  In this thesis we develop methods for many-body open quantum systems and
apply them to systems of organic polaritons. The methods employ a mean-field
approach to reduce the dimensionality of large-scale problems. Initially
assuming the absence of correlations in the many-body state, this approach is
built upon in two ways.
  First, we show how the mean-field approximation can be combined with matrix
product operator methods to efficiently simulate the non-Markovian dynamics of
a many-body system with strong coupling to multiple environments. We apply this
method to calculate the threshold and photoluminescence for a realistic model
of an organic laser.
  Second, we extend the mean-field description by systematically including
higher-order correlations via cumulant expansions of the Heisenberg equations
of motion. We investigate the validity and convergence properties of these
expansions, both with respect to expansion order and system size, for many-body
systems with many-to-one network structures. We then show how the cumulant
expansions may be used to calculate spatially resolved dynamics of organic
polaritons. This enables a study of organic polariton transport in which we
observe reversible conversion to dark exciton states and sub-group-velocity
propagation.
  The methods established in this work offer versatile tools for analysing
large, many-body open quantum systems and investigating finite-size effects.
Their application reveals the intricate dynamics of organic polaritons
resulting from the interplay of strong light-matter coupling and vibrational
effects.
