Instantons revisited: dynamical tunnelling and resonant tunnelling

Starting from trace formulae for the tunnelling splittings (or decay rates) analytically continued in the complex time domain, we obtain explicit semiclassical expansions in terms of complex trajectories that are selected with appropriate complex-time paths. We show how this instanton-like approach, which takes advantage of an incomplete Wick rotation, accurately reproduces tunnelling effects not only in the usual double-well potential but also in situations where a pure Wick rotation is insufficient, for instance dynamical tunnelling or resonant tunnelling. Even though only one-dimensional autonomous Hamiltonian systems are quantitatively studied, we discuss the relevance of our method for multidimensional and/or chaotic tunnelling.