a cold three-state atomic system,” Phys. Rev. E 72, 016617 (2005).
[CrossRef]

Dietz, E. R.

W. D. Knight, R. Monot, E. R. Dietz, and A. R. George, “Stern–Gerlach deflection of metallic-cluster beams,” Phys. Rev. Lett. 40, 1324–1326 (1978).
[CrossRef]

Eberhardt, W.

S. Peredkov, M. Neeb, W. Eberhardt, J. Meyer, M. Tombers, H. Kampschulte, and G. Niedner-Schatteburg, “Spin and orbital magnetic moments of free nanoparticles,” Phys. Rev. Lett. 107, 233401 (2011).
[CrossRef]

Emmert, J. W.

F. W. Payne, W. Jiang, J. W. Emmert, J. Deng, and L. A. Bloomfield, “Magnetic structure of free cobalt clusters studied with Stern–Gerlach deflection experiments,” Phys. Rev. B 75, 094431 (2007).
[CrossRef]

Fleischhauer, M.

M. Fleischhauer, A. Imamoğlu, and J. P. Marangos, “Electromagnetically induced transparency: optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
[CrossRef]

Gedanken, A.

A. Gedanken, N. A. Kuebler, M. B. Robin, and D. R. Herrick, “Stern–Gerlach deflection spectra of nitrogen oxide radicals,” J. Chem. Phys. 90, 3981–3993 (1989).
[CrossRef]

N. A. Kuebler, M. B. Robin, J. J. Yang, and A. Gedanken, “Fully resolved Zeeman pattern in the Stern–Gerlach deflection spectrum of O2 (Σg−3, K=1),” Phys. Rev. A 38, 737–749 (1988).
[CrossRef]

George, A. R.

W. D. Knight, R. Monot, E. R. Dietz, and A. R. George, “Stern–Gerlach deflection of metallic-cluster beams,” Phys. Rev. Lett. 40, 1324–1326 (1978).
[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.

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

Harré, R.

R. Harré, Great Scientific Experiments: 20 Experiments that Changed our View of the World (Phaidon, Oxford, 1981).

Herrick, D. R.

A. Gedanken, N. A. Kuebler, M. B. Robin, and D. R. Herrick, “Stern–Gerlach deflection spectra of nitrogen oxide radicals,” J. Chem. Phys. 90, 3981–3993 (1989).
[CrossRef]

Herzberg, G.

G. Herzberg, Atomic Spectra and Atomic Structure (Dover, 1944).

Huang, G.

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

H.-j. Li, Y.-p. Wu, and G. Huang, “Stable weak light ultraslow spatiotemporal solitons via atomic coherence,” Phys. Rev. A 84, 033816 (2011).
[CrossRef]

G. Huang, L. Deng, and M. G. Payne, “Dynamics of ultraslow optical solitons in a cold three-state atomic system,” Phys. Rev. E 72, 016617 (2005).
[CrossRef]

Imamoglu, A.

M. Fleischhauer, A. Imamoğlu, and J. P. Marangos, “Electromagnetically induced transparency: optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
[CrossRef]

Jiang, W.

F. W. Payne, W. Jiang, J. W. Emmert, J. Deng, and L. A. Bloomfield, “Magnetic structure of free cobalt clusters studied with Stern–Gerlach deflection experiments,” Phys. Rev. B 75, 094431 (2007).
[CrossRef]

Kampschulte, H.

S. Peredkov, M. Neeb, W. Eberhardt, J. Meyer, M. Tombers, H. Kampschulte, and G. Niedner-Schatteburg, “Spin and orbital magnetic moments of free nanoparticles,” Phys. Rev. Lett. 107, 233401 (2011).
[CrossRef]

Karpa, L.

L. Karpa and M. Weitz, “A Stern–Gerlach experiment for slow light,” Nat. Phys. 2, 332–335 (2006).
[CrossRef]

Kivshar, Y. S.

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).

Knight, W. D.

W. D. Knight, R. Monot, E. R. Dietz, and A. R. George, “Stern–Gerlach deflection of metallic-cluster beams,” Phys. Rev. Lett. 40, 1324–1326 (1978).
[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]

Kuebler, N. A.

A. Gedanken, N. A. Kuebler, M. B. Robin, and D. R. Herrick, “Stern–Gerlach deflection spectra of nitrogen oxide radicals,” J. Chem. Phys. 90, 3981–3993 (1989).
[CrossRef]

N. A. Kuebler, M. B. Robin, J. J. Yang, and A. Gedanken, “Fully resolved Zeeman pattern in the Stern–Gerlach deflection spectrum of O2 (Σg−3, K=1),” Phys. Rev. A 38, 737–749 (1988).
[CrossRef]

Li, H.-j.

H.-j. Li, Y.-p. Wu, and G. Huang, “Stable weak light ultraslow spatiotemporal solitons via atomic coherence,” Phys. Rev. A 84, 033816 (2011).
[CrossRef]

Li, Y.

Y. Li, C. Bruder, and C. P. Sun, “Generalized Stern–Gerlach effect for chiral molecules,” Phys. Rev. Lett. 99, 130403 (2007).
[CrossRef]

Malomed, B. A.

B. B. Baizakov, B. A. Malomed, and M. Salerno, “Multidimensional solitons in a low-dimensional periodic potential,” Phys. Rev. A 70, 053613 (2004).
[CrossRef]

Marangos, J. P.

M. Fleischhauer, A. Imamoğlu, and J. P. Marangos, “Electromagnetically induced transparency: optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
[CrossRef]

Meyer, J.

S. Peredkov, M. Neeb, W. Eberhardt, J. Meyer, M. Tombers, H. Kampschulte, and G. Niedner-Schatteburg, “Spin and orbital magnetic moments of free nanoparticles,” Phys. Rev. Lett. 107, 233401 (2011).
[CrossRef]

Monot, R.

W. D. Knight, R. Monot, E. R. Dietz, and A. R. George, “Stern–Gerlach deflection of metallic-cluster beams,” Phys. Rev. Lett. 40, 1324–1326 (1978).
[CrossRef]

Moro, R.

X. Xu, S. Yin, R. Moro, and W. A. de Heer, “Magnetic moments and adiabatic magnetization of free cobalt clusters,” Phys. Rev. Lett. 95, 237209 (2005).
[CrossRef]

Neeb, M.

S. Peredkov, M. Neeb, W. Eberhardt, J. Meyer, M. Tombers, H. Kampschulte, and G. Niedner-Schatteburg, “Spin and orbital magnetic moments of free nanoparticles,” Phys. Rev. Lett. 107, 233401 (2011).
[CrossRef]

Niedner-Schatteburg, G.

S. Peredkov, M. Neeb, W. Eberhardt, J. Meyer, M. Tombers, H. Kampschulte, and G. Niedner-Schatteburg, “Spin and orbital magnetic moments of free nanoparticles,” Phys. Rev. Lett. 107, 233401 (2011).
[CrossRef]

Nielsen, M. A.

M. A. Nielsen and I. L. Chang, Quantum Computation and Quantum Information (Cambridge, 2000).

Payne, F. W.

F. W. Payne, W. Jiang, J. W. Emmert, J. Deng, and L. A. Bloomfield, “Magnetic structure of free cobalt clusters studied with Stern–Gerlach deflection experiments,” Phys. Rev. B 75, 094431 (2007).
[CrossRef]

Payne, M. G.

G. Huang, L. Deng, and M. G. Payne, “Dynamics of ultraslow optical solitons in a cold three-state atomic system,” Phys. Rev. E 72, 016617 (2005).
[CrossRef]

Peredkov, S.

S. Peredkov, M. Neeb, W. Eberhardt, J. Meyer, M. Tombers, H. Kampschulte, and G. Niedner-Schatteburg, “Spin and orbital magnetic moments of free nanoparticles,” Phys. Rev. Lett. 107, 233401 (2011).
[CrossRef]

Pokrant, S.

S. Pokrant, “Evidence for adiabatic magnetization of cold DyN clusters,” Phys. Rev. A 62, 051201(R) (2000).
[CrossRef]

Robin, M. B.

A. Gedanken, N. A. Kuebler, M. B. Robin, and D. R. Herrick, “Stern–Gerlach deflection spectra of nitrogen oxide radicals,” J. Chem. Phys. 90, 3981–3993 (1989).
[CrossRef]

N. A. Kuebler, M. B. Robin, J. J. Yang, and A. Gedanken, “Fully resolved Zeeman pattern in the Stern–Gerlach deflection spectrum of O2 (Σg−3, K=1),” Phys. Rev. A 38, 737–749 (1988).
[CrossRef]

Sakurai, J. J.

J. J. Sakurai, Modern Quantum Mechanics, rev. ed. (Addison-Wesley, 1994).

Salerno, M.

B. B. Baizakov, B. A. Malomed, and M. Salerno, “Multidimensional solitons in a low-dimensional periodic potential,” Phys. Rev. A 70, 053613 (2004).
[CrossRef]

Silberberg, Y.

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]

Y. Li, C. Bruder, and C. P. Sun, “Generalized Stern–Gerlach effect for chiral molecules,” Phys. Rev. Lett. 99, 130403 (2007).
[CrossRef]

Tombers, M.

S. Peredkov, M. Neeb, W. Eberhardt, J. Meyer, M. Tombers, H. Kampschulte, and G. Niedner-Schatteburg, “Spin and orbital magnetic moments of free nanoparticles,” Phys. Rev. Lett. 107, 233401 (2011).
[CrossRef]

Weitz, M.

L. Karpa and M. Weitz, “A Stern–Gerlach experiment for slow light,” Nat. Phys. 2, 332–335 (2006).
[CrossRef]

Wu, Y.-p.

H.-j. Li, Y.-p. Wu, and G. Huang, “Stable weak light ultraslow spatiotemporal solitons via atomic coherence,” Phys. Rev. A 84, 033816 (2011).
[CrossRef]

Xu, X.

X. Xu, S. Yin, R. Moro, and W. A. de Heer, “Magnetic moments and adiabatic magnetization of free cobalt clusters,” Phys. Rev. Lett. 95, 237209 (2005).
[CrossRef]

Yan, J.

J. Yan, G. Zhou, and J. You, “Nonpropagating soliton and kink soliton in a mildly sloping channel,” Phys. Fluids A 4, 690–694 (1992).
[CrossRef]

Yang, J. J.

N. A. Kuebler, M. B. Robin, J. J. Yang, and A. Gedanken, “Fully resolved Zeeman pattern in the Stern–Gerlach deflection spectrum of O2 (Σg−3, K=1),” Phys. Rev. A 38, 737–749 (1988).
[CrossRef]

Yin, S.

X. Xu, S. Yin, R. Moro, and W. A. de Heer, “Magnetic moments and adiabatic magnetization of free cobalt clusters,” Phys. Rev. Lett. 95, 237209 (2005).
[CrossRef]

You, J.

J. Yan, G. Zhou, and J. You, “Nonpropagating soliton and kink soliton in a mildly sloping channel,” Phys. Fluids A 4, 690–694 (1992).
[CrossRef]

Zhou, G.

J. Yan, G. Zhou, and J. You, “Nonpropagating soliton and kink soliton in a mildly sloping channel,” Phys. Fluids A 4, 690–694 (1992).
[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]

J. Chem. Phys.

A. Gedanken, N. A. Kuebler, M. B. Robin, and D. R. Herrick, “Stern–Gerlach deflection spectra of nitrogen oxide radicals,” J. Chem. Phys. 90, 3981–3993 (1989).
[CrossRef]

Nat. Phys.

L. Karpa and M. Weitz, “A Stern–Gerlach experiment for slow light,” Nat. Phys. 2, 332–335 (2006).
[CrossRef]

Opt. Lett.

Phys. Fluids A

J. Yan, G. Zhou, and J. You, “Nonpropagating soliton and kink soliton in a mildly sloping channel,” Phys. Fluids A 4, 690–694 (1992).
[CrossRef]

Phys. Rev. A

N. A. Kuebler, M. B. Robin, J. J. Yang, and A. Gedanken, “Fully resolved Zeeman pattern in the Stern–Gerlach deflection spectrum of O2 (Σg−3, K=1),” Phys. Rev. A 38, 737–749 (1988).
[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]

B. B. Baizakov, B. A. Malomed, and M. Salerno, “Multidimensional solitons in a low-dimensional periodic potential,” Phys. Rev. A 70, 053613 (2004).
[CrossRef]

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

S. Pokrant, “Evidence for adiabatic magnetization of cold DyN clusters,” Phys. Rev. A 62, 051201(R) (2000).
[CrossRef]

H.-j. Li, Y.-p. Wu, and G. Huang, “Stable weak light ultraslow spatiotemporal solitons via atomic coherence,” Phys. Rev. A 84, 033816 (2011).
[CrossRef]

Phys. Rev. B

F. W. Payne, W. Jiang, J. W. Emmert, J. Deng, and L. A. Bloomfield, “Magnetic structure of free cobalt clusters studied with Stern–Gerlach deflection experiments,” Phys. Rev. B 75, 094431 (2007).
[CrossRef]

Phys. Rev. E

G. Huang, L. Deng, and M. G. Payne, “Dynamics of ultraslow optical solitons in a cold three-state atomic system,” Phys. Rev. E 72, 016617 (2005).
[CrossRef]

Phys. Rev. Lett.

S. Peredkov, M. Neeb, W. Eberhardt, J. Meyer, M. Tombers, H. Kampschulte, and G. Niedner-Schatteburg, “Spin and orbital magnetic moments of free nanoparticles,” Phys. Rev. Lett. 107, 233401 (2011).
[CrossRef]

X. Xu, S. Yin, R. Moro, and W. A. de Heer, “Magnetic moments and adiabatic magnetization of free cobalt clusters,” Phys. Rev. Lett. 95, 237209 (2005).
[CrossRef]

Y. Li, C. Bruder, and C. P. Sun, “Generalized Stern–Gerlach effect for chiral molecules,” Phys. Rev. Lett. 99, 130403 (2007).
[CrossRef]

W. D. Knight, R. Monot, E. R. Dietz, and A. R. George, “Stern–Gerlach deflection of metallic-cluster beams,” Phys. Rev. Lett. 40, 1324–1326 (1978).
[CrossRef]

Rev. Mod. Phys.

M. Fleischhauer, A. Imamoğlu, and J. P. Marangos, “Electromagnetically induced transparency: optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
[CrossRef]

Other

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).

R. Harré, Great Scientific Experiments: 20 Experiments that Changed our View of the World (Phaidon, Oxford, 1981).

J. J. Sakurai, Modern Quantum Mechanics, rev. ed. (Addison-Wesley, 1994).

M. A. Nielsen and I. L. Chang, Quantum Computation and Quantum Information (Cambridge, 2000).

G. Herzberg, Atomic Spectra and Atomic Structure (Dover, 1944).

The far-detuned optical field (2) will generates a magnetic field Bm(x,t)=z^(E0/c)sin(x/R⊥)sin(ωLt). In our model, we choose ωL∼1013  Hz, which is far from any resonance in atoms. It is easy to show that the magnitude of the effective magnetic field resulted from Bm(x,t) is proportional to E02/(4c2B0), which is around 6.94×10−4 Gauss when taking E0∼5×104  V/m and B0=1.0×103 Gauss. Because the second term of the SG magnetic field |B1y|≤|B1Ly|∼4.32×10−3 Gauss for Ly∼100R⊥ (Ly is the deflection distance in the y direction used below), the magnetic field Bm generated by the far-detuned optical field (2) can be safely neglected for the SG deflection considered here.

In the atomic medium, the frequency and wave number of the jth probe field are given by ωpj+ω and kpj+Kj(ω), respectively. Thus ω=0 corresponds to the center frequency of all probe fields.

D. A. Steck, “Rubidium 87 D line data,” http://steck.us/alkalidata/ .

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

Photons in vacuum have no magnetic moment, and hence experience no force and have no trajectory deflection when passing through an inhomogeneous magnetic field. However, photons may acquire effective magnetic moments in atomic media, thus experience a magnetic force and display the SG effect.

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Figures (5)

Fig. 1.
Fig. 1.

(a) Atomic levels and excitation scheme. All quantities have been defined in the text. (b) Possible arrangement of the experiment for observing the SG effect, where a SG gradient magnetic field BSG(y)=z^(B0+B1y) is applied to the system. θ1,θ2,,θn are deflection angles of n probe fields, respectively. The (green) thick arrow denotes the far-detuned optical lattice field EL(x,t)=y^E0cos(x/R)cos(ωLt) used to stabilize LBs. y^ and z^ are unit vectors along y and z directions, respectively. The probe and control fields are co-propagating laser beams to avoid Doppler shifts.

Fig. 2.
Fig. 2.

Linear dispersion relations of the three probe fields. (a) Im(Kj) and (b) Re(Kj) (j=1, 2, 3) as functions of ω. Each (blue) solid curve consists, in fact, of three curves, which cannot be resolved since they nearly coincide with each other due to the symmetry of the system. The inset in both panels show the absorption and dispersion curves near ω=0. The curves 1, 2, and 3 in the inset of panel (a) [panel (b)] are for Im(K1), Im(K2), Im(K3) (Re(K1), Re(K2), and Re(K3)), respectively. The parameters used for plotting the figure have been given in the text.

Fig. 3.
Fig. 3.

SG deflection spectrum of the 3-component ultraslow LB. (a), (b), (c), and (d) are deflections of the LB when propagating to z=2LDiff, z=4LDiff, z=6LDiff, and z=8LDiff, respectively. The bright spots from top to bottom are distributions of |Ep1|2, |Ep2|2, and |Ep3|2 in the xy plane, respectively.

Fig. 4.
Fig. 4.

(a) Deflection angles of the LB components as functions of medium length L for magnetic field gradient B1=1.2mG/mm. The solid, dashed, and dashed–dotted lines denote deflection angles θ1, θ2, and θ3, respectively. Points labeled by “×” are center positions of the LB components obtained numerically. (b) Deflection angles of the LB components as functions of B1 for L=4LDiff. The results of θ1, θ2, and θ3 are, respectively, labeled by solid, dashed, and dashed–dotted lines.

Fig. 5.
Fig. 5.

SG deflection spectrum of a 5-component ultraslow LB. (a), (b), (c), and (d) show the deflections of the LB components when propagating to z=2LDiff, z=4LDiff, z=6LDiff, and z=8LDiff, respectively.

Equations (55)

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BSG(y)=z^B(y)=z^(B0+B1y)
EL(x,t)=y^E0cos(x/R)cos(ωLt)
H^int=j=1n+2Δj|jj|[Ωp1|n+21|+Ωp2|n+22|++Ωpn|n+2n|+Ωc|n+2n+1|+H.c.].
σt=i[Hint,σ]Γ(σ),
i(z+1ct)Ωpj+c2ωpj(2x2+2y2)Ωpj+κj,n+2σn+2,j=0,
Ωpj(1)=Fjeiθj,
σ4j(1)=Ωc*σjj(0)DjFjeiθj,
σ5j(1)=(ω+d4j(0))σjj(0)DjFjeiθj,
Kj(ω)=ωc+κj5σjj(0)(ω+d4j(0))Dj.
i(z2+1Vgjt2)Fj+c2ωpj(2x12+2y12)Fjl=13Wjl|Fl|2Fje2a¯lz2+[MjB1y1+NjE02cos2(x1/R)]Fj=0,
Mj=κj5(ω+d4j(0))2μ5j+|Ωc|2μ4j3Dj2,
Nj=κj5(ω+d4j(0))2α5j+|Ωc|2α4j12Dj2,
[i(s+1vgjτ)+12(2ξ2+2η2)]ujl=13gjl|ul|2uj+Vj(ξ,η)uj=iAjuj,
Vj(ξ,η)=Mjη+Njcos2(ξ),
V˜g1=3.964×105c,
V˜g2=3.966×105c,
V˜g3=3.967×105c.
[ivgjτ+12(2ξ2+2η2)]vj12πρ0l=13gjl|vl|2vj+Vj(ξ,η)vj=0.
[ivgjτ+12(2ξ2+2η2)]vj12πρ0gjj|vj|2vj+(Mjη+NjNjξ2)vj=0.
(ivgjτ+122η2)wjNj1/423/4πρ0gjj|wj|2wj+(MjηNj2)wj=0.
wj=AjeiφjsechΘj,
uj=Aj[1/(ρ0π)]1/2(2Nj/π)1/4eiφje(svgjτ)2/(2ρ02)eNjξ2/2sechΘj,
Ωpj=U0Aj(1ρ0π)1/2(2Njπ)1/4eiφjeNjx2/(2R2)e(zV˜gjt)2/(2LDiff2ρ02)sech{(2Nj)1/4R(yMjRV˜gj22LDiff2t2)}.
Vj=(0,MjV˜gj2RLDiff2t,V˜gj).
θj=VyjV˜gj=LV˜gjμsoljpjr2B1,
(xj,yj,zj)=(0,MjL2R2LDiff2,L).
i(z2+1Vgjt2)Fj+c2ωpj(2x12+2y12)Fjl=1nWjl|Fl|2e2a¯lz2Fj+[MjB1y1+NjE02cos2(x1/R)]Fj=0,
itσ11iΓ15σ55+Ωp1*σ51Ωp1σ51*=0,
itσ22iΓ25σ55+Ωp2*σ52Ωp2σ52*=0,
itσ33iΓ35σ55+Ωp3*σ53Ωp3σ53*=0,
itσ44iΓ45σ55+Ωc*σ54Ωcσ54*=0,
i(t+Γ5)σ55+Ωp1σ51*+Ωp2σ52*+Ωp3σ53*+Ωcσ54*Ωp1*σ51Ωp2*σ52Ωp3*σ53Ωc*σ54=0,
(it+d21)σ21+Ωp2*σ51Ωp1σ52*=0,
(it+d31)σ31+Ωp3*σ51Ωp1σ53*=0,
(it+d32)σ32+Ωp3*σ52Ωp2σ53*=0,
(it+d41)σ41+Ωc*σ51Ωp1σ54*=0,
(it+d42)σ42+Ωc*σ52Ωp2σ54*=0,
(it+d43)σ43+Ωc*σ53Ωp3σ54*=0,
(it+d51)σ51+Ωp1(σ11σ55)+Ωp2σ21+Ωp3σ31+Ωcσ41=0,
(it+d52)σ52+Ωp2(σ22σ55)+Ωp1σ21*+Ωp3σ32+Ωcσ42=0,
(it+d53)σ53+Ωp3(σ33σ55)+Ωp1σ31*+Ωp2σ32*+Ωcσ43=0,
(it+d54)σ54+Ωc(σ44σ55)+Ωp1σ41*+Ωp2σ42*+Ωp3σ43*=0,
σ21(2)=Ωp1(1)σ52*(1)Ωp2*(1)σ51(1)ω+d21,
σ31(2)=Ωp1(1)σ53*(1)Ωp3*(1)σ51(1)ω+d31,
σ32(2)=Ωp2(1)σ53*(1)Ωp3*(1)σ52(1)ω+d32,
σ4j(2)=1Dj[(ω+d5j)it1σ4j(1)Ωc*it1σ5j(1)],
σ5j(2)=1Dj[(ω+d4j)it1σ5j(1)Ωcit1σ4j(1)],
σ54(2)=Ωp1(1)σ41*(1)+Ωp2(1)σ42*(1)+Ωp3(1)σ43*(1)ω+d54,
Wjj=κj5|Ωc|23Dj2(ω+d54*),(j=1,2,3)
W12=κ153D1[ω+d41ω+d21(ω+d42*D2*ω+d41D1)+|Ωc|2D2(ω+d54*)],
W13=κ153D1[ω+d41ω+d31(ω+d43*D3*ω+d41D1)+|Ωc|2D3(ω+d54*)],
W21=κ253D2[ω+d42ω+d21*(ω+d42D2ω+d41*D1*)+|Ωc|2D1(ω+d54*)],
W23=κ253D2[ω+d42ω+d32(ω+d43*D3*ω+d42D2)+|Ωc|2D3(ω+d54*)],
W31=κ353D3[ω+d43ω+d31*(ω+d43D3ω+d41*D1*)+|Ωc|2D1(ω+d54*)],
W32=κ353D3[ω+d43ω+d32*(ω+d43D3ω+d42*D2*)+|Ωc|2D2(ω+d54*)],

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