  We develop and implement a method for modeling decoherence processes on an
N-dimensional quantum system that requires only an $N^2$-dimensional quantum
environment and random classical fields. This model offers the advantage that
it may be implemented on small quantum information processors in order to
explore the intermediate regime between semiclassical and fully quantum models.
We consider in particular $\sigma_z\sigma_z$ system-environment couplings which
induce coherence (phase) damping, though the model is directly extendable to
other coupling Hamiltonians. Effective, irreversible phase-damping of the
system is obtained by applying an additional stochastic Hamiltonian on the
environment alone, periodically redressing it and thereby irreversibliy
randomizing the system phase information that has leaked into the environment
as a result of the coupling. This model is exactly solvable in the case of
phase-damping, and we use this solution to describe the model's behavior in
some limiting cases. In the limit of small stochastic phase kicks the system's
coherence decays exponentially at a rate which increases linearly with the kick
frequency. In the case of strong kicks we observe an effective decoupling of
the system from the environment. We present a detailed implementation of the
method on an nuclear magnetic resonance quantum information processor.
