# Controlled manipulation of solitons in a recirculating fiber loop using external potentials

François Copie, Pierre Suret, Stéphane Randoux

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François Copie, Pierre Suret, Stéphane Randoux. Controlled manipulation of solitons in a recirculating fiber loop using external potentials. Physical Review Research, 2025, 7 (4), pp.043232. 10.1103/qcts-whtp. hal-05391290

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Controlled manipulation of solitons in a recirculating fiber loop using external potentials

François Copie,*, Pierre Suret, and Stéphane Randoux

Univ. Lille, CNRS, UMR 8523-PhLAM-Physique des Lasers Atomes et Molécules, F-59000 Lille, France

<figure><img src="image_2.png" /></figure>

(Received 27 August 2025; accepted 22 October 2025; published 1 December 2025)

Optical solitons are self-sustained wave packets that propagate without distortion due to a balance between dispersion and nonlinearity. Their unique stability underpins key photonic applications while also playing a central role in nonlinear wave physics. However, real-time control over soliton dynamics in nondissipative systems remains a major challenge, limiting their practical applications in photonic systems. Here, we introduce a fiber-based platform for soliton manipulation, by creating programmable external potentials through synchronous arbitrary phase modulation in a recirculating optical fiber loop. We demonstrate precise soliton trapping, parametric excitation, and coupled multisoliton interactions, revealing particle-like behavior in excellent agreement with a Hamiltonian description in which solitons are treated as interacting particles. The strong analogy with matter-wave solitons in Bose-Einstein condensates highlights the broader implications of our approach, which provides a versatile experimental tool for the study of nonlinear wave dynamics and engineered soliton manipulation.

DOI: 10.1103/qcts-whtp

# I. INTRODUCTION

Solitons are localized coherent nonlinear structures involved in the complex dynamical behavior of many nonlinear wave systems [1]. Their ubiquitous nature is reflected in the fact that they occur in very diverse fields, including hydrodynamics, plasmas, quantum gas physics, or nonlinear optics. Their robustness underscores their significance in practical systems such as mode-locked lasers, supercontinuum sources [2], and frequency combs, and they have also been extensively explored for high-speed data transmission [3].

In their simplest—conservative—form, solitons are persisting wave packets that retain their shape and velocity over virtually infinite time owing to the balance between dispersive and nonlinear effects. From a mathematical perspective, these solitons represent exact solutions of the class of integrable partial differential equations that includes the one-dimensional (1D) nonlinear Schrödinger (NLS) equation. This equation that describes, at leading order, the propagation of light fields in optical fibers can be solved using the inverse scattering transform (IST) method [1]. Within this framework, solitons are identified as nonlinear eigenmodes that can be used in practical applications such as fiber telecommunications [4]. A key feature of their dynamics is the ability to interact or *collide* elastically, exhibiting particle-like behavior [1]. Even though the exact dynamics of the optical field during the collision process is complex, the only effect of

the interaction that is visible afterward is a positional shift of the colliding solitons [1,5] that has often been interpreted as resulting from an interaction force [6,7]. In an integrable setting, however, soliton interactions are highly constrained, and external control over their motion is impossible. This implies that the spatiotemporal dynamics of a set of interacting solitons, no matter how complex, is completely determined by the initial wave field and further manipulation during evolution is essentially prohibited. The lack of a general method to dynamically manipulate solitons' trajectories during propagation limits their potential for tunable optical applications as well as more fundamental investigations of complex many soliton interaction.

In contrast, matter-wave solitons in Bose-Einstein condensates (BECs) can be formed and subsequently manipulated using external optical potentials [8,9]. In this context, the degree of freedom introduced by the application of an external potential has proven particularly promising in manipulating the spatiotemporal dynamics of solitons, enabling, for instance, their repeated collision [10] or interaction with barriers that can lead to their controlled splitting [11,12]. Technical constraints in generating stable arbitrary optical traps and physical constraints such as spatial dimensionality or the collapse instability might, however, hinder part of this potential. Achieving an equivalent level of control with optical solitons would significantly benefit the study of nonlinear wave physics and could unlock functionalities for soliton-based applications.

In this work, we present a versatile fiber-optic system for soliton manipulation, based on synchronous phase modulation within a recirculating loop. By applying a configurable spatiotemporal external potential to propagating solitons, we demonstrate soliton trapping, parametric excitation, and controlled multisoliton interactions confirming the particle-like nature of solitons in our experiments. Our approach

<footnote>*Contact author: francois.copie@univ-lille.fr</footnote>
<figure><img src="image_3.png" /></figure>

<figure><img src="image_4.png" /><figcaption>FIG. 1. Experimental setup. (a) Principle of operation of the synchronously phase-modulated recirculating fiber loop. Example of experimentally recorded spatiotemporal dynamics of a single soliton in the case of (b) no external potential and (c) trapping within a truncated parabolic potential [used for describing the setup in panel (a)]. Typical peak power and duration of the solitons are ~30 mW and ~45 ps FWHM.</figcaption></figure>

establishes an original experimental platform for exploring and engineering nonlinear optical wave dynamics while further strengthening the existing parallel with matter-wave systems.

Our system relies on the recirculating optical fiber loop configuration that gives access to the space-time dynamics of the wave field [13–16] enabling single-shot observation of nonrepetitive dynamics. This has recently been used to observe the nonlinear spatiotemporal dynamics of modulation instability [17–19], soliton collision [20], and soliton gases [21,22]. This approach stands in contrast to most optical methods for accessing space-time dynamics, which typically rely on indirect techniques—such as parameter scanning to simulate different propagation distances [23–27], backscattered signal processing [28], or iterative reconstructions [29]—and are generally limited to deterministic dynamics.

## II. EXPERIMENTAL SETUP AND MODELS

The recirculating loop platform implemented in this work is entirely fiber based and its principle of operation is schematically depicted in Fig. 1(a). A single-frequency, continuous wave laser at 1555 nm is modulated in intensity through a 20 GHz bandwidth electro-optic modulator to generate the initial optical signal, which consists of a custom pattern of

pulses that realizes a specific set of experiments. The electric signal that creates the pulses pattern is obtained using the first channel of a 13.5 GHz bandwidth arbitrary waveform generator (AWG). An acousto-optic modulator is then used to ensure a high extinction of surrounding waves and to prevent the buildup of Brillouin scattering. This field is fed to the recirculating loop through a 90/10 coupler whose ports are arranged such that 90% of the power is recirculating. Consequently, 10% of the circulating field is extracted at each round trip and directed toward the detection system.

The recirculating loop itself measures approximately 5 km (round-trip time ≈24.552 µs) and is mostly comprised of a spool of commercial SMF-28 fiber [group velocity dispersion (GVD) coefficient $\beta_2 = -22 \text{ ps}^2 \text{ km}^{-1}$ and Kerr nonlinearity coefficient $\gamma = 1.23 \text{ W}^{-1} \text{ km}^{-1}$] connected at both ends to wavelength division multiplexers (WDMs). Through these, light from a 1455 nm pump laser is coupled in and out of the loop to realize backward Raman amplification. The laser’s power is gated to the duration of the experiment and finely tuned to compensate for the total dissipation experienced by the signal during a round trip. In this way, losses are effectively canceled for the signal that can then propagate over extremely long distances, typically several thousands of kilometers. A representative evolution of the optical power during a single experiment is given in the Supplemental Material [30]. We emphasize that in our experiments, even though the optical power is on average almost constant, the amplitude of the optical wave field may undergo significant evolution in the course of one round trip due to the distributed Raman amplification. Upon consecutive circulations, the wave field thus experiences periodic evolution of its power. For the range of parameters used in this work, the typical linear and nonlinear length scales associated with the pulses we consider are $L_L \approx L_{NL} \approx 30$ km, which is substantially longer than the length of a round trip (see parameters above and in the caption of Fig. 1). In this case, it is known that the nonlinear pulses behave as *effective* solitons also referred to as *guiding-center* solitons [31,32].

We create an effective external potential by using an electro-optic phase modulator (EOM$_{φ}$) embedded in the fiber loop that is driven synchronously with both the intensity pulse pattern and the round-trip time of the loop. This ensures precise and repeatable initial positioning of the pulses relative to the potential and prevents drifting of the latter relative to the circulating field during propagation. Prior to the experiment, a mapping of the targeted potential landscape is designed that consists of a sequence of waveforms (one per round trip). The second channel of the AWG sequentially plays each waveform to generate the electrical signal that drives EOM$_{φ}$ at each pass of the optical signal. In the general case, a different waveform can be played at each round trip, resulting in a dynamic potential landscape. The AWG can also be set to play the same waveform and thus create a static potential. Note that intraloop phase modulation has already been used to realize experiments in synthetic frequency dimensions [33–37] and generate optical gap solitons [38], typically relying on static harmonic modulations. For the latter, the potential consists of a fast modulation that creates an array of coupled waveguides shorter than the wave packet itself. In contrast, our experiments operate in a different parameter
regime in which the potential timescale is broader than the soliton duration. Application of an external potential is also a powerful means of tailoring the properties of Kerr cavity solitons in resonators [39,40]. In that context, however, phase modulation is constrained by the requirement for the wave-form to match specific cavity resonances, typically limiting it to periodic shapes. These constraints do not apply in our nonresonant loop system, allowing arbitrary and dynamically varying phase profiles. In a different context, it is noteworthy that static periodic potentials have also been proposed as supporting structures to stabilize the propagation of soliton trains for long-haul transmissions, although in that case the potential was generated via cross-phase modulation with an optical clock signal [41–43].

The system uses essentially polarization-maintaining (PM) fiber components except for the 5 km fiber spool and the WDMs. A polarization controller is thus placed after this non-PM section to realign the polarization of the signal after each circulation, which mitigates polarization-dependent effects.

The signal extracted from the loop at the 10% port of the coupler is detected with a 65 GHz bandwidth oscilloscope used in a sequential mode: After each round trip, the oscilloscope is triggered to record the signal in a segment. The full recording consists of up to a few thousand segments and thus contains the dynamics of the signal for more than 10 000 km of propagation. As an illustration, two typical examples of fully processed recordings are shown in Figs. 1(b) and 1(c) depicting, respectively, the spatiotemporal dynamics of a single soliton freely propagating and oscillating within a parabolic potential generated via synchronous phase modulation with a truncated quadratic profile. Additional information concerning the experimental setup and details regarding optical power calibration are given in the Supplemental Material [30].

The evolution of the wave field in the recirculating fiber loop over several round trips is well captured by the following 1D NLS equation, also referred to as 1D Gross-Pitaevskii equation in the Bose-Einstein condensate community:

$$i \frac{\partial A}{\partial Z} = \frac{\beta_2}{2} \frac{\partial^2 A}{\partial T^2} - \gamma |A|^2 A - i \frac{\alpha_{\text{eff}}}{2} A - \frac{\phi(T, Z)}{L} A, \quad (1)$$

where $A(T, Z)$ is the envelope of the electromagnetic field, $Z$ is the propagation distance, and $T$ is the time defined in the reference frame traveling at the group velocity at the carrier frequency. $\beta_2$ and $\gamma$ are the GVD and the Kerr nonlinearity coefficients, respectively. $\alpha_{\text{eff}}$ is a linear dissipation term that describes the potentially imperfect loss compensation that results in an effective exponential power variation over many round trips. $L$ is the total length of the recirculating loop. $\phi(T, Z)$ is the phase modulation applied at distance $Z = nL$, where $n$ is the round-trip number. As explicitly shown in the Supplemental Material [30], Eq. (1) derives from an iterative model accounting for the nonlinear light propagation in the loop and the periodic application of discrete recirculating conditions (local losses and phase modulation). It is valid provided that the round trip-to-round trip evolution of the field envelope $A_n$ and the phase modulation $\phi_n$ are small. Consequently, the subscript $n$ can be dropped, and the term $\phi(T, Z)/L$ plays the role of an external potential that results from the longitudinal distribution of the phase modulation $\phi(T, Z)$. It is noteworthy in this model that, since $\phi$ can

now be seen as a continuous function of $T$ and $Z$, the phase modulation applied in a discrete fashion (once per round trip) generates a distributed potential landscape. We checked numerically that in the range of parameters of all the experiments reported in this work the mean-field and iterative models yield identical results. The signal measured in our experiments is proportional to the optical power sampled at each round trip, which corresponds to $|A(T, Z = nL)|^2$.

It is possible to write Eq. (1) in a normalized form that, considering the anomalous dispersion regime (i.e., $\beta_2 < 0$), reads

$$i \frac{\partial \psi}{\partial t} = -\frac{1}{2} \frac{\partial^2 \psi}{\partial x^2} - |\psi|^2 \psi - i \epsilon \psi + V(x, t) \psi, \quad (2)$$

by introducing the following adimensional variables: $\psi = A/\sqrt{P_0}$; $x = T \sqrt{\gamma P_0 / |\beta_2|}$; $t = Z \gamma P_0$; $\epsilon = \alpha_{\text{eff}} / (2\gamma P_0)$; and $V = -\phi / (\gamma LP_0)$. Note that within this notation, the physical propagation distance $Z$ (resp. physical time $T$) is replaced by a normalized time $t$ (resp. normalized position $x$).

Considering that individual NLS solitons behave like particles, maintaining their shape at all positions in space and time, a simplified Hamiltonian description has been proposed in Refs. [44–46]. In this model, the interaction among solitons is described by an intersoliton potential that has been introduced in order to match the shift experienced by solitons upon pairwise collisions. Considering an initial wave field consisting of $N$ well-separated single solitons, the initial condition reads

$$\psi(x, t = 0) = \sum_{i=1}^{N} 2\eta_i \text{sech}[2\eta_i(x - q_{0i})] e^{i(v_{0i}x + \phi_{0i})}. \quad (3)$$

Each soliton is characterized by a set of four parameters: $\eta_i$, $q_{0i}$, $v_{0i}$, and $\phi_{0i}$, encoding its amplitude, initial position, velocity, and phase, respectively. In the particle analogy, the Hamiltonian $H$ for such system reads [44–46]

$$H = \sum_{i=1}^{N} \frac{p_i^2}{2\eta_i} + \eta_i V(q_i) - \sum_{1 \le i < j \le N} 2\eta_i \eta_j (\eta_i + \eta_j) \text{sech}^2 \left[ \frac{2\eta_i \eta_j}{\eta_i + \eta_j} (q_i - q_j) \right], \quad (4)$$

where $q_i(t)$ denotes the position of the peak of soliton $i$ and $p_i(t) = \eta_i \dot{q}_i$ its momentum. The first two terms represent the kinetic and potential energies, while the last term corresponds to the soliton-soliton interaction potential. This expression is valid for arbitrary potential shapes provided that $V(q)$ varies slowly compared to the soliton width, ensuring that each soliton experiences the potential through its local value $V(q_i)$. Note that the phase of the solitons does not appear in Eq. (4), since it does not affect the positional shift induced by the collisions. The relative phase between solitons, however, influences the exact spatiotemporal dynamics observed during collisions [20]. The simultaneous interaction of more than two solitons is not accounted for by this Hamiltonian, limiting its accuracy only to scenarios involving pairwise collisions occurring at distinct positions in space and time. Note also that this model fails to reproduce the correct shift when solitons interact with vanishing relative velocities [44]. In this work, we
<figure><img src="image_5.png" /><figcaption>FIG. 2. Dynamics of a single soliton in a trapping potential taking consecutively three different shapes. (a) Space-time dynamics of a single soliton subjected to the potential trap. Close-ups highlighting (d) sinusoidal oscillations, (c) total temporal reflections, and (b) succession of temporal reflections and parabolic trajectories. Dashed lines in panels (b)–(d) are the trajectories estimated by the particle model shifted by −80 ps for clarity. (e)–(g) Shapes of the phase modulations applied in each segment obtained from heterodyne measurement (see the Supplemental Material [30] for details regarding the measurement method). Solid lines are the averaged profiles and colored areas show the standard deviation of the phase measurement.</figcaption></figure>

have solved the Hamilton’s equation of motion $\dot{q}_i = \partial H/\partial p_i$ and $\dot{p}_i = -\partial H/\partial q_i$ to find the trajectories of solitons $q_i(t)$, within the particle analogy, subjected to external potentials $V(x, t)$ and compared them to our experimental observations.

### III. RESULTS

#### A. Single-soliton dynamics

The first set of experiments that we present focuses on the spatiotemporal dynamics of a single soliton under a trapping potential that sequentially takes three distinct shapes: a quadratic well, a square box, and an asymmetric well. In the spatiotemporal diagram of Fig. 2(a), the horizontal axis is the time in the reference frame that moves at the group velocity of the signal in the absence of external potential. The external potential is set to be stationary in this reference frame. The zero of this axis is the central position of the external potential and the soliton is initially shifted by −300 ps from the center [see Figs. 2(a) and 2(e)]. In the first 3000 km of propagation, a truncated parabolic phase modulation spanning 1 ns is synchronously applied, which creates an effective quadratic

potential. As a result, the soliton’s velocity is modulated and its temporal position oscillates sinusoidally [Fig. 2(d)] similar to what is observed in BEC experiments [10,12]. It should be noted that this oscillation of the velocity physically translates into an oscillation of the central frequency of the Fourier spectrum of the soliton. Little distortion of the soliton’s shape is observed as a consequence of the interaction with weak radiative waves that are also trapped, but this was not found to alter its trajectory in a significant way. Considering a quadratic phase modulation of the form $\phi(T)/L = -C/2 \times T^2$, the trap wave number is determined, by analogy with the quantum harmonic oscillator, to be $K_0^2 = \beta_2 C$. When applying a truncated quadratic phase modulation parametrized by its depth in radian $\phi_m$ and duration (width) W, as schematically shown in Fig. 3(d), the spatial period of oscillation $Z_0$ can be expressed as

$$Z_0 = \frac{2\pi}{K_0} = \sqrt{\frac{\pi^2 W^2 L}{2|\beta_2|\phi_m}} \quad (5)$$

The experimentally observed period $Z_0 = 835$ km is consistent with the phase modulation depth obtained via heterodyne measurement $\phi_m \approx 1.5 \pm 0.1$ rad [Fig. 2(e)]. After 3000 km of propagation, the phase modulation is abruptly switched (from one round trip to the next) to produce a square-well potential. As a result, the spatiotemporal dynamics of the soliton is modified: It experiences total reflections at the edges while crossing the flat bottom of the potential at a constant velocity without changing shape [Fig. 2(c)]. After each reflection, the soliton recovers precisely its shape. The observed dynamics is characteristic of the temporal reflection phenomenon that is known to be the analog of reflection of an optical beam at the interface between two dielectric media [47,48]. At a propagation distance of 6000 km, the phase modulation is switched to generate an asymmetric, semiparabolic well in which the soliton experiences temporal reflection at the sharp right boundary and follows a curved trajectory in the region of quadratic potential [Fig. 2(b)].

The results illustrated in Fig. 2 demonstrate the capacity of our experimental setup to manipulate individual 1D solitons through the application of an arbitrary shaped and reconfigurable external potential. The trajectory of the soliton in response to each potential shape is perfectly reproduced within the particle analogy by integrating the simplified Hamiltonian model [Eq. (4)] for a single particle subjected to corresponding potentials. This is illustrated in Figs. 2(b)–2(d), where the dashed lines (shifted for clarity) represent the trajectories numerically computed and rescaled to physical units.

The recirculating loop configuration allows the phase modulation signal applied to the circulating field to be updated at each round trip. This enables the creation of dynamic potential landscapes (external potentials varying in both T and Z), providing an additional tool for controlling optical waves. As a demonstration, we consider a parabolic trap with Z-dependent strength, with results presented in Fig. 3. In Figs. 3(a) and 3(b), the parabolic trap strength is longitudinally modulated by varying its temporal width W while keeping its depth $\phi_m$ constant [schematized in Fig. 3(d)]. In both cases, the evolution of W follows a continuous function sampled once per round trip. However, the variation of W
<figure><img src="image_6.png" /><figcaption>FIG. 3. Manipulation of solitons using spatiotemporally varying harmonic potentials. (a) Soliton in a slowly modulated trap and (b) driven at the parametric resonance. The red dashed lines denote the boundaries of the truncated parabolic traps. Outside of the region enclosed by these lines, the external potential is flat. The blue lines in panel (b) indicate the exponential growth of the amplitude of the oscillations predicted by the particle model. (c) Evolution of the instantaneous period of oscillation of the soliton in panel (a) compared to the prediction of Eq. (5) considering an adiabatic evolution. (d) Sketch of the truncated parabolic phase modulation.</figcaption></figure>

from round trip to round trip is small, ensuring that the wave
field experiences a smoothly evolving potential landscape
without discernible effects of the discrete nature of the phase
modulation. When the trap is slowly modulated [Fig. 3(a)],
the soliton's oscillation period and amplitude increase as W
increases, and vice versa. Under adiabatic evolution, Eq. (5)
indeed consistently predicts that the soliton oscillation period
is proportional to W. The experimentally measured evolution
of the period aligns well with this prediction for adiabatic
evolution [Fig. 3(c)], showcasing precise control over the
spatiotemporal dynamics of trapped solitons. This concept
 extends to scenarios involving multiple interacting solitons.
Figure 3(b) illustrates another example of soliton control,
achieved by periodically modulating the spatial frequency of
the harmonic potential at parametric resonance. It has been
shown theoretically that the motion of a soliton under such
conditions follows the equation of motion of the parametri-
cally driven pendulum [49,50]: A modulation period close
to half the oscillator's natural period induces exponential
growth in oscillation amplitude. This is realized in our case
using a truncated parabolic trap weakly modulated according

to $W(Z) = W_0(1 - 0.05 \sin(4\pi Z/Z_0))$ with $W_0 = 1$ ns and $Z_0 = 800$ km. The oscillations of the soliton initially near the trap's center ($-75$ ps) exhibit rapid amplitude growth due to parametric excitation. Our experimental results strongly agree with the particle model, as shown by the amplitude of the oscillations closely following the predicted exponential amplification [blue lines in Fig. 3(b)]. It should be noted that we focus here on the positional dynamics of the pulse, but the presence of the trapping potential eventually introduces internal dynamics in the form of width oscillations, which may lead to more complex behavior [51].

The superposition of the predicted trajectories over the
experimental observations reported in Figs. 3(a) and 3(b) is
provided in the Supplemental Material [30], showing an ex-
cellent agreement.

B. Interacting solitons

The experiments presented thus far all involve a single
soliton subjected to an effective external potential whose pri-
mary effect is to modulate the velocity of the soliton. We now
consider the scenario in which several solitons are confined in
a stationary quadratic trapping potential that makes it possible
to produce intricate multicollision dynamics. Such scenario
is usually precluded in nonlinear fiber optics experiments as
solitons with different velocities can only interact once in the
absence of potential. We focus on a regime in which the soli-
tons are well separated and distinguishable before and after
interaction (i.e., sufficiently broad trapping potential and large
initial separation between solitons). Under these conditions,
the solitons dynamics can be modeled to a good approxima-
tion using the Hamiltonian particle model [Eq. (4)].

We first present experiments realized in the two-soliton
configuration for which the particle model is integrable, which
has been shown to imply regular (nonchaotic) dynamics of the
system [45]. To better emphasize the dynamics of the soliton
pair, we designed initial conditions consisting of a soliton
placed at the center of the harmonic trap, while a second
one is generated shifted from the center [situation depicted in
Fig. 4(a)]. In the absence of interaction, the centered soliton
would remain pinned to the center of the trap. In contrast,
the coherent interaction with the second soliton results in
the soliton pair exhibiting coupled dynamics. We varied the
strength of the coupling by changing the natural period of
the trap (phase modulation depth) and the characteristics of
the solitons (amplitude and initial temporal separation) as
illustrated schematically in Fig. 4(a).

We show in Figs. 4(b)–4(d) three experiments sorted from
weakest to strongest coupling. Over the range of propagation
distances plotted, the two solitons collide in each realiza-
tion at least 20 times without significant alteration of their
initial amplitude, duration, and shape. The dashed colored
lines serve as visual guides for tracking the dynamics of
each soliton. For the weakest coupling [Fig. 4(b)], the central
soliton is gradually set into motion and the amplitude of the
oscillations of its trajectory increases at the expense of that
of the initially shifted soliton. Further increasing the coupling
strength [Fig. 4(c)], we observe after ~9000 km a point at
which the amplitude of oscillation of the shifted soliton is can-
celed, whereas the initially centered one oscillates strongly.
<figure><img src="image_7.png" /><figcaption>FIG. 4. Coherent dynamics of soliton pairs. (a) Schematic view of the initial condition of the experiments: One soliton is initially centered in the quadratic trap, while the other is detuned. (b)–(d) Spatiotemporal dynamics recorded for three increasing coupling strengths: $\Delta Z/Z_0 = -1.6\%$, $-6.9\%$, and $-11.2\%$, respectively. Dashed blue and orange lines denote the amplitude of oscillation of each soliton. Panels (b2) and (d2) are same as panels (b) and (d) rotated by $90^\circ$. Simulated particle trajectories obtained from Eq. (4) are superimposed as dashed and dotted lines.</figcaption></figure>

Figure 4(d) shows the most strongly coupled dynamics where the solitons apparently exchange their initial position twice, reaching back almost perfectly the initial configuration after 10 000 km. Although the trajectory of each soliton is less distinguishable because of their smaller separation, their coupled motion is still remarkably regular.

A convenient way to quantify the coupling strength that has been reported in Ref. [10] is to measure the relative shift of the period of oscillation of the solitons with respect to the natural period $Z_0$ of the trap (i.e., the period of oscillation experienced by a single soliton within the same trap) $\Delta Z_0/Z_0$. The collision process shifts solitons forward in their propagation direction such that the coupling between solitons effectively decreases their period of oscillation [52]. Using reference single-soliton experiments performed simultaneously to those reported in Figs. 4(b)–4(d) but not shown for clarity, we were able to estimate these shifts as $\Delta Z_0/Z_0 = -1.6\%$, $-6.9\%$, and $-11.2\%$, respectively. The latter is roughly 4 times larger than the matter-wave soliton experiments reported in Ref. [10].

Our experiments can be compared to the predictions of the Hamiltonian model that treats solitons as interacting classical particles. By integrating Hamiltonian dynamical equations with the Hamiltonian given by Eq. (4), we obtain the positional dynamics $q_1(t)$ and $q_2(t)$ of the two equivalent particles associated with the solitons of amplitude $\eta_1$ and $\eta_2$, respectively. The computed trajectories, rescaled to physical units, corresponding to the cases illustrated in Figs. 4(b) and 4(d) are superimposed on the experiments in Figs. 4(b2) and 4(d2) and reproduce quantitatively very well the coupled dynamics of the soliton pairs. For better visualization, the spatiotemporal representations are rotated by $90^\circ$ and the

colormap has been changed. It should be noted that the initial temporal detuning between the solitons in the particle model was lowered slightly relative to the experiments to account for a transient initial regime during which the external potential stabilizes.

Finally, we realized the same type of experiment with three solitons interacting in a harmonic trap. In practice, the number of solitons and their initial temporal position are easily set by tailoring the electric signal sent to the intensity EOM. The three-particle system is not integrable and supports both chaotic and regular regimes [45,46]. We demonstrate here [see Fig. 5(a)] that we are able to specifically generate soliton triplets with a regular dynamics by carefully shaping the initial state, i.e., the position and relative velocity of each soliton. To do so, three equally spaced solitons are first generated and propagate freely for 500 km before a specifically shaped phase modulation is applied for a few consecutive round trips. The profile of the phase modulation [dashed green line in Fig. 5(b)] is specifically designed such that the central soliton acquires a positive velocity, while the two surrounding ones get a velocity of opposite sign and half amplitude. This, in turn, produces a triplet of solitons characterized by regular pairwise collisions when the trapping harmonic potential is turned on.

Figure 5(c) shows an example of chaotic motion produced following a similar procedure. The central soliton is here slightly detuned and no relative velocity is imprinted before the trap is turned on. As a result, the coupled dynamics becomes very intricate, with in particular the three solitons overlapping at their first encounter. In this configuration especially, the particle model is not able to describe properly
<figure><img src="image_8.png" /><figcaption>FIG. 5. Regular and chaotic dynamics of soliton triplets. Spatiotemporal dynamics of (a) a regular and (c) a chaotic soliton triplet. The green horizontal line in panel (a) shows the range of distance over which velocities of the solitons are initiated. Dashed vertical lines indicate the boundaries of the truncated parabolic trap. (b), (d) Respective intensity profiles recorded at $z = 500$ km in blue. Solid red lines and shaded areas illustrate the trapping potential applied after 500 km, while the dashed green line in panel (b) indicates the phase modulation applied for $500 \text{ km} \le z \le 585$ km to initiate velocities of the solitons.</figcaption></figure>

the observed dynamics and exhibits an increased sensitivity to the initial setting (see the Supplemental Material [30] for a related discussion). These experiments provide a clear demonstration of the capabilities offered by this experimental system in designing arbitrary multisoliton optical wave packets. In particular, the results presented in Fig. 5 show how our system can be a powerful experimental platform for exploration of chaotic dynamics in wave mechanics.

## IV. CONCLUSION

In summary, we have experimentally realized the spatiotemporal manipulation of NLS solitons within a recirculating optical fiber loop. The implementation of intraloop synchronous phase modulation enables the generation of an effective external potential acting on the trajectories of solitons. We take advantage of the capacity to arbitrary shape the potential landscape in both space and time to demonstrate

various control scenarios, including periodic oscillation, temporal reflection, and parametric excitation. In all cases, the solitons showcase particle-like behaviors with trajectories akin to that of classical particles. In the situation where two solitons collide repeatedly within a trapping potential, the collective motion of the system is also accurately described by a simplified Hamiltonian model treating solitons as interacting particles. Precise design of the initial wave field and external potential reveals the predicted onset of chaotic motion when more than two solitons are interacting.

The photonic platform presented here benefits from a high degree of configurability in addition to truly single-shot operation, which makes it readily compatible for more advanced manipulation of optical wave fields. In particular, it constitutes a promising approach toward shaping of the IST “nonlinear” spectrum of complex multisoliton waveforms with possible implications for nonlinear “eigenvalue” telecommunications [4,53].

The weak breaking of integrability provided by an external potential offers a possible route toward the observation of thermalization of soliton gases, i.e., large random ensembles of solitons [54]. In this sense, our system proves highly valuable for testing and refining the predictions of the recent theoretical framework of generalized hydrodynamics (hydrodynamics of many-body integrable systems), which, to date, has been almost exclusively focused on quantum systems ([55] and references therein, [56]). Localization and branched flow phenomena are also expected to occur within periodic [57] or random [58,59] potential landscapes, respectively. The study of the scattering of nonlinear waves from potential barriers is one fundamental topic that could be addressed and that can greatly benefit from the strong analogy between nonlinear optics and matter waves. Although there is a wealth of literature on this topic, practical implementations in matter-wave contexts are limited [12,60–63]. Many of these may be more easily realized using photonic platforms.

## ACKNOWLEDGMENTS

The authors are grateful to H. Damart, G. Dekyndt, and the Centre d'Etudes et de Recherche Lasers et Application (CERLA) for technical support. This work was partially supported by the Agence Nationale de la Recherche through the StormWave (ANR-21-CE30-0009) and SO-GOOD (ANR-21-CE30-0061) projects, the LABEX CEMPI project (ANR-11-LABX-0007), the Ministry of Higher Education and Research, Hauts de France council, and European Regional Development Fund (ERDF) through the Contrat de Projets Etat-Région (CPER Photonics for Society P4S).

## DATA AVAILABILITY

The data that support the findings of this article are openly available [64].

<footnote>[1] J. Yang, *Nonlinear Waves in Integrable and Nonintegrable Systems*, Mathematical Modeling and Computation No. 16 (SIAM, Philadelphia, PA, 2010).</footnote>

<footnote>[2] *Supercontinuum Generation in Optical Fibers*, edited by J. M. Dudley and J. R. Taylor (Cambridge University Press, Cambridge, 2010).</footnote>
[3] A. Hasegawa and Y. Kodama, *Solitons in Optical Communications*, Oxford Series in Optical & Imaging Sciences No. 7 (Oxford University Press, Oxford, 1995).

[4] S. K. Turitsyn, J. E. Prilepsky, S. T. Le, S. Wahls, L. L. Frumin, M. Kamalian, and S. A. Derevyanko, *Nonlinear Fourier transform for optical data processing and transmission: Advances and perspectives*, *Optica* **4**, 307 (2017).

[5] V. E. Zakharov and A. B. Shabat, *Interaction between solitons in a stable medium*, Zh. Eksp. Tear. Fiz. **64**, 1627 (1973) [Sov. Phys. JETP **37**, 823 (1973)].

[6] J. P. Gordon, *Interaction forces among solitons in optical fibers*, Opt. Lett. **8**, 596 (1983).

[7] F. M. Mitschke and L. F. Mollenauer, *Experimental observation of interaction forces between solitons in optical fibers*, Opt. Lett. **12**, 355 (1987).

[8] K. E. Strecker, G. B. Partridge, A. G. Truscott, and R. G. Hulet, *Formation and propagation of matter-wave soliton trains*, Nature (London) **417**, 150 (2002).

[9] L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L. Carr, Y. Castin, and C. Salomon, *Formation of a matter-wave bright soliton*, Science **296**, 1290 (2002).

[10] J. H. V. Nguyen, P. Dyke, D. Luo, B. A. Malomed, and R. G. Hulet, *Collisions of matter-wave solitons*, Nat. Phys. **10**, 918 (2014).

[11] A. L. Marchant, T. P. Billam, T. P. Wiles, M. M. H. Yu, S. A. Gardiner, and S. L. Cornish, *Controlled formation and reflection of a bright solitary matter-wave*, Nat. Commun. **4**, 1865 (2013).

[12] O. J. Wales, A. Rakonjac, T. P. Billam, J. L. Helm, S. A. Gardiner, and S. L. Cornish, *Splitting and recombination of bright-solitary-matter waves*, Commun. Phys. **3**, 51 (2020).

[13] E. Desurvire, M. Digonnet, and H. J. Shaw, *Raman amplification of recirculating pulses in a reentrant fiber loop*, Opt. Lett. **10**, 83 (1985).

[14] L. F. Mollenauer, M. J. Neubelt, S. G. Evangelides, J. P. Gordon, J. R. Simpson, and L. G. Cohen, *Experimental study of soliton transmission over more than 10,000 km in dispersion-shifted fiber*, Opt. Lett. **15**, 1203 (1990).

[15] M. Nakazawa, E. Yamada, H. Kubota, and K. Suzuki, *10 Gbit/s soliton data transmission over one million kilometres*, Electron. Lett. **27**, 1270 (1991).

[16] J.-W. Goossens, H. Hafermann, and Y. Jaouën, *Experimental realization of Fermi-Pasta-Ulam-Tsingou recurrence in a long-haul optical fiber transmission system*, Sci. Rep. **9**, 18467 (2019).

[17] A. E. Kraych, P. Suret, G. El, and S. Randoux, *Nonlinear evolution of the locally induced modulational instability in fiber optics*, Phys. Rev. Lett. **122**, 054101 (2019).

[18] A. E. Kraych, D. Agafontsev, S. Randoux, and P. Suret, *Statistical properties of the nonlinear stage of modulation instability in fiber optics*, Phys. Rev. Lett. **123**, 093902 (2019).

[19] F. Copie, P. Suret, and S. Randoux, *Spatiotemporal observation of higher-order modulation instability in a recirculating fiber loop*, Opt. Lett. **47**, 3560 (2022).

[20] F. Copie, P. Suret, and S. Randoux, *Space-time observation of the dynamics of soliton collisions in a recirculating optical fiber loop*, Opt. Commun. **545**, 129647 (2023).

[21] P. Suret, M. Dufour, G. Roberti, G. El, F. Copie, and S. Randoux, *Soliton refraction by an optical soliton gas*, Phys. Rev. Res. **5**, L042002 (2023).

[22] L. Fache, F. Copie, P. Suret, and S. Randoux, *Perturbed non-linear evolution of optical soliton gases: Growth and decay in integrable turbulence*, Phys. Rev. Lett. **135**, 157201 (2025).

[23] D. Pierangeli, M. Flammini, L. Zhang, G. Marcucci, A. J. Agranat, P. G. Grinevich, P. M. Santini, C. Conti, and E. DelRe, *Observation of Fermi-Pasta-Ulam-Tsingou recurrence and its exact dynamics*, Phys. Rev. X **8**, 041017 (2018).

[24] G. Marcucci, D. Pierangeli, A. J. Agranat, R.-K. Lee, E. DelRe, and C. Conti, *Topological control of extreme waves*, Nat. Commun. **10**, 5090 (2019).

[25] F. Xin, F. Di Mei, L. Falsi, D. Pierangeli, C. Conti, A. J. Agranat, and E. DelRe, *Evidence of chaotic dynamics in three-soliton collisions*, Phys. Rev. Lett. **127**, 133901 (2021).

[26] L. Dieli, D. Pierangeli, E. DelRe, and C. Conti, *Observation of two-dimensional dam break flow and a gaseous phase of solitons in a photon fluid*, Phys. Rev. Lett. **133**, 183801 (2024).

[27] G. Van Simaeys, P. Emplit, and M. Haelterman, *Experimental demonstration of the Fermi-Pasta-Ulam recurrence in a modulationally unstable optical wave*, Phys. Rev. Lett. **87**, 033902 (2001).

[28] A. Mussot, C. Naveau, M. Conforti, A. Kudlinski, F. Copie, P. Szriftgiser, and S. Trillo, *Fibre multi-wave mixing combs reveal the broken symmetry of Fermi-Pasta-Ulam recurrence*, Nat. Photon. **12**, 303 (2018).

[29] A. Sheveleva, U. Andral, B. Kibler, P. Colman, J. M. Dudley, and C. Finot, *Idealized four-wave mixing dynamics in a nonlinear Schrödinger equation fiber system*, Optica **9**, 656 (2022).

[30] See Supplemental Material at http://link.aps.org/supplemental/10.1103/qcts-whtp for details about the experimental setup, the mathematical models and additional comparisons between experiments and the Hamiltonian model.

[31] A. Hasegawa and Y. Kodama, *Guiding-center soliton in optical fibers*, Opt. Lett. **15**, 1443 (1990).

[32] A. Hasegawa and Y. Kodama, *Guiding-center soliton*, Phys. Rev. Lett. **66**, 161 (1991).

[33] A. Dutt, M. Minkov, Q. Lin, L. Yuan, D. A. B. Miller, and S. Fan, *Experimental band structure spectroscopy along a synthetic dimension*, Nat. Commun. **10**, 3122 (2019).

[34] H. Chen, N. Yang, C. Qin, W. Li, B. Wang, T. Han, C. Zhang, W. Liu, K. Wang, H. Long, X. Zhang, and P. Lu, *Real-time observation of frequency Bloch oscillations with fibre loop modulation*, Light Sci. Appl. **10**, 48 (2021).

[35] N. Englebert, N. Goldman, M. Erkintalo, N. Mostaan, S.-P. Gorza, F. Leo, and J. Fatome, *Bloch oscillations of coherently driven dissipative solitons in a synthetic dimension*, Nat. Phys. **19**, 1014 (2023).

[36] D. Cheng, E. Lustig, K. Wang, and S. Fan, *Multi-dimensional band structure spectroscopy in the synthetic frequency dimension*, Light Sci. Appl. **12**, 158 (2023).

[37] A. Senanian, L. G. Wright, P. F. Wade, H. K. Doyle, and P. L. McMahon, *Programmable large-scale simulation of bosonic transport in optical synthetic frequency lattices*, Nat. Phys. **19**, 1333 (2023).

[38] C. Bersch, G. Onishchukov, and U. Peschel, *Optical gap solitons and truncated nonlinear Bloch waves in temporal lattices*, Phys. Rev. Lett. **109**, 093903 (2012).

[39] N. Englebert, C. Simon, C. M. Arabí, F. Leo, and S.-P. Gorza, *Manipulation and control of temporal cavity solitons with trapping potential*, arXiv:2406.12848.
[40] Y. Sun, P. Parra-Rivas, M. Ferraro, F. Mangini, M. Zitelli, R. Jauberteau, F. Rinaldo Talenti, and S. Wabnitz, Dissipative Kerr solitons, breathers, and chimera states in coherently driven passive cavities with parabolic potential, *Opt. Lett.* **47**, 6353 (2022).

[41] T. Widdowson, D. Malyon, A. Ellis, K. Smith, and K. Blow, Soliton shepherding: All-optical active solitoncontrol over global distances, *Electron. Lett.* **30**, 990 (1994).

[42] S. Bigo, P. Brindel, O. Leclerc, E. Brun-Maunand, and E. Desurvire, Error-free 20-Gbit/s soliton transmission over 7150 km through all-optical synchronous phase modulation, in *Proceedings of the Optical Fiber Communication Conference* 7 (Optical Society of America, Washington, DC, 1997), pp. 143-144.

[43] A. Shipulin, G. Onishchukov, and B. A. Malomed, Suppression of soliton Jitter by a copropagating support structure, *J. Opt. Soc. Am. B* **14**, 3393 (1997).

[44] R. Scharf and A. R. Bishop, Soliton chaos in the nonlinear Schrödinger equation with spatially periodic perturbations, *Phys. Rev. A* **46**, R2973 (1992).

[45] A. D. Martin, C. S. Adams, and S. A. Gardiner, Bright matter-wave soliton collisions in a harmonic trap: Regular and chaotic dynamics, *Phys. Rev. Lett.* **98**, 020402 (2007).

[46] A. D. Martin, C. S. Adams, and S. A. Gardiner, Bright solitary-matter-wave collisions in a harmonic trap: Regimes of solitonlike behavior, *Phys. Rev. A* **77**, 013620 (2008).

[47] Y. Xiao, D. N. Maywar, and G. P. Agrawal, Reflection and transmission of electromagnetic waves at a temporal boundary, *Opt. Lett.* **39**, 574 (2014).

[48] B. W. Plansinis, W. R. Donaldson, and G. P. Agrawal, What is the temporal analog of reflection and refraction of optical beams? *Phys. Rev. Lett.* **115**, 183901 (2015).

[49] Y. V. Kartashov, L. Torner, and V. A. Vysloukh, Parametric amplification of soliton steering in optical lattices, *Opt. Lett.* **29**, 1102 (2004).

[50] C. H. Tenorio, E. V. Vargas, V. N. Serkin, M. A. Granados, T. L. Belyaeva, R. P. Moreno, and L. M. Lara, Dynamics of solitons in the model of nonlinear Schrödinger equation with an external harmonic potential: I. Bright solitons, *Quantum Electron.* **35**, 778 (2005).

[51] B. Baizakov, G. Filatrella, B. Malomed, and M. Salerno, Double parametric resonance for matter-wave solitons

in a time-modulated trap, *Phys. Rev. E* **71**, 036619 (2005).

[52] A. D. Martin, Collision-induced frequency shifts in bright matter-wave solitons and soliton molecules, *Phys. Rev. A* **93**, 023631 (2016).

[53] A. Mucci, P. Suret, F. Copie, S. Randoux, R. Mullyadzhanov, and A. Gelash, Manipulation of strongly interacting solitons in optical fiber experiments, *Phys. Rev. Lett.* **134**, 193804 (2025).

[54] P. Suret, S. Randoux, A. Gelash, D. Agafontsev, B. Doyon, and G. El, Soliton gas: Theory, numerics, and experiments, *Phys. Rev. E* **109**, 061001 (2024).

[55] B. Doyon, S. Gopalakrishnan, F. Møller, J. Schmiedmayer, and R. Vasseur, Generalized hydrodynamics: A perspective, *Phys. Rev. X* **15**, 010501 (2025).

[56] A. Bastianello, A. Tikan, F. Copie, S. Randoux, and P. Suret, Observation of a generalized Gibbs ensemble in photonics, *arXiv:2503.16526*.

[57] R. Scharf and A. R. Bishop, Length-scale competition for the one-dimensional nonlinear Schrödinger equation with spatially periodic potentials, *Phys. Rev. E* **47**, 1375 (1993).

[58] A. Patsyk, U. Sivan, M. Segev, and M. A. Bandres, Observation of branched flow of light, *Nature (London)* **583**, 60 (2020).

[59] A. Patsyk, Y. Sharabi, U. Sivan, and M. Segev, Incoherent branched flow of light, *Phys. Rev. X* **12**, 021007 (2022).

[60] T. Ernst and J. Brand, Resonant trapping in the transport of a matter-wave soliton through a quantum well, *Phys. Rev. A* **81**, 033614 (2010).

[61] J. L. Helm, S. J. Rooney, C. Weiss, and S. A. Gardiner, Splitting bright matter-wave solitons on narrow potential barriers: Quantum to classical transition and applications to interferometry, *Phys. Rev. A* **89**, 033610 (2014).

[62] A. L. Marchant, T. P. Billam, M. M. H. Yu, A. Rakonjac, J. L. Helm, J. Polo, C. Weiss, S. A. Gardiner, and S. L. Cornish, Quantum reflection of bright solitary matter waves from a narrow attractive potential, *Phys. Rev. A* **93**, 021604(R) (2016).

[63] F. Lorenzi and L. Salasnich, Atomic soliton transmission and induced collapse in scattering from a narrow barrier, *Sci. Rep.* **14**, 4665 (2024).

[64] F. Copie, Data from the article "Controlled manipulation of solitons in a recirculating fiber loop using external potentials", Recherche Data Gouv (2025), https://doi.org/10.57745/JPPDGA.
Supplemental material for:
“Controlled manipulation of solitons in a recirculating fiber loop using external potentials”

François Copie,\* Pierre Suret, and Stéphane Randoux

Univ. Lille, CNRS, UMR 8523 - PhLAM - Physique des Lasers Atomes et Molécules, F-59000 Lille, France

In this Supplemental Material we provide some details about the experimental setup, the experimental methodology and the mathematical models describing our system. It also provides additional comparison between experiments and the Hamiltonian model describing solitons as classical particles. We also discuss annex question about the failure of the particle model in the case where the interaction among 3 solitons occur at the same space-time position. All equations, figures, reference numbers within this document are prepended with “S” to distinguish them from corresponding numbers in the main manuscript.

CONTENTS

I. Details of the experimental system 1

A. Description of the experimental setup 1

B. Voltage-to-Optical Power conversion 3

C. Measurement of the phase modulation profiles 3

II. Derivation of the 1D-Gross-Pitaevskii mean-field equation from the iterative map 5

Iterative model in physical units 5

Lumped model 5

Mean-field model 6

III. Comparison between experiments and Hamiltonian model in dynamic potential 6

IV. Sensitivity of the Hamiltonian model with three trapped particles 7

References 8

I. DETAILS OF THE EXPERIMENTAL SYSTEM

A. Description of the experimental setup

The recirculating loop platform implemented in this work is entirely fiber based and is schematically depicted in Fig. S1(a). The operating principle consists in three stages that are described here: (i) *Generation of the initial optical wavefield*; (ii) *Propagation in the recirculating loop and application of the external potential*; (iii) *Single-shot detection*.

(i) *Generation of the initial optical wavefield*

A single frequency, continuous wave (cw) laser at 1555 nm is modulated in intensity using a 20 GHz bandwidth electro-optic modulator (EOM, Exail MXER-LN-20) to generate a 200 ns long optical signal which consists of a custom pattern of pulses that realizes a specific set of experiments (illustrated in the top right corner of Fig. S1(a)). The electric signal that creates the pulses pattern is obtained using a dual-channel 13.5 GHz bandwidth arbitrary waveform generator (Tektronix AWG70002B). A single run contains a few tens of completely programmable independent experiments involving short solitonic pulses along with a sequence of pulses solely used for power calibration and synchronization purposes. This pattern is continuously generated and amplified using an Erbium doped fiber amplifier (EDFA) to

<footnote>\* francois.copie@univ-lille.fr</footnote>
reach Watt-level peak power. An acousto-optic modulator (AOM) is used to pick a pulses pattern that will constitute the initial wavefield for the experiment and ensures a high extinction of surrounding waves to prevent the buildup of Brillouin scattering. The AOM is driven by the signal from an arbitrary waveform generator (AFG₁). The field is then fed to the recirculating loop through a 90/10 coupler whose ports are arranged such that 90% of the power is recirculating. Consequently, 10% of the circulating field is extracted at each roundtrip and directed towards the detection system.

(ii) Propagation in the recirculating loop and application of the external potential

The recirculating loop itself measures approximately 5 km (roundtrip time ≈ 24.552 µs) and is mostly comprised of a spool of commercial SMF-28 fiber (GVD coefficient $\beta_2$ = -22 ps²/km, Kerr nonlinearity coefficient $\gamma$ = 1.23 W⁻¹ km⁻¹) connected at both ends to wavelength division multiplexers (WDMs). Through these, light from a 1455 nm pump laser (Keopsys CRFL) is coupled in and out of the loop to realize backward Raman amplification. The laser's power is gated to the duration of the experiment using an AOM and finely tuned with a mechanical variable attenuator (VA) to compensate for the total dissipation experienced by the signal during a roundtrip. This way, losses are effectively canceled for the signal which can then propagate over extremely long distances. Figure S1(b) shows the normalised optical power evolution in a single experiment, namely, the regular three-solitons dynamics illustrated in Fig. 5(a) of the main manuscript.

<figure><img src="image_9.png" /><figcaption>FIG. S1. **Experimental system.** (a) Schematic of the setup. (b) Longitudinal normalised evolution of the optical power in a single experiment. AFG: arbitrary function generator, AOM: acousto-optic modulator, AWG: arbitrary waveform generator, cw: continuous wave, EDFA: Erbium doped fiber amplifier, EOM: electro-optic modulator, ISO: isolator, OSC: oscilloscope, PC: polarisation controller, PD: photodetector, Pow: power meter, VA: variable attenuator, WDM: wavelength division multiplexer.</figcaption></figure>

An effective external potential is obtained by using an electro-optic phase modulator ($EOM_\phi$ Exail MPZ-LN-20) embedded in the fiber loop. Prior to the experiment, a mapping of the targeted potential landscape is designed which consists of a sequence of waveforms (one per roundtrip). The 2^nd^ channel of the AWG sequentially plays each waveform to generate the electrical signal that drives $EOM_\phi$ at each pass of the signal. To do so, the AWG is triggered by a burst signal from AFG₁ whose period is finely tuned to match the roundtrip time of the fiber loop. In the general case, a different waveform can be played at each roundtrip resulting in a dynamic potential landscape. The AWG can also be set to play the same waveform and thus create a static potential.
A tap coupler extracts 1% of the field which is used to monitor in real-time the evolution of the optical power to finely tune the parameters of the Raman amplification. Note that the system uses essentially polarization maintaining (PM) fiber components except for the 5 km fiber spool and the WDMs. A polarization controller is thus placed after this non-PM section to realign the polarization of the signal after each circulation which mitigates polarization dependent effects.

### (iii) Single-shot detection

The signal extracted from the loop at the 10% port of the main coupler is detected using a 50 GHz bandwidth photodiode (Finisar XPDV2120R) coupled to a 65 GHz bandwidth oscilloscope (LeCroy Labmaster 10-Zi-A) used in a sequential mode: after each roundtrip, the oscilloscope is triggered to record the pattern of pulses in a segment. The full recording consists of up to a few thousands segments and can thus contains the dynamics of the signal for more than 10 000 km of propagation. Importantly, the same burst signal that triggers application of the synchronous phase modulation is used to trigger the acquisition of each segment, such that the optical signal is naturally recorded in the reference frame of the external potential.

## B. Voltage-to-Optical Power conversion

The signal recorded consists of ~ 10% of the optical power extracted from the recirculating fiber loop at each roundtrip. The optical signal is detected and converted to an electric voltage via an XPDV2120R 50 GHz photodetector coupled to a LeCroy Labmaster 10 Zi-A 65 GHz oscilloscope. A calibration is thus needed to convert the measured voltage into optical power circulating within the fiber loop. This consists in the determination of a conversion factor that is applied a posteriori during the processing of the experimental data. The calibration procedure takes advantage of the process of spontaneous modulation instability (MI) that occurs in the propagation of flat-top pulses. This fundamental nonlinear phenomenon materializes in the destabilization of an initially noisy quasi-continuous pulse which entails the formation at intermediate propagation distance of a quasi-periodic pulse train. The small signal gain associated to the process reads

$$g(\Delta f) = 4\pi^2|\beta_2\Delta f|\sqrt{\Delta f_c^2 - \Delta f^2} \quad (S1)$$

with $\Delta f_c = \frac{1}{\pi}\sqrt{\gamma P_0/\beta_2}$. $\Delta f$ is the relative frequency detuning with respect to the pump frequency, $\beta_2$ and $\gamma$ are the group velocity dispersion coefficient and Kerr nonlinearity respectively, and $P_0$ is the average optical power of the initial quasi-continuous pump. The Fourier spectrum of the optical waves sees the growth of sidebands symmetrically located around the pump according to the gain curve (S1) with power-dependent cutoffs at $\pm \Delta f_c$ and peak positions $\pm \Delta f_{MI} = \pm \Delta f_c/\sqrt{2}$.

The initial signals used in our experiments consists in arbitrarily designed optical patterns enabling several independent experimental runs to be realized in the same recording. Alongside the experimental runs, a portion of the pattern is dedicated to the power calibration and the jitter correction procedure. Taking for instance the recording associated to Fig. 4 of the main manuscript, a flat-top pulse of 10 ns-long duration is generated and after 800 km of propagation, it has turned into a high contrast train of pulses as a consequence of spontaneous MI (see Fig. S2(a, b)). The Fourier spectrum of the optical signal at an earlier stage ($z_{cal} = 430$ km, Fig. S2(c)), exhibits MI induced sidebands with peak position $|\Delta f_{MI}| \approx 4.5$ GHz. The small signal MI gain spectrum $\exp[g(\Delta f)z_{cal}]$ can be adjusted to the experimental spectrum with $P_0$ as fitting parameter. Figure S2(e) shows the best fit for this recording obtained for $P_0 = 7$ mW. The mean voltage recorded at the first roundtrip for the flat-top part of the pulse is $V_0 = 13.5$ mW (Fig. S2(d)). The Volt-to-Optical Power conversion factor is consequently $P_0/V_0 = 0.52$ W/V for this specific recording. The voltage measured experimentally is then simply multiplied by this factor to be converted to optical power.

This procedure is performed systematically for each recording and shows negligible variation.

## C. Measurement of the phase modulation profiles

The electro-optic phase modulator (EOM$_\phi$) embedded within the fiber loop enables application of arbitrary shaped phase modulation synchronous to the circulating field resulting in an effective external potential. We measured the profile of the phase modulations independently of the soliton propagation experiments reported in this work by performing heterodyne measurements described here. This is useful for two main reasons: (i) Confirm the arbitrary
<figure><img src="image_10.png" /></figure>

FIG. S2. Calibration procedure for Volt-to-Optical Power conversion based on spontaneous MI. (a) Experimentally recorded spatiotemporal diagram of spontaneous MI on a 10 ns-long flat-top pulse. (b-d) Voltage recorded at $z = 800$ km, 430 km, and 0 km showing the fully developed MI pattern, earlier stage spontaneous modulation, and initial pulse profile respectively. (e) FFT analysis of the flat-top part of the pulse at $z = 430$ km in red and fitted small signal MI gain spectrum in green.

shape of the external potentials (i.e. minimal distortion with respect to the targeted shape) ; (ii) Separately confirm the depth of the potential traps which can otherwise be inferred directly in the experiments from the oscillation period of the solitons.

An initial condition consisting in a long optical pulse in generated and sent to the recirculating fiber loop. During the first circulation, the phase modulation to be characterized is applied to the flat-top part of the pulse. Heterodyne detection is realized by mixing the optical signal exiting the fiber loop with a continuous wave laser detuned by $\Delta f = -45$ GHz (accordingly $\Delta\lambda \sim 0.36$ nm) before being recorded by the same detection system as the one used for the experiments. The applied phase modulation is retrieved from the recording via a simple numerical procedure: (i) frequency components centered around the beating frequency are selected using a passband filter and centered around null frequency; (ii) in the time domain, the phase is obtained as the argument of the complex array containing the filtered data. This single-shot phase measurement is repeated 1000 times which enables computation of a mean profile and standard deviation as shown in Fig. 2(e-g) of the manuscript. Figure S3(a, b) shows an example of raw heterodyne signal in the case of quadratic modulation and the corresponding single-shot phase profile retrieved respectively.

<figure><img src="image_11.png" /></figure>

FIG. S3. Heterodyne measurement of the phase modulation. (a) Sample of raw heterodyne recording in the case of quadratic modulation and (b) corresponding single-shot phase modulation retrieved. The gray shaded area illustrates the standard deviation computed from 1000 single-shot recordings. The dashed red line is an illustrative quadratic fit. The average profile in plotted in Fig. 2(e) of the manuscript.
## II. DERIVATION OF THE 1D-GROSS-PITAEVSKII MEAN-FIELD EQUATION FROM THE ITERATIVE MAP

### Iterative model in physical units

Experiments realised in the recirculating fiber loop are described by an iterative model consisting of a propagation equation and a boundary (recirculating) condition.

$$ \frac{\partial A_n}{\partial Z} = -i \left( \frac{\beta_2}{2} \frac{\partial^2 A_n}{\partial T^2} - \gamma |A_n|^2 A_n \right) + \frac{g_r}{2} A_n \quad (S2) $$

$$ A_{n+1}(T, Z=0) = \rho A_n(T, Z=L) e^{i\phi_n(T)} \quad (S3) $$

Propagation over the roundtrip $n$ follows a NLSE with gain [Eq. (S2)]. The field at the beginning of roundtrip $n+1$ as a function of the one at the end of roundtrip $n$ is given by Eq. (S3). $\rho$ is a constant coefficient that account for local losses.

$$ \rho = e^{-\frac{(g_r + \alpha_{\text{eff}})}{2} L} \quad (S4) $$

In this expression, $g_r$ accounts for the distributed Raman amplification while $\alpha_{\text{eff}}$ traduces the imperfect loss compensation which results in an effective exponential power decay. $L$ is the length of the recirculating loop.

The phase modulation $\phi_n(T)$ can additionally be applied and appears in the recirculating condition.

Figure S4 schematically depicts the power evolution over 3 roundtrips.

<figure><img src="image_12.png" /><figcaption>FIG. S4. Illustration of the power evolution in the iterative model. The blue curve illustrates the periodic amplification and local losses encountered across consecutive roundtrips while the gray dashed line illustrates a potential slow effective evolution of the mean power. Periodic application of a phase modulation at each roundtrip is depicted by the red arrows.</figcaption></figure>

### Lumped model

Equations (S2) and (S3) can be reduced to a single PDE by expressing the discrete recirculating condition as a function of a Dirac delta comb.

$$ \left. \frac{\partial A_n}{\partial Z} \right|_{Z=nL} = \sum_n \ln \left( \rho e^{i\phi_n(T)} \right) \delta(Z - nL) A_n \quad (S5) $$

Including the boundary condition (S3) into (S2) consists in adding the right hand side of (S5) to (S2):
$$ \frac{\partial A_n}{\partial Z} = -i \left( \frac{\beta_2}{2} \frac{\partial^2 A_n}{\partial T^2} - \gamma |A_n|^2 A_n \right) + \frac{g_r}{2} A_n + \sum_n \left( -\frac{g_r + \alpha_{\text{eff}}}{2} L + i \phi_n(T) \right) \delta(Z - nL) A_n, \quad (S6) $$

where the expression of $\rho$ (S4) has been explicit.

### Mean-field model

Integrating Eq. (S6) from $z_{n-} = z_n - \epsilon$ to $z_{n-} = z_n + \epsilon$, where $0 < \epsilon \ll 1$ and assuming continuity of $A$ and its temporal derivatives we find that the field $A_n(z_{n+})$ just after the boundary is expressed as a function of the one just before $A_n(z_{n-})$ by

$$ A_n(z_{n+}) = A_n(z_{n-}) \left[ 1 - \frac{g_r + \alpha_{\text{eff}}}{2} L + i \phi_n(T) \right], \quad (S7) $$

which is a rewriting of Eq. (S3).

Assuming small variation of the field $A$ during propagation from $Z = 0$ to $Z = L$, i.e. in the course of a roundtrip, we can write (first order Taylor expansion):

$$ A_{n+1}(0, T) = \underbrace{\left( A_n(0, T) + \frac{\partial A_n}{\partial Z} \bigg|_{Z=0} \times L \right)}_{A_n(L,T)} \left[ 1 - \frac{g_r + \alpha_{\text{eff}}}{2} L + i \phi_n(T) \right], \quad (S8) $$

by noting that $A_n(z_{n-})$ corresponds to $A_n(L, T)$ and $A_n(z_{n+})$ corresponds to $A_{n+1}(0, T)$.

The partial derivative which describes the small variation of $A$ during one roundtrip is then evaluated with Eq. (S2). By keeping only first order terms and identifying $\frac{\partial A}{\partial Z} = (A_{n+1} - A_n)/L$ we find the mean-field model

$$ \frac{\partial A}{\partial Z} = -i \frac{\beta_2}{2} \frac{\partial^2 A}{\partial T^2} + i \gamma |A|^2 A - \frac{\alpha_{\text{eff}}}{2} A + i \frac{\phi(Z, T)}{L} A, \quad (S9) $$

which is Eq. (1) of the manuscript. This takes the form of a damped NLSE with an external potential played by the discrete phase modulation that is distributed. The roundtrip number $n$ has been dropped and $Z$ is the propagation distance that is not restricted to the roundtrip length $[0; L]$ anymore. It should be noted that the model remains valid for phase modulation varying slowly from roundtrip to roundtrip which enables a $Z$ dependence of $\phi$ provided that $\partial\phi/\partial Z \ll 1$.

### III. COMPARISON BETWEEN EXPERIMENTS AND HAMILTONIAN MODEL IN DYNAMIC POTENTIAL

In the manuscript we present experimental recordings of the dynamics of single solitons subjected to dynamically varying external potentials, namely a truncated parabolic trap with $Z$-dependent strength. In these experiments, the depth of the phase modulation is fixed at $\sim 1.75$ rad and the strength is modulated by varying the duration $W$. Here, we compare the observed trajectories of the solitons reported in this article to the corresponding predictions of the Hamiltonian model.

In the case of the slowly modulated trap illustrated in Fig. S5(c), the duration of the truncated modulation $W$ follows the expression:

$$ W(Z) = W_0 - 2 \frac{\Delta W}{\Delta Z} \exp(0.5) \times (Z - Z_c) \exp \left[ \frac{\sqrt{2}(Z - Z_c)}{\Delta Z} \right]^2, \quad (S10) $$
with $W_0 = 1$ ns, $\Delta W = 0.4$ ns, $\Delta Z = 3000$ km and $Z_c = 4500$ km. This evolution is depicted in Fig. S5(b) and Fig. 3(a) of the manuscript (dashed red lines). From Eq. (S10) and Eq. (5) of the main manuscript it is possible to compute the evolution of the trap wavenumber $K_0(Z)$ which is then used to compute the particle trajectory in the Hamiltonian model. The result is superimposed on the experiment as a dashed blue line in Fig. S5(c) and shows excellent agreement.

In the case of parametrically modulated trap illustrated in Fig. S5(f), the duration of the truncated modulation $W$ follows the expression:

$$W(Z) = W_0 \left[ 1 - \frac{\Delta W}{W_0} \sin(2K_0 Z) \right], \qquad (S11)$$

with $W_0 = 1$ ns, $\Delta W = 50$ ps and $K_0 = 2\pi/Z_0$ with the natural spatial period of the trap $Z_0 = 800$ km. The factor 2 in the sine function denotes the parametric forcing at half the natural period of the trap. This evolution is depicted in Fig. S5(e) and Fig. 3(b) of the manuscript (dashed red lines). Again, Eq. (S11) and Eq. (5) of the main manuscript are used to compute the evolution of the trap wavenumber $K_0(Z)$ then used to compute the particle trajectory in the Hamiltonian model. The result is superimposed on the experiment as a dashed blue line in Fig. S5(f) and shows a remarkable agreement, in particular in reproducing the exponential growth of the oscillation amplitude.

<figure><img src="image_13.png" /><figcaption>FIG. S5. Dynamically modulated potentials: Experiments vs. Hamiltonian model. (a-c) Slowly modulated and (d-f) parametrically modulated parabolic traps. (a, d) 3D views of the external potentials, (b, e) Longitudinal evolutions of the parabolic trap duration $W(Z)$. (c, f) Spatiotemporal dynamics recorded experimentally and particle trajectories predicted by the Hamiltonian model superimposed as dashed blue lines.</figcaption></figure>

## IV. SENSITIVITY OF THE HAMILTONIAN MODEL WITH THREE TRAPPED PARTICLES

In the manuscript we report the observation of regular and chaotic three-solitons dynamics in a harmonic trap (see Fig. 5) and mention the failure of the Hamiltonian model in the latter case because of the simultaneous interaction of the 3 “particles”, a situation mentioned in Ref. [S1]. Additionally, the discrepancy between the model and an actual realization (physical or numerical) builds up from the accumulation of small error originating from the fact that trapped solitons do not reach asymptotic separation after collision as discussed in Refs. [S2, S3]. This is all the more significant when larger number of solitons are in strong interaction.
Here we illustrate how a quantitative agreement between our observations and the particle model is particularly hard to reach in the three-solitons situation by comparing the experimental result with predicted particle trajectories with 3 slightly different initial conditions.

The initial setting is depicted in Fig. S6(a) and consists of 3 unequally spaced solitons with no differential velocity (but with a weak velocity with respect to the trap). By finely tuning the parameters of the particle model (position and amplitude of the solitons) a fairly good overall agreement with the experiment is reached as can be seen in Fig. S6(b) (Note that we impose the same amplitude for the 3 solitons for simplicity which is not exactly realized in practice). We stress however the sensitivity of this agreement to the initial positioning of the central particle. Indeed, a small displacement of the latter in the initial state rapidly alters the computed trajectories (Fig. S6(c)) up to a very significant qualitative change where the particle model predicts the formation and complex evolution of two-solitons bound-state subsequent to the simultaneous interaction of the 3 components (Fig. S6(d) at 2300 km).

<figure><img src="image_14.png" /><figcaption>FIG. S6. **Three soliton dynamics: Experiment vs. Hamiltonian model.** (a) Sketch of the configuration when the trapping potential is turned on ($z = 500$ km). (b-d) Spatiotemporal dynamics recorded experimentally with the 3 computed particle trajectories superimposed as colored lines. The 3 panels show trajectories computed for 3 slightly different positions of the central “particle/soliton” at $z = 500$ km depicted in (a).</figcaption></figure>

Such behaviour is not observed nor expected in our work because the integrable features of soliton collision are globally preserved within the range of parameters of the experiments. Specifically, the outcome of collisional processes involving 3 solitons even simultaneously is in agreement with the sum of the effect of the 3 constitutive pairwise interactions [S4].

Finally, we note that several details of the experiments might easily hinder accurate correspondence with the Hamiltonian model including the fact that the pulses are not exactly fundamental solitons, the continuous interaction with radiative content, and non-measurable deformation of the external potential.

[S1] R. Scharf and A. R. Bishop, Soliton chaos in the nonlinear Schrödinger equation with spatially periodic perturbations, Physical Review A **46**, R2973 (1992).

[S2] A. D. Martin, C. S. Adams, and S. A. Gardiner, Bright Matter-Wave Soliton Collisions in a Harmonic Trap: Regular and Chaotic Dynamics, Physical Review Letters **98**, 020402 (2007).

[S3] A. D. Martin, C. S. Adams, and S. A. Gardiner, Bright solitary-matter-wave collisions in a harmonic trap: Regimes of solitonlike behavior, Physical Review A **77**, 013620 (2008).

[S4] F. Copie, P. Suret, and S. Randoux, Space-time observation of the dynamics of soliton collisions in a recirculating optical fiber loop, Optics Communications **545**, 129647 (2023).