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Frontiers in multidimensional self-trapping of nonlinear fields and matter

Abstract

2D and 3D solitons and related states, such as quantum droplets, can appear in optical systems, atomic Bose–Einstein condensates (BECs) and liquid crystals, among other physical settings. However, multidimensional solitary states supported by the standard cubic nonlinearity tend to be strongly unstable — a property far less present in 1D systems. Thus, the central challenge is to stabilize multidimensional states, and to that end numerous approaches have been proposed over the years. Most strategies involve non-cubic nonlinearities or using various potentials, including periodic ones. Completely new directions have recently emerged in two-component BECs with spin–orbit coupling, which have been predicted to support stable 2D and metastable 3D solitons. A recent breakthrough is the creation of 3D quantum droplets. These are self-sustained states existing in two-component BECs, stabilized by the quantum fluctuations around the underlying mean-field states. Here, we review recent results in this field and outline outstanding current challenges.

Key points

  • We provide a brief summary of recent theoretical and experimental advances in the study of multidimensional solitons, chiefly in nonlinear optics, ultracold bosonic gases and liquid-crystal and magnetic media.

  • We cover results for fundamental nonlinear modes and for topologically non-trivial states, such as vortex solitons, hopfions and skyrmions.

  • The experimental realization of multidimensional solitons has proved to be more challenging than for 1D solitons owing to the propensity to instabilities of 2D and 3D states, both fundamental and topologically structured. We address different stabilization mechanisms that have been put forward to potentially observe multidimensional solitons, such as competing nonlinearities, linear and nonlinear potentials, spin–orbit coupling, quantum corrections and dissipative effects.

  • Special attention is paid to recent theoretical and experimental results that produced stable 3D solitons in the form of quantum droplets in ultracold bosonic gases, the stability of which is secured by a macroscopic effect of quantum fluctuations around mean-field states.

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Fig. 1: Observed excitation of short-lived fundamental and vortex light bullets in hexagonal waveguiding arrays.
Fig. 2: Predicted light bullets in radially symmetric and complex potentials.
Fig. 3: Predicted hopfions and hybrid 3D vortex solitons.
Fig. 4: Predicted stable multidimensional solitons in Bose–Einstein condensates with spin–orbit coupling.
Fig. 5: Predicted vortex solitons in the binary Bose–Einstein condensates coupled by a microwave field.
Fig. 6: Observed self-sustained multidimensional quantum droplets.
Fig. 7: Observed topological solitons with differently knotted nematic fields in a liquid crystal.

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Acknowledgements

The authors greatly appreciate many valuable collaborations and discussions with S. K. Adhikari, G. Dong, A. Gammal, R. G. Hulet, V. V. Konotop, O. D. Lavrentovich, Y. Li, D. Mihalache, D. S. Petrov, H. Sakaguchi, L. Salasnich, E. Y. Sherman, Y. Shnir, D. V. Skryabin and L. Tarruell. L.T. and Y.V.K. acknowledge support from the Severo Ochoa program (SEV-2015-0522) of the Government of Spain, Fundació Cellex, Fundació Mir-Puig, Generalitat de Catalunya and Centres de Recerca de Catalunya (CERCA). The work of B.A.M. is supported, in part, by the joint programme in physics between the US National Science Foundation (NSF) and Binational (US–Israel) Science Foundation through project no. 2015616 and by the Israel Science Foundation through grant no. 1286/17. B.A.M. appreciates the hospitality of the Institute of Photonic Sciences (ICFO) during the preparation of this Review. G.E.A. acknowledges financial support from the Ministry of Science, Innovation and Universities (MICINN, Spain), grant no. FIS2017-84114-C2-1-P.

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Glossary

Hopfions

A class of 3D localized modes, in the form of tori carrying the global vorticity, which are additionally twisted in the torus cross section. This mode carries two independent topological charges (winding numbers), one representing the overall vorticity and the other accounting for the intrinsic twist.

Skyrmions

Complex 3D states in various two-component field-theory systems, which carry two independent topological numbers. They were introduced by Skyrme as a classical field model, which effectively describes baryons, and may be derived as a low-energy (semi-classical) limit of quantum chromodynamics.

Spinor Bose–Einstein condensates

(spinor BECs). Condensates composed of two or several components, which may be considered as a set forming a spinor wavefunction, corresponding to a pseudo spin of 1/2 (two components), 1 (three components) or 2 (five components).

Kerr nonlinearities

Universal optical nonlinearities occurring in dielectric media that yield a correction to the local refractive index proportional to the intensity of the electromagnetic wave. They are represented by the cubic self-attractive term in the corresponding nonlinear Schrödinger equation.

Tandem structures

Optical tandem structures represent periodic stacks of materials with different parameters, such as the refractive index, nonlinearity and dispersion, in which widths of individual layers are usually small in comparison with the average diffraction and dispersion lengths.

Vortex solitons

2D or 3D solitons represented by a complex wavefunction whose phase carries an integer winding number (vorticity, also known as the topological charge) and has an amplitude that vanishes at the central pivot.

Parity–time (PT )-symmetry

Special symmetry of the evolution equation or non-Hermitian Hamiltonian governing a dissipative system under the transformation of time reversal and parity inversion (flip of the sign of the spatial coordinate). In the so-called unbroken PT phase, such Hamiltonian shows an entirely real energy spectrum despite being non-Hermitian.

X-wave

A delocalized linear or nonlinear 2D wave, with the local power featuring an X-shaped profile, which may be supported by defocusing nonlinear optical material when the signs of dispersion and diffraction coefficients are opposite.

Bessel and Airy beams

Bessel beams represent 2D non-diffracting solutions of the Helmholtz equation in circular cylindrical coordinates, in which this equation is separable. Airy beams are non-diffracting 1D or 2D beams that bend along a parabolic trajectory upon propagation while maintaining their functional shapes. Their combinations can be used to construct non-diffracting 3D wavepackets.

Spin–orbit coupling

(SOC). Originally, it referred to the coupling between the spin of electrons in semiconductors and their motion through the crystalline electrostatic field. In the context of the Bose–Einstein condensate, SOC is realized as linear mixing between two components of a binary condensate through first spatial derivatives of the respective wavefunctions.

Topological insulators

Originally, dielectric materials (insulators) possessing a complete gap in the bulk, but admitting conductance through in-gap edge states existing as a result of peculiarities of the intrinsic topological structure of the material. This name is also used for photonic settings that emulate the same phenomenology in terms of light transmission.

Pseudopotentials

Effective potentials that are induced by the nonlinearity whose local strength is subject to spatial modulation.

Feshbach resonances

The effects that make it possible to change the magnitude and sign of the scattering length characterizing collisions between atoms in quantum gases. The Feshbach resonance is a powerful experimental tool enabling control of the strength and sign of the effective nonlinearity in Bose–Einstein condensates (as concerns both self-interactions and cross interactions, in the case of two-component condensates).

Semi-vortex

A stable two-component 2D or 3D soliton in a two-component spin–orbit-coupled Bose–Einstein condensate, in which, unlike mixed modes, one component has zero vorticity, whereas the other one carries a vorticity of 1.

Mixed-mode

Stable 2D and 3D solitons in a two-component Bose–Einstein condensate with the spin–orbit coupling between the components. Unlike semi-vortices, each component of such a mode is a mixture of terms with zero vorticity and vorticities of ±1.

Torons

Toroidal localized modes that may be created in liquid crystals and ferrofluids. A toron is organized in essentially the same way as a hopfion (a twisted torus, which may carry the overall vorticity).

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Kartashov, Y.V., Astrakharchik, G.E., Malomed, B.A. et al. Frontiers in multidimensional self-trapping of nonlinear fields and matter. Nat Rev Phys 1, 185–197 (2019). https://doi.org/10.1038/s42254-019-0025-7

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