Competing tunneling trajectories in a 2D potential with variable topology as a model for quantum bifurcations

We present a path - integral approach to treat a 2D model of a quantum bifurcation. The model potential has two equivalent minima separated by one or two saddle points, depending on the value of a continuous parameter. Tunneling is therefore realized either along one trajectory or along two equivalent paths. Zero point fluctuations smear out the sharp transition between these two regimes and lead to a certain crossover behavior. When the two saddle points are inequivalent one can also have a first order transition related to the fact that one of the two trajectories becomes unstable. We illustrate these results by numerical investigations. Even though a specific model is investigated here, the approach is quite general and has potential applicability for various systems in physics and chemistry exhibiting multi-stability and tunneling phenomena.