  In this work we introduce a phase-space description based on the positive P
representation for bosonic fields interacting with a system of quantum
emitters. The formalism is applicable to collective light-matter interactions
and open quantum systems with decoherence. Conservation of particle numbers is
considered, and a Jordan-Schwinger transformation enables the representation of
multi-level quantum emitters. The evolution of the phase-space description of
the combined system of emitters and field is formulated in terms of stochastic
trajectories and we derive the rules of mapping from traditional quantum
mechanics to this stochastic formalism. The resulting equations of motion
encode deterministic, classical evolution with quantum effects incorporated by
stochastic noise terms. The framework's equations and properties are provided
without specifying the Hamiltonian, aiming for broad applicability in diverse
research domains. A potential future application is the quantum mechanical
description of collective spontaneous emission of an incoherently pumped
ensemble of atoms.
