  Stern-Gerlach interferometer (SGI) is a kind of matter-wave interferometer
driven by magnetic field and has been proposed for many gravity experiments.
Acceleration noises such as vibration and inertial forces, together with
higher-order noises like the fluctuation of the gravity gradient or the
magnetic field, can cause decoherence problems of SGI, including dephasing,
loss of contrast and position localization decoherence. In this paper, I will
theoretically study these mechanisms of decoherence based on the analytical
time-evolution operator of an SGI modeled as a harmonic oscillator under
acceleration noises described by Gaussian stochastic processes. As will be
proved, for a single arm of an SGI, the shape of the Wigner function keeps
invariant under an acceleration noise, although the phase and the coordinate in
the classical phase space fluctuate as linear responses to the noise, and its
density matrix also experiences a position localization decoherence due to the
ensemble average. For the witness constructed in the spin space, the degrees of
freedom in the classical phase space have to be traced out. Then acceleration
noises can lead to dephasing effects on the witness, while the fluctuation on
the trajectories in the phase space and position localization decoherence of
the two arms will be canceled with each other. On the other hand, higher-order
noises can be treated as non-uniform acceleration noises in the leading order,
and they will cause decoherence in the spin space due to the contrast loss
besides the dephasing. Remarkably, the randomness of the noise is essential for
dephasing and position-localization decoherence, and these two mechanisms don't
cause purity loss or entropy increase if the noise is monitored as a
deterministic process during the experiment.
