## Abstract

Solitons are nonlinear solitary waves which maintain their shape over time and through collisions, occurring in a variety of nonlinear media from plasmas to optics. We present an experimental and theoretical study of hydrodynamic phenomena in a two-component atomic Bose-Einstein condensate where a soliton array emerges from the imprinting of a periodic spin pattern by a microwave pulse-based winding technique. We observe the ensuing dynamics which include shape deformations, the emergence of dark-antidark solitons, apparent spatial frequency tripling, and decay and revival of contrast related to soliton collisions. For the densest arrays, we obtain soliton complexes where solitons undergo continued collisions for long evolution times providing an avenue towards the investigation of soliton gases in atomic condensates.

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## Introduction

Since their first observation in water waves^{1,2}, the dynamics of solitary wave structures has evolved into a major thrust within nonlinear science. These dispersionless, localized, coherent structures, which can undergo collisions without changing shape, are found in a wide range of integrable and near-integrable systems with broad applicability in optics^{3}, atomic physics^{4}, plasmas^{5}, fluids^{6}, and other fields^{6,7}.

Dilute gas Bose-Einstein condensates (BECs)^{8,9} offer a highly flexible and controllable platform for investigating the nonlinear dynamics and interactions of such solitary wave structures^{4}. The experimental realization of multi-component BECs, is discussed in refs. ^{10,11}, has led to an additional wealth of nonlinear states including dark-bright (DB), dark-dark, and dark–antidark vector solitons, among many others summarized, e.g., in ref. ^{12}, as well as three-component^{13,14} and magnetic solitons^{15}. While these works merely represent a small fraction of recent experimental and theoretical developments, they exemplify the remarkable flexibility offered by ultracold atomic systems for engineering and interrogating superfluid hydrodynamics.

Despite the intense research efforts directed towards solitons and their dynamics, most of the associated experimental BEC studies have concentrated on individual solitons or very small clusters (or molecules) thereof^{16} and their interactions^{14}. However, over the past few years, there has been a substantial interest devoted to the realization and exploration of soliton gases, given their intriguing generalized hydrodynamic properties^{17}. First theoretically introduced in 1971 as a dilute soliton gas^{18}, the concept was later extended to the dense soliton gas in the theory works of refs. ^{19,20,21}. However, experimental evidence for the realization of a soliton gas has been obtained only recently in the setting of shallow water waves^{22,23}. The relevant theory for integrable systems has been summarized in ref. ^{24}. These investigations, along with earlier efforts, namely on light pulses in optical fiber ring resonators^{25} and also ones in soliton turbulence in shallow water waves^{26}, has changed our perspective on soliton lattices, soliton fluids, and soliton gases, as well as transitions between them^{27}. Each of these experimental platforms provides a different framework for probing the physics underlying nonlinear hydrodynamics, each with its own advantages and limitations. In this work, we will make use of the tunability, reproducibility, and mature numerical modeling methods of ultracold atomic gases to provide a different perspective on these topics.

We present here a combined experimental and numerical study of hydrodynamic excitations arising from a periodic phase winding in a two-component spinor BEC. Previously, experimental efforts along these lines have led to different examples of pattern formation in atomic BECs including Faraday waves^{28,29,30,31}, space-time crystals^{32}, and bright soliton trains produced from a dynamical (modulational) instability^{33,34,35}. In contrast to these earlier works, our method provides an initial condition from which exceptionally regular, highly tunable solitonic arrays will develop.

We experimentally produce the initial state through a Ramsey pulse sequence in the presence of a small magnetic field gradient which produces an alternating magnetization pattern as shown in Fig. 1. The periodicity of the pattern depends on the winding time *τ* between the two Ramsey pulses, with more time producing a finer pattern. The experimental observations are made through state-selective absorption imaging as shown in Fig. 1a. Complementary numerical simulations are produced by solving the nonlinear Schrödinger equation in three dimensions, using a phase wound initial state as shown in Fig. 1b. This sinusoidal magnetization pattern then evolves under nonlinear interactions into different hydrodynamic phenomena depending on the spacing of the initial winding pattern. In our exposition of the observed dynamics, we demonstrate how a broad magnetization pattern, under the influence of a magnetic gradient, leads to the emergence of solitons from an array of hydrodynamic shock fronts. We then study finer initial windings where we observe the formation of an array of dark–anitdark solitons which subsequently interact and evolve. For our tightest arrays, we obtain dense, long-lived collisional soliton complexes spanning essentially the full extent of the BEC.

## Results

We begin discussing the dynamics in mixtures with a wide initial winding pattern as shown in Fig. 2a, which corresponds to an experimental winding time of 10 ms. In these and all following experimental images, only one of the two spin states is shown as the second component forms the complementary pattern producing a nearly unmodulated total density (see Supplementary Note 1). While later in this work we remove the axial magnetic gradient after the winding process to observe the unbiased evolution, for the data shown in Fig. 2 we maintain the magnetic gradient of 5 mG/cm along the long axis of the BEC to induce a small amount of counterflow between the two components. During the in-trap evolution, the initial sinusoidal shapes of the windings, seen in Fig. 2a, d, steepen on their right side, approaching a gradient catastrophe^{36}. The steepening is arrested by the formation of dark notches which first emerge at the steep edges [Fig. 2b, e] and then spread throughout the cloud [Fig. 2c, f], similarly to previous work in superfluid counterflow^{37}. Such dynamics are a hallmark phenomenon of dispersive hydrodynamics where, in the absence of viscosity, a gradient catastrophe is regularized by the formation of dispersive shock waves^{36}. The shock structures involve the formation of solitons, which persist for a long time. These observations lay the foundation for the following dynamics in more tightly wound clouds, where the size of each wound domain offers less space for the shock-like dynamics to unfold.

Next, we consider a case where the initial pattern size is too small for a full dispersive shock-wave train to evolve, but wide enough for the initial stages of such dynamics to still emerge. This case, showcased in Fig. 3, is reached after 20 ms winding (resulting in an initial pattern periodicity of 47 μm). Here, no magnetic gradient is applied during the evolution following the initial winding, and dynamics set in symmetrically on either side of each winding. The sinusoidal shape of the initial windings begins by deforming and becoming triangular [Fig. 3c]. This evolution transitions to the formation of plateaus with a high-density peak in the center of each plateau as depicted in Fig. 3d, taken after an evolution time of 50 ms. The corresponding density profile is shown in Fig. 3a and is compared against the prediction of the 3D Gross-Pitaevskii equation for this winding configuration, shown in Fig. 3b. These peaks, corroborated through our simulations, have the character of antidark solitons: bright solitonic peaks sitting on a finite background with a corresponding dark soliton in the other spin component^{16,38}. On both sides of each antidark peak, the steep edges of the plateaus lead to further evolution, as intuitively expected from the edge dynamics seen in Fig. 2 (or, mathematically, from the well-known Riemann problem of nonlinear differential equations^{36}). Notches appear and deepen, leading in this case to a tripling of the initial spatial period [Fig. 3e after 80 ms]. The width of these features is approximately on the length scale that we experimentally observe for DB solitons in our system, as demonstrated in Fig. 2, restricting the space against further soliton formation. Instead, the pattern becomes more irregular in amplitude at longer evolution times [Fig. 3f at 135 ms]. A noticeable observation is the occasional occurrence of peaks that are higher than the typical height of modulations in the cloud. We interpret these peaks as structures where adjacent solitons have constructively interfered—an effect that is familiar from the collision of bright solitons^{35}, in which the apparent number and height of peaks can vary through interactions.

For a winding time of 60 ms, a pattern periodicity of 16 μm is produced. This pattern periodicity is too fine to allow for the formation of pronounced plateaus that were observed in the previous case. Here, the pattern remains virtually unchanged for the first 70 ms, as dynamics gradually set in, shown in Fig. 4a, d. A prominent and reproducible feature of the evolution is the emergence of a fuzzy stage in which the contrast of the peak structure is strongly diminished (90% suppression of the spectral pattern) in large areas of the BEC, nearly disappearing into a uniform Thomas-Fermi profile [Fig. 4b, e], followed by a revival of the peaks (to 62% of the original spectral strength) [Fig. 4c, f]. This peculiar stage is also reproduced qualitatively in our full 3D simulations when slight residual counterflow induces currents in the two components (see Supplementary Note 3). We interpret this stage as a nearly simultaneous collision among the nonlinear waves throughout the array, enabled by the highly ordered initial conditions of the system and the good matching between the array periodicity and the natural length scale of the solitonic features.

Finally, as the culmination of the progression described above, we arrive at the case of a dense pattern obtained after a winding time of 100 ms (with a pattern spacing of about 9 μm) presented in Fig. 5. Here we observe dense collisional soliton complexes that maintain their qualitative character for long times. Figure 5a, d show the initial magnetization pattern. The pattern contrast appears reduced due to imaging effects including imaging resolution, expansion of the atomic cloud during time of flight, and residual thermal fraction. After approximately 25 ms of evolution, the pattern becomes irregular. However, comparing the density profile after an evolution of 45 ms [Fig. 5b, e] with that after 400 ms [Fig. 5c, f] reveals that the pattern remains qualitatively unchanged in its overall characteristics (e.g., typical feature widths, peak heights, cloud size etc.) for surprisingly long times.

A close inspection reveals that the disordered pattern maintains some structure: overlaying integrated cross sections with different evolution times (such as the characteristic example in Fig. 6 with a cross section obtained after 150 ms) with the original pattern demonstrates that in many instances the original pattern periodicity is still maintained [Fig. 6a-I] but the amplitude of the peaks has changed. A frequent observation is also that individual original peaks seem to be missing or strongly reduced and their neighboring peaks have grown in amplitude [Fig. 6a-II]. Further corroborating evidence of this perspective is provided through one-dimensional simulations of the system provided in Fig. 6b, where the time evolution of the location and phase of individual solitons can be traced through time.

Such dynamics are familiar from colliding bright solitons and are intrinsic to a dense soliton gas. We also note in passing that it is interesting to compare, on a qualitative level, the cross sections in Fig. 5e, f to those published in ref. ^{39}. There, a 128-soliton solution was numerically calculated for the focusing, one-dimensional, nonlinear Schrödinger equation—a challenging task involving arbitrary-precision techniques to achieve reliable accuracy. What sets our observations apart is that the experimentally observed collisional complexes are maintained for long periods of time as the solitons are held together by the harmonic trap confining the BEC. This provides an experimental platform for the future experiments focusing on the intricate dynamics of interacting soliton complexes, including aspects of “thermalization” of the initial regular pattern.

## Conclusion

Having reached the regime of dense collisional soliton complexes, our experiments provide a path to the study of dense soliton gases and soliton condensates within the framework of hydrodynamics in ultracold quantum gases. Recently, two pioneering experiments have reported the formation of a soliton gas in a long water tank^{22,23}. In the realm of BECs, a unique aspect is the possibility to dynamically change an axial harmonic confinement of the gas to modify the relevant “stress”. An analysis of the types of configurations experimentally realized herein involving the development of the inverse scattering transform for the two-component nonlinear Schrödinger type problem^{21}, while out of the scope of the present manuscript, appears to us to be fully within reach. Excitations evolving on top of a regular soliton background, in terms of hypersolitons^{40} or topological breathers^{41,42}, are another exciting research direction that will provide connections to condensed-matter systems.

## Methods

### Experimental methods

Our experimental technique for the generation of dense nonlinear excitations is based on a two-pulse Ramsey sequence in the presence of a small magnetic gradient. We begin with an elongated BEC of approximately 9 × 10^{5} ^{87}Rb atoms in the \(\left\vert F,{m}_{F}\right\rangle =\left\vert 1,-1\right\rangle\) hyperfine state held in an optical trap with harmonic trap frequencies of ** ω**/2

*π*= {2.5, 245, 258} Hz. In the presence of a 10 G magnetic bias field oriented in the vertical direction, a fast microwave

*π*/2 pulse is applied which coherently creates an equal superposition of atoms in the \(\left\vert 1,-1\right\rangle\) and the \(\left\vert 2,-2\right\rangle\) states. This is followed by a wait time, referred to as the winding time

*τ*, during which the 10 G bias field is supplemented by a slight gradient of 5.12(1) mG/cm along the long axis of the BEC. This leads to an accumulation of phase difference between the spin components which varies linearly across the length of the cloud. Then a second

*π*/2 pulse is applied which, depending on the phase between the precessing spins and the microwave pulse, transfers atoms into the \(\left\vert 2,-2\right\rangle\) state or back into the \(\left\vert 1,-1\right\rangle\) state. The intra- and intercomponent scattering lengths in the system are all very similar, but not exactly equal (see Supplementary Note 2), which contributes to weakly miscible longtime dynamics. In all cases, the magnetic field is small and away from any Feshbach resonances, allowing the scattering lengths to be modeled with their zero field values.

This process produces a sinusoidal magnetization pattern which, crucially for the subsequent dynamics, contains domains with *π* phase differences as shown in Fig. 1b. Notice that contrary to earlier works, e.g., ref. ^{43}, there is no continuous application of the Rabi drive as we are interested in the undriven dynamics of the system after an initial pattern has been established. Subsequent mean-field dynamics allow the sinusoidal phase pattern to relax into an array of alternating nonlinear waves. The spatial periodicity of the produced pattern can be experimentally adjusted over a wide range by varying the winding time between the two microwave pulses.

Finally, we directly image the atomic density profile through absorption imaging of the \(\left\vert 2,-2\right\rangle\) spin state. The two spin components add together to form a nearly unmodulated Thomas-Fermi distribution because the energy associated with deforming the overall density of the cloud far exceeds the energy scale for spin-mixing in this system. The second spin state forms a complementary pattern to the observed state (see Fig. 1a) and therefore is omitted in the reported figures.

### Numerical methods

To simulate the dynamics of the nonlinear phenomena of interest, we utilize the following system of coupled 3D Gross-Pitaevskii equations^{8,44}:

Here, *g*_{jk} = 4*π**ℏ*^{2}*a*_{jk}*N*_{j}/*m* with *j* = 1, 2 indexing the relevant spin states, *N*_{j} = 4.5 × 10^{5} is the particle number per component, *a*_{jk} are the 3D intra- (*j* = *k*) and intercomponent (*j* ≠ *k*) scattering lengths, and *m* is the mass of a ^{87}Rb atom. Additionally, the trapping potential \(V({{{{{{{\bf{r}}}}}}}})={\sum }_{\xi = x,y,z}m{\omega }_{\xi }^{2}{\xi }^{2}/2\) is characterized by the aforementioned experimental trapping frequencies whose aspect ratio leads to a cigar-shaped geometry and hence precludes transverse instabilities^{4,9}.

To initialize the dynamics, we first obtain the ground state of the self-interacting, yet intercomponent-decoupled, time-independent system of Eqs. (1). Then, we imprint the desired complementary configuration between the two components with wave number *k*_{0} and a phase jump of *π* between adjacent domains (see Supplementary Note 2) as shown in Fig. 1b, while switching on the intercomponent coupling. Subsequently, the resulting waveform is evolved in time.

## Data availability

The data associated with this work are available from the corresponding author upon reasonable request.

## Code availability

The code associated with this work are available from the corresponding author upon reasonable request.

## Change history

### 17 July 2024

A Correction to this paper has been published: https://doi.org/10.1038/s42005-024-01744-0

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## Acknowledgements

We acknowledge Maren Mossman for early contributions to the experimental investigation as well as Gino Biondini and Barbara Prinari for helpful theory discussions. We acknowledge fruitful discussions lending context to this work at the Dispersive Hydrodynamics Program (2022) hosted by the Issac Newton Institute for Mathematical Sciences. S.M., T.B., and P.E. acknowledge funding from NSF through Grant No. PHY-2207588. P.E. acknowledges support from the Ralph G. Yount Distinguished Professorship at WSU. S.I.M. acknowledges support from the NSF through a grant for ITAMP at Harvard University. This research was also supported in part by the National Science Foundation under Grant No. NSF PHY-1748958 (S.I.M. and P.S.). This work (P.S.) has been funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—SFB-925—project 170620586. This material is based upon work supported by the US National Science Foundation under Grants DMS-2204702 and PHY-2110030 (P.G.K.).

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S.M., T.B., and P.E. contributed to acquiring the experimental data and performing the analysis. G.K., S.I.M., and A.R. produced numerical simulation data and associated analysis. Funding for this work was acquired by P.G.K., P.S., and P.E. All authors contributed ideas and worked on preparing the manuscript.

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Mossman, S.M., Katsimiga, G.C., Mistakidis, S.I. *et al.* Observation of dense collisional soliton complexes in a two-component Bose-Einstein condensate.
*Commun Phys* **7**, 163 (2024). https://doi.org/10.1038/s42005-024-01659-w

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DOI: https://doi.org/10.1038/s42005-024-01659-w

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