  We study theoretically nonequilibrium Landau-Zener-St\"uckelberg (LZS)
dynamics in a driven double quantum dot (DQD) including dephasing and,
importantly, energy relaxation due to environmental fluctuations. We derive
effective nonequilibrium Bloch equations. These allow us to identify clear
signatures for LZS oscilations observed but not recognized as such in
experiments [Petersson et al., Phys. Rev. Lett. 105, 246804, 2010] and to
identify the full environmental fluctuation spectra acting on a DQD given
experimental data as in [Petersson et al., Phys. Rev. Lett. 105, 246804, 2010].
Herein we find that super-Ohmic fluctuations, typically due to phonons, are the
main relaxation channel for a detuned DQD whereas Ohmic fluctuations dominate
at zero detuning.
