Importance of the Wick rotation on Tunnelling

A continuous complex rotation of time $t\mapsto t\text{EXP}-i\theta$ is shown to smooth out the huge fluctuations that characterise chaotic tunnelling. This is illustrated in the kicked rotor model (quantum standard map) where the period of the map is complexified: the associated chaotic classical dynamics, if significant for $\theta =0$, is blurred out long before the Wick rotation is completed ($\theta =\pi \mathrm{/}2$). The influence of resonances on tunnelling rates weakens exponentially as $\theta$ increases from zero, all the more rapidly the sharper the fluctuations. The long range fluctuations can therefore be identified in a deterministic way without ambiguity. When the last ones have been washed out, tunnelling recovers the (quasi-)integrable exponential behaviour governed by the action of a regular instanton.