## Probing the interaction of solitons

As a pulse of light propagates through a medium, scattering and dispersion processes usually result in the pulse diffusing. However, under certain circumstances, the dispersion processes can be balanced by nonlinearities to produce localized structures known as solitons or optical bullets. Herink *et al.* used spectral interferometry to image and track the formation of soliton complexes as they propagated in a laser cavity. Real-time access to the formation processes and complex interaction dynamics could help in modeling other nonlinear systems.

*Science*, this issue p. 50

## Abstract

Solitons, particle-like excitations ubiquitous in many fields of physics, have been shown to exhibit bound states akin to molecules. The formation of such temporal soliton bound states and their internal dynamics have escaped direct experimental observation. By means of an emerging time-stretch technique, we resolve the evolution of femtosecond soliton molecules in the cavity of a few-cycle mode-locked laser. We track two- and three-soliton bound states over hundreds of thousands of consecutive cavity roundtrips, identifying fixed points and periodic and aperiodic molecular orbits. A class of trajectories acquires a path-dependent geometrical phase, implying that its dynamics may be topologically protected. These findings highlight the importance of real-time detection in resolving interactions in complex nonlinear systems, including the dynamics of soliton bound states, breathers, and rogue waves.

Despite their overwhelming complexity, nonlinear systems exhibit universal features that facilitate an understanding of their dynamics, including periodic attractors and chaos. The prototypical excitations of many nonlinear systems are solitons, localized structures balanced by nonlinearity and dispersion. Soliton dynamics attract considerable attention in numerous contexts, including fluids, Bose-Einstein condensates, plasmas, polymers, and optical systems (*1*–*5*). The soliton’s stability against perturbations endows it with particle-like characteristics, which may follow from topological protection, as manifested in skyrmions or edge states (e.g., in topological insulators).

Interactions between individual solitons create the possibility of bound states, which were theoretically predicted and successfully demonstrated in various physical forms and degrees of freedom. The self-trapping of multiple co-propagating modes was discovered, for example, in optical fiber for temporal solitons of different polarization (*6*) or wavelength (*7*) and for multicomponent spatial solitons in photorefractive media (*8*, *9*). Stable and dynamically evolving bound states were found to arise from various coupling mechanisms (*3*, *6*–*12*), resulting in distinct relationships between mutual amplitudes, phases, and separations(*3*)–*9*). In one-dimensional or single-modal propagation, attractive and repulsive interactions between temporal optical solitons result in bound states, which are frequently referred to as soliton molecules (*13*–*18*).

In dissipative nonlinear systems, a twofold balance of energy loss with gain and dispersion with nonlinearity (*19*) allows for large families of bound states between multiple solitons. Beyond stationary solutions, these systems also support soliton molecules with time-varying properties. Access to such dynamics has been gained mostly by numerical simulations. In particular, in various types of lasers, numerical studies predict stationary, periodic, or chaotic bound-state evolutions (*19*–*25*). Experimentally, time-averaged measurements have resolved static soliton molecules, and internal motions have been inferred from partial coherence losses (*26*–*30*).

Whereas the femtosecond or even attosecond time scales typically associated with the formation and dissociation of atomic bonds can only be traced via temporal reconstruction (pump-probe techniques), the dynamics of femtosecond soliton molecules often span the nanosecond to microsecond range. However, bound states form at unpredictable times, and their subsequent evolution may be nonrepetitive. Thus, observation of these dynamics requires real-time detection of the timing and relative phase within femtosecond molecules over long recording intervals.

In our experimental setup, we resolve the formation and internal dynamics of a diverse set of femtosecond soliton molecules in a broadband Kerr lens mode-locked Ti:sapphire laser with chirped-mirror dispersion compensation. Although single-pulse operation is commonly preferred, it is well known that such systems support double or multisoliton complexes (*26*–*29*, *31*). We prepare dynamic soliton molecules (Fig. 1A), and resolve their evolution using a relatively simple but powerful technique, known as the time-stretch dispersive Fourier transform (TS-DFT) (*32*): Spectral information is mapped into the time domain—using chromatic dispersion in a long optical fiber—and is detected via a high-speed photodetector and real-time digitization. This method is increasingly used for measurements of rapid signals (*33*–*35*) and, in recent experiments with mode-locked sources, has been applied to record the build-up of femtosecond mode locking (*36*) and soliton instabilities in a fiber oscillator (*37*).

Typically, the TS-DFT is used to obtain spectral dynamics rather than temporal information on ultrashort time scales. Yet an ensemble of closely spaced pulses exhibits spectral interference, which encodes both precise timing and phase information (Fig. 1B). The extraction of timing information of bound states in nonlinear cavities and fiber oscillators via TS-DFT was recently discussed (*38*, *39*). Generally, both timing and phase can be obtained from the interferogram by considering a bound state as a superposition of temporally separated individual solitons. For example, the bound state field of a doublet with the soliton envelopes *E*_{1}(*t*),* E*_{2}(*t*) at a common carrier frequency ω_{0} can be expressed as(1)(*16*, *20*). In the case of identical envelopes,* E*_{2}(*t*) can be replaced by *E*_{1}(*t* + τ) exp(*i*Δϕ) with soliton separation τ and a possible relative phase Δ#x03C6;. In the frequency domain, the temporal separation translates to a phase factor exp(*i*ωτ), which modulates the spectrum* E*(ω) with a fringe period of 1/τ. Thus, the pulse separation is mapped into a modulation observed as an interferogram *S*(ω) = |*E*(ω)|^{2}, and the phase of the fringe pattern at ω_{0} encodes the relative phase of the pulses in the doublet:(2)We implement the TS-DFT with an optical fiber of group velocity dispersion (GVD) parameter β = 4.7 × 10^{–2} ps^{2}/m and length *L* = 400 m (Fig. 1C). The spectral fringe period 1/τ encoding the bound-state separation is mapped into the time domain and stretched to Δ*t* = 2πβ*L*/τ. The signal is detected with a fast photodiode (bandwidth > 8 GHz) and a high-speed real-time oscilloscope (bandwidth 16 GHz). Thus, we can record spectral interferograms as a continuous real-time series at the repetition rate of the mode-locked laser (78 MHz) over an interval of typically *T* = 4 ms, spanning ~300,000 consecutive roundtrips.

## Formation of soliton molecules

In the real-time series of interferograms tracking the formation of a stable soliton molecule from two individual solitons (Fig. 2, A and B), the state is prepared by adjusting the cavity alignment and applying a slight mechanical perturbation(see supplementary materials). The real-time data exhibit interferences with increasing fringe period and a complex temporal evolution. The Fourier transform of each single-shot spectrum (Fig. 2C) corresponds to a field autocorrelation, which directly represents the separation between the two solitons. During the transition, solitons approach each other via steps toward a stable binding separation of 180 fs, accompanied by a characteristic sequence in the relative phase evolution and a locked final phase relation (compare to interference fringes in Fig. 2A).

The natural representation of the bound-state configuration space is the polar diagram or “interaction plane,” assigning the pulse separation to the radius and the relative phase to the angle (*20*). Constructed over 6000 roundtrips prior to the establishment of the bound state in Fig. 2D, the trajectory illustrates the evolution through periodic metastable soliton separations. Finally, the system reaches a fixed point with locked relative phase, settling at a constant binding separation. The continuous tracing of successive interferograms yields the direction of the phase change: We observe a predominant shift of the fringe pattern toward shorter wavelengths, corresponding to a decreasing relative phase and a clockwise rotation in the interaction plane. This finding can be explained by a slight intensity difference of the two constituents of the bound state, which have different phase velocities in the gain medium due to the intensity-dependent refractive index (Kerr effect). A persistent intensity difference of the two constituents during the approach results in different carrier envelope phase shifts after each roundtrip and, in turn, leads to a long-term evolution of the relative phase, as theoretically described (*21*, *30*).

The present interferometric detection over long record intervals enables sensitivity to very small phase shifts at the single-shot level. The observation of discrete separations during the bond formation may be connected to the periodic spatial dependence of the interaction potential, as previously reported theoretically and experimentally for stable bound states (*15*, *28*). However, the detailed origin of this strong transient stabilization remains a subject for further study.

## Doublet and triplet states with dynamical phase evolution

To induce dynamic bound states that continuously evolve, we initially generate a soliton molecule with a separation of ~170 fs and locked relative phase, similar to the previous final state (Fig. 3A). Then, reducing the pump power below a critical threshold, we observe a highly stable stepwise evolution of the relative phase, evident in the dynamic fringe pattern in Fig. 3B. Each step is composed of a steady decrease in relative phase by a small fraction of π, followed by a rapid completion of a (–2π)-cycle. This evolution of the relative phase is plotted on a normalized time axis for two periods in Fig. 3F. Such steplike progressions of the phase in mode-locked lasers were attributed to gain saturation dynamics in numerical studies (*25*). From the observed phase slip per roundtrip, we can readily infer that the intensity difference of the soliton molecule’s constituents is ~0.5%, which would be extremely difficult to measure in any other way.

A further reduction in pump power (Fig. 3, C and D) not only accelerates the phase dynamics but also results in the transition from a stepwise evolution toward a linear phase shift over time (Fig. 3F). The decreasing number of roundtrips per period is depicted in the inset of Fig. 3F. The interaction planes for all pump powers in Fig. 3D reveal that below a threshold pump value of about 4.5 W, the system evolves on limit cycles of fixed intersoliton separation. The bound state traces orbits around the singularity at τ = 0 with the relative phase constantly accumulating Δϕ = –2π in each revolution. This finite phase may be regarded as a geometric or Pancharatnam-Berry phase. The topology of these dynamics in the configuration space can be characterized by a winding number *m* = = –1, where *C* denotes a closed integration path.

These findings directly relate to numerical results predicting periodic bound-state dynamics, which, although not discussed in terms of a winding number, would imply solutions of* m* = –1 (*25*) or *m* = –2 (*21*). One may speculate to what degree these different dynamic bound states may be topologically protected against perturbation, adding another topological aspect to the field of photonics and soliton interactions (*11*, *40*). In the spectral domain, the continuous phase revolution by *m*2π within *k* roundtrips (*k* need not be integer) implies an overall geometrical shift of the corresponding frequency comb (*41*) by *f*_{geo} = *f*_{rep}/*k*, adding to the carrier envelope offset frequency *f*_{CEO} and the fundamental repetition rate *f*_{rep}, such that the frequency of the *n*th comb line will be *f*(*n*) = *nf*_{rep} + *f*_{CEO} + *f*_{geo}. In our observations, this contribution accounts for a downward shift by up to 0.4 MHz. Further investigations may address possible couplings between these periodic changes in the relative phase and absolute carrier envelope phase of such multisoliton complexes (i.e., relating *f*_{CEO} and *f*_{geo}).

In addition, it is also possible to induce bound triplet states for identical laser adjustment at enhanced pump power. At 4.8 W, we observe triplets with fixed separations and dynamically evolving relative phases. The recorded interferograms in Fig. 3G exhibit three superimposed fringe periodicities arising from the three temporal delays present in the state (the corresponding autocorrelation is shown in Fig. 3H). Each relative phase continuously evolves in time (Fig. 3I). The rate of phase slip is larger between pulses with greater separation, indicating that the three intensities of the bound solitons represent a monotonic sequence. Via reduction of the pump power, the triplet decays into the doublet states shown in Fig. 3, A to E.

## Soliton molecule vibrations

Finally, we present another set of qualitatively different dynamic soliton molecules with shorter binding separations. Shown in the left panels of Fig. 4, A to D, is a series of experimental real-time interferograms of soliton doublets, prepared via reducing the pump power in each case. Here, both the pulse separation and their relative phase evolve rapidly on a time scale of a few hundred roundtrips. The extracted binding separations and relative phases are shown in the center panels, with corresponding interaction planes in the right panels. In this set of measurements, all trajectories evolve between separations of 95 fs and 115 fs in configuration space (dashed radii in Fig. 3, right). For the highest pump power (Fig. 4A), the bound state periodically traces a closed orbit in the interaction plane, given by two partial circles over phases of 3π/2. In contrast to previous measurements with larger, invariant separation (Fig. 3), here the relative phase does not monotonically increase over time but oscillates between two turning points, reminiscent of theoretical predictions (*24*, *25*). In those studies, the regular internal motion of the bound state was attributed to a flipping between two unstable solutions, facilitated by gain dynamics and soliton interaction.

By reducing the pump power, the trajectories extend over phases above 4π (Fig. 4D). Specifically, in this state, we can readily observe that the closed orbits comprise interleaving spiral segments. The outer spiral corresponds to the stage of growing soliton separation with increasing relative phase (counterclockwise rotation). At the turning point, the trajectory passes into the inner spiral, both solitons approach, and the relative phase decreases (clockwise rotation). These observations are consistent with a Kerr-mediated interaction: Throughout the inner spiral, a dominant leading pulse experiences nonlinear phase retardation and propagates with reduced velocity [compare to the “heavy pulse” of (*27*)]; as a result, the relative phase and separation decrease. At the turning points, the maximum intensity passes from one pulse to the other, effectively reversing the motion and yielding the complementary spiral in the interaction plane.

Upon reduction of the pump level, the initial periodic orbit in Fig. 4A exhibits increased sensitivity to fluctuations, resulting in aperiodic dynamics (Fig. 4B). In the interaction plane, the bound state features several turning points consisting of partially periodic segments that twist because of aperiodic shifts in the phase. As a consequence, the configuration space progressively fills over time. Further reduction of the pump level, however, leads to a stabilization on closed orbits (*21*) (Fig. 4C, right). Contrary to the closed orbits in Fig. 4, A and D (*m* = 0), the relative phase continuously advances over time, yielding a winding number of *m* = –1. In this regard, the dynamics are related to the step-like phase evolution of the bound states in Fig. 3 with equal winding number. In contrast, the present state (Fig. 4C) exhibits extensive internal motion, demonstrated by six turning points in the configuration space. This multifaceted closed orbit demonstrates complicated, periodic trajectories as a result of intricate multisoliton interactions emerging from the interplay of gain dynamics and Kerr nonlinearity.

## Conclusions

Real-time access to multipulse interactions in a femtosecond laser oscillator allows us to track the formation of stable soliton molecules and uncover rapid internal motions for a diverse set of bound states. Highly complex excitations, both periodic and aperiodic, are resolved via the direct mapping of kinematic orbits in configuration space. In particular, we find oscillatory and progressively evolving bound states whose topologies are characterized by different winding numbers, pointing toward a possible topological protection of the dynamic excitations. The approach is expected to provide real-time insight into a wider class of phenomena in nonlinear systems, including the dynamics of breathers, the scattering of solitons or rogue waves, and, generally, transient interactions in the emergence of rare and nonrepetitive events (*4*, *42*).