Research paperOne-dimensional localized modes of spin-orbit-coupled Bose-Einstein condensates with spatially periodic modulated atom-atom interactions: Nonlinear lattices
Introduction
Since their first observations in rubidium-87 [1], sodium-23 [2] and Lithium-7 [3] atoms more than two decades ago, Bose-Einstein condensates (BECs) or ultracold quantum gases, due to their peculiar properties and interesting nonlinear effects, have aroused widespread attention of a great number of theoretical and experimental studies [4], [5], [6], [7], [8]. The emergence of matter-wave localized structures of various solitonic excitations such as dark, bright and grey solitons, soliton trains, and their interactions, is among one of the very fascinating nonlinear effects in the trapped ultracold atoms. Under the mean-field approximation, the dynamics of BECs can be well described by the Gross-Pitaevskii equation, which gives rise to analytical solutions of one-dimensional (1D) bright and dark solitons depending, respectively, on the self-focusing and self-defocusing interactions of the atomic BECs [4], [5], [6], [7]. In two- and three-dimensional spaces, without the action of any external potential, however, the stability of bright solitons upheld by self-attractive (focusing) cubic nonlinearity, arose by atom-atom contact interactions, is a destructive problem because of the inherent critical collapse and supercritical one of the soliton structures in respective coordinates [9], [10], [11].
In order to overcome such a very serious problem of collapse, many solutions were thus proposed in the past decades [7], [8]. One flourishing way relies on the exploitation of linear periodic potentials—optical lattices, highly configurable structures of standing waves formed by multiple counterpropagating laser beams, with which scientists can stabilize various kinds of solitons (and vortical structures) in the context of BECs including, particularly, the gap solitons dwelling in the finite band gaps of the corresponding linear spectrum can be supported by the self-defocusing nonlinearity [7], [8], [12], [13], [14], [15], [16]. It is worth mentioning that the 1D bright Bose-Einstein gap solitons for rubidium-87 atoms with repulsive atom-atom interaction have been observed [17] more than a decade ago, which is an enormously important development. Another promising method is called nonlinear management [18] assisted by a commonly used advanced technology—Feshbach resonance [19], direct use of which can adjust the variation of the local strengths and even the sign of the nonlinearity (atom-atom interactions). Periodic potential, engineered by the Feshbach resonance technique, has its extension called nonlinear lattices [7], into which the atomic condensate was loaded and stable localized modes containing matter-wave solitons can be created. In past years, the linear-nonlinear lattices, the commensuration of combined linear (optical lattices) and nonlinear lattices [20], [21], [22], [23], were also introduced to stabilize solitons in BECs and beyond (especially in nonlinear optics) [24], [25], [26].
Because the mean-field description of BECs, to which we referred above, is in the framework of the Gross–Pitaevskii equation (alias nonlinear Schrödinger equation) whose nonlinear term, induced by two-body or three-body contact interactions, can be tuned via the Feshbach resonance, the purely nonlinear models were thus elaborated to generate stable diversiform solitons. Particularly worth indicating is the recent prediction of purely self-defocusing optical and matter-wave media with spatially inhomogeneous nonlinearity, whose local strength grows quickly enough from the center to the periphery, can give rise to a good diversity of robust self-trapped states (localized modes) [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], typical results of which include 1D fundamental solitons and high-order nonlinear excitations (dipole and multipole solitons) [27], [28], [29], 2D robust vortex solitons imprinted with randomly high topological charges [27], [28], 2D localized dark solitons and vortices [38], intricate 3D structures appearing as soliton gyroscopes and skyrmions as well as Hopfions [33], [34], and flat-top solitons of several types such as fundamental solitons, 1D multipoles and 2D vortices as well as their transitions to ordinary Gaussian-like ones [41], [42]. The purely nonlinear media with competing attractive-repulsive nonlinear terms, including the nonlinear Schrödinger equations with cubic-quintic, quadratic-cubic and cubic-quartic nonlinearities, have demonstrated the nonlinear self-trapping ability of optical beams and matter waves [8]. The self-bound quantum droplets [43], [44], a new phase of quantum matter in ultracold atoms systems formed as a result of balanced competing nonlinearities interplay: mean-field nonlinearity and a beyond-mean-field nonlinear term (with an opposite sign)—the well-known Lee-Huang-Yang (LHY) correction induced by quantum fluctuations [45], have been predicted theoretically [43], [44] and observed experimentally [46], [47], [48], [49] in binary BECs very recently.
Meanwhile, there is another way of stabilizing diverse self-trapped modes in the context of BECs via the emulation of various effects typical for condensed-matter physics and other systems, such as the spin-orbit coupling (SOC) in semiconductor originated from the interaction between a quantum particle’s (generally an electron) spin and its momentum when crossing the underlying ionic lattice. Such way is based on SOC emulation in atomic condensate, achieved by using a two-component pseudo-spinor wave function of a binary BEC whose constituents—the atoms are being prepared in two different hyperfine states, into which the electron’s wave functions of spin-up and spin-down components were mapped; the BEC under such case has been viewed as spin-orbit-coupled BEC or BEC with SOC [50], [51], [52], [53], [54], [55], [56], [57], [58] [SOC Fermi gases for degenerate Fermi gases [59]]. The implementation of SOC in the 1D binary condensate of rubidium-87 atoms has been realized experimentally in 2011 [60], based on the relevant theoretical proposal [61]; experimental observation of SOC in 2D setting was announced too in 2016 [62]. The SOC in itself is a linear effect—the linear coupling between the two-component wave functions (of two different atomic states) of a binary BEC guided by properly designed laser field, and its combination with the immanent nonlinearity in the binary mixtures of bosonic condensate within the mean-field theory has opened a new avenue to explore diverse nonlinear effects, such as fundamental solitons [63], [64], [65], [66], [67], [68], [69], gap solitons backed by optical lattice potentials [70], [71], [72], [73], [74], [75], 2D vortices and vortex lattices [76], [77], [78], [79], [80], [81], [82], [83], [84], [85], [86], Skyrmions [87] and Dirac monopole [88]. As stated above, the nonlinear management technique based on Feshbach resonance could lead to a periodic variation of the strength and sign of the scattering length in spatial and in time, making it possible to stabilize soliton modes that may otherwise unstable under constant nonlinearity [19]. Based on such approach, despite 1D topological solitons in a spin-orbit-coupled BEC with the relative strength of the cross-attraction being changed periodically in time have been researched very recently [89], the investigation of quantum solitons specific to the binary BEC with SOC and with periodic spatial modulation of the attractive cubic nonlinearities is still missing, which is an incentive of this article.
Through the numerical results and theoretical analysis, we survey localized modes of spin-orbit-coupled BECs stabilized by spatially periodic modulated strengths of the self-attraction and cross-attraction terms, thus forming the nonlinear lattices mentioned above, and demonstrate their existence and dynamic stability in 1D setting. Such setting is described by a pair of Gross-Pitaevskii equations, which readily give rise to 1D soliton solutions of two types: fundamental solitons, representing ground state of the system, and composite solitons constituted by dipole solitons or two-peak solitons which are, basically, two out-of-phase or in-phase fundamental solitons respectively. By exploitation of the linear-stability analysis and direct perturbed simulations, we reveal that the three relevant physical variables—chemical potential of the setting, local strengths of the Rashba SOC and of atom-atom interactions to the intracomponent and intercomponent couplings—have great impact on the presence and stability properties of the predicted localized modes. Our findings give further theoretical insight into the formation and evolutional dynamics of matter-wave solitons in the ultracold atoms systems with the emulation of synthetic SOC.
The rest of this article is structured as follows. In Section 2, the theoretical model describing the spin-orbit-coupled BECs is first introduced in Section 2.1, and the calculated methods to solve it include the linear stability analysis and direct numerical computation methods are given in Section 2.2; and then Section 3 reports the numerical results of the localized modes we studied such as the fundamental solitons and composite solitons (dipole solitons and two-peak modes) in Sections 3.1 and 3.2, the corresponding physical interpretations are also given out in both subsections; finally, a conclusion is summarized in Section 4.
Section snippets
Theoretical model
Our starting point is the underlying physical model describing the binary spin-orbit-coupled BECs, which, under mean-field theory, yields 1D coupled Gross–Pitaevskii equations for the two-component pseudo-spinor wave function, whose scaled form can be written as [50], [51], [52], [53], [54]Where ψ ± denote the wave functions for the spin-up and spin-down components of the considered
Fundamental solitons: ground state
In what follows, we proceed to present numerical results. The localized modes appeared as fundamental solitons of the physical model, describing spin-orbit-coupled BECs trapped by a nonlinear lattice, are first explored. Fig. 1(a) and (b), respectively, depict the typical shapes of spin-up wavefunction and spin-down component of such ground-state solutions, under different strengths of cross-interaction γ while at settled chemical potential (μ) and strength (σ) of spin-orbit
Conclusion
In summarizing, we have theoretically and numerically investigated the generation of one-dimensional (1D) solitons in spin-orbit-coupled Bose-Einstein condensates (BECs), in which the cubic nonlinear contact interactions induced by atom-atom collisions are configured, via Feshbach resonance technique, to form as nonlinear lattices, realized by spatially periodic modulations of the nonlinear strengths. Such two-component setting is described by a set of two coupled Gross–Pitaevskii equations and
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
CRediT authorship contribution statement
Junbo Chen: Methodology, Software, Validation, Formal analysis, Writing - original draft, Writing - review & editing, Visualization. Jianhua Zeng: Conceptualization, Methodology, Validation, Formal analysis, Writing - original draft, Writing - review & editing, Visualization, Supervision, Project administration, Funding acquisition.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (NSFC) (Nos. 61690224, 61690222), and by the Youth Innovation Promotion Association of the Chinese Academy of Sciences (No. 2016357).
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