Research paper
One-dimensional localized modes of spin-orbit-coupled Bose-Einstein condensates with spatially periodic modulated atom-atom interactions: Nonlinear lattices

https://doi.org/10.1016/j.cnsns.2020.105217Get rights and content

Highlights

  • We address the one-dimensional (1D) localized modes of spin-orbit-coupled Bose-Einstein condensates in nonlinear lattices.

  • The setting supports two classes of 1D localized modes : fundamental solitons (with a single peak), and soliton pairs conisiting of dipole solitons (anti-phase) or two-peak solitons (in-phase).

  • The existence and stability of the localized modes are tested by linear-stability analysis and direct perturbed simulations.

Abstract

Bose-Einstein condensates (BECs) provide a clear and controllable platform to study diverse intriguing emergent nonlinear effects that appear too in other physical settings, such as bright and dark solitons in mean-field theory as well as many-body physics. Various ways have been elaborated to stabilize bright solitons in BECs, three promising schemes among which are: optical lattices formed by counterpropagating laser beams, nonlinear managements mediated by Feshbach resonance, spin-orbit coupling engineered by dressing atomic spin states (hyperfine states of spinor atomic BECs) with laser beams. By combing the latter two schemes, we discover, from theory to calculations, that the two-component BECs with a spin-orbit coupling and cubic atom-atom interactions, whose nonlinear distributions exhibit a well-defined spatially periodic modulation (nonlinear lattice), can support one-dimensional localized modes of two kinds: fundamental solitons (with a single peak), and soliton pairs comprised of dipole solitons (anti-phase) or two-peak solitons (in-phase). The influence of three physical parameters: chemical potential of the system, strengths of both the Rashba spin-orbit coupling and atom-atom interactions, on the existence and stability of the localized modes is investigated based on linear-stability analysis and direct perturbed simulations. In particular, we demonstrate that the localized modes can be stable objects provided always that both the inter- and intraspecies interactions are attractive.

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, ψ=(ψ+,ψ), whose scaled form can be written as [50], [51], [52], [53], [54]iψ+t=122ψ++σψxcos2(2x)(g|ψ+|2+γ|ψ|2)ψ+,iψt=122ψσψ+xcos2(2x)(g|ψ|2+γ|ψ+|2)ψ.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 ϕ+(x) and spin-down component ϕ(x) 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).

References (94)

  • P.G. Kevrekidis et al.

    Emergent Nonlinear Phenomena in Bose-Einstein Condensates: Theory and Experiment

    (2007)
  • Y.V. Kartashov et al.

    Solitons in nonlinear lattices

    Rev Mod Phys

    (2011)
  • Y. Kartashov et al.

    Frontiers in multidimensional self-trapping of nonlinear fields and matter

    Nat Rev Phys

    (2019)
  • G. Fibich

    The nonlinear Schrödinger equation: singular solutions and optical collapse

    (2015)
  • B.A. Malomed

    Multidimensional solitons: well-established results and novel findings

    Eur Phys J

    (2016)
  • V.A. Brazhnyi et al.

    Theory of nonlinear matter waves in optical lattices

    Mod Phys Lett B

    (2004)
  • O. Morsch et al.

    Dynamics of Bose-Einstein condensates in optical lattices

    Rev Mod Phys

    (2006)
  • M. Lewenstein et al.

    Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond

    Adv Phys

    (2007)
  • L. Zeng et al.

    Gap-type dark localized modes in a Bose-Einstein condensate with optical lattices

    Adv Photon

    (2019)
  • L. Zeng et al.

    Preventing critical collapse of higher-order solitons by tailoring unconventional optical diffraction and nonlinearities

    Commun Phys

    (2020)
  • B. Eiermann et al.

    Bright Bose-Einstein gap solitons of atoms with repulsive interaction

    Phys Rev Lett

    (2004)
  • B.A. Malomed

    Soliton management in periodic systems

    (2006)
  • C. Chin et al.

    Feshbach resonances in ultracold gases

    Rev Mod Phys

    (2010)
  • J. Zeng et al.

    Stabilization of one-dimensional solitons against the critical collapse by quintic nonlinear lattices

    Phys Rev A

    (2012)
  • X. Gao et al.

    Two-dimensional matter-wave solitons and vortices in competing cubic-quintic nonlinear lattices

    Front Phys

    (2018)
  • L. Zeng et al.

    One-dimensional solitons in fractional Schrödinger equation with a spatially periodical modulated nonlinearity: nonlinear lattice

    Opt Lett

    (2019)
  • J. Shi et al.

    1D solitons in saturable nonlinear media with space fractional derivatives

    Ann Phys (Berlin)

    (2020)
  • H. Sakaguchi et al.

    Solitons in combined linear and nonlinear lattice potentials

    Phys Rev A

    (2010)
  • J. Zeng et al.

    Two-dimensional solitons and vortices in media with incommensurate linear and nonlinear lattice potentials

    Phys Scr

    (2012)
  • J. Shi et al.

    Self-trapped spatially localized states in combined linear-nonlinear periodic potentials

    Front Phys

    (2020)
  • O.V. Borovkova et al.

    Bright solitons from defocusing nonlinearities

    Phys Rev E

    (2011)
  • O.V. Borovkova et al.

    Algebraic bright and vortex solitons in defocusing media

    Opt Lett

    (2011)
  • J. Zeng et al.

    Bright solitons in defocusing media with spatial modulation of the quintic nonlinearity

    Phys Rev E

    (2012)
  • Y.V. Kartashov et al.

    Asymmetric solitons and domain walls supported by inhomogeneous defocusing nonlinearity

    Opt Lett

    (2012)
  • L.E. Young et al.

    Self-trapping of fermi and Bose gases under spatially modulated repulsive nonlinearity and transverse confinement

    Phys Rev A

    (2013)
  • W.B. Cardoso et al.

    Bright solitons from the nonpolynomial Schrödinger equation with inhomogeneous defocusing nonlinearities

    Phys Rev E

    (2013)
  • R. Driben et al.

    Soliton gyroscopes in media with spatially growing repulsive nonlinearity

    Phys Rev Lett

    (2014)
  • Y.V. Kartashov et al.

    Twisted toroidal vortex solitons in inhomogeneous media with repulsive nonlinearity

    Phys Rev Lett

    (2014)
  • R. Driben et al.

    Three-dimensional hybrid vortex solitons

    New J Phys

    (2014)
  • P.G. Kevrekidis et al.

    Solitons and vortices in two-dimensional discrete nonlinear Schrödinger systems with spatially modulated nonlinearity

    Phys Rev E

    (2015)
  • R. Driben et al.

    Multipoles and vortex multiplets in multidimensional media with inhomogeneous defocusing nonlinearity

    New J Phys

    (2015)
  • J. Zeng et al.

    Localized dark solitons and vortices in defocusing media with spatially inhomogeneous nonlinearity

    Phys Rev E

    (2017)
  • C. Huang et al.

    Excited states of two-dimensional solitons supported by spin-orbit coupling and field-induced dipole-dipole repulsion

    Phys Rev A

    (2018)
  • R. Zhong et al.

    Self-trapping under two-dimensional spin-orbit coupling and spatially growing repulsive nonlinearity

    Front Phys

    (2018)
  • L. Zeng et al.

    Purely kerr nonlinear model admitting flat-top solitons

    Opt Lett

    (2019)
  • L. Zeng et al.

    Gaussian-like and flat-top solitons of atoms with spatially modulated repulsive interactions

    J Opt Soc Am B

    (2019)
  • D.S. Petrov

    Quantum mechanical stabilization of a collapsing Bose–Bose mixture

    Phys Rev Lett

    (2015)
  • Cited by (11)

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

      2022, iScience
      Citation Excerpt :

      One solving scheme relies on competing self-focusing and defocusing terms, including cubic-quintic (Zeng and Malomed, 2012; Gao and Zeng, 2018) and cubic-quartic (Kartashov et al., 2018) nonlinearities. Quantum mechanical stabilization of a collapsing Bose-Bose mixture leads to quantum droplets (Petrov, 2015; Petrov and Astrakharchik, 2016; Cabrera et al., 2018; Semeghini., et al., 2018; Zhang et al., 2019), which, actually, derive from the competition of attractive mean-field term and repulsive beyond mean-field Lee-Huang-Yang correction (Lee et al., 1957) characterizing the quantum many-body effects (quantum fluctuations); in degenerate quantum gases, the nonlinearity is induced by atom-atom interactions and can be tuned by Feshbach resonance (Chin et al., 2010; Chen and Zeng, 2020; Kengne et al., 2021). Besides, linear periodic potentials are widely exploited for generating multidimensional optical and atomic matter waves (Ostrovskaya and Kivshar, 2003; Ostrovskaya and Kivshar, 2004; Alexander et al., 2005; Leblond et al., 2009; Zeng and Zeng, 2019; Li and Zeng, 2021; Chen and Zeng, 2021).

    • Overcoming the snaking instability and nucleation of dark solitons in nonlinear Kerr media by spatially inhomogeneous defocusing nonlinearity

      2022, Chaos, Solitons and Fractals
      Citation Excerpt :

      Take the nonlinearity mentioned above for example, it accounts for the atom-atom interactions that can be freely manipulated in ultracold atoms experiments by using the Feshbach resonance technique [30] and which, interestingly, can adjust not only the strength and even the sign of the corresponding nonlinearity, but also the spatially or temporal (or their combination) form of the nonlinear distribution. The latter feature of nonlinear patterns can be tuned as both periodic form like nonlinear lattices [31,32] or nonuniform shape with inhomogeneous nonlinearity [33–35]; both types of nonlinearities have shown great power for generating diverse nonlinear localized wave structures like solitons, see very recent reviews [8,36] and references therein. The optical lattices, which are formed by launching pairs of counterpropagating laser beams that configure as standing waves, have aided to the discovery of bright gap solitons in shallow lattices [23] and quantum phase transition (quantum simulations) by switching the lattice depth [37].

    • One-dimensional quantum droplets under space-periodic nonlinear management

      2021, Results in Physics
      Citation Excerpt :

      In particular, the gap solitons and vortical ones are localized states supported by linear lattices—periodic potentials, including optical lattices in BECs [5–7] and photonic crystals and lattices in nonlinear optics [8–11]. The nonlinear lattices regulated by Feshbach resonances [12], or made with special nonlinear photonic crystals, have been employed to support solitons [13–21]. A combination of the two, the combined linear-nonlinear lattices, shows a great freedom to generate stable solitons [22–24].

    View all citing articles on Scopus
    View full text