An Accordion Lattice Playing a Soliton Tune
Localized wave patterns in a lattice or other periodic media have been observed using arrays of coupled torsion pendula, chains of Josephson junctions, and arrays of optical waveguides. Joining this diverse repertoire is a recent experiment by Robbie Cruickshank of the University of Strathclyde in the UK and his collaborators [1]. Starting from a Bose-Einstein condensate (BEC) of cesium atoms, the researchers used an ingenious combination of experimental methods to realize, visualize, and theoretically explore coherent wave structures known as discrete solitons. These nonlinear waveforms have long been theorized to exist, and their implications have been extensively studied. In my view, Cruickshank and company’s experiment constitutes the clearest manifestation of discrete solitons so far achieved in ultracold atomic systems, paving the way for a variety of future explorations.
Solitons are localized wave packets that emerge from the interplay of dispersion and nonlinearity. Dispersion tends to make wave packets spread, and nonlinearity tends to localize them. The interplay can be robust and balanced, resulting in long-lived structures. The presence of a lattice introduces a new dimensional unit, the lattice constant, to the interplay, enabling a potential competition between the lattice constant and the scale of the soliton. When the latter is much larger than the former, the soliton is effectively insensitive to the lattice, which it experiences as a continuum. But as the two scales approach one another, lattice effects become more pronounced, and the associated waveforms become discrete solitons. In nonlinear variants of the Schrödinger equation, discreteness typically favors standing waves rather than traveling ones. That’s because the lattice-induced energy barrier known as the Peierls-Nabarro barrier makes discrete solitons less mobile.
Discrete solitons made their experimental debut in nonlinear optics in the pioneering work of Hagai Eisenberg and his collaborators, who observed them in optical waveguide arrays [2]. Relevant studies and further work in photorefractive crystal lattices in the 1990s and 2000s have been summarized in [3]. Subsequent manifestations include, among many others, discrete breathers in Josephson junctions [4] and in coupled pendula [5] and dispersive shock-like structures in mechanical [6] and magnetic metamaterials [7]. Now, thanks to the work of Cruickshank and his collaborators, we have a systematic realization and a direct visualization of the dynamics of discrete solitons in a BEC. With it comes the prospect of investigating these remarkable nonlinear structures with the tunability and precision of cold-atom techniques.
Accordion optical lattices trap atoms in a periodic potential, whose spacing can be continuously stretched or squeezed, like the musical instrument. Cruickshank and his collaborators started with a BEC of cesium atoms in an accordion lattice and gradually reduced the number of trapped atoms to a few thousand. Using the accordion lattice, they could tailor the atomic density to engender discrete solitons. Additionally, the researchers used microwaves to further remove atoms, leaving just those in a few central wells of the lattice. By deepening the lattice, the team could “freeze” the atoms in place and image them. Discrete solitons appeared as stable concentrations of atoms spread over one or more lattice sites. Thanks to the experiment’s flexibility, the researchers gained an unprecedented glimpse into discrete solitons over a wide range of conditions.
Cruickshank and his collaborators also obtained theoretical insights. By representing the discrete solitons as simple Gaussian wave packets, the team derived an effective energy landscape as a function of the strength of the nonlinearity and the characteristics of the lattice, such as the potential depth and the spacing between the wells. This analysis enabled the researchers to predict the local and global minimizers of the relevant energy “surface.” It also enabled them to anticipate how extended the solitons were. When the nonlinearity was sufficiently weak, the system had no localized states. At stronger nonlinearities, discrete solitons localized predominantly in a single well or across several adjacent wells, a case the researchers dubbed multisite solitons (Fig. 1). Lastly, they identified when the values of the two parameters (typically weak nonlinearities and wide or deep lattices) conspired to produce a multistable regime.
The stability of discrete solitons is often analyzed using variants of the nonlinear Schrödinger model [8]. The multistability discovered by Cruickshank and company may require reevaluating our understanding of robust, observable discrete wave patterns. Applying the team’s experimental arsenal to higher-dimensional discrete solitary waves could lead to the discovery of structures—such as vortex cubes and vortex diamonds—that have so far existed only as theoretical speculations [9]. Decades after their experimental realization in optics and elsewhere, nonlinear dynamical lattices and their discrete solitary waves seem to have the ability to continue to surprise us in new, unexpected formulations, posing intriguing physical questions to this day!
References
- R. Cruickshank et al., “Experimental observation of single- and multisite matter-wave solitons in an optical accordion lattice,” Phys. Rev. Lett. 135, 263404 (2025).
- H. S. Eisenberg et al., “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81, 3383 (1998).
- F. Lederer et al., “Discrete solitons in optics,” Phys. Rep. 463, 1 (2008).
- E. Trías et al., “Discrete breathers in nonlinear lattices: Experimental detection in a Josephson array,” Phys. Rev. Lett. 84, 741 (2000).
- J. Cuevas et al., “Discrete breathers in a forced-damped array of coupled pendula: Modeling, computation, and experiment,” Phys. Rev. Lett. 102, 224101 (2009).
- H. Kim et al., “Demonstration of dispersive rarefaction shocks in hollow elliptical cylinder chains,” Phys. Rev. Lett. 120, 194101 (2018).
- J. Li et al., “Observation of ultraslow shock waves in a tunable magnetic lattice,” Phys. Rev. Lett. 127, 014302 (2021).
- P. G. Kevrekidis, The Discrete Nonlinear Schrödinger Equation, Springer Tracts in Modern Physics Vol. 232 (Springer-Verlag, Berlin Heidelberg, 2009)[Amazon][WorldCat].
- M. Lukas et al., “Lyapunov–Schmidt reduction algorithm for three-dimensional discrete vortices,” Phys. D 237, 339 (2008).