Two-dimensional anisotropic vortex quantum droplets in dipolar Bose−Einstein condensates

doi:10.1007/s11467-023-1338-7

Front. Phys.

2024, Vol. 19

Issue (2) : 22202 https://doi.org/10.1007/s11467-023-1338-7

RESEARCH ARTICLE

Two-dimensional anisotropic vortex quantum droplets in dipolar Bose−Einstein condensates

Guilong Li¹, Xunda Jiang¹, Bin Liu¹(

), Zhaopin Chen², Boris A. Malomed^3,⁴, Yongyao Li^1,⁵(

)

¹. School of Physics and Optoelectronic Engineering, Foshan University, Foshan 528225, China
². Physics Department and Solid-State Institute, Technion, Haifa 32000, Israel
³. Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel
⁴. Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica, Chile
⁵. Guangdong-Hong Kong-Macao Joint Laboratory for Intelligent Micro−Nano Optoelectronic Technology, Foshan University, Foshan 528225, China

Download: PDF(3982 KB) HTML
Export: BibTeX | EndNote | Reference Manager | ProCite | RefWorks

Abstract

Creation of stable intrinsically anisotropic self-bound states with embedded vorticity is a challenging issue. Previously, no such states in Bose−Einstein condensates (BECs) or other physical settings were known. Dipolar BEC suggests a unique possibility to predict stable two dimensional anisotropic vortex quantum droplets (2D-AVQDs). We demonstrate that they can be created with the vortex axis oriented perpendicular to the polarization of dipoles. The stability area and characteristics of the 2D-AVQDs in the parameter space are revealed by means of analytical and numerical methods. Further, the rotation of the polarizing magnetic field is considered, and the largest angular velocities, up to which spinning 2D-AVQDs can follow the rotation in clockwise and anti-clockwise directions, are found. Collisions between moving 2D-AVQDs are studied too, demonstrating formation of bound states with a vortex−antivortex−vortex structure. A stability domain for such stationary bound states is identified. Unstable dipolar states, that can be readily implemented by means of phase imprinting, quickly transform into robust 2D-AVQDs, which suggests a straightforward possibility for the creation of these states in the experiment.

Keywords dipolar Bose−Einstein condensate anisotropic vortex quantum droplets

Corresponding Author(s): Bin Liu,Yongyao Li

Issue Date: 26 September 2023

Cite this article:

Guilong Li,Xunda Jiang,Bin Liu, et al. Two-dimensional anisotropic vortex quantum droplets in dipolar Bose−Einstein condensates[J]. Front. Phys. , 2024, 19(2): 22202.

URL:

https://academic.hep.com.cn/fop/EN/10.1007/s11467-023-1338-7
https://academic.hep.com.cn/fop/EN/Y2024/V19/I2/22202

Fig.1 (a1, a2) A typical example of stable 2D-AVQDs, with

(N, g) = (1000, 0.25)

. The panels (a1, a2) display, severally, density and phase patterns of the droplet. (b) In the plane of

(N, g)

, 2D-AVQDs (

S = 1

), fundamental QDs (

S = 0

) are stable in the orange area. In the green area, only dipole mode and fundamental QDs can be stable.

Fig.2 (a1, a2) Density and phase patterns of the stable vortex?antivortex?vortex bound state with

(N, g) = (1000,

0.25)

x 0 = 13.6

and

μ = ? 0.8482

. (b, c) The chemical potential of the states of this type versus

g

(at

N = 1000

) and

N

(at

g = 0.25

), respectively. Solid and dashed parts of the curves represent, severally, stable and unstable states. (d1?d4) Steady spinning of a vortex?antivortex?vortex bound state with

(N, g) = (1000, 0.25)

, under the action of the polarizing magnetic field rotating with angular velocity

ω = 0.2 π × 10 ? 3

. The shape of the bound state is displayed at

t = 0

(d1),

1250

(d2)

2500

(d3),

3750

(d4).

Fig.3 The peak density (

I P

), chemical potential (

μ

), effective area (

A e f f

), aspect ratio (

E l

), and average angular momentum (

L ˉ z

), see Eq. (9), vs.

N

(a1?a5) and

g

(b1?b5). In panels (a1?a5),

g = 0.25

is fixed, while in panels (b1?b5)

N = 1000

. Red dashed curves in panels (a1?a3, b1?b3) represent the analytical approximation given by Eqs. (7) and (8), respectively.

Fig.4 (a?d) Steady spinning of 2D-AVQDs with

(N, g) = (1000, 0.25)

, which follows the rotation of the polarizing magnetic field with angular velocity

ω = 0.25 π × 10 ? 3

. The shape of 2D-AVQDs is displayed at

t = 0

(a),

1000

(b)

2000

(c),

3000

(d). (e, f) The largest angular velocities,

| ω c r (±) |

, which admit stable spinning of 2D-AVQDs in two opposite directions, vs.

g

(at

N = 1000

) and

N

(at

g = 0.25

), respectively.

Fig.5 The collision of two 2D-AVQDs with identical vorticities, initiated, at

t = 0

, by input

? (x ? x 0, y) e ? i η x +

? (x + x 0, y) e i η x

with

x 0 = 64

η = 0.025

g = 0.25

, and norm of each 2D-AVQDs

N = 1000

. (a1?d1) Density patterns at

t = 400

(a1),

635

(b1),

760

(c1), and

900

(d1). (a2?d2) The corresponding phase diagrams for panels (a2?d2).

Fig.6 (a?c) Spontaneous transformation of the unstable dipole mode with

N = 1000

and

g = 0.25

into a stable 2D-AVQDs is represented by density snapshots taken at

t = 0

(a),

50

(b), and

700

(c). (d) The phase pattern at

t = 700

1	Fibich G. , Papanicolaou G. . Self-focusing in the perturbed and unperturbed nonlinear Schrӧdinger equation in critical dimension. SIAM J. Appl. Math., 1999, 60(1): 183 https://doi.org/10.1137/S0036139997322407
2	Bergé L. . Wave collapse in physics: Principles and applications to light and plasma waves. Phys. Rep., 1998, 303(5−6): 259 https://doi.org/10.1016/S0370-1573(97)00092-6
3	A. Kuznetsov E. , Dias F. . Bifurcations of solitons and their stability. Phys. Rep., 2011, 507(2−3): 43 https://doi.org/10.1016/j.physrep.2011.06.002
4	A. Malomed B. . Two-dimensional solitons in nonlocal media: A brief review. Symmetry (Basel), 2022, 14(8): 1565 https://doi.org/10.3390/sym14081565
5	A. Malomed B., Multidimensional Solitons, American Institute of Physics: Melville, NY, 2022
6	Heidemann R. , Raitzsch U. , Bendkowsky V. , Butscher B. , Löw R. , Pfau T. . Rydberg excitation of Bose–Einstein condensates. Phys. Rev. Lett., 2008, 100(3): 033601 https://doi.org/10.1103/PhysRevLett.100.033601
7	Maucher F. , Henkel N. , Saffman M. , Królikowski W. , Skupin S. , Pohl T. . Rydberg-induced solitons: Three-dimensional self-trapping of matter waves. Phys. Rev. Lett., 2011, 106(17): 170401 https://doi.org/10.1103/PhysRevLett.106.170401
8	O’Dell D. , Giovanazzi S. , Kurizki G. , M. Akulin V. . Bose–Einstein condensates with 1/r interatomic attraction: Electromagnetically induced “gravity”. Phys. Rev. Lett., 2000, 84(25): 5687 https://doi.org/10.1103/PhysRevLett.84.5687
9	Qin J. , Dong G. , A. Malomed B. . Stable giant vortex annuli in microwave-coupled atomic condensates. Phys. Rev. A, 2016, 94(5): 053611 https://doi.org/10.1103/PhysRevA.94.053611
10	Lahaye T. , Menotti C. , Santos L. , Lewenstein M. , Pfau T. . The physics of dipolar bosonic quantum gases. Rep. Prog. Phys., 2009, 72(12): 126401 https://doi.org/10.1088/0034-4885/72/12/126401
11	Pedri P. , Santos L. . Two-dimensional bright solitons in dipolar Bose–Einstein condensates. Phys. Rev. Lett., 2005, 95(20): 200404 https://doi.org/10.1103/PhysRevLett.95.200404
12	Nath R. , Pedri P. , Santos L. . Stability of dark solitons in three dimensional dipolar Bose–Einstein condensates. Phys. Rev. Lett., 2008, 101(21): 210402 https://doi.org/10.1103/PhysRevLett.101.210402
13	Tikhonenkov I. , A. Malomed B. , Vardi A. . Anisotropic solitons in dipolar Bose–Einstein condensates. Phys. Rev. Lett., 2008, 100(9): 090406 https://doi.org/10.1103/PhysRevLett.100.090406
14	Raghunandan M. , Mishra C. , Lakomy K. , Pedri P. , Santos L. , Nath R. . Two-dimensional bright solitons in dipolar Bose–Einstein condensates with tilted dipoles. Phys. Rev. A, 2015, 92(1): 013637 https://doi.org/10.1103/PhysRevA.92.013637
15	B. Blakie P. . Axial collective mode of a dipolar quantum droplet. Photonics, 2023, 10(4): 393 https://doi.org/10.3390/photonics10040393
16	Zhao Y. , Lei Y. , Xu Y. , Xu S. , Triki H. , Biswas A. , Zhou Q. . Vector spatiotemporal solitons and their memory features in cold Rydberg gases. Chin. Phys. Lett., 2022, 39(3): 034202 https://doi.org/10.1088/0256-307X/39/3/034202
17	Tikhonenkov I. , A. Malomed B. , Vardi A. . Vortex solitons in dipolar Bose–Einstein condensates. Phys. Rev. A, 2008, 78(4): 043614 https://doi.org/10.1103/PhysRevA.78.043614
18	Jiang X. , Fan Z. , Chen Z. , Pang W. , Li Y. , A. Malomed B. . Two-dimensional solitons in dipolar Bose-Einstein condensates with spin–orbit coupling. Phys. Rev. A, 2016, 93(2): 023633 https://doi.org/10.1103/PhysRevA.93.023633
19	Liao B. , Ye Y. , Zhuang J. , Huang C. , Deng H. , Pang W. , Liu B. , Li Y. . Anisotropic solitary semivortices in dipolar spinor condensates controlled by the twodimensional anisotropic spin–orbit coupling. Chaos Solitons Fractals, 2018, 116: 424 https://doi.org/10.1016/j.chaos.2018.10.001
20	Liao B. , Li S. , Huang C. , Luo Z. , Pang W. , Tan H. , A. Malomed B. , Li Y. . Anisotropic semivortices in dipolar spinor condensates controlled by Zeeman splitting. Phys. Rev. A, 2017, 96(4): 043613 https://doi.org/10.1103/PhysRevA.96.043613
21	Schmitt M. , Wenzel M. , Böttcher F. , Ferrier-Barbut I. , Pfau T. . Self-bound droplets of a dilute magnetic quantum liquid. Nature, 2016, 539(7628): 259 https://doi.org/10.1038/nature20126
22	Chomaz L. , Baier S. , Petter D. , J. Mark M. , Wächtler F. , Santos L. , Ferlaino F. . Quantum-fluctuation-driven crossover from a dilute Bose–Einstein condensate to a macrodroplet in a dipolar quantum fluid. Phys. Rev. X, 2016, 6(4): 041039 https://doi.org/10.1103/PhysRevX.6.041039
23	R. Cabrera C. , Tanzi L. , Sanz J. , Naylor B. , Thomas P. , Cheiney P. , Tarruell L. . Quantum liquid droplets in a mixture of Bose–Einstein condensates. Science, 2018, 359(6373): 301 https://doi.org/10.1126/science.aao5686
24	Semeghini G. , Ferioli G. , Masi L. , Mazzinghi C. , Wolswijk L. , Minardi F. , Modugno M. , Modugno G. , Inguscio M. , Fattori M. . Self-bound quantum droplets of atomic mixtures in free space. Phys. Rev. Lett., 2018, 120(23): 235301 https://doi.org/10.1103/PhysRevLett.120.235301
25	D’Errico C. , Burchianti A. , Prevedelli M. , Salasnich L. , Ancilotto F. , Modugno M. , Minardi F. , Fort C. . Observation of quantum droplets in a heteronuclear bosonic mixture. Phys. Rev. Res., 2019, 1(3): 033155 https://doi.org/10.1103/PhysRevResearch.1.033155
26	S. Petrov D. . Quantum mechanical stabilization of a collapsing Bose–Bose mixture. Phys. Rev. Lett., 2015, 115(15): 155302 https://doi.org/10.1103/PhysRevLett.115.155302
27	Luo Z. , Pang W. , Liu B. , Li Y. , A. Malomed B. . A new kind form of liquid matter: Quantum droplets. Front. Phys., 2021, 16(3): 32201 https://doi.org/10.1007/s11467-020-1020-2
28	Böttcher F. , N. Schmidt J. , Hertkorn J. , S. H. Ng K. , D. Graham S. , Guo M. , Langen T. , Pfau T. . New states of matter with fine-tuned interactions: Quantum droplets and dipolar supersolids. Rep. Prog. Phys., 2021, 84(1): 012403 https://doi.org/10.1088/1361-6633/abc9ab
29	Guo M. , Pfau T. . A new state of matter of quantum droplets. Front. Phys., 2021, 16(3): 32202 https://doi.org/10.1007/s11467-020-1035-8
30	A. Malomed B. . The family of quantum droplets keeps expanding. Front. Phys., 2021, 16(2): 22504 https://doi.org/10.1007/s11467-020-1024-y
31	Li Y. , Chen Z. , Luo Z. , Huang C. , Tan H. , Pang W. , A. Malomed B. . Two-dimensional vortex quantum droplets. Phys. Rev. A, 2018, 98(6): 063602 https://doi.org/10.1103/PhysRevA.98.063602
32	V. Kartashov Y. , A. Malomed B. , Tarruell L. , Torner L. . Three-dimensional droplets of swirling superfluids. Phys. Rev. A, 2018, 98(1): 013612 https://doi.org/10.1103/PhysRevA.98.013612
33	Zhang X. , Xu X. , Zheng Y. , Chen Z. , Liu B. , Huang C. , A. Malomed B. , Li Y. . Semidiscrete quantum droplets and vortices. Phys. Rev. Lett., 2019, 123: 113901
34	Cidrim A. , E. A. dos Santos F. , A. L. Henn E. , Macri T. . Vortices in self-bound dipolar droplets. Phys. Rev. A, 2018, 98(2): 023618 https://doi.org/10.1103/PhysRevA.98.023618
35	Baillie D. , M. Wilson R. , N. Bisset R. , B. Blakie P. . Self-bound dipolar droplet: A localized matter wave in free space. Phys. Rev. A, 2016, 94: 021602(R) https://doi.org/10.1103/PhysRevA.94.021602
36	Sinha S. , Santos L. . Cold dipolar gases in quasi-one-dimensional geometries. Phys. Rev. Lett., 2007, 99(14): 140406 https://doi.org/10.1103/PhysRevLett.99.140406
37	Cuevas J. , A. Malomed B. , G. Kevrekidis P. , J. Frantzeskakis D. . Solitons in quasi-one-dimensional Bose-Einstein condensates with competing dipolar and local interactions. Phys. Rev. A, 2009, 79(5): 053608 https://doi.org/10.1103/PhysRevA.79.053608
38	G. Vakhitov N. , A. Kolokolov A. . Stationary solutions of the wave equation in a medium with nonlinearity saturation. Radiophys. Quantum Electron., 1973, 16(7): 783 https://doi.org/10.1007/BF01031343
39	S. Kivshar Yu. , A. Malomed B. . Dynamics of solitons in nearly integrable systems. Rev. Mod. Phys., 1989, 61(4): 763 https://doi.org/10.1103/RevModPhys.61.763
40	Burger S. , Bongs K. , Dettmer S. , Ertmer W. , Sengstock K. , Sanpera A. , V. Shlyapnikov G. , Lewenstein M. . Dark solitons in Bose−Einstein condensates. Phys. Rev. Lett., 1999, 83(25): 5198 https://doi.org/10.1103/PhysRevLett.83.5198
41	P. Anderson B. , C. Haljan P. , A. Regal C. , L. Feder D. , A. Collins L. , W. Clark C. , A. Cornell E. . Watching dark solitons decay into vortex rings in a Bose–Einstein condensate. Phys. Rev. Lett., 2001, 86(14): 2926 https://doi.org/10.1103/PhysRevLett.86.2926
42	J. Shen Y. , J. Wang X. , W. Xie Z. , J. Min C. , Fu X. , Liu Q. , L. Gong M. , C. Yuan X. . Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities. Light Sci. Appl., 2019, 8(1): 90 https://doi.org/10.1038/s41377-019-0194-2
43	Ramachandhran B. , Opanchuk B. , J. Liu X. , Pu H. , D. Drummond P. , Hu H. . Half-quantum vortex state in a spin–orbit-coupled Bose–Einstein condensate. Phys. Rev. A, 2012, 85(2): 023606 https://doi.org/10.1103/PhysRevA.85.023606
44	Li Y. , Liu J. , Pang W. , A. Malomed B. . Matter-wave solitons supported by field-induced dipole–dipole repulsion with spatially modulated strength. Phys. Rev. A, 2013, 88(5): 053630 https://doi.org/10.1103/PhysRevA.88.053630
45	Li Y. , Liu Y. , Fan Z. , Pang W. , Fu S. , A. Malomed B. . Two-dimensional dipolar gap solitons in free space with spin−orbit coupling. Phys. Rev. A, 2017, 95(6): 063613 https://doi.org/10.1103/PhysRevA.95.063613
46	Huang C. , Ye Y. , Liu S. , He H. , Pang W. , A. Malomed B. , Li Y. . Excited states of two-dimensional solitons supported by spin–orbit coupling and field-induced dipole–dipole repulsion. Phys. Rev. A, 2018, 97(1): 013636 https://doi.org/10.1103/PhysRevA.97.013636
47	Boudjemâa A. . Fluctuations and quantum self-bound droplets in a dipolar Bose–Bose mixture. Phys. Rev. A, 2018, 98(3): 033612 https://doi.org/10.1103/PhysRevA.98.033612
48	N. Bisset R. , A. P. Ardila L. , Santos L. . Quantum droplets of dipolar mixtures. Phys. Rev. Lett., 2021, 126(2): 025301 https://doi.org/10.1103/PhysRevLett.126.025301

Viewed

Full text

Abstract

Cited

Shared

Discussed