Complex time method for quantum dynamics when an exceptional point is encircled in the parameter space

We revisit the complex time method for application to quantum dynamics as an exceptional point is encircled in parameter space of Hamiltonian. The basic idea using contour integration perform first-order adiabatic perturbation integral. In this way, dynamical problem transformed a study singularities plane – transition points which represent degeneracies Hamiltonian time-dependent parameters defining encircling are analytically continued plane. As underlying illustration approach we discuss switch between Rabi oscillations and rapid passage occurs upon special time-symmetric case.