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We follow up the work, where in light of the Picard-Lefschetz thimbleapproach, we split up the real-time path integral into two parts: the initialdensity matrix part which can be represented via an ensemble of initialconditions, and the dynamic part of the path integral which corresponds to theintegration over field variables at all later times. This turns the pathintegral into a two-stage problem where, for each initial condition, thereexits one and only one critical point and hence a single thimble in the complexspace, whose existence and uniqueness are guaranteed by the characteristics ofthe initial value problem. In this paper, we test the method for a fullyquantum mechanical phenomenon, quantum tunnelling in quantum mechanics. Wecompare the method to solving the Schr\"odinger equation numerically, and tothe classical-statistical approximation, which emerges naturally in awell-defined limit. We find that the Picard-Lefschetz result matches theexpectation from quantum mechanics and that, for this application, theclassical-statistical approximation does not.

Quantum tunnelling, real-time dynamics and Picard-Lefschetz thimbles