Can vortex quantum droplets be realized experimentally?

Graphical abstract

Creation of stable self-trapped vortex states (alias vortex solitons) in Bose−Einstein condensates (BECs) in free space is a challenging problem [1]. To date, an experimental demonstration of this possibility is missing. A new powerful platform for the realization of various self-trapped states is offered by quantum droplets (QDs), i.e., stable localized modes maintained by the balance of mean-field (MF) interactions and corrections to them induced by quantum fluctuations [2]. Recent theoretical predictions [3, 4] and experimental demonstrations [5–10] suggest various possibilities for the creation of novel fundamental and vortical patterns, as summarized in reviews [11–13]. In particular, QDs with embedded vorticity were theoretically investigated in detail [14–21].

This story starts with the original experimental realization of BECs [22–24], which has made it possible to probe properties of quantum matter that are otherwise difficult to access. By observing macroscopic quantum phenomena manifested by BECs, researchers gain valuable insights into the underlying physical phenomena and mechanisms [25–36].

In experimental settings, the BEC lifetime is usually short, as the condensates are prone to spatial expansion or collapse, in the cases of the repulsive and attractive inter-atomic interactions, respectively [37]. The lifetime may be substantially extended by means of a trapping potential which confines the condensate [1, 25, 26].

It is relevant to stress that the natural interaction between atoms is repulsive, due to the fact that colliding hard particles bounce back from each other. The interaction may be switched to attraction by dint of the Feshbach-resonance technique, implemented by dc magnetic field or uniform laser illumination applied to BEC [38, 39]. However, in the multidimensional geometry the attractive interaction readily leads to the critical or supercritical collapse (alias blowup), in the two- and three-dimensional (2D and 3D) cases, respectively [1, 40, 41]. In terms of theoretical modeling, the respective Gross−Pitaevskii equations (GPEs) [25, 26] produce a true singularity after a finite evolution time, while in the experiment the collapse leads to destruction of the BEC state. The collapse tends to destabilize the multidimensional solitons, unlike their 1D counterparts, which are usually stable [42] (in BEC, the stability of the effectively 1D matter-wave solitons has been demonstrated in well-known experiments [43–45]).

Therefore, the creation of physically relevant self-trapped 2D and 3D states in BECs, which should be stable against the expansion and collapse, is a challenging topic, with important implications not only for BEC but also with interesting analogies in the fields of nonlinear optics and photonics, plasma physics, etc. [1]. In this context, BECs offer an ideal platform for the prediction and observation of the dynamics and stability of nonlinear self-trapped states in the multidimensional space. In particular, the multidimensional space offers possibilities for building states with integer intrinsic vorticity (alias the topological charge or winding number) [46]. However, vortex self-trapped states (vortex solitons) are subject to azimuthal modulational instability which leads to spontaneous splitting of vortices into separating fragments, prior to the onset of the collapse, as the splitting instability is stronger than its collapse-driven counterpart (later, the fragments may suffer the intrinsic blowup) [1, 46–55].

Under the action of specific potential traps, quantized vortices were experimentally observed in BEC with the self-repulsive nonlinearity [56–62], which is obviously impossible in the free space. On the other hand, the interplay of a spatially periodic optical-lattice potential with the MF cubic self-repulsion was predicted to support stable 2D gap solitons with embedded vorticity [63, 64].

Promising theoretical predictions of matter-wave solitons with embedded vorticity were produced making use of the linear spin-orbit coupling in binary BEC. As a result, stable 2D [65–67] and metastable 3D [68] solitons have been predicted, in the form of semi-vortices, as only one component carries the vorticity in them. Also predicted were stable vortex solitons in ultracold gases of Rydberg atoms [69, 70], and in microwave-coupled two-component BEC [71].

The theoretical and experimental studies of multidimensional self-trapped objects (both fundamental (zero-vorticity) and vortex solitons) have been significantly propelled by the above-mentioned QD concept, realized in two-component (binary) BEC in the three-dimensional geometry, with the cubic MF attraction between the components slightly dominating over the MF intra-component self-repulsion. The overall stabilization is provided by the higher-order (quartic) beyond-MF (BMF) repulsion. The balance between the residual MF attraction and BMF repulsion gives rise to QDs filled by the ultradilute superfluid [3]. The QDs were observed experimentally in monoatomic binary BECs realized in the ultracold gas of {}^{39}K atoms [7–9], and heteroatomic {}^{41}K–{}^{87}Rb [10] and {}^{23}Na–{}^{87}Rb [72] mixtures. The balance corresponds to a minimum of the total energy of the binary BEC for a given number of particles in the condensate [73]. In addition to the scheme involving binary condensates, single-component BECs of magnetic atoms can also form dipolar QDs, as demonstrated experimentally in the BEC of {}^{164}Dy [5, 6], {}^{166}Er [74], and {}^{162}Dy [75] atoms (in the absence of the BMF stabilization, dipole−dipole interactions between magnetic atoms in the 3D free space give rise to the d-wave collapse [76]).

The above-mentioned achievements in the experimental creation of stable quasi-2D [7, 8] and 3D [9] QDs strongly suggest to investigate the possibility of predicting stably self-trapped two-component QDs with embedded vorticity, in the 2D [16] and 3D [14] free space (on the contrary to these results, single-component vortex QDs in the 3D dipolar BEC with the simplest isotropic structure are unstable [15]). Also predicted were stable vortex QDs with “exotic” anisotropic shapes [17, 18, 77].

Here we aim to briefly review the theoretical results for the QDs with embedded vorticity (a somewhat broader review on a related topic was very recently published in Ref. [78]). First, on the basis of the BMF amendment to the MF theory, which arises from the Bogoliubov excitations around the MF state, as elaborated long ago by Lee, Huang, and Yang [2], the corresponding correction to the BEC energy density was derived by Petrov [3]:

(1)E_{\mathrm{B}\mathrm{M}\mathrm{F}}={\displaystyle \frac{128}{30\sqrt{\pi }}}g{n}^2\sqrt{n{a}_s^3} ,

where a_s is the s-wave scattering length, n are equal density of the two components of the binary BEC, g=4\pi {\hslash }^{2}m/a_s represents the strength of the contact interaction, and m is the atomic mass.

Using the systems of LHY-amended GPEs for the binary condensate, 3D vortex-QD solutions with the toroidal structure, carrying the topological charge S=1 and 2 were constructed in Ref. [14]. An example of the stable 3D self-trapped vortex torus with S=1 is displayed in Fig.1(a).

Further, the dimensional reduction 3D \to 2D, imposed by strong confinement (trapping potential) acting in the third direction, produces the BMF correction to the 2D energy density in the form of [4]

(2)E_{2\mathrm{D}\mathrm{B}\mathrm{M}\mathrm{F}}={\displaystyle \frac{8\pi n^2}{{\ln }^2(\left|a_{\uparrow \downarrow }\right|/a)}}[\ln (n /(n_0{)}_{2\mathrm{D}})- 1],

where n_{\uparrow \uparrow }=n_{\downarrow \downarrow }\equiv n are equal densities of the “top” and “bottom” BEC components (if the MF wave function is construed as a spinor), a_{\uparrow \uparrow }=a_{\downarrow \downarrow }\equiv a>0 are equal scattering lengths of the repulsive intra-component interactions, is the negative scattering accounting for the inter-component attraction, and (n_0{)}_{2\mathrm{D}} is the equilibrium density. The change of the sign of expression (2) from negative to positive with the increase of n implies the arrest of the critical collapse, and it was demonstrated in Ref. [16] that the corresponding BMF-amended 2D GPE produced vortex-QD solutions which may be stabilized not only against the collapse, but also against the azimuthal modulational instability (splitting), see an example of a stable vortex mode with S=1 in Fig.1(b). Due to the presence of the logarithmic factor in Eq. (2), these QDs tend to exhibit a flat-top shape, expanding with the increase of the norm and vorticity. The results demonstrate that such vortex QDs keep a stability area in the respective parameter area up to S=5, which is unusual for vortex solitons in nonlinear media. It should be mentioned that in addition to vortex QDs condensates with LHY quantum correction may support rich variety of 2D or 3D dynamical cluster-like states with nontrivial phase distributions that exhibit rotation and pulsations in the course of evolution [79–81].

The use of dipolar BECs in ultracold gases of magnetic atoms opens new perspective for the creation of stable vortex QDs. The energy of the long-range dipole−dipole interaction (DDI) for polarized magnetic moments \mu is given by the usual expression [82, 83]:

(3)V_{\mathrm{D}\mathrm{D}\mathrm{I}}={\displaystyle \frac{{\mu }_0{\mu }^2}{4\pi }}{\displaystyle \frac{1-3{\cos }^2\theta }{r^2}},

where \theta is the angle between the vector of length r connecting the dipoles and the polarization direction, {\mu }_0 being the vacuum permeability. This interaction potential demonstrates the anisotropic nature of DDIs, paving the way for the emergence of novel phenomena and anisotropic structures.

As mentioned above, the simplest isotropic (cylindrically symmetric) 3D vortex-QD states in the dipolar BEC, with the vorticity axis aligned with the polarization direction of the atomic magnetic moments (imposed by externally applied magnetic field) and winding number S=1, are unstable against splitting or the onset of vortex-line corrugation caused by Kelvin-wave excitations [15]. Anisotropic QDs with the vorticity axis directed perpendicular to the polarization of the magnetic moments were investigated in Refs. [17, 18]. This configuration transforms the hollow core of the vortex from a state associated with lower negative attraction energy to one that removes some positive repulsion energy, resulting in a lower overall energy and, eventually, stability of the resulting vortex QDs with the anisotropic shape.

In Ref. [17] it was proposed, leveraging the established experimental technique [84–88], to create 2D anisotropic vortex QDs by phase-imprinting a dark soliton onto an original zero-vorticity dipolar QD, allowing it to transform into a vortex. Subsequently, the analysis was extended to the 3D free space, predicting stable strongly anisotropic vortex QDs [18], see an example in Fig.1(c, d). Even in the presence of loss effects induced by three-body collisions, the 3D vortex persists in the course of long evolution. The robustness of the so predicted anisotropic vortex QDs is highly encouraging for their creation in the experiment.

To date, a clear picture has emerged: the persistent challenge of seeking for stable self-trapped vortices in BECs may be resolved using the powerful QD platform. The above-mentioned demonstrations of the fundamental (zero-vorticity) QDs in various experimental setups and theoretical predictions of the stable QDs with embedded stability provide a strong incentive for the further work in this direction. A possible experimental approach may rely on injecting the topological charge into fundamental QDs, using available technologies such as phase imprinting [85–88] and magnetosrirring [61, 62]. If the prediction of the stable vortex QDs is experimentally validated, it will imply the first realization of self-trapped vortices in the free-space BEC.

References

[1] B. A. Malomed, Multidimensional Solitons, AIP Publishing, 2022

[2] T. D. Lee, K. Huang, and C. N. Yang, Eigenvalues and eigenfunctions of a Bose system of hard spheres and its low-temperature properties, Phys. Rev. 106(6), 1135 (1957)

[3] D. S. Petrov, Quantum mechanical stabilization of a collapsing Bose‒Bose mixture, Phys. Rev. Lett. 115(15), 155302 (2015)

[4] D. S. Petrov and G. E. Astrakharchik, Ultradilute low-dimensional liquids, Phys. Rev. Lett. 117(10), 100401 (2016)

[5] I. Ferrier-Barbut, H. Kadau, M. Schmitt, M. Wenzel, and T. Pfau, Observation of quantum droplets in a strongly dipolar Bose gas, Phys. Rev. Lett. 116(21), 215301 (2016)

[6] M. Schmitt, M. Wenzel, F. Bӧttcher, I. Ferrier-Barbut, and T. Pfau, Self-bound droplets of a dilute magnetic quantum liquid, Nature 539(7628), 259 (2016)

[7] C. R. Cabrera, L. Tanzi, J. Sanz, B. Naylor, P. Thomas, P. Cheiney, and L. Tarruell, Quantum liquid droplets in a mixture of Bose‒Einstein condensates, Science 359(6373), 301 (2018)

[8] P. Cheiney, C. R. Cabrera, J. Sanz, B. Naylor, L. Tanzi, and L. Tarruell, Bright soliton to quantum droplet transition in a mixture of Bose‒Einstein condensates, Phys. Rev. Lett. 120(13), 135301 (2018)

[9] G. Semeghini, G. Ferioli, L. Masi, C. Mazzinghi, L. Wolswijk, F. Minardi, M. Modugno, G. Modugno, M. Inguscio, and M. Fattori, Self-bound quantum droplets of atomic mixtures in free space, Phys. Rev. Lett. 120(23), 235301 (2018)

[10] C. D’Errico, A. Burchianti, M. Prevedelli, L. Salasnich, F. Ancilotto, M. Modugno, F. Minardi, and C. Fort, Observation of quantum droplets in a heteronuclear bosonic mixture, Phys. Rev. Res. 1(3), 033155 (2019)

[11] Z. H. Luo,W. Pang,B. Liu,Y. Y. Li,B. A. Malomed, A new form of liquid matter: Quantum droplets, Front. Phys. 16(3), 32201 (2021)

[12] M. Guo,T. Pfau, A new state of matter of quantum droplets, Front. Phys. 16(3), 32202 (2021)

[13] F. Böttcher, J. N. Schmidt, J. Hertkorn, K. S. H. Ng, S. D. Graham, M. Guo, T. Langen, and T. Pfau, New states of matter with fine-tuned interactions: Quantum droplets and dipolar supersolids, Rep. Prog. Phys. 84(1), 012403 (2021)

[14] Y. V. Kartashov, B. A. Malomed, L. Tarruell, and L. Torner, Three-dimensional droplets of swirling superfluids, Phys. Rev. A 98(1), 013612 (2018)

[15] A. Cidrim, F. E. A. dos Santos, E. A. L. Henn, and T. Macrì, Vortices in self-bound dipolar droplets, Phys. Rev. A 98(2), 023618 (2018)

[16] Y. Li, Z. Chen, Z. Luo, C. Huang, H. Tan, W. Pang, and B. A. Malomed, Two-dimensional vortex quantum droplets, Phys. Rev. A 98(6), 063602 (2018)

[17] G. Li, X. Jiang, B. Liu, Z. Chen, B. A. Malomed, and Y. Li, Two-dimensional anisotropic vortex quantum droplets in dipolar Bose‒Einstein condensates, Front. Phys. 19(2), 22202 (2024)

[18] G. Li, Z. Zhao, X. Jiang, Z. Chen, B. Liu, B. A. Malomed, and Y. Li, Strongly anisotropic vortices in dipolar quantum droplets, Phys. Rev. Lett. 133(5), 053804 (2024)

[19] L. Dong and M. Fan, Stable higher-charge vortex droplets governed by quantum fluctuations in three dimensions, Chaos Solitons Fractals 173, 113728 (2023)

[20] L. Dong, M. Fan, and B. A. Malomed, Stable higher order vortex quantum droplets in an annular potential, Chaos Solitons Fractals 179, 114472 (2024)

[21] J. R. Salgueiro, A. Paredes, J. Guerra-Carmenate, and H. Michinel, On the stability of vortex quantum droplets, Results Phys. 64, 107923 (2024)

[22] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Observation of Bose‒Einstein condensation in a dilute atomic vapor, Science 269(5221), 198 (1995)

[23] K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. Van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, Bose‒Einstein condensation in a gas of sodium atoms, Phys. Rev. Lett. 75(22), 3969 (1995)

[24] C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet, Evidence of Bose‒Einstein condensation in an atomic gas with attractive interactions, Phys. Rev. Lett. 75(9), 1687 (1995)

[25] C. Pethick,H. Smith, Bose-Einstein Condensation in Dilute Gases, Cambridge University Press, New York, 2002

[26] L. Pitaevskii,S. Stringari, Bose−Einstein Condensation and Superfluidity, Oxford University Press, 2016

[27] Y. Zhang, F. Maucher, and T. Pohl, Supersolidity around a critical point in dipolar Bose‒Einstein condensates, Phys. Rev. Lett. 123(1), 015301 (2019)

[28] Y. Zhang, T. Pohl, and F. Maucher, Phases of supersolids in confined dipolar Bose‒Einstein condensates, Phys. Rev. A 104(1), 013310 (2021)

[29] Y. Ma, C. Peng, and X. Cui, Borromean droplet in three-component ultracold Bose gases, Phys. Rev. Lett. 127(4), 043002 (2021)

[30] M. Guo, F. Böttcher, J. Hertkorn, J. N. Schmidt, M. Wenzel, H. P. Büchler, T. Langen, and T. Pfau, The low-energy Goldstone mode in a trapped dipolar supersolid, Nature 574(7778), 386 (2019)

[31] L. J. He, B. Liu, and Y. C. Zhang, Quantum droplets in dipolar condensate mixtures with arbitrary dipole orientations, Phys. Rev. A 110(1), 013308 (2024)

[32] J. Pan, S. Yi, and T. Shi, Quantum phases of self-bound droplets of Bose‒Bose mixtures, Phys. Rev. Res. 4(4), 043018 (2022)

[33] R. Liu, W. Wang, and X. Cui, Quartet superfluid in two-dimensional mass-imbalanced fermi mixtures, Phys. Rev. Lett. 131(19), 193401 (2023)

[34] Q. Gu and X. Cui, Self-bound vortex lattice in a rapidly rotating quantum droplet, Phys. Rev. A 108(6), 063302 (2023)

[35] Y. Deng and S. Yi, Self-ordered supersolid phase beyond Dicke superradiance in a ring cavity, Phys. Rev. Res. 5(1), 013002 (2023)

[36] L. Peng, J. Pan, S. Yi, and T. Shi, Unveiling quantum phases in quasi-one-dimensional dipolar gases using continuous matrix product states, Phys. Rev. A 110(2), 023330 (2024)

[37] E. A. Donley, N. R. Claussen, S. L. Cornish, J. L. Roberts, E. A. Cornell, and C. E. Wieman, Dynamics of collapsing and exploding Bose‒Einstein condensates, Nature 412(6844), 295 (2001)

[38] C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Feshbach resonances in ultracold gases, Rev. Mod. Phys. 82(2), 1225 (2010)

[39] P. Courteille, R. S. Freeland, D. J. Heinzen, F. A. van Abeelen, and B. J. Verhaar, Observation of a Feshbach resonance in cold atom scattering, Phys. Rev. Lett. 81(1), 69 (1998)

[40] L. Bergé, Wave collapse in physics: Principles and applications to light and plasma waves, Phys. Rep. 303(5−6), 259 (1998)

[41] G. Fibich, The Nonlinear Schrödinger Equation: Singular Solutions and Optical Collapse, Springer, Heidelberg, 2015

[42] V. E. Zakharov,S. V. Manakov,S. P. Novikov,L. P. Pitaevskii, Solitons: The Inverse Scattering Method, Nauka Publishers, Moscow, 1980

[43] K. E. Strecker, G. B. Partridge, A. G. Truscott, and R. G. Hulet, Formation and propagation of matter-wave soliton trains, Nature 417(6885), 150 (2002)

[44] L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L. D. Carr, Y. Castin, and C. Salomon, Formation of a matter-wave bright soliton, Science 296(5571), 1290 (2002)

[45] S. L. Cornish, S. T. Thompson, and C. E. Wieman, Formation of bright matter-wave solitons during the collapse of attractive Bose‒Einstein condensates, Phys. Rev. Lett. 96(17), 170401 (2006)

[46] B. A. Malomed, Vortex solitons: Old results and new perspectives, Physica D 399, 108 (2019)

[47] B. A. Malomed, L. C. Crasovan, and D. Mihalache, Stability of vortex solitons in the cubic‒quintic model, Physica D 161(3-4), 187 (2002)

[48] L. Torner and D. V. Petrov, Azimuthal instabilities and self-breaking of beams into sets of solitons in bulk second harmonic generation, Electron. Lett. 33(7), 608 (1997)

[49] W. J. Firth and D. V. Skryabin, Optical solitons carrying orbital angular momentum, Phys. Rev. Lett. 79(13), 2450 (1997)

[50] J. P. Torres, J. M. Soto-Crespo, L. Torner, and D. V. Petrov, Solitary-wave vortices in quadratic nonlinear media, J. Opt. Soc. Am. B 15(2), 625 (1998)

[51] V. I. Kruglov,R. A. Vlasov, Spiral self-trapping propagation of optical beams in media with cubic nonlinearity, Phys. Lett. A 111(8–9), 401 (1985)

[52] M. S. Bigelow, P. Zerom, and R. W. Boyd, Breakup of ring beams carrying orbital angular momentum in sodium vapor, Phys. Rev. Lett. 92(8), 083902 (2004)

[53] D. V. Petrov, L. Torner, J. Martorell, R. Vilaseca, J. P. Torres, and C. Cojocaru, Observation of azimuthal modulational instability and formation of patterns of optical solitons in a quadratic nonlinear crystal, Opt. Lett. 23(18), 1444 (1998)

[54] S. Minardi, G. Molina-Terriza, P. Di Trapani, J. P. Torres, and L. Torner, Soliton algebra by vortex-beam splitting, Opt. Lett. 26(13), 1004 (2001)

[55] V. Tikhonenko, J. Christou, and B. Luther-Daves, Spiraling bright spatial solitons formed by the breakup of an optical vortex in a saturable self-focusing medium, J. Opt. Soc. Am. B 12(11), 2046 (1995)

[56] E. A. L. Henn, Quantum vortices get stretched, Front. Phys. 19(3), 31301 (2024)

[57] M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman, and E. A. Cornell, Vortices in a Bose‒Einstein condensate, Phys. Rev. Lett. 83(13), 2498 (1999)

[58] K. W. Madison, F. Chevy, W. Wohlleben, and J. Dalibard, Vortex formation in a stirred Bose‒Einstein condensate, Phys. Rev. Lett. 84(5), 806 (2000)

[59] M. J. H. Ku, W. Ji, B. Mukherjee, E. Guardado-Sanchez, L. W. Cheuk, T. Yefsah, and M. W. Zwierlein, Motion of a solitonic vortex in the BEC‒BCS crossover, Phys. Rev. Lett. 113(6), 065301 (2014)

[60] S. Donadello, S. Serafini, M. Tylutki, L. P. Pitaevskii, F. Dalfovo, G. Lamporesi, and G. Ferrari, Observation of solitonic vortices in Bose‒Einstein condensates, Phys. Rev. Lett. 113(6), 065302 (2014)

[61] L. Klaus, T. Bland, E. Poli, C. Politi, G. Lamporesi, E. Casotti, R. N. Bisset, M. J. Mark, and F. Ferlaino, Observation of vortices and vortex stripes in a dipolar condensate, Nat. Phys. 18(12), 1453 (2022)

[62] E. Casotti,E. Poli,L. Klaus,A. Litvinov,C. Ulm, C. Politi, Observation of vortices in a dipolar supersolid, arXiv: 2403.18510 (2024)

[63] H. Sakaguchi and B. A. Malomed, Two-dimensional loosely and tightly bound solitons in optical lattices and inverted traps, J. Phys. B: At. Mol. Opt. Phys. 37(11), 2225 (2004)

[64] E. A. Ostrovskaya and Y. S. Kivshar, Matter-wave gap vortices in optical lattices, Phys. Rev. Lett. 93(16), 160405 (2004)

[65] 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)

[66] X. Jiang, Z. Fan, Z. Chen, W. Pang, Y. Li, and B. A. Malomed, Two-dimensional solitons in dipolar Bose‒Einstein condensates with spin‒orbit coupling, Phys. Rev. A 93(2), 023633 (2016)

[67] H. Deng, J. Li, Z. Chen, Y. Liu, D. Liu, C. Jiang, C. Kong, and B. A. Malomed, Semi-vortex solitons and their excited states in spin‒orbit-coupled binary bosonic condensates, Phys. Rev. E 109(6), 064201 (2024)

[68] Y. C. Zhang, Z. W. Zhou, B. A. Malomed, and H. Pu, Stable solitons in three dimensional free space without the ground state: Self-trapped Bose‒Einstein condensates with spin‒orbit coupling, Phys. Rev. Lett. 115(25), 253902 (2015)

[69] Z. Y. Bai, W. B. Li, and G. X. Huang, Stable single light bullets and vortices and their active control in cold Rydberg gases, Optica 6(3), 309 (2019)

[70] L. Qin, C. Hang, and G. X. Huang, High-fidelity and controllable cloning of high-dimensional optical beams with a Rydberg atomic gas, Phys. Rev. A 102(6), 063707 (2020)

[71] J. Qin, G. Dong, and B. A. Malomed, Stable giant vortex annuli in microwave-coupled atomic condensates, Phys. Rev. A 94(5), 053611 (2016)

[72] Z. Guo, Quantum droplet in a mixture of Rb and Na Bose‒Einstein condensates, Doctoral dissertation, The Chinese University of Hong Kong, 2022

[73] I. Ferrier-Barbut and T. Pfau, Quantum liquids get thin, Science 359(6373), 274 (2018)

[74] L. Chomaz, S. Baier, D. Petter, M. J. Mark, F. Wächtler, L. Santos, and F. Ferlaino, Quantum-fluctuation-driven crossover from a dilute Bose‒Einstein condensate to a macrodroplet in a dipolar quantum fluid, Phys. Rev. X 6(4), 041039 (2016)

[75] F. Böttcher, M. Wenzel, J. Schmidt, M. Guo, T. Langen, I. Ferrier-Barbut, T. Pfau, R. Bombín, J. Sánchez-Baena, J. Boronat, and F. Mazzanti, Dilute dipolar quantum droplets beyond the extended Gross‒Pitaevskii equation, Phys. Rev. Res. 1(3), 033088 (2019)

[76] T. Lahaye,J. Metz,B. Fröhlich,T. Koch,M. Meister,A. Griesmaier,T. Pfau,H. Saito, Y. Kawaguchi,M. Ueda, D-wave collapse and explosion of a dipolar Bose‒Einstein condensate, Phys. Rev. Lett. 101(8), 080401 (2008)

[77] Y. V. Kartashov, B. A. Malomed, and L. Torner, Structured heterosymmetric quantum droplets, Phys. Rev. Res. 2(3), 033522 (2020)

[78] G. H. Chen, H. C. Wang, H. M. Deng, and B. A. Malomed, Vortex quantum droplets under competing nonlinearities, Chin. Phys. Lett. 41, 020501 (2024)

[79] Y. V. Kartashov, B. A. Malomed, and L. Torner, Metastability of quantum droplet clusters, Phys. Rev. Lett. 122(19), 193902 (2019)

[80] L. Dong and Y. V. Kartashov, Rotating multidimensional quantum droplets, Phys. Rev. Lett. 126(24), 244101 (2021)

[81] M. N. Tengstrand, P. Stürmer, E. Ö. Karabulut, and S. M. Reimann, Rotating binary Bose‒Einstein condensates and vortex clusters in quantum droplets, Phys. Rev. Lett. 123(16), 160405 (2019)

[82] 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)

[83] J. Stuhler, A. Griesmaier, T. Koch, M. Fattori, T. Pfau, S. Giovanazzi, P. Pedri, and L. Santos, Observation of dipole‒dipole interaction in a degenerate quantum gas, Phys. Rev. Lett. 95(15), 150406 (2005)

[84] S. Burger, K. Bongs, S. Dettmer, W. Ertmer, K. Sengstock, A. Sanpera, G. V. Shlyapnikov, and M. Lewenstein, Dark solitons in Bose‒Einstein condensates, Phys. Rev. Lett. 83(25), 5198 (1999)

[85] A. E. Leanhardt, A. Görlitz, A. P. Chikkatur, D. Kielpinski, Y. Shin, D. E. Pritchard, and W. Ketterle, Imprinting vortices in a Bose‒Einstein condensate using topological phases, Phys. Rev. Lett. 89(19), 190403 (2002)

[86] G. Andrelczyk, M. Brewczyk, L. Dobrek, M. Gajda, and M. Lewenstein, Optical generation of vortices in trapped Bose‒Einstein condensates, Phys. Rev. A 64(4), 043601 (2001)

[87] P. Scherpelz, K. Padavić, A. Rançon, A. Glatz, I. S. Aranson, and K. Levin, Phase imprinting in equilibrating fermi gases: The transience of vortex rings and other defects, Phys. Rev. Lett. 113(12), 125301 (2014)

[88] Y. Shen, X. Wang, Z. Xie, C. Min, X. Fu, Q. Liu, M. Gong, and X. Yuan, Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities, Light Sci. Appl. 8(1), 90 (2019)

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