The level splitting distribution in chaos-assisted tunnelling

Abstract

A compound tunneling mechanism from one integrable region to another mediated by a delocalized state in an intermediate chaotic region of phase space was recently introduced to explain peculiar features of tunneling in certain two-dimensional systems. This mechanism is known as chaos-assisted tunneling. We study its consequences for the distribution of the level splittings and obtain a general analytical form for this distribution under the assumption that chaos assisted tunneling is the only operative mechanism. We have checked that the analytical form we obtain agrees with splitting distributions calculated numerically for a model system in which chaos-assisted tunneling is known to be the dominant mechanism. The distribution depends on two parameters: The first gives the scale of the splittings and is related to the magnitude of the classically forbidden processes, the second gives a measure of the efficiency of possible barriers to classical transport which may exist in the chaotic region. If these are weak, this latter parameter is irrelevant; otherwise it sets an energy scale at which the splitting distribution crosses over from one type of behavior to another. The detailed form of the crossover is also obtained and found to be in good agreement with numerical results for models for chaos-assisted tunneling.

Influence of classical resonances on chaotic tunneling

“…[19], a fruitful approach of quantum chaos is to replace a chaotic but deterministic Hamiltonian by a random element of an ensemble of matrices that only encapsulates the global symmetries. These hybrid techniques, with both semiclassical and statistical ingredients, first allowed us to qualitatively understand the so-called chaosassisted tunnelling, i.e the observation [20] that tunnelling is increased on average as the transport through chaotic regions grows [21,22,23]. However, the extreme sensitivity of tunnelling renders the predictions very difficult even if just an order of magnitude is required.…”

Chaos-Assisted Tunneling and1/fαSpectral Fluctuations in the Order-Chaos Transition

“…This process can be modeled with random matrix ensembles, in which independent Gaussian random variables connect chaotic with integrable subspaces [5,16 -18]. These models describe the level splitting distribution [16] and the powerlaw level repulsion [18,19] in mixed systems. Recently, a semiclassical formalism has been derived to calculate the variance of the connecting elements [17,20].…”

“…The variance of these elements is usually taken as a constant, estimated by means of semiclassical arguments [17,19,20] or fitted as a free parameter [5,16,18,21]. Such a model reproduces the fractional level repulsion, but, as we will see below, it does not give rise to 1=f noise in the transition form order to chaos.…”

Resonance- and Chaos-Assisted Tunneling in Mixed Regular-Chaotic Systems

“…Indeed, chaosassisted tunneling processes arise in a number of physical systems, e.g. in the ionization of resonantly driven hydrogen [7], in microwave or optical cavities [8,9], as well as in the effective pendulum dynamics describing tunneling experiments of cold atoms in optical lattices [10,11].While the statistical properties of the chaos-assisted tunneling rates are well reproduced by a random matrix description of the chaotic part of the Hamiltonian [12], the formulation of a tractable and reliable semiclassical theory for the average tunneling rate is still an open problem. Promising progress in this direction was reported by Shudo and coworkers [13] who obtain a good quantitative reproduction of classically forbidden propagation processes in mixed systems by incorporating complex trajectories into the semiclassical propagator Their approach requires, however, the study of highly nontrivial structures in complex phase space, and cannot be straightforwardly connected to single coupling matrix elements between regular and chaotic states.…”