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In generic Hamiltonian systems tori of regular motion are dynamically separated from regions of chaotic motion in phase space. Quantum mechanically these phase-space regions are coupled by dynamical tunneling. We introduce a semiclassical approach based on complex paths for the prediction of dynamical tunneling rates from regular tori to the chaotic region. This approach is demonstrated for the standard map giving excellent agreement with numerically determined tunneling rates.PACS numbers: 05.45.Mt, 03.65.Sq Tunneling is a fundamental manifestation of quantum mechanics. Its basic features are established in standard textbooks [1,2] for particles confined by one-dimensional energy barriers: While the particle is classically trapped, it can escape quantum mechanically if the energy barrier is finite. This process typically exhibits an exponential decay exp (−γt), which is characterized by the tunneling rate γ. This rate describes the inverse life-time of the confined state and captures the relevant information of the tunneling process. Since time-independent onedimensional systems have integrable dynamics, the tunneling rates through the energy barrier can be computed from complex WKB-paths in the classically forbidden re-Here, the imaginary part of the action S = p dq, which increases with the width and the height of the barrier, is divided by Planck's constant h = 2π . In the semiclassical limit this ratio increases and tunneling vanishes exponentially.In contrast to one-dimensional potential wells, many systems of physical relevance have non-integrable Hamiltonians such as driven atoms and molecules [5][6][7], coldatom systems [8,9], optical as well as microwave cavities [10][11][12][13][14], and nano-wires [15]. These systems generically have a mixed phase space in which classically disjoint regions of regular and chaotic motion coexist. Quantum mechanical transitions between such regions are called dynamical tunneling [16,17]. In the above systems dynamical tunneling is the key to understanding life-times and decay channels of long-lived states associated to a regular region. Furthermore, dynamical tunneling has important consequences for the structure of eigenfunctions [18,19] and spectral statistics [20][21][22][23] in mixed systems. Due to this broad interest a lot of effort has been made to investigate dynamical tunneling experimentally [8][9][10][11] and theoretically [24][25][26][27][28][29][30][31][32][33][34][35][36][37][38].In this paper we derive a semiclassical prediction of dynamical tunneling rates γ for the ubiquitous situa-tion of regular-to-chaotic tunneling. The focus is on the experimentally relevant regime, in which direct tunneling to the chaotic region dominates. We generalize the WKB-formula, Eq. (1), to mixed systems by unifying the semiclassical time evolution method of complex paths [28] with the fictitious integrable system approach [36]. This demonstrates that direct regular-to-chaotic tunneling rates are determined by complex paths, which connect regular tori to the boundary bet...

For systems with a mixed phase space we demonstrate that dynamical tunneling universally leads to a fractional power law of the level-spacing distribution P(s) over a wide range of small spacings s. Going beyond Berry-Robnik statistics, we take into account that dynamical tunneling rates between the regular and the chaotic region vary over many orders of magnitude. This results in a prediction of P(s) which excellently describes the spectral data of the standard map. Moreover, we show that the power-law exponent is proportional to the effective Planck constant h(eff).

For generic Hamiltonian systems we derive predictions for dynamical tunneling from regular to chaotic phase-space regions. In contrast to previous approaches, we account for the resonanceassisted enhancement of regular-to-chaotic tunneling in a non-perturbative way. This provides the foundation for future semiclassical complex-path evaluations of resonance-assisted regular-to-chaotic tunneling. Our approach is based on a new class of integrable approximations which mimic the regular phase-space region and its dominant nonlinear resonance chain in a mixed regular-chaotic system. We illustrate the method for the standard map.