  A quantum object can accumulate a geometric phase when it is driven along a
trajectory in a parameterized state space with non-trivial gauge structures.
Inherent to quantum evolutions, a system can not only accumulate a quantum
phase but may also experience dephasing, or quantum diffusion. Here we show
that the diffusion of quantum trajectories can also be of geometric nature as
characterized by the imaginary part of the geometric phase. Such an imaginary
geometric phase results from the interference of geometric phase dependent
fluctuations around the quantum trajectory. As a specific example, we study the
quantum trajectories of the optically excited electron-hole pairs, driven by an
elliptically polarized terahertz field, in a material with non-zero Berry
curvature near the energy band extremes. While the real part of the geometric
phase leads to the Faraday rotation of the linearly polarized light that
excites the electron-hole pair, the imaginary part manifests itself as the
polarization ellipticity of the terahertz sidebands. This discovery of
geometric quantum diffusion extends the concept of geometric phases.
