Many-body dynamics with time-dependent interaction

In this work we study the many-body dynamics of Bose-Einstein condensates subject to an arbitrary time-varying scattering length. By employing a variational ansatz which assumes the majority of the particles are condensed, we derive an effective Bogoliubov-like Hamiltonian that governs the dynamics of thermal particles. Crucially, we show that there exists a hidden symmetry in this Hamiltonian that can map the many-body dynamics to the precession of an SU(1,1) "spin" and also allows an exact dynamical solution for this precession in an arbitrary "magnetic field." As a demonstration, we calculate the situation where the scattering length is sinusoidally modulated. We show that the noncompactness of the SU(1,1) group naturally leads to solutions with exponentially growth of Bogoliubov modes and causes instabilities.