  Many-body approaches to open quantum systems have recently become powerful
tools for investigating the detailed role of dissipative environments in
diverse non-equilibrium molecular and condensed matter processes. Here, we
report the development of an efficient algorithm that utilises a time-dependent
variational principle for matrix product states to evolve large
system-environment states. By thus capturing all system-environment
correlations, we reproduce the highly non-perturbative, quantum-critical
dynamics of the zero temperature spin-boson model, and then exploit the
many-body information to output a complete time-frequency spectrum of the
environmental excitations. We highlight how theoretical 'environmental spectra'
could yield valuable insights into a wide range of complex dissipative
processes, by showing that correlated motion of modes entangled with the spin
can appear with persistent vibrational coherence, in spite of incoherent spin
relaxation.
