  We quantify the non-adiabatic contributions to the vibronic sidebands of
equilibrium and explicitly time-resolved non-equilibrium photoelectron spectra
for a vibronic model system of Trans-Polyacetylene. Using exact
diagonalization, we directly evaluate the sum-over-states expressions for the
linear-response photocurrent. We show that spurious peaks appear in the
Born-Oppenheimer approximation for the vibronic spectral function, which are
not present in the exact spectral function of the system. The effect can be
traced back to the factorized nature of the Born-Oppenheimer initial and final
photoemission states and also persists when either only initial, or final
states are replaced by correlated vibronic states. Only when correlated initial
and final vibronic states are taken into account, the spurious spectral weights
of the Born-Oppenheimer approximation are suppressed. In the non-equilibrium
case, we illustrate for an initial Franck-Condon excitation and an explicit
pump-pulse excitation how the vibronic wavepacket motion of the system can be
traced in the time-resolved photoelectron spectra as function of the pump-probe
delay.
