Semidiscrete Quantum Droplets and Vortices

Vortex solitons in quasi-phase-matched photonic crystals with competing quadratic and cubic nonlinearity

We introduce a methodology for investigating vortex solitons (VSs) within quasi-phase-matched (QPM) photonic crystals, featuring competing quadratic and cubic nonlinearities. The photonic crystal is introduced with a checkerboard structure, which is feasible through contemporary technological advancements. The VS family is constructed as quadrupole and 8-pole configurations, with the quadrupole solitons displaying rhombus and square modes depending on different phase-matching conditions. Notably, an intriguing transformation from the quadrupole to the 8-pole configuration can be achieved by modulating the cubic nonlinear coefficient. Additionally, the square-shaped VSs would transfer to rhombus patterns by adjusting the power and size of the checkerboard cell in the framework of cubic nonlinearity. This work presents a versatile and powerful tool for exploring and manipulating vortex solitons in QPM photonic structures, with potential applications in optical signal processing, optical communications, and nonlinear optics research.

Shell-Shaped Quantum Droplet in a Three-Component Ultracold Bose Gas

Shell-shaped Bose-Einstein condensate is a typical quantum system in curved geometry. Here, we propose a new type of shell-shaped Bose-Einstein condensate with a self-bound character, thereby liberating it from stringent conditions such as microgravity or a fine-tuned trap. Specifically, we consider a three-component (1, 2, 3) ultracold Bose gas where (1, 2) and (2, 3) both form quantum droplets. The two droplets are mutually immiscible due to strong 1-3 repulsion, while still linked by component-2 to form a globally self-bound object. The outer droplet then naturally develops a shell structure without any trapping potential. It is shown that the shell structure can significantly modify the equilibrium density of the core, and lead to unique collective excitations highlighting the core-shell correlation. All results have been demonstrated in a realistic ^{23}Na-^{39}K-^{41}K mixture. By extending quantum droplets from flat to curved geometries, this Letter paves the way for future explorations of the interplay of quantum fluctuations and nontrivial real-space topologies in ultracold gases.

Stable quantum droplets with high-order vorticity in zero-order Bessel lattice

A theoretical framework is presented to investigate the stability of novel two-dimensional quantum droplets within zeroth-order Bessel lattices. The evolution of quantum droplets is studied by the Gross-Pitaevskii equations with Lee-Huang-Yang corrections. The circular groove structure inherent in the zeroth-order Bessel lattice potential facilitates the formation of distinct configurations, including stable zero-vorticity annular quantum droplets and annular quantum droplets featuring embedded vorticity. The stability region of these quantum droplets is achieved through direct numerical simulations. It is found that the lower limit of the stability range for quantum droplets under this optical lattice remains unaffected by vorticity. Conversely, the upper limit exhibits a discernible dependence on vorticity. Subsequently, the study extends to the construction of stable composite states, manifesting as nested concentric multiring structures. Numerical results not only validate the feasibility of nesting vortical quantum droplets under the influence of a zeroth-order Bessel lattice potential but also establish that the vorticity of the smaller droplet within nested vortical quantum droplets does not surpass half of that observed in the larger droplet. Moreover, a comparative analysis highlights the enhanced stability of nested quantum droplets with varying vorticities when contrasted with their counterparts possessing identical vorticities.

Three-dimensional matter wave soliton transformation between different optical lattices in a cold Rydberg atomic system

We present a novel approach to realize three-dimensional (3D) matter wave solitons (MWSs) transformation between different optical potential wells by manipulating their depths and centers. The 3D MWSs are obtained by the square operator method, and transformed to other types (elliptical/ring/necklace) by performing time evolution with the split-step Fourier method. The effectiveness and reliability of our approach is demonstrated by comparing the transformed solitons with those obtained iteratively using the square operator method. Redistribution of MWSs are observed due to the modulation of potentials. In certain complex optical potential wells, we demonstrate the generation of exotic MWSs, such as double gyration modes, through this transformation approach. Overall, the controllable soliton transformation provides an excellent opportunity for all-optical switching, optical information processing, and various other applications.

Generation of patterns and higher harmonics in 1D quantum droplets in tilted and driven quasi-periodic confinements

The generation of patterns by breaking the spatial symmetry in external confinement is a captivating area of physics. The emergence of patterns is a fundamental inquiry spanning various disciplines such as nonlinear optics, condensed matter physics, and fluid dynamics. The article investigates the generation of a variety of patterns in a one-dimensional binary mixture of Bose-Einstein condensate forming quantum droplets. By solving the extended Gross-Pitaevskii equation in the presence of tilted and driven engineered bi-chromatic optical lattices (BOL), the out-of-equilibrium dynamics of droplets under strong dc and ac fields are illustrated. Under the influence of a dc field, a stripe-like pattern emerges in the temporal domain, while the scenario with ac fields demonstrates temporal periodic and bi-periodic oscillations of density waves. The width and period of formed patterns are directly correlated with the strength of ac and dc fields. Moreover, temporal modulation of the BOL potential depth yields various harmonics in the oscillations of the condensate density pattern. Through Fast Fourier Transform (FFT) analysis, it is confirmed that these harmonics encompass multiple and combinational frequencies, suggesting potential applications in generating desired frequency combs within quantum droplets. We have also carried out a thorough numerical stability check of the obtained solutions and found them sufficiently stable.

Strongly Anisotropic Vortices in Dipolar Quantum Droplets

We construct strongly anisotropic quantum droplets with embedded vorticity in the 3D space, with mutually perpendicular vortex axis and polarization of atomic magnetic moments. Stability of these anisotropic vortex quantum droplets (AVQDs) is verified by means of systematic simulations. Their stability area is identified in the parametric plane of the total atom number and scattering length of the contact interactions. We also construct vortex-antivortex-vortex bound states and find their stability region in the parameter space. The application of a torque perpendicular to the vorticity axis gives rise to robust intrinsic oscillations or rotation of the AVQDs. The effect of three-body losses on the AVQD stability is considered too. The results show that the AVQDs can retain the topological structure (vorticity) for a sufficiently long time if the scattering length exceeds a critical value.

Optical vortex-antivortex crystallization in free space

Stable vortex lattices are basic dynamical patterns which have been demonstrated in physical systems including superconductor physics, Bose-Einstein condensates, hydrodynamics and optics. Vortex-antivortex (VAV) ensembles can be produced, self-organizing into the respective polar lattices. However, these structures are in general highly unstable due to the strong VAV attraction. Here, we demonstrate that multiple optical VAV clusters nested in the propagating coherent field can crystallize into patterns which preserve their lattice structures over distance up to several Rayleigh lengths. To explain this phenomenon, we present a model for effective interactions between the vortices and antivortices at different lattice sites. The observed VAV crystallization is a consequence of the globally balanced VAV couplings. As the crystallization does not require the presence of nonlinearities and appears in free space, it may find applications to high-capacity optical communications and multiparticle manipulations. Our findings suggest possibilities for constructing VAV complexes through the orbit-orbit couplings, which differs from the extensively studied spin-orbit couplings.

Dynamics of discrete solitons in the fractional discrete nonlinear Schrödinger equation with the quasi-Riesz derivative

We elaborate a fractional discrete nonlinear Schrödinger (FDNLS) equation based on an appropriately modified definition of the Riesz fractional derivative, which is characterized by its Lévy index (LI). This FDNLS equation represents a novel discrete system, in which the nearest-neighbor coupling is combined with long-range interactions, that decay as the inverse square of the separation between lattice sites. The system may be realized as an array of parallel quasi-one-dimensional Bose-Einstein condensates composed of atoms or small molecules carrying, respectively, a permanent magnetic or electric dipole moment. The dispersion relation (DR) for lattice waves and the corresponding propagation band in the system's linear spectrum are found in an exact form for all values of LI. The DR is consistent with the continuum limit, differing in the range of wave numbers. Formation of single-site and two-site discrete solitons is explored, starting from the anticontinuum limit and continuing the analysis in the numerical form up to the existence boundary of the discrete solitons. Stability of the solitons is identified in terms of eigenvalues for small perturbations, and verified in direct simulations. Mobility of the discrete solitons is considered too, by means of an estimate of the system's Peierls-Nabarro potential barrier, and with the help of direct simulations. Collisions between persistently moving discrete solitons are also studied.

Semivortex solitons and their excited states in spin-orbit-coupled binary bosonic condensates

It is known that two-dimensional two-component fundamental solitons of the semivortex (SV) type, with vorticities (s_{+},s_{-})=(0,1) in their components, are stable ground states (GSs) in the spin-orbit-coupled (SOC) binary Bose-Einstein condensate with the contact self-attraction acting in both components, in spite of the possibility of the critical collapse in the system. However, excited states (ESs) of the SV solitons, with the vorticity set (s_{+},s_{-})=(S_{+},S_{+}+1) and S_{+}=1,2,3,..., are unstable in the same system. We construct ESs of SV solitons in the SOC system with opposite signs of the self-interaction in the two components. The main finding is stability of the ES-SV solitons, with the extra vorticity (at least) up to S_{+}=6. The threshold value of the norm for the onset of the critical collapse, N_{thr}, in these excited states is higher than the commonly known critical value, N_{c}≈5.85, associated with the single-component Townes solitons, N_{thr} increasing with the growth of S_{+}. A velocity interval for stable motion of the GS-SV solitons is found too. The results suggest a solution for the challenging problem of the creation of stable vortex solitons with high topological charges.

Discrete and Semi-Discrete Multidimensional Solitons and Vortices: Established Results and Novel Findings

This article presents a concise survey of basic discrete and semi-discrete nonlinear models, which produce two- and three-dimensional (2D and 3D) solitons, and a summary of the main theoretical and experimental results obtained for such solitons. The models are based on the discrete nonlinear Schrödinger (DNLS) equations and their generalizations, such as a system of discrete Gross-Pitaevskii (GP) equations with the Lee-Huang-Yang corrections, the 2D Salerno model (SM), DNLS equations with long-range dipole-dipole and quadrupole-quadrupole interactions, a system of coupled discrete equations for the second-harmonic generation with the quadratic (χ(2)) nonlinearity, a 2D DNLS equation with a superlattice modulation opening mini-gaps, a discretized NLS equation with rotation, a DNLS coupler and its PT-symmetric version, a system of DNLS equations for the spin-orbit-coupled (SOC) binary Bose-Einstein condensate, and others. The article presents a review of the basic species of multidimensional discrete modes, including fundamental (zero-vorticity) and vortex solitons, their bound states, gap solitons populating mini-gaps, symmetric and asymmetric solitons in the conservative and PT-symmetric couplers, cuspons in the 2D SM, discrete SOC solitons of the semi-vortex and mixed-mode types, 3D discrete skyrmions, and some others.

Controllable orbital-angular-momentum Hall effect by engineering intrinsic orbit-orbit interaction

We report both theoretically and experimentally a process of optical intrinsic orbit-orbit interaction with a vortex-antivortex structure nested in a freely propagating light field. The orbit-orbit interaction is originating from the coupling between different vortices and antivortices. Based on this process, we reveal the resultant controllable orbital-angular-momentum Hall effect by considering a typical structure, which comprises a vortex-antivortex pair and another vortex (or antivortex) as a controllable knob. The intrinsic Hall effect can be spatially manipulated by appropriately engineering the orbit-orbit interaction, namely arranging the initial distribution of these elements. This work can find interesting potential applications. For example, it provides an effective technique for controllable paired photon generation.

Semidiscrete optical vortex droplets in quasi-phase-matched photonic crystals

What we believe is a new scheme for producing semidiscrete self-trapped vortices ("swirling photon droplets") in photonic crystals with competing quadratic (χ(2)) and self-defocusing cubic (χ(3)) nonlinearities is proposed. The photonic crystal is designed with a striped structure, in the form of spatially periodic modulation of the χ(2) susceptibility, which is imposed by the quasi-phase-matching technique. Unlike previous realizations of semidiscrete optical modes in composite media, built as combinations of continuous and arrayed discrete waveguides, the semidiscrete vortex "droplets" are produced here in the fully continuous medium. This work reveals that the system supports two types of semidiscrete vortex droplets, viz., onsite- and intersite-centered ones, which feature, respectively, odd and even numbers of stripes, N. Stability areas for the states with different values of N are identified in the system's parameter space. Some stability areas overlap with each other, giving rise to the multistability of states with different N. The coexisting states are mutually degenerate, featuring equal values of the Hamiltonian and propagation constant. An experimental scheme to realize the droplets is outlined, suggesting new possibilities for the long-distance transmission of nontrivial vortex beams in nonlinear media.

Ring-shaped quantum droplets with hidden vorticity in a radially periodic potential

We study the stability and characteristics of two-dimensional circular quantum droplets (QDs) with embedded hidden vorticity (HV), i.e., opposite angular momenta in two components, formed by binary Bose-Einstein condensates (BECs) trapped in a radially periodic potential. The system is modeled by the Gross-Pitaevskii equations with the Lee-Huang-Yang terms, which represent the higher-order self-repulsion induced by quantum fluctuations around the mean-field state, and a potential which is a periodic function of the radial coordinate. Ring-shaped QDs with high winding numbers (WNs) of the HV type, which are trapped in particular circular troughs of the radial potential, are produced by means of the imaginary-time-integration method. Effects of the depth and period of the potential on these QD states are studied. The trapping capacity of individual circular troughs is identified. Stable compound states in the form of nested multiring patterns are constructed too, including ones with WNs of opposite signs. The stably coexisting ring-shaped QDs with different WNs can be used for the design of BEC-based data-storage schemes.

Stable higher-charge vortex solitons in the cubic-quintic medium with a ring potential

We put forward a model for trapping stable optical vortex solitons (VSs) with high topological charges m. The cubic-quintic nonlinear medium with an imprinted ring-shaped modulation of the refractive index is shown to support two branches of VSs, which are controlled by the radius, width, and depth of the modulation profile. While the lower-branch VSs are unstable in their nearly whole existence domain, the upper branch is completely stable. Vortex solitons with m ≤  12 obey the anti-Vakhitov-Kolokolov stability criterion. The results suggest possibilities for the creation of stable narrow optical VSs with a low power, carrying higher vorticities.

Matter-wave solitons in an array of spin-orbit-coupled Bose-Einstein condensates

We investigate matter-wave solitons in a binary Bose-Einstein condensate (BEC) with spin-orbit (SO) coupling, loaded in a one-dimensional (1D) deep optical lattice and a three-dimensional anisotropic magnetic trap, which creates an array of elongated sub-BECs with transverse tunneling. We show that the system supports 1D continuous and discrete solitons localized in the longitudinal (along the array) and the transverse (across the array) directions, respectively. In addition, such solitons are always unpolarized in the zero-momentum state but polarized in finite-momentum states. We also show that the system supports stable two-dimensional semidiscrete solitons, including single- and multiple-peaked ones, localized in both the longitudinal and transverse directions. Stability diagrams for single-peaked semidiscrete solitons in different parameter spaces are identified. The results reported here are beneficial not only for understanding the physical property of SO-coupled BECs but also for generating new types of matter-wave solitons.

Triangular bright solitons in nonlinear optics and Bose-Einstein condensates

We demonstrate what we believe to be novel triangular bright solitons that can be supported by the nonlinear Schrödinger equation with inhomogeneous Kerr-like nonlinearity and external harmonic potential, which can be realized in nonlinear optics and Bose-Einstein condensates. The profiles of these solitons are quite different from the common Gaussian or sech envelope beams, as their tops and bottoms are similar to the triangle and inverted triangle functions, respectively. The self-defocusing nonlinearity gives rise to the triangle-up solitons, while the self-focusing nonlinearity supports the triangle-down solitons. Here, we restrict our attention only to the lowest-order fundamental triangular solitons. All such solitons are stable, which is demonstrated by the linear stability analysis and also clarified by direct numerical simulations. In addition, the modulated propagation of both types of triangular solitons, with the modulated parameter being the strength of nonlinearity, is also presented. We find that such propagation is strongly affected by the form of the modulation of the nonlinearity. For example, the sudden change of the modulated parameter causes instabilities in the solitons, whereas the gradual variation generates stable solitons. Also, a periodic variation of the parameter causes the regular oscillation of solitons, with the same period. Interestingly, the triangle-up and triangle-down solitons can change into each other, when the parameter changes the sign.

Formations and dynamics of two-dimensional spinning asymmetric quantum droplets controlled by a PT-symmetric potential

In this paper, vortex solitons are produced for a variety of 2D spinning quantum droplets (QDs) in a PT-symmetric potential, modeled by the amended Gross-Pitaevskii equation with Lee-Huang-Yang corrections. In particular, exact QD states are obtained under certain parameter constraints, providing a guide to finding the respective generic family. In a parameter region of the unbroken PT symmetry, different families of QDs originating from the linear modes are obtained in the form of multipolar and vortex droplets at low and high values of the norm, respectively, and their stability is investigated. In the spinning regime, QDs become asymmetric above a critical rotation frequency, most of them being stable. The effect of the PT-symmetric potential on the spinning and nonspinning QDs is explored by varying the strength of the gain-loss distribution. Generally, spinning QDs trapped in the PT-symmetric potential exhibit asymmetry due to the energy flow affected by the interplay of the gain-loss distribution and rotation. Finally, interactions between spinning or nonspinning QDs are explored, exhibiting elastic collisions under certain conditions.

Matter-wave gap solitons and vortices in three-dimensional parity-time-symmetric optical lattices

Past decades have witnessed the emergence and increasing expansion of parity-time (PT)-symmetric systems in diverse physical fields and beyond as they manifest entirely all-real spectra, although being non-Hermitian. Nonlinear waves in low-dimensional PT-symmetric non-Hermitian systems have recently been explored broadly; however, understanding these systems in higher dimensions remains abstruse and has yet to be revealed. We survey, theoretically and numerically, matter-wave nonlinear gap modes of Bose-Einstein condensates with repulsive interparticle interactions in three-dimensional PT optical lattices with emphasis on multidimensional gap solitons and vortices. Utilizing direct perturbed simulations, we address the stability and instability areas of both localized modes in the underlying linear band gap spectra. Our study provides deep and consistent understandings of the formation, structural property, and dynamics of coherent localized matter waves supported by PT optical lattices in multidimensional space, thus opening a way for exploring and stabilizing three-dimensional localized gap modes in non-Hermitian systems

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3D parity-time (PT)-symmetric optical lattices are used to overcome the collapse of 3D ultracold atoms.

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3D matter-wave gap solitons and vortices are found in PT-symmetric optical lattices.

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Rich properties and dynamics of 3D matter-wave localized modes are disclosed.

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In-depth soliton physics is provided in 3D non-Hermitian periodic physical systems.

Localization-to-delocalization transition of light in frequency-tuned photonic moiré lattices

We demonstrate in a numerical manner the intriguing localization-to-delocalization transition of light in frequency-tuned photonic moiré lattices, both in the zero-order and the higher-order regimes of light waves. We present a different technique to realize the composite photonic lattices, by means of two relatively twisted sublattices with different modulated lattice constants. Even though various kinds of photonic patterns including the commensurable and the incommensurable lattices can be well constructed, the observed transition between the localization and the delocalization of light field is moiré angle-independent. This angle-insensitive property was not reported before, and cannot be achieved by those photonic moiré lattices that are all moiré angle-dependent. We reveal that the obtained phase transition of light is robust to the changes of refractive index modulation of the photonic lattices. Moreover, we reveal that the effect of moiré angle-independent transition of light can be extended to the higher-order vortex light field, hence allowing prediction, for the first time to our knowledge, of both the localization and the delocalization of the vortex light field in the photonic lattices.

Rotating Multidimensional Quantum Droplets

We predict a new type of two- and three-dimensional stable quantum droplets persistently rotating in broad external two-dimensional and weakly anharmonic potential. Their evolution is described by the system of the Gross-Pitaevskii equations with Lee-Huang-Yang quantum corrections. Such droplets resemble whispering-gallery modes localized in the polar direction due to nonlinear interactions and, depending on their chemical potential and rotation frequency, they appear in rich variety of shapes, ranging from nearly flat-top or strongly localized rotating wave packets, to crescentlike objects extending nearly over the entire range of polar angles. Above critical rotation frequency quantum droplets transform into vortex droplets (in two dimensions) or vortex tori (in three dimensions), whose topological charge gradually increase with the increase of the modulus of chemical potential, and therefore they belong to the family of nonlinear modes connecting fundamental and vortex quantum droplets. Rotating quantum droplets are exceptionally robust objects, stable practically in the entire range of their existence.

Modulational Instability, Inter-Component Asymmetry, and Formation of Quantum Droplets in One-Dimensional Binary Bose Gases

Quantum droplets are ultradilute liquid states that emerge from the competitive interplay of two Hamiltonian terms, the mean-field energy and beyond-mean-field correction, in a weakly interacting binary Bose gas. We relate the formation of droplets in symmetric and asymmetric two-component one-dimensional boson systems to the modulational instability of a spatially uniform state driven by the beyond-mean-field term. Asymmetry between the components may be caused by their unequal populations or unequal intra-component interaction strengths. Stability of both symmetric and asymmetric droplets is investigated. Robustness of the symmetric solutions against symmetry-breaking perturbations is confirmed.