  By extending the quantum evolution of a scalar field in time-dependent
backgrounds to the complex-time plane and transporting the in-vacuum along a
closed path, we argue that the geometric transition from the simple pole at
infinity determines the multi-pair production depending on the winding number.
We apply the geometric transition to Schwinger mechanism in the time-dependent
vector potential for a constant electric field and to Gibbons-Hawking particle
production in the planar coordinates of a de Sitter space.
