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.

The desire to exert active optical control over matter is a unifying theme across multiple scientific disciplines, as exemplified by all-optical magnetic switching 1,2 , light-induced metastable or exotic phases of solids 3-9 and the coherent control of chemical reactions 10,11 . Typically, these approaches dynamically steer a system towards states or reaction products far from equilibrium. In solids, metal-insulator transitions are an important target for optical manipulation, offering dramatic and ultrafast changes of the electronic 4,5 and lattice [12][13][14][15][16][17][18] properties. In this context, essential questions concern the role of coherence in the efficiencies and thresholds of such transitions. Here, we demonstrate coherent vibrational control over a metal-insulator structural phase transition in a quasi-one-dimensional solid-state surface system. An optical double-pulse excitation scheme [19][20][21][22] is used to drive the system from the insulating to a metastable metallic state, and the corresponding structural changes are monitored by ultrafast low-energy electron diffraction [23][24][25] . We observe strong oscillations in the switching efficiency as a function of the double-pulse delay, revealing the importance of vibrational coherence in two key structural modes governing the transition on a femtosecond timescale. This mode-selective coherent control of solids and surfaces could open new routes to switching chemical and physical functionalities, facilitated by metastable and non-equilibrium states.Femtochemistry entails the search for understanding and control of ultrafast reaction pathways 10,22 . To this end, coherences in the electronic and vibrational states of reactants are employed to guide the system across a complex, generally multidimensional energy landscape 10,26 . Established for small molecules, a possible transfer of this concept to extended systems and solids is complicated, e.g. due to a high electronic and vibrational density of states, and couplings to an external heat bath 27 . Low-dimensional and strongly correlated systems represent a promising intermediate between molecules and solids, with phase transitions assuming the role of a "reaction". A number of these transitions can be driven optically -either by means of transient heating 24,28 , electronic excitation [15][16][17][18]29,30 or direct resonant coupling to certain vibrational degrees of freedom [4][5][6][7]31 . The prototypical case of a phase transition governed

The emergence of confined structures and pattern formation are exceptional manifestations of concurring nonlinear interactions found in a variety of physical, chemical and biological systems 1 . Optical solitons are a hallmark of extreme spatial or temporal confinement enabled by a variety of nonlinearities. Such particle-like structures can assemble in complex stable arrangements, forming "soliton molecules" 2,3 . Recent works revealed oscillatory internal motions of these bound states, akin to molecular vibrations 4-8 . These observations beg the question as to how far the "molecular" analogy reaches, whether further concepts from molecular spectroscopy apply in this scenario, and if such intra-molecular dynamics can be externally driven or manipulated. Here, we probe and control such ultrashort bound-states in an optical oscillator, utilizing real-time spectroscopy and time-dependent external perturbations. We introduce two-dimensional spectroscopy of the linear and nonlinear bound-state response and resolve anharmonicities in the soliton interaction leading to overtone and sub-harmonic generation. Employing a non-perturbative interaction, we demonstrate all-optical switching between distinct states with different binding separation, opening up novel schemes of ultrafast spectroscopy, optical logic operations and all-optical memory.Mirroring the hierarchy of the atomic and molecular composition of matter, fundamental solitons constitute stable entities which can aggregate to form structures of increased complexity. Such condensation manifests in soliton fluids, molecules and crystals, as observed in diverse physical systems 2,9,10 . The underlying forces are responsible for various emergent phenomena, such as self-organization of Bose-Einstein condensates, soliton crystallization in micro-cavities, soliton fission in supercontinua or rogue waves in optical fiber [10][11][12][13][14][15] . It is wellknown that ultrashort solitons can form highly-stable bound-states, and these optical soliton molecules arise from balanced forces during propagation in passive fibers and in dissipative laser resonators [2][3][4][5]8,[16][17][18][19][20][21] . Recent real-time studies have revealed rapid transients, internal vibrations and even chaotic dynamics 6,7,[22][23][24] .In this study, we transfer the concepts of optical spectroscopy to the case of ultrafast soliton molecules by driving them with an external perturbation and monitoring their response in real time. We discover a resonance in the response of soliton molecules to external perturbation and probe the associated anharmonic binding potential (Fig. 1a, approach I). Applying stronger, non-perturbative stimuli, we also demonstrate reliable and reversible all-optical switching between two different bound states (Fig. 1a, approach II)suggesting applications in fast optical sampling and pulse control.

We present a passively mode-locked Yb:CALGO oscillator with harmonic repetition rate operation up to the third order. It is operated in the solitary regime with a fundamental roundtrip rate of 94 MHz and pulse durations between 200 fs and 600 fs. Harmonic operation was observed being stable for several days. The harmonic mode-locking regions are analyzed depending on intra-cavity dispersion. The transient pulsing dynamics converging to the stable harmonic modes is tracked and a theoretical model describing the pulse moving mechanisms is presented.

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