A. Isomers of soliton triplets

Soliton triplets, constituted by three constituents, are regarded as the fundamental MSMs, and the temporal distribution and internal dynamics can be characterized by (τ12, φ12) and (τ13, φ13) routinely. Considering the symmetry of the temporal distribution in the soliton triplets, two kinds of optical isomers, namely the unequally spaced (US) triplet and equally spaced (ES) triplet, are introduced naturally to manifest their distinctive structure. As demonstrated in Fig. 2, by setting the pump power at 242 mW, we initially capture a US triplet with a fundamental repetition rate of 55.16 MHz, which corresponds to the laser length of 3.7 m. As depicted in Fig. 2(a), the obvious periodic evolution could be observed from the consecutive real-time interferograms over 10 000 round trips. By taking the fast Fourier transform of the consecutive interferograms, the corresponding 2D contour plot of the first-order autocorrelation traces (FATs) could be acquired, as shown in Fig. 2(b). Considering the correlation principle and the asymmetrical temporal distribution of the soliton triplet, seven bright fringes are symmetrically distributed, where the information about the temporal separations and relative phases is involved. In particular, a typical single-shot spectrum at the 3000th round trip, highlighted by the white dashed line in Fig. 2(a), is presented in Fig. 2(c), displaying a clear superimposed interference modulation. In Fig. 2(d), three temporal delays are displayed in the single-shot FAT and (τ12, φ12) and (τ13, φ13) could be retrieved from the given peaks directly. Although one set of the FATs is consistent with two possible solutions of the temporal distribution, only one possibility, where the front part is denser than the latter part in the multi-pulse structures, is taken into consideration due to the gain dynamics within the oscillator. The temporal separations are retrieved as depicted in Fig. 2(e). It is found that the second soliton and the third soliton vibrate with an identical period of 3750 round trips, albeit with different amplitudes. As exhibited in Fig. 2(f), the retrieved relative phases φ12 and φ13 both generate a decrease of 2π in each period, declaring the sliding phase dynamics within the US triplet. In the present case, in light of the intensity-dependent index in the Kerr medium, the intensity of the first soliton remains larger than the other two solitons so that the relative phases φ12 and φ13 keep decreasing during the propagation. Moreover, there is no persistent intensity difference between the second soliton and the third soliton. Visualized insights into the sliding phase dynamics of the US triplet over one period can be gained from the 2D interaction plane as pictured in Fig. 2(f). Two almost circular trajectories of (τ12, φ12) and (τ13, φ13) are represented due to the continuous accumulation of the relative phases.Another isomer of the soliton triplets, the ES triplet, is prepared by tuning the laser settings. Figure 3(a) shows the 2D contour plots of the consecutive interferograms over 5000 round trips, and the corresponding calculated FATs are represented in Fig. 3(b), where the wobbling fringes declare the oscillating structures of the isomer with the period of 1537 round trips. The spectral frame at the 2700th round trip in Fig. 3(c) displays two superimposed fringe periodicities, which is consistent with the two temporal delays in its single-shot FAT in Fig. 3(e). Five peaks of the single-shot FAT declare the symmetry of the temporal distribution, that is τ12 = τ23. However, the asymmetry of the temporal distribution could be induced due to the oscillation of the constituents. Accordingly, the typical single-shot interferogram and FAT at the 400th round trip are shown, respectively, in Figs. 3(d) and 3(f), emphasizing the variation of the temporal distribution. The temporal separations, retrieved from the given fringes of the FATs, are illustrated in Fig. 3(g). Different from the situation of Fig. 2(e), the variation of τ12 is remarkably larger than that of τ13. As demonstrated in Fig. 3(h), both the retrieved relative phases φ12 and φ13 generate a decrease of 2π in each period, implying the intensity relationship among the three solitons. The transient dynamics within the ES triplet over one period is visualized in Fig. 3(i) in the form of the 2D interaction plane. Particularly, the trajectory of (τ12, φ12) evolves with an obvious offtrack from the circular orbit due to the large variation of τ12. Based on the evolving characterization of the temporal separations and the relative phases, the transient dynamics of the isomers could be revealed, as well as the energy flow inside the entities, which remains an arduous task in any other way. In light of the existing knowledge about the soliton triplets, the intriguing light–matter analogy is further extended in regard to the isomeric dynamics.

B. Isomers of soliton quadruplets

Beyond the fundamental level of soliton triplets, the MSMs with four constituents should exhibit more complex temporal distributions and serve as key objects to exhibit the versality of the internal motions. Compared to a soliton triplet, one additional soliton extends the degrees of freedom to portray the soliton quadruplet. Considering the relationship among the adjacent separations (τ12, τ23, and τ34) within the soliton quadruplets, four kinds of isomers are introduced naturally, namely US quadruplet, 3+1 quadruplet, 2+2 quadruplet, and ES quadruplet. The relationship between the adjacent separations within the US quadruplet can be interpreted as τ12 ≠ τ23 ≠ τ34. When τ12 = τ23 or τ23 = τ34, the soliton quadruplet could be considered as the combination of an ES triplet and a soliton singlet and, thus, termed 3+1 quadruplet. Moreover, the 2+2 quadruplet could be regarded as the assembly of two soliton pairs with identical temporal separations and characterized with τ12 = τ34. The ES quadruplet, consisting of four symmetrically located solitons, is a common observation in the nonlinear system. Since the FATs serve as the effective tools for us to resolve the temporal distributions of the MSMs, the simulation analysis on the FATs for the four stable isomers is performed, giving us a support to identify the isomers experimentally. In this simulation, the adjacent separations of the four isomers are assumed and summarized in Table I and each soliton is chosen to have the same intensity and a Gaussian profile with a temporal width of 1.5 ps.

TABLE I. Assumed temporal separations of the isomers in the simulation.

Isomerτ12 (ps)τ23 (ps)τ34 (ps)US94293+166302+26306ES666As presented in Fig. 4, the simulation results of the single-shot FAT and the corresponding 2D contour plot of the consecutive FATs for the four isomers are provided. For the US quadruplet, 13 peaks are located in its FAT. Considering the symmetrical structure of the specific three solitons in the 3+1 quadruplet, the corresponding FATs are featured with 11 peaks. The 2+2 quadruplet and the ES quadruplet are characterized with nine and seven peaks on the FATs, respectively. Focusing on the specific fringes in the 2D contour plot of the FATs, the pairwise interrelations of the isomers can be readily resolved. Apart from the simple notion for the isomers of the soliton quadruplets, the real-time transient dynamics within the isomers should be followed to draw the parallel between the matter isomers and the optical isomers.By adjusting the pump power and PCs, a set of different dynamical isomers of soliton quadruplets are experimentally prepared and the real-time interferograms are depicted as the first column in Fig. 5, exhibiting the diversity of the transient dynamics inside the isomers. The consecutive interferograms of a US quadruplet is displayed in Fig. 5(a), and the periodicities of the blurred fringes can be unveiled with the help of the FATs in Fig. 5(b). Considering that the front part should be denser than the latter part in the soliton molecule, the temporal separations (τ12, τ13, and τ14) and the relative phases (φ12, φ13, and φ14) are retrieved by aiming at the given fringes of the 2D contour plot of the FATs and the calculated results are demonstrated in Figs. 5(c) and 5(d). The variations of the temporal separations corroborate the vibration within the US quadruplet with the period of 148 round trips. The continuous decrease in the sliding phase φ13 is generated by the positive intensity difference between the first soliton and the third soliton, and the relative phase φ12 oscillates in line with the temporal separation τ12, implying the recurrent reversion of the intensity relationship between the first soliton and the second soliton. In addition, the intensity reversion between the first soliton and the fourth soliton can be certified from the in-phase oscillation of φ14 and τ14. Thus, the intensity relationship among the four constituents inside the isomer is revealed according to the relative phases. This case witnesses the coexistence of the vibration dynamics and the sliding phase dynamics inside a soliton molecule, exhibiting the independence among the real-time motions of the individual solitons. In addition, the consecutive interferograms of a dynamical 3+1 quadruplet with the evolving period of 2203 round trips is depicted in Fig. 5(f), and the calculated 2D contour plot of the FATs is presented in Fig. 5(g). As shown in Figs. 5(h) and 5(i), the retrieved results of temporal separations and the relative phases illustrate the sliding dynamics within the soliton molecule. Although φ12 and φ14 both represent the features of sliding dynamics, φ14 decreases more rapidly than φ12, which could be attributed to the persistent intensity difference between the second and the fourth soliton. Moreover, a dynamical 2+2 quadruplet with the evolving period of 3561 round trips is also prepared, as displayed in Figs. 5(k)–5(o). The retrieved relative phases of φ12, φ13, and φ14 all generate a 2π-decrease in each period, demonstrating that the first soliton keeps the largest one among the four solitons and there is no persistent intensity difference among the other three ones. Apart from that, Figs. 5(p)–5(t) manifest an ES quadruplet evolving with the period of 560 round trips. Particularly, according to the relative phases φ12 and φ13, the asynchronism of the vibration of the second and the third soliton is observed, extending the independence among the motions of the solitons. All these complex real-time dynamics over one period could be visualized in the form of the 2D interplane, as depicted in the last column of Fig. 5.

In the context of the isomers, the investigations about the real-time dynamics of the soliton quadruplets reveal the interactions among the constituents from the perspective of the energy flow and the vibrations of the individual solitons. On the other hand, different coexistences of the vibration dynamics and sliding phase dynamics are explored and even the asynchronism of the vibration dynamics is also demonstrated, exhibiting the independence of the constituents within a soliton quadruplet. It is manifested that each isomer could involve different combinations of the transient dynamics due to the independence, providing more degrees of freedom for the artificial-assembly of the MSMs. Moreover, by combining the real-time measuring methods of the DFT technique, the time-lens technique, and the balanced optical cross correlation technique, the isomeric dynamics are believed to be further demonstrated from the perspective of the temporal distribution, pulse intensity, and interactions.