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The level splitting distribution in chaos-assisted tunnelling

Leyvraz, François Ullmo, Denis
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    A compound tunnelling 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 tunnelling in certain two-dimensional systems. This mechanism is known as chaos-assisted tunnelling. 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 tunnelling 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 tunnelling 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 behaviour 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 tunnelling.


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