Solution for the dynamics of the BCS and central spin problems

We develop an explicit description of a time-dependent response of fermionic condensates to perturbations. The dynamics of Cooper pairs at times shorter than the energy relaxation time can be described by the BCS model. We obtain a general explicit solution for the dynamics of the BCS model. We also solve a closely related dynamical problem - the central spin model, which describes a localized spin coupled to a "spin bath". Here, we focus on presenting the solution and describing its general properties, but also mention some applications, e.g. to nonstationary pairing in cold Fermi gases and to the issue of electron spin decoherence in quantum dots. A typical dynamics of the BCS and central spin models is quasi-periodic with a large number of frequencies and stable under small perturbations. We show that for certain special initial conditions the number of frequencies decreases and the solution simplifies. In particular, periodic solutions correspond to the ground state and excitations of the BCS model.