Tunnelling out of a time-dependent well

Solutions to explicit time-dependent problems in quantum mechanics are rare. In fact, all known solutions are coupled to specific properties of the Hamiltonian and may be divided into two categories: one class consists of time-dependent Hamiltonians which are not higher than quadratic in the position operator, like e.g. the driven harmonic oscillator with time-dependent frequency. The second class is related to the existence of additional invariants in the Hamiltonian, which can be used to map the solution of the time-dependent problem to that of a related time-independent one. In this paper we discuss and develop analytic methods for solving time-dependent tunnelling problems, which cannot be addressed by using quadratic Hamiltonians. Specifically, we give an analytic solution to the problem of tunnelling from an attractive time-dependent potential which is embedded in a long-range repulsive potential. Recent progress in atomic physics makes it possible to observe experimentally time-dependent phenomena and record the probability distribution over a long range of time. Of special interest is the observation of macroscopical quantum-tunnelling phenomena in Bose-Einstein condensates with time-dependent trapping potentials. We apply our model to such a case in the last section.