Abstract

Quantum topological defects usually are excited on density. In this study, we extend the topological defect to another dimensionality: the spin. We indicate that the spin singularity, where the spin density |S| vanishes locally, occurs commonly in the two-dimensional spin-1 Bose-Einstein condensates. We assume a coreless half-quantum vortex solution in the spin-1 Bose-Einstein condensates according to the spin singularity. This half-quantum vortex can further induce a half-quantum spin defect, which is similar to the general half-quantum Skyrmion but its centre proposes singularity on the spin density. We use the variational method to obtain a possible stable solution under the combined interaction of the spin-orbit coupling and the external magnetic field. Our calculation shows that the isotropic spin-orbit coupling provides local energy minimum to stabilize the coreless half-quantum vortex and the corresponding singular half-quantum spin texture excitation. The combined restriction of the isotropic spin-orbit coupling and the perpendicular magnetic field is a key factor to obtain this kind of excitation. Our study also provides the stability phase diagram and the most possible size of the half-quantum vortex. Furthermore, our analysis shows that adjusting of the spin interaction hardly affects the stability of the half-quantum vortex.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
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References

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  1. J. Ruostekoski and J. R. Anglin, “Monopole core instability and Alice rings in spinor Bose-Einstein condensates,” Phys. Rev. Lett. 91, 190402 (2003).
    [Crossref] [PubMed]
  2. H. T.C. Stoof, E. Vliegen, and U. Al Khawaja, “Monopoles in an antiferromagnetic Bose-Einstein condensate,” Phys. Rev. Lett. 87, 120407 (2001).
    [Crossref] [PubMed]
  3. J.-P. Martikainen, A. Collin, and K.-A. Suominen, “Creation of a monopole in a spinor condensate,” Phys. Rev. Lett. 88, 090404 (2002).
    [Crossref] [PubMed]
  4. Juan J. Garcia-Ripoll, Juan I. Cirac, J. Anglin, Victor M. Perez-Garcia, and P. Zoller, “Spin monopoles with Bose-Einstein condensates,” Phys. Rev. A 61, 053609 (2000).
    [Crossref]
  5. T. Busch and J. R. Anglin, “Wave-function monopoles in Bose-Einstein condensates,” Phys. Rev. A 60, R2669–R2672 (1999).
    [Crossref]
  6. C. F. Liu, H. Fan, S.C. Gou, and W. M. Liu, “Crystallized and amorphous vortices in rotating atomic-molecular Bose-Einstein condensates,” Sci. Rep. 4, 4224 (2014).
    [Crossref] [PubMed]
  7. C. F. Liu, H. Fan, Y. C. Zhang, D. S. Wang, and W. M. Liu, “Circular-hyperbolic skyrmion in rotating pseudo-spin-1/2 Bose-Einstein condensates with spin-orbit coupling,” Phys. Rev. A 86, 053616 (2012).
    [Crossref]
  8. Y.-J. Lin, K. Jiménez-García, and I. B. Spielman, “Spin-orbit-coupled Bose-Einstein condensates,” Nature 471, 83–88 (2011).
    [Crossref] [PubMed]
  9. C. Wang, C. Gao, C.-M. Jian, and H. Zhai, “Spin-orbit coupled spinor Bose-Einstein condensates,” Phys. Rev. Lett. 105, 160403 (2010).
    [Crossref]
  10. T.-L. Ho and S. Z. Zhang, “Bose-Einstein condensates with spin-orbit interaction,” Phys. Rev. Lett. 107, 150403 (2011).
    [Crossref] [PubMed]
  11. C.-M. Jian and H. Zhai, “Paired superfluidity and fractionalized vortices in systems of spin-orbit coupled bosons,” Phys. Rev. B 84, 060508(R) (2011).
    [Crossref]
  12. S. Sinha, R. Nath, and L. Santos, “Trapped two-dimensional condensates with synthetic spin-orbit coupling,” Phys. Rev. Lett. 107, 270401 (2011).
    [Crossref]
  13. H. Hu, B. Ramachandhran, H. Pu, and X.-J. Liu, “Spin-orbit coupled weakly interacting Bose-Einstein condensates in harmonic traps,” Phys. Rev. Lett. 108, 010402 (2012).
    [Crossref] [PubMed]
  14. C. Wu, I. Mondragon-Shem, and X. F. Zhou, “Unconventional Bose-Einstein condensations from spin-orbit coupling,” Chin. Phys. Lett.,  28, 097102 (2011).
    [Crossref]
  15. C. F. Liu, Y. M. Yu, S. C. Gou, and W. M. Liu, “Vortex chain in anisotropic spin-orbit-coupled spin-1 Bose-Einstein condensates,” Phys. Rev. A 87, 063630 (2013).
    [Crossref]
  16. C. F. Liu, G. Juzeliûnas, and W. M. Liu, “Spin-orbit coupling manipulating composite topological spin textures in atomic-molecular Bose-Einstein condensates,” Phys. Rev. A 95, 023624 (2017).
    [Crossref]
  17. S. Gautam and S. K. Adhikari, “Vortex-bright solitons in a spin-orbit-coupled spin-1 condensate,” Phys. Rev. A 95, 013608 (2017).
    [Crossref]
  18. S. Gautam and S. K. Adhikari, “Fractional-charge vortex in a spinor Bose-Einstein condensate,” Phys. Rev. A 93, 013630 (2016).
    [Crossref]
  19. S. Gautam and S. K. Adhikari, “Mobile vector soliton in a spin-orbit coupled spin-1 condensate,” Laser Phys. Lett. 12, 045501 (2015).
    [Crossref]
  20. X.-Q. Xu and J. H. Han, “Spin-orbit coupled Bose-Einstein condensate under rotation,” Phys. Rev. Lett. 107, 200401 (2011).
    [Crossref] [PubMed]
  21. X.-F. Zhou, J. Zhou, and C. Wu, “Vortex structures of rotating spin-orbit-coupled Bose-Einstein condensates,” Phys. Rev. A 84, 063624 (2011).
    [Crossref]
  22. C. F. Liu and W. M. Liu, “Spin-orbit-coupling-induced half-skyrmion excitations in rotating and rapidly quenched spin-1 Bose-Einstein condensates,” Phys. Rev. A 86, 033602 (2012).
    [Crossref]
  23. S. W. Su, I. K. Liu, Y. C. Tsai, W. M. Liu, and S. C. Gou, “Crystallized half-skyrmions and inverted half-skyrmions in the condensation of spin-1 Bose gases with spin-orbit coupling,” Phys. Rev. A 86, 023601 (2012).
    [Crossref]
  24. T. Mizushima, K. Machida, and T. Kita, “Mermin-Ho vortex in ferromagnetic spinor Bose-Einstein condensates,” Phys. Rev. Lett. 89, 030401 (2002).
    [Crossref] [PubMed]
  25. T. Mizushima, N. Kobayashi, and K. Machida, “Coreless and singular vortex lattices in rotating spinor Bose-Einstein condensates,” Phys. Rev. A 70, 043613 (2004).
    [Crossref]
  26. A. S. Bradley and B. P. Anderson, “Energy spectra of vortex distributions in two-dimensional quantum turbulence,” Phys. Rev. X 2, 041001 (2012).

2017 (2)

C. F. Liu, G. Juzeliûnas, and W. M. Liu, “Spin-orbit coupling manipulating composite topological spin textures in atomic-molecular Bose-Einstein condensates,” Phys. Rev. A 95, 023624 (2017).
[Crossref]

S. Gautam and S. K. Adhikari, “Vortex-bright solitons in a spin-orbit-coupled spin-1 condensate,” Phys. Rev. A 95, 013608 (2017).
[Crossref]

2016 (1)

S. Gautam and S. K. Adhikari, “Fractional-charge vortex in a spinor Bose-Einstein condensate,” Phys. Rev. A 93, 013630 (2016).
[Crossref]

2015 (1)

S. Gautam and S. K. Adhikari, “Mobile vector soliton in a spin-orbit coupled spin-1 condensate,” Laser Phys. Lett. 12, 045501 (2015).
[Crossref]

2014 (1)

C. F. Liu, H. Fan, S.C. Gou, and W. M. Liu, “Crystallized and amorphous vortices in rotating atomic-molecular Bose-Einstein condensates,” Sci. Rep. 4, 4224 (2014).
[Crossref] [PubMed]

2013 (1)

C. F. Liu, Y. M. Yu, S. C. Gou, and W. M. Liu, “Vortex chain in anisotropic spin-orbit-coupled spin-1 Bose-Einstein condensates,” Phys. Rev. A 87, 063630 (2013).
[Crossref]

2012 (5)

C. F. Liu and W. M. Liu, “Spin-orbit-coupling-induced half-skyrmion excitations in rotating and rapidly quenched spin-1 Bose-Einstein condensates,” Phys. Rev. A 86, 033602 (2012).
[Crossref]

S. W. Su, I. K. Liu, Y. C. Tsai, W. M. Liu, and S. C. Gou, “Crystallized half-skyrmions and inverted half-skyrmions in the condensation of spin-1 Bose gases with spin-orbit coupling,” Phys. Rev. A 86, 023601 (2012).
[Crossref]

A. S. Bradley and B. P. Anderson, “Energy spectra of vortex distributions in two-dimensional quantum turbulence,” Phys. Rev. X 2, 041001 (2012).

C. F. Liu, H. Fan, Y. C. Zhang, D. S. Wang, and W. M. Liu, “Circular-hyperbolic skyrmion in rotating pseudo-spin-1/2 Bose-Einstein condensates with spin-orbit coupling,” Phys. Rev. A 86, 053616 (2012).
[Crossref]

H. Hu, B. Ramachandhran, H. Pu, and X.-J. Liu, “Spin-orbit coupled weakly interacting Bose-Einstein condensates in harmonic traps,” Phys. Rev. Lett. 108, 010402 (2012).
[Crossref] [PubMed]

2011 (7)

C. Wu, I. Mondragon-Shem, and X. F. Zhou, “Unconventional Bose-Einstein condensations from spin-orbit coupling,” Chin. Phys. Lett.,  28, 097102 (2011).
[Crossref]

T.-L. Ho and S. Z. Zhang, “Bose-Einstein condensates with spin-orbit interaction,” Phys. Rev. Lett. 107, 150403 (2011).
[Crossref] [PubMed]

C.-M. Jian and H. Zhai, “Paired superfluidity and fractionalized vortices in systems of spin-orbit coupled bosons,” Phys. Rev. B 84, 060508(R) (2011).
[Crossref]

S. Sinha, R. Nath, and L. Santos, “Trapped two-dimensional condensates with synthetic spin-orbit coupling,” Phys. Rev. Lett. 107, 270401 (2011).
[Crossref]

X.-Q. Xu and J. H. Han, “Spin-orbit coupled Bose-Einstein condensate under rotation,” Phys. Rev. Lett. 107, 200401 (2011).
[Crossref] [PubMed]

X.-F. Zhou, J. Zhou, and C. Wu, “Vortex structures of rotating spin-orbit-coupled Bose-Einstein condensates,” Phys. Rev. A 84, 063624 (2011).
[Crossref]

Y.-J. Lin, K. Jiménez-García, and I. B. Spielman, “Spin-orbit-coupled Bose-Einstein condensates,” Nature 471, 83–88 (2011).
[Crossref] [PubMed]

2010 (1)

C. Wang, C. Gao, C.-M. Jian, and H. Zhai, “Spin-orbit coupled spinor Bose-Einstein condensates,” Phys. Rev. Lett. 105, 160403 (2010).
[Crossref]

2004 (1)

T. Mizushima, N. Kobayashi, and K. Machida, “Coreless and singular vortex lattices in rotating spinor Bose-Einstein condensates,” Phys. Rev. A 70, 043613 (2004).
[Crossref]

2003 (1)

J. Ruostekoski and J. R. Anglin, “Monopole core instability and Alice rings in spinor Bose-Einstein condensates,” Phys. Rev. Lett. 91, 190402 (2003).
[Crossref] [PubMed]

2002 (2)

J.-P. Martikainen, A. Collin, and K.-A. Suominen, “Creation of a monopole in a spinor condensate,” Phys. Rev. Lett. 88, 090404 (2002).
[Crossref] [PubMed]

T. Mizushima, K. Machida, and T. Kita, “Mermin-Ho vortex in ferromagnetic spinor Bose-Einstein condensates,” Phys. Rev. Lett. 89, 030401 (2002).
[Crossref] [PubMed]

2001 (1)

H. T.C. Stoof, E. Vliegen, and U. Al Khawaja, “Monopoles in an antiferromagnetic Bose-Einstein condensate,” Phys. Rev. Lett. 87, 120407 (2001).
[Crossref] [PubMed]

2000 (1)

Juan J. Garcia-Ripoll, Juan I. Cirac, J. Anglin, Victor M. Perez-Garcia, and P. Zoller, “Spin monopoles with Bose-Einstein condensates,” Phys. Rev. A 61, 053609 (2000).
[Crossref]

1999 (1)

T. Busch and J. R. Anglin, “Wave-function monopoles in Bose-Einstein condensates,” Phys. Rev. A 60, R2669–R2672 (1999).
[Crossref]

Adhikari, S. K.

S. Gautam and S. K. Adhikari, “Vortex-bright solitons in a spin-orbit-coupled spin-1 condensate,” Phys. Rev. A 95, 013608 (2017).
[Crossref]

S. Gautam and S. K. Adhikari, “Fractional-charge vortex in a spinor Bose-Einstein condensate,” Phys. Rev. A 93, 013630 (2016).
[Crossref]

S. Gautam and S. K. Adhikari, “Mobile vector soliton in a spin-orbit coupled spin-1 condensate,” Laser Phys. Lett. 12, 045501 (2015).
[Crossref]

Al Khawaja, U.

H. T.C. Stoof, E. Vliegen, and U. Al Khawaja, “Monopoles in an antiferromagnetic Bose-Einstein condensate,” Phys. Rev. Lett. 87, 120407 (2001).
[Crossref] [PubMed]

Anderson, B. P.

A. S. Bradley and B. P. Anderson, “Energy spectra of vortex distributions in two-dimensional quantum turbulence,” Phys. Rev. X 2, 041001 (2012).

Anglin, J.

Juan J. Garcia-Ripoll, Juan I. Cirac, J. Anglin, Victor M. Perez-Garcia, and P. Zoller, “Spin monopoles with Bose-Einstein condensates,” Phys. Rev. A 61, 053609 (2000).
[Crossref]

Anglin, J. R.

J. Ruostekoski and J. R. Anglin, “Monopole core instability and Alice rings in spinor Bose-Einstein condensates,” Phys. Rev. Lett. 91, 190402 (2003).
[Crossref] [PubMed]

T. Busch and J. R. Anglin, “Wave-function monopoles in Bose-Einstein condensates,” Phys. Rev. A 60, R2669–R2672 (1999).
[Crossref]

Bradley, A. S.

A. S. Bradley and B. P. Anderson, “Energy spectra of vortex distributions in two-dimensional quantum turbulence,” Phys. Rev. X 2, 041001 (2012).

Busch, T.

T. Busch and J. R. Anglin, “Wave-function monopoles in Bose-Einstein condensates,” Phys. Rev. A 60, R2669–R2672 (1999).
[Crossref]

Cirac, Juan I.

Juan J. Garcia-Ripoll, Juan I. Cirac, J. Anglin, Victor M. Perez-Garcia, and P. Zoller, “Spin monopoles with Bose-Einstein condensates,” Phys. Rev. A 61, 053609 (2000).
[Crossref]

Collin, A.

J.-P. Martikainen, A. Collin, and K.-A. Suominen, “Creation of a monopole in a spinor condensate,” Phys. Rev. Lett. 88, 090404 (2002).
[Crossref] [PubMed]

Fan, H.

C. F. Liu, H. Fan, S.C. Gou, and W. M. Liu, “Crystallized and amorphous vortices in rotating atomic-molecular Bose-Einstein condensates,” Sci. Rep. 4, 4224 (2014).
[Crossref] [PubMed]

C. F. Liu, H. Fan, Y. C. Zhang, D. S. Wang, and W. M. Liu, “Circular-hyperbolic skyrmion in rotating pseudo-spin-1/2 Bose-Einstein condensates with spin-orbit coupling,” Phys. Rev. A 86, 053616 (2012).
[Crossref]

Gao, C.

C. Wang, C. Gao, C.-M. Jian, and H. Zhai, “Spin-orbit coupled spinor Bose-Einstein condensates,” Phys. Rev. Lett. 105, 160403 (2010).
[Crossref]

Garcia-Ripoll, Juan J.

Juan J. Garcia-Ripoll, Juan I. Cirac, J. Anglin, Victor M. Perez-Garcia, and P. Zoller, “Spin monopoles with Bose-Einstein condensates,” Phys. Rev. A 61, 053609 (2000).
[Crossref]

Gautam, S.

S. Gautam and S. K. Adhikari, “Vortex-bright solitons in a spin-orbit-coupled spin-1 condensate,” Phys. Rev. A 95, 013608 (2017).
[Crossref]

S. Gautam and S. K. Adhikari, “Fractional-charge vortex in a spinor Bose-Einstein condensate,” Phys. Rev. A 93, 013630 (2016).
[Crossref]

S. Gautam and S. K. Adhikari, “Mobile vector soliton in a spin-orbit coupled spin-1 condensate,” Laser Phys. Lett. 12, 045501 (2015).
[Crossref]

Gou, S. C.

C. F. Liu, Y. M. Yu, S. C. Gou, and W. M. Liu, “Vortex chain in anisotropic spin-orbit-coupled spin-1 Bose-Einstein condensates,” Phys. Rev. A 87, 063630 (2013).
[Crossref]

S. W. Su, I. K. Liu, Y. C. Tsai, W. M. Liu, and S. C. Gou, “Crystallized half-skyrmions and inverted half-skyrmions in the condensation of spin-1 Bose gases with spin-orbit coupling,” Phys. Rev. A 86, 023601 (2012).
[Crossref]

Gou, S.C.

C. F. Liu, H. Fan, S.C. Gou, and W. M. Liu, “Crystallized and amorphous vortices in rotating atomic-molecular Bose-Einstein condensates,” Sci. Rep. 4, 4224 (2014).
[Crossref] [PubMed]

Han, J. H.

X.-Q. Xu and J. H. Han, “Spin-orbit coupled Bose-Einstein condensate under rotation,” Phys. Rev. Lett. 107, 200401 (2011).
[Crossref] [PubMed]

Ho, T.-L.

T.-L. Ho and S. Z. Zhang, “Bose-Einstein condensates with spin-orbit interaction,” Phys. Rev. Lett. 107, 150403 (2011).
[Crossref] [PubMed]

Hu, H.

H. Hu, B. Ramachandhran, H. Pu, and X.-J. Liu, “Spin-orbit coupled weakly interacting Bose-Einstein condensates in harmonic traps,” Phys. Rev. Lett. 108, 010402 (2012).
[Crossref] [PubMed]

Jian, C.-M.

C.-M. Jian and H. Zhai, “Paired superfluidity and fractionalized vortices in systems of spin-orbit coupled bosons,” Phys. Rev. B 84, 060508(R) (2011).
[Crossref]

C. Wang, C. Gao, C.-M. Jian, and H. Zhai, “Spin-orbit coupled spinor Bose-Einstein condensates,” Phys. Rev. Lett. 105, 160403 (2010).
[Crossref]

Jiménez-García, K.

Y.-J. Lin, K. Jiménez-García, and I. B. Spielman, “Spin-orbit-coupled Bose-Einstein condensates,” Nature 471, 83–88 (2011).
[Crossref] [PubMed]

Juzeliûnas, G.

C. F. Liu, G. Juzeliûnas, and W. M. Liu, “Spin-orbit coupling manipulating composite topological spin textures in atomic-molecular Bose-Einstein condensates,” Phys. Rev. A 95, 023624 (2017).
[Crossref]

Kita, T.

T. Mizushima, K. Machida, and T. Kita, “Mermin-Ho vortex in ferromagnetic spinor Bose-Einstein condensates,” Phys. Rev. Lett. 89, 030401 (2002).
[Crossref] [PubMed]

Kobayashi, N.

T. Mizushima, N. Kobayashi, and K. Machida, “Coreless and singular vortex lattices in rotating spinor Bose-Einstein condensates,” Phys. Rev. A 70, 043613 (2004).
[Crossref]

Lin, Y.-J.

Y.-J. Lin, K. Jiménez-García, and I. B. Spielman, “Spin-orbit-coupled Bose-Einstein condensates,” Nature 471, 83–88 (2011).
[Crossref] [PubMed]

Liu, C. F.

C. F. Liu, G. Juzeliûnas, and W. M. Liu, “Spin-orbit coupling manipulating composite topological spin textures in atomic-molecular Bose-Einstein condensates,” Phys. Rev. A 95, 023624 (2017).
[Crossref]

C. F. Liu, H. Fan, S.C. Gou, and W. M. Liu, “Crystallized and amorphous vortices in rotating atomic-molecular Bose-Einstein condensates,” Sci. Rep. 4, 4224 (2014).
[Crossref] [PubMed]

C. F. Liu, Y. M. Yu, S. C. Gou, and W. M. Liu, “Vortex chain in anisotropic spin-orbit-coupled spin-1 Bose-Einstein condensates,” Phys. Rev. A 87, 063630 (2013).
[Crossref]

C. F. Liu and W. M. Liu, “Spin-orbit-coupling-induced half-skyrmion excitations in rotating and rapidly quenched spin-1 Bose-Einstein condensates,” Phys. Rev. A 86, 033602 (2012).
[Crossref]

C. F. Liu, H. Fan, Y. C. Zhang, D. S. Wang, and W. M. Liu, “Circular-hyperbolic skyrmion in rotating pseudo-spin-1/2 Bose-Einstein condensates with spin-orbit coupling,” Phys. Rev. A 86, 053616 (2012).
[Crossref]

Liu, I. K.

S. W. Su, I. K. Liu, Y. C. Tsai, W. M. Liu, and S. C. Gou, “Crystallized half-skyrmions and inverted half-skyrmions in the condensation of spin-1 Bose gases with spin-orbit coupling,” Phys. Rev. A 86, 023601 (2012).
[Crossref]

Liu, W. M.

C. F. Liu, G. Juzeliûnas, and W. M. Liu, “Spin-orbit coupling manipulating composite topological spin textures in atomic-molecular Bose-Einstein condensates,” Phys. Rev. A 95, 023624 (2017).
[Crossref]

C. F. Liu, H. Fan, S.C. Gou, and W. M. Liu, “Crystallized and amorphous vortices in rotating atomic-molecular Bose-Einstein condensates,” Sci. Rep. 4, 4224 (2014).
[Crossref] [PubMed]

C. F. Liu, Y. M. Yu, S. C. Gou, and W. M. Liu, “Vortex chain in anisotropic spin-orbit-coupled spin-1 Bose-Einstein condensates,” Phys. Rev. A 87, 063630 (2013).
[Crossref]

C. F. Liu and W. M. Liu, “Spin-orbit-coupling-induced half-skyrmion excitations in rotating and rapidly quenched spin-1 Bose-Einstein condensates,” Phys. Rev. A 86, 033602 (2012).
[Crossref]

C. F. Liu, H. Fan, Y. C. Zhang, D. S. Wang, and W. M. Liu, “Circular-hyperbolic skyrmion in rotating pseudo-spin-1/2 Bose-Einstein condensates with spin-orbit coupling,” Phys. Rev. A 86, 053616 (2012).
[Crossref]

S. W. Su, I. K. Liu, Y. C. Tsai, W. M. Liu, and S. C. Gou, “Crystallized half-skyrmions and inverted half-skyrmions in the condensation of spin-1 Bose gases with spin-orbit coupling,” Phys. Rev. A 86, 023601 (2012).
[Crossref]

Liu, X.-J.

H. Hu, B. Ramachandhran, H. Pu, and X.-J. Liu, “Spin-orbit coupled weakly interacting Bose-Einstein condensates in harmonic traps,” Phys. Rev. Lett. 108, 010402 (2012).
[Crossref] [PubMed]

Machida, K.

T. Mizushima, N. Kobayashi, and K. Machida, “Coreless and singular vortex lattices in rotating spinor Bose-Einstein condensates,” Phys. Rev. A 70, 043613 (2004).
[Crossref]

T. Mizushima, K. Machida, and T. Kita, “Mermin-Ho vortex in ferromagnetic spinor Bose-Einstein condensates,” Phys. Rev. Lett. 89, 030401 (2002).
[Crossref] [PubMed]

Martikainen, J.-P.

J.-P. Martikainen, A. Collin, and K.-A. Suominen, “Creation of a monopole in a spinor condensate,” Phys. Rev. Lett. 88, 090404 (2002).
[Crossref] [PubMed]

Mizushima, T.

T. Mizushima, N. Kobayashi, and K. Machida, “Coreless and singular vortex lattices in rotating spinor Bose-Einstein condensates,” Phys. Rev. A 70, 043613 (2004).
[Crossref]

T. Mizushima, K. Machida, and T. Kita, “Mermin-Ho vortex in ferromagnetic spinor Bose-Einstein condensates,” Phys. Rev. Lett. 89, 030401 (2002).
[Crossref] [PubMed]

Mondragon-Shem, I.

C. Wu, I. Mondragon-Shem, and X. F. Zhou, “Unconventional Bose-Einstein condensations from spin-orbit coupling,” Chin. Phys. Lett.,  28, 097102 (2011).
[Crossref]

Nath, R.

S. Sinha, R. Nath, and L. Santos, “Trapped two-dimensional condensates with synthetic spin-orbit coupling,” Phys. Rev. Lett. 107, 270401 (2011).
[Crossref]

Perez-Garcia, Victor M.

Juan J. Garcia-Ripoll, Juan I. Cirac, J. Anglin, Victor M. Perez-Garcia, and P. Zoller, “Spin monopoles with Bose-Einstein condensates,” Phys. Rev. A 61, 053609 (2000).
[Crossref]

Pu, H.

H. Hu, B. Ramachandhran, H. Pu, and X.-J. Liu, “Spin-orbit coupled weakly interacting Bose-Einstein condensates in harmonic traps,” Phys. Rev. Lett. 108, 010402 (2012).
[Crossref] [PubMed]

Ramachandhran, B.

H. Hu, B. Ramachandhran, H. Pu, and X.-J. Liu, “Spin-orbit coupled weakly interacting Bose-Einstein condensates in harmonic traps,” Phys. Rev. Lett. 108, 010402 (2012).
[Crossref] [PubMed]

Ruostekoski, J.

J. Ruostekoski and J. R. Anglin, “Monopole core instability and Alice rings in spinor Bose-Einstein condensates,” Phys. Rev. Lett. 91, 190402 (2003).
[Crossref] [PubMed]

Santos, L.

S. Sinha, R. Nath, and L. Santos, “Trapped two-dimensional condensates with synthetic spin-orbit coupling,” Phys. Rev. Lett. 107, 270401 (2011).
[Crossref]

Sinha, S.

S. Sinha, R. Nath, and L. Santos, “Trapped two-dimensional condensates with synthetic spin-orbit coupling,” Phys. Rev. Lett. 107, 270401 (2011).
[Crossref]

Spielman, I. B.

Y.-J. Lin, K. Jiménez-García, and I. B. Spielman, “Spin-orbit-coupled Bose-Einstein condensates,” Nature 471, 83–88 (2011).
[Crossref] [PubMed]

Stoof, H. T.C.

H. T.C. Stoof, E. Vliegen, and U. Al Khawaja, “Monopoles in an antiferromagnetic Bose-Einstein condensate,” Phys. Rev. Lett. 87, 120407 (2001).
[Crossref] [PubMed]

Su, S. W.

S. W. Su, I. K. Liu, Y. C. Tsai, W. M. Liu, and S. C. Gou, “Crystallized half-skyrmions and inverted half-skyrmions in the condensation of spin-1 Bose gases with spin-orbit coupling,” Phys. Rev. A 86, 023601 (2012).
[Crossref]

Suominen, K.-A.

J.-P. Martikainen, A. Collin, and K.-A. Suominen, “Creation of a monopole in a spinor condensate,” Phys. Rev. Lett. 88, 090404 (2002).
[Crossref] [PubMed]

Tsai, Y. C.

S. W. Su, I. K. Liu, Y. C. Tsai, W. M. Liu, and S. C. Gou, “Crystallized half-skyrmions and inverted half-skyrmions in the condensation of spin-1 Bose gases with spin-orbit coupling,” Phys. Rev. A 86, 023601 (2012).
[Crossref]

Vliegen, E.

H. T.C. Stoof, E. Vliegen, and U. Al Khawaja, “Monopoles in an antiferromagnetic Bose-Einstein condensate,” Phys. Rev. Lett. 87, 120407 (2001).
[Crossref] [PubMed]

Wang, C.

C. Wang, C. Gao, C.-M. Jian, and H. Zhai, “Spin-orbit coupled spinor Bose-Einstein condensates,” Phys. Rev. Lett. 105, 160403 (2010).
[Crossref]

Wang, D. S.

C. F. Liu, H. Fan, Y. C. Zhang, D. S. Wang, and W. M. Liu, “Circular-hyperbolic skyrmion in rotating pseudo-spin-1/2 Bose-Einstein condensates with spin-orbit coupling,” Phys. Rev. A 86, 053616 (2012).
[Crossref]

Wu, C.

X.-F. Zhou, J. Zhou, and C. Wu, “Vortex structures of rotating spin-orbit-coupled Bose-Einstein condensates,” Phys. Rev. A 84, 063624 (2011).
[Crossref]

C. Wu, I. Mondragon-Shem, and X. F. Zhou, “Unconventional Bose-Einstein condensations from spin-orbit coupling,” Chin. Phys. Lett.,  28, 097102 (2011).
[Crossref]

Xu, X.-Q.

X.-Q. Xu and J. H. Han, “Spin-orbit coupled Bose-Einstein condensate under rotation,” Phys. Rev. Lett. 107, 200401 (2011).
[Crossref] [PubMed]

Yu, Y. M.

C. F. Liu, Y. M. Yu, S. C. Gou, and W. M. Liu, “Vortex chain in anisotropic spin-orbit-coupled spin-1 Bose-Einstein condensates,” Phys. Rev. A 87, 063630 (2013).
[Crossref]

Zhai, H.

C.-M. Jian and H. Zhai, “Paired superfluidity and fractionalized vortices in systems of spin-orbit coupled bosons,” Phys. Rev. B 84, 060508(R) (2011).
[Crossref]

C. Wang, C. Gao, C.-M. Jian, and H. Zhai, “Spin-orbit coupled spinor Bose-Einstein condensates,” Phys. Rev. Lett. 105, 160403 (2010).
[Crossref]

Zhang, S. Z.

T.-L. Ho and S. Z. Zhang, “Bose-Einstein condensates with spin-orbit interaction,” Phys. Rev. Lett. 107, 150403 (2011).
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Zhang, Y. C.

C. F. Liu, H. Fan, Y. C. Zhang, D. S. Wang, and W. M. Liu, “Circular-hyperbolic skyrmion in rotating pseudo-spin-1/2 Bose-Einstein condensates with spin-orbit coupling,” Phys. Rev. A 86, 053616 (2012).
[Crossref]

Zhou, J.

X.-F. Zhou, J. Zhou, and C. Wu, “Vortex structures of rotating spin-orbit-coupled Bose-Einstein condensates,” Phys. Rev. A 84, 063624 (2011).
[Crossref]

Zhou, X. F.

C. Wu, I. Mondragon-Shem, and X. F. Zhou, “Unconventional Bose-Einstein condensations from spin-orbit coupling,” Chin. Phys. Lett.,  28, 097102 (2011).
[Crossref]

Zhou, X.-F.

X.-F. Zhou, J. Zhou, and C. Wu, “Vortex structures of rotating spin-orbit-coupled Bose-Einstein condensates,” Phys. Rev. A 84, 063624 (2011).
[Crossref]

Zoller, P.

Juan J. Garcia-Ripoll, Juan I. Cirac, J. Anglin, Victor M. Perez-Garcia, and P. Zoller, “Spin monopoles with Bose-Einstein condensates,” Phys. Rev. A 61, 053609 (2000).
[Crossref]

Chin. Phys. Lett. (1)

C. Wu, I. Mondragon-Shem, and X. F. Zhou, “Unconventional Bose-Einstein condensations from spin-orbit coupling,” Chin. Phys. Lett.,  28, 097102 (2011).
[Crossref]

Laser Phys. Lett. (1)

S. Gautam and S. K. Adhikari, “Mobile vector soliton in a spin-orbit coupled spin-1 condensate,” Laser Phys. Lett. 12, 045501 (2015).
[Crossref]

Nature (1)

Y.-J. Lin, K. Jiménez-García, and I. B. Spielman, “Spin-orbit-coupled Bose-Einstein condensates,” Nature 471, 83–88 (2011).
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Phys. Rev. A (11)

Juan J. Garcia-Ripoll, Juan I. Cirac, J. Anglin, Victor M. Perez-Garcia, and P. Zoller, “Spin monopoles with Bose-Einstein condensates,” Phys. Rev. A 61, 053609 (2000).
[Crossref]

T. Busch and J. R. Anglin, “Wave-function monopoles in Bose-Einstein condensates,” Phys. Rev. A 60, R2669–R2672 (1999).
[Crossref]

C. F. Liu, Y. M. Yu, S. C. Gou, and W. M. Liu, “Vortex chain in anisotropic spin-orbit-coupled spin-1 Bose-Einstein condensates,” Phys. Rev. A 87, 063630 (2013).
[Crossref]

C. F. Liu, G. Juzeliûnas, and W. M. Liu, “Spin-orbit coupling manipulating composite topological spin textures in atomic-molecular Bose-Einstein condensates,” Phys. Rev. A 95, 023624 (2017).
[Crossref]

S. Gautam and S. K. Adhikari, “Vortex-bright solitons in a spin-orbit-coupled spin-1 condensate,” Phys. Rev. A 95, 013608 (2017).
[Crossref]

S. Gautam and S. K. Adhikari, “Fractional-charge vortex in a spinor Bose-Einstein condensate,” Phys. Rev. A 93, 013630 (2016).
[Crossref]

X.-F. Zhou, J. Zhou, and C. Wu, “Vortex structures of rotating spin-orbit-coupled Bose-Einstein condensates,” Phys. Rev. A 84, 063624 (2011).
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C. F. Liu and W. M. Liu, “Spin-orbit-coupling-induced half-skyrmion excitations in rotating and rapidly quenched spin-1 Bose-Einstein condensates,” Phys. Rev. A 86, 033602 (2012).
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S. W. Su, I. K. Liu, Y. C. Tsai, W. M. Liu, and S. C. Gou, “Crystallized half-skyrmions and inverted half-skyrmions in the condensation of spin-1 Bose gases with spin-orbit coupling,” Phys. Rev. A 86, 023601 (2012).
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T. Mizushima, N. Kobayashi, and K. Machida, “Coreless and singular vortex lattices in rotating spinor Bose-Einstein condensates,” Phys. Rev. A 70, 043613 (2004).
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C. F. Liu, H. Fan, Y. C. Zhang, D. S. Wang, and W. M. Liu, “Circular-hyperbolic skyrmion in rotating pseudo-spin-1/2 Bose-Einstein condensates with spin-orbit coupling,” Phys. Rev. A 86, 053616 (2012).
[Crossref]

Phys. Rev. B (1)

C.-M. Jian and H. Zhai, “Paired superfluidity and fractionalized vortices in systems of spin-orbit coupled bosons,” Phys. Rev. B 84, 060508(R) (2011).
[Crossref]

Phys. Rev. Lett. (9)

S. Sinha, R. Nath, and L. Santos, “Trapped two-dimensional condensates with synthetic spin-orbit coupling,” Phys. Rev. Lett. 107, 270401 (2011).
[Crossref]

H. Hu, B. Ramachandhran, H. Pu, and X.-J. Liu, “Spin-orbit coupled weakly interacting Bose-Einstein condensates in harmonic traps,” Phys. Rev. Lett. 108, 010402 (2012).
[Crossref] [PubMed]

C. Wang, C. Gao, C.-M. Jian, and H. Zhai, “Spin-orbit coupled spinor Bose-Einstein condensates,” Phys. Rev. Lett. 105, 160403 (2010).
[Crossref]

T.-L. Ho and S. Z. Zhang, “Bose-Einstein condensates with spin-orbit interaction,” Phys. Rev. Lett. 107, 150403 (2011).
[Crossref] [PubMed]

J. Ruostekoski and J. R. Anglin, “Monopole core instability and Alice rings in spinor Bose-Einstein condensates,” Phys. Rev. Lett. 91, 190402 (2003).
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H. T.C. Stoof, E. Vliegen, and U. Al Khawaja, “Monopoles in an antiferromagnetic Bose-Einstein condensate,” Phys. Rev. Lett. 87, 120407 (2001).
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J.-P. Martikainen, A. Collin, and K.-A. Suominen, “Creation of a monopole in a spinor condensate,” Phys. Rev. Lett. 88, 090404 (2002).
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T. Mizushima, K. Machida, and T. Kita, “Mermin-Ho vortex in ferromagnetic spinor Bose-Einstein condensates,” Phys. Rev. Lett. 89, 030401 (2002).
[Crossref] [PubMed]

X.-Q. Xu and J. H. Han, “Spin-orbit coupled Bose-Einstein condensate under rotation,” Phys. Rev. Lett. 107, 200401 (2011).
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Phys. Rev. X (1)

A. S. Bradley and B. P. Anderson, “Energy spectra of vortex distributions in two-dimensional quantum turbulence,” Phys. Rev. X 2, 041001 (2012).

Sci. Rep. (1)

C. F. Liu, H. Fan, S.C. Gou, and W. M. Liu, “Crystallized and amorphous vortices in rotating atomic-molecular Bose-Einstein condensates,” Sci. Rep. 4, 4224 (2014).
[Crossref] [PubMed]

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Figures (12)

Fig. 1
Fig. 1 Several schematics of the spin density and the spin singularity in the spin-1 BECs. (a) The formation of wave function is Ψ = (sin(D), cos(D)sin(A), cos(D)cos(A)) T . In (b), (c) and (d), the formation of the wave function is Ψ = (sin(D), cos(D)sech[C|rr0|], cos(D) tanh[C|rr0|]e 0 ) T . (b) β0 = 0, (c) β0 = π/2 and (d) β0 = π. The color denotes the value of |S| and the mauve dots indicate the positions of the spin singularities.
Fig. 2
Fig. 2 Energy of the spin-1 BECs with a possible half-quantum vortex. (a) The effect of |F = 1, mF = 1〉 component on the energy of the BECs as the variational parameter C changes. Other parameters are gn = 1, gs = 0, ω = 0.1, Bz = 0.5, B = 0, κx = κy = κ = 1 and (x0, y0) = (0, −0.5). (b) The effect of the SOC on the energy of the BECs as the variational parameter C varies. (c) The effect of the Bz on the energy of the BECs as the variational parameter C varies. (d) The effect of the B on the energy of the BECs as the variational parameter C varies. The energy, magnetisation, length, strength of the SOC and strength of the interaction are in units of ħω, ħω, ξ, μ / m , and ξ3μ, respectively.
Fig. 3
Fig. 3 Approximate half-quantum vortex solution in the spin-1 BEC and the corresponding singular spin texture. (a),(b),(c) and (d) show an example of the half-quantum vortex. The parameters are gn = 1, gs = 0, ω = 0.1, sin(D) = 0, C = 0.38, Bz = 0.5, B = 0, κx = κy = κ = 1 and (x0, y0) = (0, −0.5). (a) and (b) are the densities of the |F = 1, mF = 0〉 and |F = 1, mF = −1〉 components, respectively. (c) and (d) are the corresponding phases. (e) shows the profile of the half-quantum vortex. The green curve is the profile of a normal vortex with Eq. 6. (f) Spin density |S|. (g) The profile of the spin density |S|. (h) Spin texture. (i) Topological charge density q(x, y). The energy, magnetisation, length, strength of the SOC and strength of the interaction are in units of ħω, ħω, ξ, μ / m , and ξ3μ, respectively.
Fig. 4
Fig. 4 (a) The stable minimum energy Emin of the BECs as a function of the position of the vortex (x, y). We set Bz = 0.5 and κx = κy = κ = 1. (b) The corresponding parameter C E min as a function of the position of the vortex (x, y). (c) The unstable minimum energy Emin of the BECs as a function of the position of the vortex (x, y). We set Bz = 1.5 and κ = 0.5. (d) Stability of the half-quantum vortex under the magnetic field Bz and the SOC κ. The energy, magnetisation, length, strength of the SOC and strength of the interaction are in units of ħω, ħω, ξ, μ / m , and ξ3μ, respectively.
Fig. 5
Fig. 5 The SOC causes the energy break as the position of the vortex changes. The strength of the SOC is κx = κy = κ. (a) The total energy of the system as the position of the vortex (x0 = 0, y0 = Y) varies. (b) The corresponding energy of the SOC term, i.e., ESOC. We set the half-quantum vortex with C = 0.4 in the BECs, and strength of the magnetic field Bz = 0.5. The energy, magnetisation, length, strength of the SOC and strength of the interaction are in units of ħω, ħω, ξ, μ / m , and ξ3μ, respectively.
Fig. 6
Fig. 6 (a) The most stable position of the half-quantum vortex with the parameter C E min in the BECs as the SOC κ (κx = κy = κ) varies. | r | = x s 2 + y s 2 . (b) The corresponding parameter C E min as a function of κ. (c) The most stable position of the half-quantum vortex with C E min in the BECs as the magnetic field Bz varies. (d) The corresponding parameter C E min as a function of Bz. The energy, magnetisation, length, strength of the SOC and strength of the interaction are in units of ħω, ħω, ξ, μ / m , and ξ3μ, respectively.
Fig. 7
Fig. 7 The effect of the SOC on the SOC energy eSOC(x, y) of the BECs under the parameter C E min (0, −0.5). (a) κx = κy = κ = 0. (b) κx = κy = κ = 0.5. (c) κx = κy = κ = 1. (d) The minimum values of eSOC as a function of the SOC κ. (e) The sum of the SOC energy ESOC [ESOC = ∬ eSOC(x, y)dxdy] as a function of the SOC κ. We set Bz = 0.5 and the position of the vortex to be (0, −0.5). The energy, magnetisation, length, strength of the SOC and strength of the interaction are in units of ħω, ħω, ξ, μ / m , and ξ3μ, respectively.
Fig. 8
Fig. 8 The effect of the magnetic field Bz on the SOC energy eSOC(x, y) of the BECs under the parameter C E min (0, −0.5). (a) Bz = 0. (b) Bz = 0.5. (c) Bz = 1. (d) The minimum values of eSOC as a function of the magnetic field Bz. (e) The sum of the SOC energy ESOC as a function of the magnetic field Bz. We set the SOC κx = κy = κ = 0.5 and the position of the vortex to be (0, −0.5). The energy, magnetisation, length, strength of the SOC and strength of the interaction are in units of ħω, ħω, ξ, μ / m , and ξ3μ, respectively.
Fig. 9
Fig. 9 The total energy of BECs as gs and C change. We set the position of the vortex to be (0, −0.5). (a) κx = κy = κ = 1 and Bz = 0.5. (b) κx = κy = κ = 1 and Bz = 0. (c) κx = κy = κ = 0 and Bz = 0. (d) κx = κy = κ = 0 and Bz = 0.5. The stars indicate the best C to create the lowest energy as gs varies. The energy, magnetisation, length, strength of the SOC and strength of the interaction are in units of ħω, ħω, ξ, μ / m , and ξ3μ, respectively.
Fig. 10
Fig. 10 The total energy of BECs as the position of the vortex (x0 = 0, y0 = Y) and C change. (a) gs = 0.2, κx = κy = κ = 1 and Bz = 0.5. (b) gs = 0.2, κx = κy = κ = 1 and Bz = 0. (c) gs = −0.2, κx = κy = κ = 0 and Bz = 0. (d) gs = −0.2, κx = κy = κ = 0 and Bz = 0.5. The arrow indicates the stablest region, where the energy is the lowest. The energy, magnetisation, length, strength of the SOC and strength of the interaction are in units of ħω, ħω, ξ, μ / m , and ξ3μ, respectively.
Fig. 11
Fig. 11 The minimum energy Emin of the BECs as a function of the position of the vortex (x, y). We set Bz = 0.5 and κx = κy = κ = −1. The energy, magnetisation, length, strength of the SOC and strength of the interaction are in units of ħω, ħω, ξ, μ / m , and ξ3μ, respectively.
Fig. 12
Fig. 12 (a1)–(a4) are the results of the spin vortex with C = 0.4 at (0, −0.5). (a1) phase of Ψ 1 ( 0 , 0.5 ) . (a2) the real part of x Ψ 1 ( 0 , 0.5 ) ( x , y ) . (a3) the imaginary part of x Ψ 1 ( 0 , 0.5 ) ( x , y ) . (a4) e soc ( 0 , 0.5 ) ( x , y ) . (b1)–(b4) are the results of the spin vortex with C = 0.4 at (0, 0.5). (b1) phase of Ψ 1 ( 0 , 0.5 ) . (b2) the real part of x Ψ 1 ( 0 , 0.5 ) ( x , y ) . (a3) the imaginary part of x Ψ 1 ( 0 , 0.5 ) ( x , y ) . (a4) e soc ( 0 , 0.5 ) ( x , y ) . For all other parameters, we use Bz = 0.5, B = 0 and κx = κy = κ = 1. The energy, magnetisation, length, strength of the SOC and strength of the interaction are in units of ħω, ħω, ξ, μ / m , and ξ3μ, respectively.

Equations (8)

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S α = m , n = 0 , ± 1 Ψ m * ( F ^ α ) m , n Ψ n / | Ψ | 2 ( α = x , y , z ) .
| S | 2 = 2 Ψ 0 * 2 Ψ 1 Ψ 1 / | Ψ | 4 + 2 Ψ 0 2 Ψ 1 * Ψ 1 * / | Ψ | 4 + | Ψ 0 | 2 ( | Ψ 1 + Ψ 1 | 2 + | Ψ 1 Ψ 1 | 2 ) / | Ψ | 4 + ( | Ψ 1 | 2 | Ψ 1 | 2 ) 2 / | Ψ | 4 .
i Ψ j t = [ 2 2 2 m + m ω 2 ( x 2 + y 2 ) 2 + g n | Ψ | 2 ] Ψ j + g s α = x , y , z n , k , l = 0 ± 1 ( F ^ α ) j n ( F ^ α ) k l Ψ n Ψ k * Ψ l + n = 0 ± 1 B j n Ψ n + α = x , y n = 0 ± 1 κ α ( F ^ α ) j n p α Ψ n ,
B = ( B z B e i θ 2 0 B e i θ 2 0 B e i θ 2 0 B e i θ 2 B z )
E = d 2 r n = 1 , 0 , 1 Ψ n * ( 2 2 2 m + m ω 2 ( x 2 + y 2 ) 2 ) Ψ n + g n 2 ( | Ψ 1 | 2 + | Ψ 0 | 2 + | Ψ 1 | 2 ) 2 + κ m [ Ψ 1 * ( i x y ) Ψ 0 + Ψ 0 * ( i x y ) Ψ 1 + H . c . ] + B z ( | Ψ 1 | 2 | Ψ 1 | 2 ) + g s 2 [ ( | Ψ 1 | 2 | Ψ 1 | 2 ) 2 + 2 | Ψ 1 * Ψ 0 + Ψ 0 * Ψ 1 | 2 ] + B 2 ( Ψ 1 * Ψ 0 e i θ + Ψ 0 * Ψ 1 e i θ + H . c . ) .
ϕ = n 0 | r r k | e ± i θ k ( r r k ) 2 + Λ 2 ,
Ψ = ( Ψ 1 Ψ 0 Ψ 1 ) = n 0 ( r ) ( sin ( D ) cos ( D ) sech [ C | r r 0 | ] cos ( D ) tanh [ C | r r 0 | ] e i β ) ,
e soc ( 0 , 0.5 ) ( x , y ) = κ m Ψ 0 ( 0 , 0.5 ) ( x , y ) [ i x Ψ 1 ( 0 , 0.5 ) ( x , y ) y Ψ 1 ( 0 , 0.5 ) ( x , y ) ] + κ m ( Ψ 1 ( 0 , 0.5 ) ( x , y ) ) * [ i x Ψ 0 ( 0 , 0.5 ) ( x , y ) + y Ψ 0 ( 0 , 0.5 ) ( x , y ) ] = κ m Ψ 0 ( 0 , 0.5 ) ( x , y ) [ i x Ψ 1 ( 0 , 0.5 ) ( x , y ) + y Ψ 1 ( 0 , 0.5 ) ( x , y ) ] + κ m ( Ψ 1 ( 0 , 0.5 ) ( x , y ) ) * [ i x Ψ 0 ( 0 , 0.5 ) ( x , y ) y Ψ 0 ( 0 , 0.5 ) ( x , y ) ] = e soc ( 0 , 0.5 ) ( x , y ) .

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