We study the spectral function of two-leg Hubbard ladders with the time-dependent density matrix renormalization group method (tDMRG). The high-resolution spectrum displays features of spin-charge separation and a scattering continuum of excitations with coherent bands of bound states `leaking from it. As the inter-leg hopping is increased, the continuum in the bonding channel moves to higher energies and spinon and holon branches merge into a single coherent quasi-particle band. Simultaneously, the spectrum undergoes a crossover from a regime with two minima at incommensurate values of $k_x$ (a Mott insulator), to one with a single minimum at $k_x=pi$ (a band insulator). We identify the presence of a continuum of scattering states consisting of a triplon and a polaron. We analyze the processes leading to quasiparticle formation by studying the time evolution of charge and spin degrees of freedom in real space after the hole is created. At short times, incoherent holons and spinons are emitted but after a characteristic time $tau$ charge and spin form polarons that propagate coherently.
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