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Based on the Heisenberg-picture analog of the master equation, we develop amethod for computing the exact time dependence of noise-averaged observablesfor general noninteracting fermionic systems with noisy fluctuations. Uponnoise averaging, these fluctuations generate effective interactions, limitinganalytical approaches. While the short-time dynamics can be studied withLangevin-type numerical simulations, the long-time limit is not amenable tosuch simulations. Our results provide access to this long-time limit. As asimple example, we examine the fate of the fractional charge in cold-atomemulations of polyacetylene after stochastic driving. We find that in a quantumquench to a fluctuating hopping Hamiltonian, the fractional charge remainsrobust for hopping between different sublattices, while it becomes unstable inthe presence of noisy hopping on the same sublattice.

Dynamics of noisy quantum systems in the Heisenberg picture: application to the stability of fractional charge