Dipolar Bose gas with SU(3) spin−orbit coupling held under a toroidal trap
Fang Wang, Jia Liu, Si-Lin Chen, Lin Wen, Xue-Ying Yang, Xiao-Fei Zhang
Dipolar Bose gas with SU(3) spin−orbit coupling held under a toroidal trap
We consider a dipolar spin-1 Bose gas with SU(3) spin−orbit coupling trapped in a two-dimensional toroidal trap. Due to the combined effects of SU(3) spin−orbit coupling, dipole−dipole interaction, and spin−exchange interaction, the system exhibits a rich variety of ground-state phases and topological defects, including modified stripe, azimuthal distributed petal and triangular lattice, double-quantum spin vortices, and so on. In particular, by studying the spin texture of such a system, it is found that the formation and transformation between meron and skyrmion topological spin textures can be realized by a choice of dipole−dipole interaction, SU(3) spin−orbit coupling, and spin−exchange interaction. We also give an experimental protocol to observe such novel states within current experimental capacity.
Bose−Einstein condensate / spin−orbit coupling / dipolar condensate / quantum vortex
[1] |
I. Žutić, J. Fabian, and S. Das Sarma, Spintronics: Fundamentals and applications, Rev. Mod. Phys. 76(2), 323 (2004)
CrossRef
ADS
Google scholar
|
[2] |
J. Wunderlich, B. Kaestner, J. Sinova, and T. Jungwirth, Experimental observation of the spin-Hall effect in a two-dimensional spin–orbit coupled semiconductor system, Phys. Rev. Lett. 94(4), 047204 (2005)
CrossRef
ADS
Google scholar
|
[3] |
S. Murakami, Quantum spin Hall effect and enhanced magnetic response by spin–orbit coupling, Phys. Rev. Lett. 97(23), 236805 (2006)
CrossRef
ADS
Google scholar
|
[4] |
M. Z. Hasan and C. L. Kane, Colloquium: Topological insulators, Rev. Mod. Phys. 82(4), 3045 (2010)
CrossRef
ADS
arXiv
Google scholar
|
[5] |
X. L. Qi and S. C. Zhang, Topological insulators and superconductors, Rev. Mod. Phys. 83(4), 1057 (2011)
CrossRef
ADS
arXiv
Google scholar
|
[6] |
Y. J. Lin, K. Jiménez-García, and I. B. Spielman, Spin–orbit-coupled Bose–Einstein condensates, Nature 471(7336), 83 (2011)
CrossRef
ADS
arXiv
Google scholar
|
[7] |
P. Wang, Z. Q. Yu, Z. Fu, J. Miao, L. Huang, S. Chai, H. Zhai, and J. Zhang, Spin–orbit coupled degenerate Fermi gases, Phys. Rev. Lett. 109(9), 095301 (2012)
CrossRef
ADS
arXiv
Google scholar
|
[8] |
L. Huang, Z. Meng, P. Wang, P. Peng, S. L. Zhang, L. Chen, D. Li, Q. Zhou, and J. Zhang, Experimental realization of two-dimensional synthetic spin–orbit coupling in ultracold Fermi gases, Nat. Phys. 12(6), 540 (2016)
CrossRef
ADS
arXiv
Google scholar
|
[9] |
J. R. Li,J. Lee,W. Huang,S. Burchesky,B. Shteynas,F. Ç. Top,A. O. Jamison,W. Ketterle, A stripe phase with supersolid properties in spin−orbit-coupled Bose−Einstein condensates, Nature 543(7643), 91 (2017)
|
[10] |
J. Zhong, B. Tang, X. Chen, and L. Zhou, Quantum gravimetry going toward real applications, Innovation 3(3), 100230 (2022)
CrossRef
ADS
Google scholar
|
[11] |
S. C. Guo, Y. M. Xu, R. Cheng, J. S. Zhou, and X. Chen, Thermal Hall effect in insulating quantum materials, Innovation 3(5), 100290 (2022)
CrossRef
ADS
Google scholar
|
[12] |
R. Barnett,G. R. Boyd,V. Galitski, SU(3) spin–orbit coupling in systems of ultracold atoms, Phys. Rev. Lett. 109(23), 235308 (2012)
|
[13] |
W. Han, X. F. Zhang, S. W. Song, H. Saito, W. Zhang, W. M. Liu, and S. G. Zhang, Double-quantum spin vortices in SU(3) spin−orbit-coupled Bose gases, Phys. Rev. A 94(3), 033629 (2016)
CrossRef
ADS
arXiv
Google scholar
|
[14] |
K. Sun, C. Qu, Y. Xu, Y. Zhang, and C. Zhang, Interacting spin–orbit-coupled spin-1 Bose–Einstein condensates, Phys. Rev. A 93(2), 023615 (2016)
CrossRef
ADS
arXiv
Google scholar
|
[15] |
C. Wu, I. Mondragon-Shem, and X. F. Zhou, Unconventional Bose–Einstein condensations from spin–orbit coupling, Chin. Phys. Lett. 28(9), 097102 (2011)
CrossRef
ADS
arXiv
Google scholar
|
[16] |
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(1), 010402 (2012)
CrossRef
ADS
arXiv
Google scholar
|
[17] |
H. Sakaguchi, B. Li, and B. A. Malomed, Creation of two-dimensional composite solitons in spin–orbit-coupled self-attractive Bose–Einstein condensates in free space, Phys. Rev. E 89(3), 032920 (2014)
CrossRef
ADS
arXiv
Google scholar
|
[18] |
W. Han, G. Juzeliūnas, W. Zhang, and W. M. Liu, Supersolid with nontrivial topological spin textures in spin–orbit-coupled Bose gases, Phys. Rev. A 91(1), 013607 (2015)
CrossRef
ADS
arXiv
Google scholar
|
[19] |
H. Wang, L. H. Wen, H. Yang, C. X. Shi, and J. H. Li, Vortex states and spin textures of rotating spin–orbit-coupled Bose–Einstein condensates in a toroidal trap, J. Phys. At. Mol. Opt. Phys. 50(15), 155301 (2017)
CrossRef
ADS
arXiv
Google scholar
|
[20] |
G. I. Martone, Y. Li, L. P. Pitaevskii, and S. Stringari, Anisotropic dynamics of a spin–orbit-coupled Bose–Einstein condensate, Phys. Rev. A 86(6), 063621 (2012)
CrossRef
ADS
arXiv
Google scholar
|
[21] |
Z. Wu, L. Zhang, W. Sun, X. T. Xu, B. Z. Wang, S. C. Ji, Y. Deng, S. Chen, X. J. Liu, and J. W. Pan, Realization of two-dimensional spin–orbit coupling for Bose–Einstein condensates, Science 354(6308), 83 (2016)
CrossRef
ADS
arXiv
Google scholar
|
[22] |
Y. Xu, Y. Zhang, and B. Wu, Bright solitons in spin–orbit-coupled Bose–Einstein condensates, Phys. Rev. A 87(1), 013614 (2013)
CrossRef
ADS
arXiv
Google scholar
|
[23] |
T. Graß, R. W. Chhajlany, C. A. Muschik, and M. Lewenstein, Spiral spin textures of a bosonic Mott insu-9 lator with SU(3) spin–orbit coupling, Phys. Rev. B 90(19), 195127 (2014)
CrossRef
ADS
arXiv
Google scholar
|
[24] |
C. Ryu, M. Andersen, P. Cladé, V. Natarajan, K. Helmerson, and W. Phillips, Observation of persistent flow of a Bose–Einstein condensate in a toroidal trap, Phys. Rev. Lett. 99(26), 260401 (2007)
CrossRef
ADS
arXiv
Google scholar
|
[25] |
X. F. Zhang, M. Kato, W. Han, S. G. Zhang, and H. Saito, Spin–orbit-coupled Bose–Einstein condensates held under a toroidal trap, Phys. Rev. A 95(3), 033620 (2017)
CrossRef
ADS
arXiv
Google scholar
|
[26] |
A. C. White, Y. P. Zhang, and T. Busch, Odd-petal-number states and persistent flows in spin–orbit coupled Bose–Einstein condensates, Phys. Rev. A 95(4), 041604(R) (2017)
CrossRef
ADS
arXiv
Google scholar
|
[27] |
J. G. Wang, L. L. Xu, and S. J. Yang, Ground-state phases of the spin–orbit-coupled spin-1 Bose gas in a toroidal trap, Phys. Rev. A 96(3), 033629 (2017)
CrossRef
ADS
Google scholar
|
[28] |
K. Liu, H. He, C. Wang, Y. Chen, and Y. Zhang, Spin–orbit-coupled spin-1 Bose–Einstein condensates in a toroidal trap: Even-petal-number necklace-like state and persistent flow, Phys. Rev. A 105(1), 013323 (2022)
CrossRef
ADS
arXiv
Google scholar
|
[29] |
A. Griesmaier, J. Werner, S. Hensler, J. Stuhler, and T. Pfau, Bose–Einstein condensation of chromium, Phys. Rev. Lett. 94(16), 160401 (2005)
CrossRef
ADS
Google scholar
|
[30] |
M. W. Ray, E. Ruokokoski, K. Tiurev, M. Möttönen, and D. S. Hall, Observation of isolated monopoles in a quantum field, Science 348(6234), 544 (2015)
CrossRef
ADS
Google scholar
|
[31] |
L. Tanzi, E. Lucioni, F. Famà, J. Catani, A. Fioretti, C. Gabbanini, R. N. Bisset, L. Santos, and G. Modugno, Observation of a dipolar quantum gas with metastable supersolid properties, Phys. Rev. Lett. 122(13), 130405 (2019)
CrossRef
ADS
arXiv
Google scholar
|
[32] |
T. Lahaye, C. Menotti, L. Santos, M. Lewenstein, and T. Pfau, The physics of dipolar bosonic quantum gases, Rep. Prog. Phys. 72(12), 126401 (2009)
CrossRef
ADS
arXiv
Google scholar
|
[33] |
K. Góral, K. Rzążewski, and T. Pfau, Bose–Einstein condensation with magnetic dipole–dipole forces, Phys. Rev. A 61(5), 051601 (2000)
CrossRef
ADS
Google scholar
|
[34] |
L. Santos, G. V. Shlyapnikov, and M. Lewenstein, Roton–maxon spectrum and stability of trapped dipolar Bose–Einstein condensates, Phys. Rev. Lett. 90(25), 250403 (2003)
CrossRef
ADS
Google scholar
|
[35] |
F. Malet, T. Kristensen, S. M. Reimann, and G. M. Kavoulakis, Rotational properties of dipolar Bose–Einstein condensates confined in anisotropic harmonic potentials, Phys. Rev. A 83(3), 033628 (2011)
CrossRef
ADS
arXiv
Google scholar
|
[36] |
B. Liu, X. Li, L. Yin, and W. V. Liu, Weyl superfluidity in a three-dimensional dipolar fermi gas, Phys. Rev. Lett. 114(4), 045302 (2015)
CrossRef
ADS
arXiv
Google scholar
|
[37] |
S. Y. Chä and U. R. Fischer, Probing the scale invariance of the inflationary power spectrum in expanding quasitwo-dimensional dipolar condensates, Phys. Rev. Lett. 118(13), 130404 (2017)
CrossRef
ADS
arXiv
Google scholar
|
[38] |
M. O. Borgh, J. Lovegrove, and J. Ruostekoski, Internal structure and stability of vortices in a dipolar spinor Bose–Einstein condensate, Phys. Rev. A 95(5), 053601 (2017)
CrossRef
ADS
arXiv
Google scholar
|
[39] |
R. N. Bisset, P. B. Blakie, and S. Stringari, Staticresponse theory and the roton–maxon spectrum of a flattened dipolar Bose–Einstein condensate, Phys. Rev. A 100(1), 013620 (2019)
CrossRef
ADS
arXiv
Google scholar
|
[40] |
Y. Deng, J. Cheng, H. Jing, C. P. Sun, and S. Yi, Spin–orbit-coupled dipolar Bose–Einstein condensates, Phys. Rev. Lett. 108(12), 125301 (2012)
CrossRef
ADS
arXiv
Google scholar
|
[41] |
S. Gopalakrishnan, I. Martin, and E. A. Demler, Quantum quasicrystals of spin–orbit-coupled dipolar bosons, Phys. Rev. Lett. 111(18), 185304 (2013)
CrossRef
ADS
arXiv
Google scholar
|
[42] |
Y. Xu, Y. P. Zhang, and C. Zhang, Bright solitons in a two-dimensional spin–orbit-coupled dipolar Bose–Einstein condensate, Phys. Rev. Lett. 92(1), 013633 (2015)
|
[43] |
R. M. Wilson, B. M. Anderson, and C. W. Clark, Meron ground state of Rashba spin–orbit-coupled dipolar bosons, Phys. Rev. Lett. 111(18), 185303 (2013)
CrossRef
ADS
arXiv
Google scholar
|
[44] |
G. Chen, J. Ma, and S. T. Jia, Long-range superfluid order in trapped Bose–Einstein condensates with spin–orbit coupling, Phys. Rev. A 86(4), 045601 (2012)
CrossRef
ADS
Google scholar
|
[45] |
N. Q. Burdick, Y. Tang, and B. L. Lev, Long-lived spin–orbit-coupled degenerate dipolar Fermi gas, Phys. Rev. X 6(3), 031022 (2016)
CrossRef
ADS
arXiv
Google scholar
|
[46] |
X. J. Feng, J. X. Li, L. Qin, Y. Y. Zhang, S. Q. Xia, L. Zhou, C. J. Yang, Z. L. Zhu, W. M. Liu, and X. D. Zhao, Itinerant ferromagnetism entrenched by the anisotropy of spin–orbit coupling in a dipolar Fermi gas, Front. Phys. 18(5), 52303 (2023)
CrossRef
ADS
Google scholar
|
[47] |
N. Su, Q. B. Wang, J. G. Hu, X. H. Su, and L. H. Wen, Topological defects in rotating spin–orbit-coupled dipolar spin-1 Bose−Einstein condensates, J. Phys. At. Mol. Opt. Phys. 53(21), 215301 (2020)
CrossRef
ADS
arXiv
Google scholar
|
[48] |
Y. Y. Li, Y. Liu, Z. W. Fan, W. Pang, S. H. Fu, and B. A. Malomed, Two-dimensional dipolar gap solitons in free space with spin–orbit coupling, Phys. Rev. A 95(6), 063613 (2017)
CrossRef
ADS
arXiv
Google scholar
|
[49] |
T. Oshima and Y. Kawaguchi, Spin Hall effect in a spinor dipolar Bose–Einstein condensate, Phys. Rev. A 93(5), 053605 (2016)
CrossRef
ADS
arXiv
Google scholar
|
[50] |
M. Kato, X. F. Zhang, D. Sasaki, and H. Saito, Twisted spin vortices in a spin-1 Bose–Einstein condensate with Rashba spin–orbit coupling and dipole–dipole interaction, Phys. Rev. A 94(4), 043633 (2016)
CrossRef
ADS
arXiv
Google scholar
|
[51] |
Y. Kawaguchi and M. Ueda, Spinor Bose–Einstein condensates, Phys. Rep. 520(5), 253 (2012)
CrossRef
ADS
arXiv
Google scholar
|
[52] |
Y. Kawaguchi, H. Saito, K. Kudo, and M. Ueda, Spontaneous magnetic ordering in a ferromagnetic spinor dipolar Bose–Einstein condensate, Phys. Rev. A 82(4), 043627 (2010)
CrossRef
ADS
arXiv
Google scholar
|
[53] |
D. M. Stamper-Kurn and M. Ueda, Spinor Bose gases: Symmetries, magnetism, and quantum dynamics, Rev. Mod. Phys. 85(3), 1191 (2013)
CrossRef
ADS
Google scholar
|
[54] |
D. S. Wang, Y. R. Shi, K. W. Chow, Z. X. Yu, and X. G. Li, Matter-wave solitons in a spin-1 Bose–Einstein condensate with time-modulated external potential and scattering lengths, Eur. Phys. J. D 67(11), 242 (2013)
CrossRef
ADS
Google scholar
|
[55] |
G. B. Arfken,H. J. Weber,F. E. Harris, Mathematical Methods for Physicists, 7th Ed., Academic Press, New York, 2000
|
[56] |
Z. F. Xu, Y. Kawaguchi, L. You, and M. Ueda, Symmetry classification of spin–orbit-coupled spinor Bose–Einstein condensates, Phys. Rev. A 86(3), 033628 (2012)
CrossRef
ADS
arXiv
Google scholar
|
[57] |
Z. Lan and P. Öhberg, Raman-dressed spin-1 spin–orbit coupled quantum gas, Phys. Rev. A 89(2), 023630 (2014)
CrossRef
ADS
arXiv
Google scholar
|
[58] |
Y. Zhang, L. Mao, and C. Zhang, Mean-field dynamics of spin–orbit coupled Bose–Einstein condensates, Phys. Rev. Lett. 108(3), 035302 (2012)
CrossRef
ADS
arXiv
Google scholar
|
[59] |
F. Dalfovo and S. Stringari, Bosons in anisotropic traps: Ground state and vortices, Phys. Rev. A 53(4), 2477 (1996)
CrossRef
ADS
Google scholar
|
[60] |
M. L. Chiofalo, S. Succi, and M. P. Tosi, Ground state of trapped interacting Bose–Einstein condensates by an explicit imaginary-time algorithm, Phys. Rev. E 62(5), 7438 (2000)
CrossRef
ADS
Google scholar
|
[61] |
W. Bao,I. L. Chern,F. Y. Lim, Efficient and spectrally accurate numerical methods for computing ground and first excited states in Bose–Einstein condensates, J. Comput. Phys. 219(2), 836 (2006)
|
[62] |
C. Wang, C. Gao, C. M. Jian, and H. Zhai, Spin-orbit coupled spinor Bose–Einstein condensates, Phys. Rev. Lett. 105(16), 160403 (2010)
CrossRef
ADS
arXiv
Google scholar
|
[63] |
T. Mizushima, N. Kobayashi, and K. Machida, Coreless and singular vortex lattices in rotating spinor Bose–Einstein condensates, Phys. Rev. A 70(4), 043613 (2004)
CrossRef
ADS
Google scholar
|
[64] |
K. Kasamatsu, M. Tsubota, and M. Ueda, Spin textures in rotating two-component Bose–Einstein condensates, Phys. Rev. A 71(4), 043611 (2005)
CrossRef
ADS
Google scholar
|
[65] |
S. Heinze, K. von Bergmann, M. Menzel, J. Brede, A. Kubetzka, R. Wiesendanger, G. Bihlmayer, and S. Blügel, Spontaneous atomic-scale magnetic skyrmion lattice in two dimensions, Nat. Phys. 7(9), 713 (2011)
CrossRef
ADS
Google scholar
|
[66] |
S. Banerjee, J. Rowland, O. Erten, and M. Randeria, Enhanced stability of skyrmions in two-dimensional chiral magnets with Rashba spin–orbit coupling, Phys. Rev. X 4(3), 031045 (2014)
CrossRef
ADS
arXiv
Google scholar
|
[67] |
B. Dong, Q. Sun, W. M. Liu, A. C. Ji, X. F. Zhang, and S. G. Zhang, Multiply quantized and fractional skyrmions in a binary dipolar Bose–Einstein condensate under rotation, Phys. Rev. A 96(1), 013619 (2017)
CrossRef
ADS
Google scholar
|
[68] |
H. Saito, Y. Kawaguchi, and M. Ueda, Breaking of chiral symmetry and spontaneous rotation in a spinor Bose–Einstein condensate, Phys. Rev. Lett. 96(6), 065302 (2006)
CrossRef
ADS
Google scholar
|
[69] |
X. Z. Yu, W. Koshibae, Y. Tokunaga, K. Shibata, Y. Taguchi, N. Nagaosa, and Y. Tokura, Transformation between meron and skyrmion topological spin textures in a chiral magnet, Nature 564(7734), 95 (2018)
CrossRef
ADS
Google scholar
|
[70] |
C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Feshbach resonances in ultracold gases, Rev. Mod. Phys. 82(2), 1225 (2010)
CrossRef
ADS
arXiv
Google scholar
|
/
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