Synthesis of parallel soliton molecules
To investigate the dynamics of soliton-molecule formation, we prepared a stable soliton supramolecule (see Methods) as the initial state (Fig. 2a), consisting of 195 time-slots, all with two long-range-bound solitons, except for a reference slot containing a single soliton (see Fig. 2b). In the absence of repulsive forces, the solitons within a single time-slot will tend to collide. This is prevented in practice by repulsive forces that arise from the shedding of dispersive waves5,46,47, leading to formation of a secondary trapping potential (Fig. 1d) that causes long-range binding at ~ 60 ps separation (Figs. 2b and 2c)—much longer that the ~ 1 ps duration of the solitons (see SI Section II). As a result the two trapped solitons have negligible field overlap and are thus uncorrelated in phase5,15.
Synthesis is initiated by abruptly changing the pump power or the intra-cavity loss, which causes a decrease in the inter-soliton repulsive force (see SI Section II) and a gradual reduction in soliton spacing in all the reactors, culminating in multiple soliton collisions (Fig. 2a). We studied the real-time motion of the reacting solitons by time-domain recording (limited by the bandwidth of the photodiode, so suitable only for long-range binding) and DFT (suitable for short-range (< 14 ps) binding) in all the reactors (see Methods). The entire reaction process over all time-slots, initiated by pump perturbation, was first recorded at 5 kHz frame rate and is shown in Fig. 2b in cylindrical coordinates for 5 selected frames (see Supplementary Movie 1 and SI Section III). Recordings over the initial 49,000 round-trips (~ 5 ms) in 8 consecutive reactors (out of 195) are plotted in Fig. 2c and 2d, showing the soliton dynamics on finer time-scales.
The experimental results indicate that the formation of a stable soliton-pair, resembling a molecule, generally requires multiple collisions within the reactors before an effective collision establish a stable spacing/phase relation between the solitons. As shown in Fig. 2c and Supplementary Movie 1, while in some reactors a stable soliton pair formed within 5 ms after only a few soliton collisions, in many other reactors hundreds of collisions were required. In average, an effective collision occurred out of ~ 25 collisions as we estimate from the recording. To examine the statistics of the synthesis, the cumulative collision numbers in all the reactors is plotted in Fig. 2e against the total number of soliton molecules, measured at time intervals of 50 µs. The plot shows an approximately linear dependence, consistent with the collision theory of chemical kinetics which states that the rate of a gas-phase reaction is proportional to the collision frequency48. (See SI Section IV).
Once formed, the soliton molecules would propagate as single entities with precisely synchronized phase and group velocities between the bound solitons. However, they would generally differ40–42 from single solitons in group velocity, and as a consequence, gradually shift to slightly different positions within their time-slots, while remaining trapped by the optoacoustic trapping potential, as seen in Fig. 2c; see also the reference slot in final state in Fig. 2b. The optomechanical lattice is robust enough to host all the soliton reactions until they are completed, without destabilization.
By retrieving the soliton spacings and phases from the DFT signal11, the complex nonlinear dynamics after the initiation in hundreds of parallel soliton reactors can be resolved, revealing the stochasticity of soliton-molecule formation (see Fig. 3). Panel (i) – (iii) in Fig. 3a (with corresponding trajectories in Figs. 3b – 3e) show reaction processes in 3 parallel reactors in which the two solitons in each time-slot attempt to transit from a phase-uncorrelated long-range bound state (~ 60 ps spacing) to a phase-locked soliton-molecule (3.8 ps spacing and π-phase difference22) after the pump power is perturbed (same as in Fig. 2). While in panel (i) the formation process is completed within 1000 round-trips, following a rather simple trajectory (spacing and phase evolution in Fig. 3b and interaction plane21 in Fig. 3c), the reaction shown in panel (ii) lasted more than 3000 cavity round-trips and followed a more complex trajectory (see Fig. 3d). Panel (i) and (ii) show how effective collisions could took place in a reactor, while Panel (iii) shows a more frequently observed case of soliton collision which lasts thousands of cavity round-trips without, however, giving birth to a soliton molecule (see Fig. 3e). Instead, the two solitons strongly repel each other, before drifting towards the next collision.
We observed that the soliton motion at most separations is stochastic, reminiscent of a one-dimensional random walk with fixed step-length49. This is probably caused by phase-dependent inter-soliton forces4,19 that are constantly varying in strength and direction, weakly perturbed by non-solitonic components47. The initial phase difference between two solitons before binding is also random15, probably accounting for the widely different trajectories from reactor to reactor. When the soliton separation is less than the molecular spacing (< 3.8 ps), however, a strong repulsive force emerges4 which quickly pushes the solitons apart. This is a ubiquitous feature not only in the synthesis of soliton-molecules but also in their dissociation, as described below (see SI Section V.). Only in effective collisions, the colliding solitons would enter a converging trajectory with quickly damped spacing and phase relation (see DFT signal in Fig. 3b and 3d).
Synthesis between different solitonic elements have also been realized in controlled manner using the parallel-reactors, which was previously challenging due to their intrinsic velocity discrepancy that cause uncontrolled collisions. In experiment, we first prepared a soliton supramolecule in which most time-slots hosted a single soliton bound with a soliton-pair molecule, their group velocities being synchronized by long-range interactions15 such that they would not collide freely before the initiation45. Then we abruptly increased the intra-cavity loss using the fast tunable attenuator, which weakened the dispersive-wave perturbation (See SI Section II). Consequently, attraction overcame repulsion, initiating soliton reactions. Two examples of three-soliton reactions are shown in panels (iv) and (v) of Fig. 3a. In panel (iv), collision between the soliton pair and the single soliton resulted in formation of a phase-locked soliton triplet. The measured trajectories between neighbouring solitons during synthesis in panel (iv) are shown in Fig. 3 f. The single soliton collides strongly with the soliton-pair, resulting in strong disturbance to the soliton-pair (highlighted in yellow in Fig. 3f) before the establishment of a second molecular bond (the reaction process is sketched in Fig. 3g). In panel (v), however, a similar collision result in dissociation of the soliton-pair molecule, followed by strong repulsion between all three solitons, highlighting the complexity during the three-soliton reactions. (See SI Section VI)
Dissociation of parallel soliton molecules
Phase-locked soliton molecules can also dissociate into single solitons under global control (Fig. 4a). A typical example is recorded and plotted in cylindrical coordinates in Fig. 4b for 5 selected frames (for full recording see Supplementary Movie 2). The reaction is initiated by a slight decrease in pump power, which enhances the dispersive-wave perturbations, causing rapid break-up of the soliton-molecules. This process is much faster than soliton-molecule formation that generally requires multiple collisions. The dissociation follows highly diverse trajectories from reactor to reactor, as seen in Fig. 4c (time-domain) and Fig. 4d (DFT). We attribute the stochastic fluctuations during the early stages of dissociation to noise-like repulsive forces between the solitons exerted by randomly excited dispersive waves5,46. After dissociation, long-range binding between the solitons is gradually established, eventually settling down after a few milliseconds.
Since soliton molecules dissociate over diverse trajectories, the dissociation rate can only be determined statistically. We first define criteria for determining the completion of a reaction: full dissociation for a separation of 14 ps or greater (the maximum spacing retrieved from DFT signal), and long-range soliton binding for a separation of 55 ps or greater. We plot the total number of soliton molecules and long-range double-solitons during dissociation against the number of round-trips (Fig. 4e). Both measurements are roughly exponential, indicating that the dissociation rate is proportional to the number of un-dissociated reactants, i.e. following first-order reaction48. This allows us to estimate a soliton-molecule “half-life” of ~ 1200 round-trips (~ 120 µs) using the 14 ps criterion. (See SI Section IV for details)
The soliton motion during dissociation retrieved from the DFT signal are found to share many characteristics with that of synthesis. Figure 5a shows a few dissociation trajectories recorded in the parallel reactors initiated by perturbing the pump power. Panel (i) shows fast dissociation (< 1000 round-trips) with a relatively smooth trajectory, as indicated by the retrieved spacing and phase (Fig. 5b and c). Panel (ii) shows another trajectory recorded within the parallel reactors, which however exhibit a rather long dissociation lasting > 10000 round-trips with a random-walk-like trajectory (see Fig. 5e). Within the trajectory, we can notice a “metastable state” at spacing of ~ 11 ps at which the two solitons temporally reside with quasi-stable relative phases (~ π), which might corresponds to fixed-point attractor with weak stability. Similar phenomena also appeared in other reactors. In addition, we can notice within the trajectory that a strong repulsion4 occurred (indicated by the blue dashed arrow) when solitons temporally reached a spacing below the initial value (marked by the dashed horizontal line) during the random walk. This phenomenon are found to be universal during the reactions and are also observed in synthesis dynamics (e.g. Figure 3(e)). Such universal soliton repulsion probably result from a mutual frequency shift that occurred spontaneously during significant soliton-overlapping, which quickly flipped their phase relation and turned the inter-soliton force from attraction to repulsion.
In a few reactors, such inter-soliton repulsion can be so radical that the dissociation can terminate immediately. One example is shown in Panel (iii) in Fig. 5) (with DFT signal in Fig. 5e), in which the abrupt drop in soliton spacing triggered strong repulsion between the solitons within only a few round-trips, followed by quick establishment of long-range binding. Occasionally, one (or both) of the interacting solitons are extinguished after such radical repulsion, as shown by the example in panel (iv) of Fig. 5a (with DFT signal in Fig. 5f). This is probably due to the fact that the carrier frequency of both solitons are significantly shifted during the radical repulsion, which affects their gain/loss balance in the laser cavity with limited gain bandwidth. The diverging soliton then experience net loss and failed to recover before the long-range forces (trapping potential) shift the frequency back (See SI Section V for more details.)
Dissociation of soliton-triplets follows even more complex dynamics, as seen when the system is loaded with soliton-triplets in each reactor and then perturbed by decreasing the cavity-loss. Three examples recorded within the parallel reactors are shown in Fig. 6. Panel (i) and (ii) show dissociations that breaks either of the two molecular-bonds within the triplet, leading to different orientations between the soliton pair and the single soliton in the final long-range bound state (see DFT signal in Fig. 6b and c). In panel (iii), both molecular-bonds between the three solitons are severed, resulting in three phase-uncorrelated single solitons (See SI Section VI for more examples.)
All-optical control of soliton reactions in selective reactors
Soliton interactions within selected reactors can be controlled by launching a sequence of precisely timed optical pulses into the laser cavity (Fig. 1a)12,14,31, permitting individual solitonic elements to be edited by XPM (see Methods and SI Section VII). To demonstrate this, we first prepared a soliton supramolecule in which a mixture of long-range double-solitons and phase-locked soliton-pairs exist in the time-slots. To convert two long-range-bound solitons into a soliton molecule, we launched a train of ~ 200-ps pulses at the cavity round-trip frequency, precisely timed to interact with targeted time-slots over ~ 3000 round trips (see Fig. 7a and 7b). Since the two solitons initially ride on different amplitude upon each addressing pulse, the two solitons would see different XPM-induced nonlinear index and tend to move closer. Therefore, an effective “attraction force” was applied to the solitons that exceed their long-range repulsion, leading to soliton collisions and formation of stable soliton-molecules (as shown DFT signal and retrieved trajectory in Fig. 7c from selected time-slots). Note that each addressing pulse only accompanied the intra-cavity soliton over a several-meter SMF section before getting absorbed by the intra-cavity polarizer (to prevent laser gain depletion). Therefore, many addressing pulses (typically a few thousands) are required to be launch repetitively to the same time-slot to ensure sufficient overlapping time with the solitons (See SI Section VII for details).
Conversely, to break apart soliton molecules in selected reactors, we developed a special trick that made use of the same train of addressing pulses as above with however a slight offset in repetition rate. In this case, the addressing pulses would effectively “walk through” the soliton molecule and impose a mutual frequency shift through a time-domain varying XPM 50 (see Fig. 7d). The externally-induced frequency-shift would flip the phase-relation of the soliton molecule from π to 0, causing strong attraction and thus compressed spacing. Then a following frequency-shift is triggered (similar to the case in Fig. 5e), which flipped the phase-relation and thus the force direction back, leading to radical repulsion that pushed the solitons apart. The addressing pulses in this case operate like an optical “scissors”, severing the molecular bond while traversing. By suitable choice of repetition-rate offset (fext – fcav ≈ 20 Hz, see Fig. 7e), we achieved deterministic dissociation of selected soliton-molecules. The trajectories retrieved from the DFT signal are shown in Fig. 7f, revealing that the traversing pulse first compresses the soliton separation to below ~ 5 ps, triggering strong repulsion within 2 ~ 3 round trips, and eventually leading to the collapse of the molecular bond and establishment of long-range binding (See SI Section VII for details).