We study the real-time dynamics of a highly excited charge carrier coupled to quantum phonons via a Holstein-type electron-phonon coupling. This is a prototypical example for the non-equilibrium dynamics in an interacting many-body system where excess energy is transferred from electronic to phononic degrees of freedom. We use diagonalization in a limited functional space (LFS) to study the non-equilibrium dynamics on a finite one-dimensional chain. This method agrees with exact diagonalization and the time-evolving block decimation method, in both the relaxation regime and the long-time stationary state, and among these three methods it is the most efficient and versatile one for this problem. We perform a comprehensive analysis of the time evolution by calculating the electron, phonon and electron-phonon coupling energies, and the electronic momentum distribution function. The numerical results are compared to analytical solutions for short times, for a small hopping amplitude and for a weak electron-phonon coupling. In the latter case, the relaxation dynamics obtained from the Boltzmann equation agrees very well with the LFS data. We also study the time dependence of the eigenstates of the single-site reduced density matrix, which defines so-called optimal phonon modes. We discuss their structure in non-equilibrium and the distribution of their weights. Our analysis shows that the structure of optimal phonon modes contains very useful information for the interpretation of the numerical data.
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