Julia set describes quantum tunneling in the presence of chaos

We find that characteristics of quantum tunneling in the presence of chaos can be regarded as a manifestation of the Julia set of the complex dynamical system. Several numerical evidences for the standard map together with a rigorous statement for the Hénon map are presented demonstrating that the complex classical paths which contribute to the semiclassical propagator are dense in the Julia set. Chaotic tunneling can thus be characterized by the transitivity of the dynamics and high density of the trajectories on the Julia set. Recent studies on tunneling in multidimensions have revealed that the existence of chaos affects the signature of quantum tunneling. The observation of purely quantum mechanical calculation in chaotic systems shows that tunneling can become chaotic or chaos seems to assist The idea capturing such a novel aspect of tunneling looks very attractive, but a direct connection between chaos and tunneling can only be accomplished by interpreting the quantum phenomenon by the trajectory of the classical dynamics [6, 7, 8, 10, 11]. When one is particularly interested in the tunneling process, the use of complex trajectories is essential since the transition due to tunneling occurs where the real-valued classical trajectories cannot reach. The most well-known technique using the complex space is so-called instanton method in which the tunneling penetration is evaluated mainly by a single classical path moving on the reversed potential [12, 13]. On the other hand, in chaotic systems, it has been found in the time-domain semiclassical analysis that a bunch of complex paths almost equally contribute to the tunneling transition between classically forbidden regions [10, 11]. They typically form a tree-like fractal structure in the complex initial value plane, and its outstanding appearance compels us to prepare some concept which controls dominating complex paths in the semiclassical sum of contributing candidates [10, 11]. All the characteristic structures appearing in the tunneling wavefunction in chaotic systems originate from it.