  Quantum decoherence is the disappearance of simple phase relations within a
discrete quantum system as a result of interactions with an environment. For
many applications, the question is not necessarily how to avoid (inevitable)
system-environment interactions, but rather how to design environments that
optimally preserve a system's phase relations in spite of such interactions.
The formation of system-environment entanglement is a major driving mechanism
for decoherence, and a detailed understanding of this process could inform
strategies for conserving coherence optimally. This requires scalable,
flexible, and systematically improvable quantum dynamical methods that retain
detailed information about the entanglement properties of the environment, yet
very few current methods offer this combination of features. Here, we address
this need by introducing a theoretical framework wherein we combine the
truncated Wigner approximation with standard time-dependent perturbation theory
allowing for computing expectation values of operators in the combined
system-environment Hilbert space. We demonstrate the utility of this framework
by applying it to the spin-boson model, representative of qubits and simple
donor-acceptor systems. For this model, our framework provides an analytical
description of perturbative contributions to expectation values. We monitor how
quantum decoherence at zero temperature is accompanied by entanglement
formation with individual environmental degrees of freedom. Based on this
entanglement behavior, we find that the selective suppression of low-frequency
environmental modes is particularly effective for mitigating quantum
decoherence.
