The trapping potential is a necessary ingredient of any setup used in experiments with atomic gases in the BEC state (otherwise, the gas would escape). Similarly, the presence of an effective trap is necessary for building various optical and photonic cavities. Therefore, the analysis of models combining the intrinsic nonlinearity acting in multidimensional settings with the trapping potential is a natural step in the theoretical work. The present chapter aims to summarize basic theoretical results produced by the work in this direction. The results are presented for 2D fundamental (zero-voticity) and vortical states trapped in the HO potential and for similar 3D states trapped in the HO potential, which may be anisotropic. Also considered are two-component 2D states with hidden vorticity, i.e., with opposite vorticities if the two components have equal norms. Stability regions are identified for all these settings. The chapter additionally reports findings for two-component 2D fundamental and vortex states in a linearly coupled system, in which the potential acting on one component is trapping, while the other component is subjected to the action of an expulsive potential. As for experimental results demonstrating the creation of 2D or 3D solitons in trapping potentials, very few of them are available. A recent experimental finding that may be relevant in this respect is the creation of quasi-stable effectively two-dimensional Townes solitons in atomic BEC loaded in a trapping potential [Bakkali-Hassani et al., Phys. Rev. Lett. 127, 023603 (2021)].
The trapping potential is a necessary ingredient of any setup used in experiments with atomic gases in the BEC state (otherwise, the gas would escape). Similarly, the presence of an effective trap is necessary for building various optical and photonic cavities. Therefore, the analysis of models combining the intrinsic nonlinearity acting in multidimensional settings with the trapping potential is a natural step in the theoretical work. The present chapter aims to summarize basic theoretical results produced by the work in this direction. The results are presented for 2D fundamental (zero-voticity) and vortical states trapped in the HO potential and for similar 3D states trapped in the HO potential, which may be anisotropic. Also considered are two-component 2D states with hidden vorticity, i.e., with opposite vorticities if the two components have equal norms. Stability regions are identified for all these settings. The chapter additionally reports findings for two-component 2D fundamental and vortex states in a linearly coupled system, in which the potential acting on one component is trapping, while the other component is subjected to the action of an expulsive potential. As for experimental results demonstrating the creation of 2D or 3D solitons in trapping potentials, very few of them are available. A recent experimental finding that may be relevant in this respect is the creation of quasi-stable effectively two-dimensional Townes solitons in atomic BEC loaded in a trapping potential [Bakkali-Hassani et al., Phys. Rev. Lett. 127, 023603 (2021)].
This concise review aims to provide a summary of the most relevant recent experimental and theoretical results for solitons, i.e., self-trapped bound states of nonlinear waves, in two- and three-dimensional (2D and 3D) media. In comparison with commonly known one-dimensional solitons, which are, normally, stable modes, a challenging problem is the propensity of 2D and 3D solitons to instability, caused by the occurrence of the critical or supercritical wave collapse (catastrophic self-compression) in the same spatial dimension. A remarkable feature of multidimensional solitons is their ability to carry vorticity; however, 2D vortex rings and 3D vortex tori are subject to strong splitting instability. Therefore, it is natural to categorize the basic results according to physically relevant settings which make it possible to maintain stability of fundamental (non-topological) and vortex solitons against the collapse and splitting, respectively. The present review is focused on schemes that were recently elaborated in terms of Bose-Einstein condensates and similar photonic setups. These are two-component systems with spin-orbit coupling, and ones stabilized by the beyond-mean-field Lee-Huang-Yang effect. The latter setting has been implemented experimentally, giving rise to stable self-trapped quasi-2D and 3D "quantum droplets".
We review the state of the art and recently obtained theoretical and experimental results for two- and three-dimensional (2D and 3D) solitons and related states, such as quantum droplets, in optical systems, atomic Bose-Einstein condensates (BECs), and other fields - in particular, liquid crystals. The central challenge is avoiding the trend of 2D and 3D solitary states supported by the ubiquitous cubic nonlinearity to be strongly unstable - a property far less present in one-dimensional systems. Many possibilities for the stabilization of multi-dimensional states have been theoretically proposed over the years. Most strategies involve non-cubic nonlinearities or using different sorts of potentials, including periodic ones. New important avenues have arisen recently in systems based on two-component BEC with spin-orbit coupling, which have been predicted to support stable 2D and metastable 3D solitons. An important recent breakthrough is the creation of 3D quantum droplets. These are self-sustained states existing in two-component BECs, which are stabilized by the presence of Lee-Hung-Yang quantum fluctuations around the underlying mean-field states. By and large, multi-dimensional geometries afford unique opportunities to explore the existence of complex self-sustained states, including topologically rich ones, which by their very nature are not possible in one-dimensional geometries. In addition to vortex solitons, these are hopfions, skyrmions, and hybrid vortex-antivortex complexes, which have been predicted in different models. Here we review recent landmark findings in this field and outline outstanding open challenges.
We demonstrate the existence of confined states in one- and two-dimensional (1D and 2D) systems of two linearly coupled components, with the confining harmonic-oscillator (HO) potential acting upon one component and an expulsive anti-HO potential acting upon the other. The systems can be implemented in optical and BEC dual-core waveguides. In the 1D linear system, codimension-one solutions are found in an exact form for the ground state (GS) and dipole mode (the first excited state). Generic solutions are produced by means of the variational approximation and are found in a numerical form. Exact codimension-one solutions and generic numerical ones are also obtained for the GS and vortex states in the 2D system (the exact solutions are found for all values of the vorticity). Both the trapped and antitrapped components of the bound states may be dominant ones, in terms of the norm. The localized modes may be categorized as bound states in continuum, as they coexist with delocalized ones. The 1D states, as well as the GS in 2D, are weakly affected and remain stable if the self-attractive or repulsive nonlinearity is added to the system. The self-attraction makes the vortex states unstable against splitting, while they remain stable under the action of the self-repulsion.
Bose-Einstein condensation(BEC) has been a research hotspot in cold atom physics for the past few decades. Recently, experiments on BECs have reported the observation of liquid-like quantum droplets, attracting significant interest. The physics for forming quantum droplets can be explained by zero-point quantum fluctuation which effectively prohibit the collapse caused by the mean-field interactions between the atoms. The stabilization mechanism beyond the mean-field theory, induced by quantum fluctuations, can be described by the Lee–Huang–Yang(LHY) correction. Normally, forming stable 2D or 3D solitons in free space is a challenging problem. Additionally, an intriguing question is how to generate 2D or 3D solitons with embedded vorticity. The LHY stability mechanism offers a new approach to producing stable solitons and vortex solitons in BECs. In this thesis, two-dimensional vortex quantum droplets in Bose-Einstein condensates, including those with and without external potential, are examined. The study of solitons and vortex solitons in BECs under mean-field interactions and external potential has yielded significant results. This thesis expands this field by introducing the LHY correction to study stable quantum droplets and vortex droplets in BECs. The work chapters begin with studying binary quantum droplets in free space, considering the case where interspecies interactions are weakly attractive and intraspecies interactions are weakly repulsive. The zero-vorticity, semi-vortex, and binary vortex quantum droplets are studied in Chapter 3. Stable semi-vortex quantum droplets with embedded vorticity up to 8 are found in this system. This result gives a way to form stable vortex droplets with high topological charges in free space by controlling one component with zero vorticity and the other component with embedded vorticity. The stability and properties of these droplets are systematically studied. Chapter 4 explores the vortex quantum droplets and hidden vortex droplets confined within a radial lattice trap. It is found that the existence curves of the droplets confined within this optical trap could violate the Vakhitov-Kolokolov(VK) criterion, which is a necessary condition to form a stable soliton. A remarkable finding is that the vortex and hidden vortex quantum droplets in this radial lattice remain stable when they carry the embedded vorticity up to 11. The double-ring vortex and hidden vortex quantum droplets confined within this radial lattice are also discussed. The work expands from non-magnetic atoms to magnetic atoms in Chapter 5, which needs to consider the dipole-dipole interactions(DDIs). The anisotropic dipolar vortex quantum droplets trapped in an annular potential are studied. Under the effect of the ring shape potential, the double peak ring-shaped vortex droplets remain stable when they carry the embedded vorticity up to 11. It is the first time stable droplets with high topological charges have been formed in a dipolar system. Additionally, the shape transformation of dipolar vortex droplets induced by the strength of the mean-field interaction is examined. Finally, the behavior of dipolar vortex droplets confined within a weak-depth annular potential is also discussed. The results of this thesis may prove highly useful in creating stable vortex quantum droplets in Bose-Einstein condensates and advancing the broader field of ultracold atoms.
The study of soliton-like states in systems with nonlocal nonlinearity is a traditional topic in optics and related areas. Some results obtained in these studies (such as solitons supported by thermal nonlinearity in optical glasses and orientational nonlinearity that affects light propagation in liquid crystals) are well known and have been properly reviewed in the literature; therefore, the respective models are outlined in the present chapter in a brief form. Some other studies such as those addressing models with fractional diffraction, which is represented by a linear nonlocal operator, have also been started more recently; therefore, it should be relevant to review them in detail when more results will be accumulated and this chapter includes a short outline of the latter topic. After providing an introduction to the general area of nonlocal nonlinearities, the chapter offers a summary of results obtained for multidimensional solitons in some specific nonlocal nonlinear models originating in studies of BEC, which are sufficiently mature but have not been reviewed previously. These are anisotropic quasi-2D solitons supported by long-range dipole–dipole interactions in a condensate of magnetic atoms [Tikhonenkov et al., Phys. Rev. Lett. 100, 090406(2008a)] and giant vortex solitons, which are stable for high values of the winding number [Qin et al., Phys. Rev. A 94, 053611 (2016)], as well as 2D vortex solitons moving with self-acceleration [Qin et al., Phys. Rev. A 99, 023610 (2019)]. These are solitons of a hybrid type, which include matter-wave and electromagnetic-wave components. They are supported, in a binary BEC composed of two different atomic states, by the resonant interaction of the two-component matter waves with a resonant microwave field which couples the two atomic states.
Bose-Einstein condensation(BEC) has been a research hotspot in cold atom physics for the past few decades. Recently, experiments on BECs have reported the observation of liquid-like quantum droplets, attracting significant interest. The physics for forming quantum droplets can be explained by zero-point quantum fluctuation which effectively prohibit the collapse caused by the mean-field interactions between the atoms. The stabilization mechanism beyond the mean-field theory, induced by quantum fluctuations, can be described by the Lee–Huang–Yang(LHY) correction. Normally, forming stable 2D or 3D solitons in free space is a challenging problem. Additionally, an intriguing question is how to generate 2D or 3D solitons with embedded vorticity. The LHY stability mechanism offers a new approach to producing stable solitons and vortex solitons in BECs. In this thesis, two-dimensional vortex quantum droplets in Bose-Einstein condensates, including those with and without external potential, are examined. The study of solitons and vortex solitons in BECs under mean-field interactions and external potential has yielded significant results. This thesis expands this field by introducing the LHY correction to study stable quantum droplets and vortex droplets in BECs. The work chapters begin with studying binary quantum droplets in free space, considering the case where interspecies interactions are weakly attractive and intraspecies interactions are weakly repulsive. The zero-vorticity, semi-vortex, and binary vortex quantum droplets are studied in Chapter 3. Stable semi-vortex quantum droplets with embedded vorticity up to 8 are found in this system. This result gives a way to form stable vortex droplets with high topological charges in free space by controlling one component with zero vorticity and the other component with embedded vorticity. The stability and properties of these droplets are systematically studied. Chapter 4 explores the vortex quantum droplets and hidden vortex droplets confined within a radial lattice trap. It is found that the existence curves of the droplets confined within this optical trap could violate the Vakhitov-Kolokolov(VK) criterion, which is a necessary condition to form a stable soliton. A remarkable finding is that the vortex and hidden vortex quantum droplets in this radial lattice remain stable when they carry the embedded vorticity up to 11. The double-ring vortex and hidden vortex quantum droplets confined within this radial lattice are also discussed. The work expands from non-magnetic atoms to magnetic atoms in Chapter 5, which needs to consider the dipole-dipole interactions(DDIs). The anisotropic dipolar vortex quantum droplets trapped in an annular potential are studied. Under the effect of the ring shape potential, the double peak ring-shaped vortex droplets remain stable when they carry the embedded vorticity up to 11. It is the first time stable droplets with high topological charges have been formed in a dipolar system. Additionally, the shape transformation of dipolar vortex droplets induced by the strength of the mean-field interaction is examined. Finally, the behavior of dipolar vortex droplets confined within a weak-depth annular potential is also discussed. The results of this thesis may prove highly useful in creating stable vortex quantum droplets in Bose-Einstein condensates and advancing the broader field of ultracold atoms.